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BKM LIE SUPERALGEBRAS IN N = 4<br />

SUPERSYMMETRIC<br />

Ý<br />

STRING THEORY<br />

ÌÁÒ<strong>×</strong>ØØÙØÓÅØÑØÐËÒ<strong>×</strong>¸ÒÒº úÓÔÐÃÖ<strong>×</strong>Ò<br />

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ÇÌÇÊÇÈÀÁÄÇËÇÈÀ ÓÖØÖÓ<br />

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ÇØÓÖ¾¼¼

ÀÓÑÆØÓÒÐÁÒ<strong>×</strong>ØØÙØ<br />

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Ø ÈÐÓ<strong>×</strong>ÓÔݺ ÓÖÝÑÝÔØ<strong>×</strong>ÙÐÐÐÒØ<strong>×</strong><strong>×</strong>ÖØØÓÒÖÕÙÖÑÒØÓÖØÖÓÓØÓÖÓ<br />

ÖÑÒ Ø ÓÒÚÒÖ Ø ÅÑÖ ÅÑÖ Ø<br />

ÅÑÖ Ø<br />

ÅÑÖ Ø<br />

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ÙËÙÖ<strong>×</strong>ÓÚÒÖÒÒ˺ÃÐÝÒÊÑ Ø

ÄÊÌÁÇÆ<br />

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úÓÔÐÃÖ<strong>×</strong>Ò

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<strong>×</strong>ÙÔÔÓÖغÁÛÓÙÐÐ<strong>×</strong>ÓÐØÓØÒÈÖÓºËÒÎݺ ÚÖÝ<strong>×</strong>ØÓÑÝÖÙØ<strong>×</strong>ØÙݺÁÛÓÙÐÐØÓØÒÈÖÓºÐËØÔÐÒÓÖ<strong>×</strong> ÁÛÓÙÐÐØÓØÒÈÖӺ̺ʺÓÚÒÖÒÓÖÒÐÔÙÐÒÒÓÙÖÒØ<br />

<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>º ÁÛ<strong>×</strong>ØÓÒÓÛÐÖºÐÓÄÒÈÖӺúƺÊÚÒÓÖÑÒÝÐÔÙÐ ÁÛÓÙÐÐØÓØÒØÁÅ˺ÓÑÑÙÒØÝÓÖÑÒÑÝ<strong>×</strong>ØÝÔÐ<strong>×</strong>ÒØÓÒº<br />

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4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÓÙÖ¹ÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ØÖÒ<br />

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Ö<strong>×</strong>ºÌÓÙÒØÒÓ1<br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒN =<br />

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4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖ<br />

ÔÖÓÚÒÓÖÑÓÙ<strong>×</strong>ÑÔØÙ<strong>×</strong>ØÓØÓÙÒØÒÓÈË<strong>×</strong>ØØ<strong>×</strong>ÒÚÖØÝÓ<strong>×</strong>ØØÒ<strong>×</strong>ºÎÎ¸Ò ÒÎÖÐÒ´Îε½℄ÒØÓØÖÝËØÖÓÑÒÖÒÎ℄ºËØÖÓÑÒÖÒÎ ÔÖÓÚÑÖÓ<strong>×</strong>ÓÔ<strong>×</strong>ÖÔØÓÒÓØÒØÖÓÔÝÓ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÐÓÐ<strong>×</strong>¸Û<strong>×</strong><br />

Ì<strong>×</strong>ØÖØÒÔÓÒØÒØÔÖÓÐÑÓÓÙÒØÒÝÓÒ<strong>×</strong>ØØ<strong>×</strong>ÒN =<br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØØÖÓØ<br />

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ÖÑÖÐÐÔÓÒØÙØÓÒ¸ÔÖÓÔÓ<strong>×</strong>ØØØÒÖÝÓ1<br />

ÖÙÙ<strong>×</strong>ÝÑÑØÖÝÒÓÛÒ<strong>×</strong>ØÀÄÓÑÔØØÓÒ<strong>×</strong>℄ºÁÒÔÖØÙÐÖ¸ØÝÚ<br />

ØÒ¸Φ 10<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÙØÛØ<br />

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4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖØÝÔÁÁ<strong>×</strong>ØÖÒ<br />

ÓÒT 6ÐÒØÓØÖÓØÓÑÔØØÓÒ<strong>×</strong>ØØÔÖ<strong>×</strong>ÖÚN =<br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>Ò<br />

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ØÓÖ<strong>×</strong>ºÌÝÚÐ<strong>×</strong>Ó<strong>×</strong>ØÙØÝÓÒ<strong>×</strong>ÔØÖÙÑÒN =<br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÓÙÖ<strong>×</strong><br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>º ÓØ<strong>×</strong>ÐÖ<strong>×</strong>ØÖÙØÙÖÑØÔÖÓÚÑÓÖÒ<strong>×</strong>ØÒØÓØÔÝ<strong>×</strong><strong>×</strong>Ó1<br />

(Z)º

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2¹ÈË<strong>×</strong>ØØ<strong>×</strong><strong>×</strong>ÒÓÙÒÒØ<strong>×</strong>Ñ<br />

NºÌÙØÓÖ¸ кÒØÒÖÝÓÐØÖÐÐÝÖ1<br />

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4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØ<br />

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a¸ÓØÛÓÖй γÒgÒ<br />

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ÛÖκ<strong>×</strong>ÔÖÓÔÓÖØÓÒÐØÝÓÒ<strong>×</strong>ØÒØÐÐØ<strong>×</strong>ØÖÒØÒ<strong>×</strong>ÓÒºÓÖØØÓÒØÓ ´½º½µ 2Ò ¿ ÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong><strong>×</strong>ÕÙÒØØÝØ<strong>×</strong>ØÖÒØÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓÙÐÚÑÒ<strong>×</strong>ÓÒ<strong>×</strong>(ÐÒØ) −2 = (Ñ<strong>×</strong><strong>×</strong>)<br />

Z = ∑ Σ<br />

∑<br />

e −S[X,Σ] =<br />

X<br />

∫<br />

S = −κ<br />

γ,<br />

∞∑<br />

∫<br />

(g s ) 2h−2<br />

h=0<br />

Σ<br />

DXDγe −S[X,γ] =<br />

∞∑<br />

Z h (g s ) 2h−2 .<br />

h=0<br />

dτdσ(−γ) 1/2 γ αβ ∂ α X µ ∂ β X ν g µν ,

′<strong>×</strong>ØÊ<strong>×</strong>ÐÓÔº ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÐØÓÒиÓÖØÝÓÒиÙØÒØ<strong>×</strong>ÖØÖÑÓ<strong>×</strong>ØÐÝÑ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÛÓÒÓØ Ì<strong>×</strong>¸ÓÛÚÖ¸<strong>×</strong>ÒÓØØÑÓ<strong>×</strong>ØÒÖÐØÓÒØØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÓÚÑÒØÓÒÖØÖº<br />

ÓÒ<strong>×</strong>Ö<strong>×</strong>ÙØÖÑ<strong>×</strong>º ÌÖÖÓØÖÔÓ<strong>×</strong><strong>×</strong>ÐØÖÑ<strong>×</strong>ØØÓÒÒ¸ÐØÒع<strong>×</strong>ÝÑÑØÖØÒ<strong>×</strong>ÓÖиÓÖ ÒØÙÖÐÙÒØ<strong>×</strong>ºÁØ<strong>×</strong>ØÒØÓκ = −1¸ÛÖα ØÒÔÒ<strong>×</strong>ÓÒÐÝÓÒØÑÒµÒØ<strong>×</strong>Ô¹ØÑÑÒÓдØØÒ<strong>×</strong>ÓÖÒ<strong>×</strong>Ö ÓØÛÓÖй<strong>×</strong>Ø<strong>×</strong>ÒÔÒÒØÓØÔÖÑØÖÞØÓÒÓØÛÓÖй<strong>×</strong>ØÙ<strong>×</strong>ØÓÑ<strong>×</strong>ÙÖ ÌÈÓÐÝÓÚØÓÒ¸ÝÓÒ<strong>×</strong>ØÖÙØÓÒ¸<strong>×</strong>Ø<strong>×</strong>ÝÑÑØÖ<strong>×</strong>ÓØÛÓÖй<strong>×</strong>Ø´ØÖ<br />

ÔÖÓÔÖÐÝÓÒØÖØØÓÑØÈÓÒÖÒÚÖÒصÙÐØÒØÓغÌ<strong>×</strong>Ô¹ØÑÑÒÓÐ <strong>×</strong>Ù<strong>×</strong>ÙÐÐÝÔ<strong>×</strong>ÙÓ¹ÊÑÒÒÒ<strong>×</strong>Ô¸ÛÒÓÙÖ<strong>×</strong>¸ÛØØÓØÅÒÓÛ<strong>×</strong> <strong>×</strong>Ô¹ØѸÛÓ<strong>×</strong><strong>×</strong>ÝÑÑØÖ<strong>×</strong>ÖØD¹ÑÒ<strong>×</strong>ÓÒÐÈÓÒÖÒÚÖÒ<br />

´½º¾µ<br />

X ′µ (σ, τ) = Λ µ ν X ν (σ, τ) + a µ ,<br />

ÓÖfºÌÒÛÑØÖ<strong>×</strong>ØÔÙÐÐÓØÓÐÓÒÒ<strong>×</strong>ÚÒÝ ÌÛÓÖй<strong>×</strong>Ø<strong>×</strong>ØÛÓ¹ÑÒ<strong>×</strong>ÓÒÐÑÒÓиÒ<strong>×</strong>ÒØ<strong>×</strong>ÖÓÙÔÓ<strong>×</strong>ÝÑÑØÖ<strong>×</strong>Ø<br />

(ξ)ØÓÓÖÒØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ ÛÖΛ ∈ SO(1, D)¸Òa ÖÓÙÔÓÓÑÓÖÔ<strong>×</strong>Ñ<strong>×</strong>f : Σ g → Σ gÓΣºÄØξ ´½º¿µ<br />

→ ξ ′a<br />

ÌÑÒØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong><strong>×</strong>X αβ<strong>×</strong>ÒÓÒ¹ÝÒÑÐÒØØÓÒ¸ÒÒÑÔÓ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÓÒØ ´½ºµ<br />

<strong>×</strong>3ÒÔÒÒØÓÑÔÓÒÒØ<strong>×</strong>ºÌØÛÓ¹ÑÒ<strong>×</strong>ÓÒÐÓÓÖÒØÖÔÖÑØÖÞØÓÒ<strong>×</strong>ÔÒ <strong>×</strong>Ý<strong>×</strong>ØѺÍÒÐ<strong>×</strong><strong>×</strong>ÛÒÙÛÝÐÐØÒÔÒÒØÖ<strong>×</strong><strong>×</strong>ÓØÑØÖ¸ÛÒÒÓØÑ<br />

′µ (σ ′ , τ ′ ) → f ∗ X µ = X µ (σ, τ) .<br />

αβÒØÛÓÑÒ<strong>×</strong>ÓÒ<strong>×</strong> ÌÑØÖγ Ø<strong>×</strong>ÓØÛÓÑÒ<strong>×</strong>ÓÒ<strong>×</strong>¸ØÖÓÙÖ<strong>×</strong>ÓÒÑÓÖÐÓÐ<strong>×</strong>ÝÑÑØÖÝÐÓÐÖ<strong>×</strong>ÐÒ<strong>×</strong>ÓØ Ù<strong>×</strong>ÛØÓÒÒÔÒÒØÔÖÑØÖÒØÑØÖØÒ<strong>×</strong>ÓÖØÓܺÁØØÙÖÒ<strong>×</strong>ÓÙظØØÙ<strong>×</strong>ØÓÖ ÓÒØÛÓÖÙÒØÓÒ<strong>×</strong>ÒÛÒÐÑÒØØÛÓÓØÓÑÔÓÒÒØ<strong>×</strong>Ù<strong>×</strong>ÒØ<strong>×</strong>ºÌ<strong>×</strong>ÐÚ<strong>×</strong><br />

ÑØÖØØ<strong>×</strong>ÒÒÚÖÒÓØÐ<strong>×</strong><strong>×</strong>ÐØÓÒºÁØ<strong>×</strong>ÐÐØÏÝÐÒÚÖÒÓØ<br />

<strong>×</strong>Ò<strong>×</strong>ÐÒØÖÔÖØØÓÒÓØÔÝ<strong>×</strong>ÐØÓÖݺÌ<strong>×</strong>ÝÑÑØÖØÒ<strong>×</strong>ÓÖγ <br />

µ<br />

(4πα ′ )<br />

Dº<br />

γ ′ αβ(σ, τ) = γ αβ (σ, τ) ,<br />

∈ R<br />

a<br />

γ αβ → f ∗ γ αβ = ∂ξγ<br />

∂ξ ′ α<br />

∂ξ δ<br />

∂ξ ′ β γ γδ .

ÑØÖÒ<strong>×</strong>ÚÒÝ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÒÚÖÒØÒØÛÓ<strong>×</strong>Ô¹ØÑÑÒ<strong>×</strong>ÓÒ<strong>×</strong>¸ÒØÙ<strong>×</strong>¸ØØÓÒÖÑÒ<strong>×</strong>ÒÚÖÒØÙÒÖغ ´½ºµ<br />

γ<br />

τ)ºÍÒÖÏÝÐÖ<strong>×</strong>ÐÒÓØÑØÖ¸ØÓÑÒØÓÒ√ αβºÌ<strong>×</strong><strong>×</strong>ÒÓÛÒ αβ<strong>×</strong><br />

ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong> <strong>×</strong>ØÓÒÓÖÑÐÙºÌÑÒÓØ<strong>×</strong>ØÖÒÒM<strong>×</strong>ÒÓØØÝØ<strong>×</strong>Ò<br />

τ)ÙÒÖØÏÝÐ<br />

ÓÖÖØÖÖÝω(σ, γγ<br />

ÇÒÒÙ<strong>×</strong>Ø<strong>×</strong>ÖÓÑØÓÜγ<br />

ÌÏÝÐÒÚÖÒÓØØÓÒ¸ÒØÛÓ<strong>×</strong>Ô¹ØÑÑÒ<strong>×</strong>ÓÒ<strong>×</strong>¸<strong>×</strong>ÚÖÝÒØÖ<strong>×</strong>ØÒÒ<br />

αβ´ØÐ<strong>×</strong>ØÐÓÐÐݵØÓÔÖÓÔÓÖØÓÒÐØÓη <strong>×</strong><strong>×</strong>ÖØÝØØÖÒ<strong>×</strong>ÓÖÑØÓÒÔÖÓÔÖØ<strong>×</strong>ÓØÙÒØÓÒ<strong>×</strong>X ÑÔÓÖØÒØÓÒ<strong>×</strong>ÕÙÒ<strong>×</strong>ÓÖØØÓÖÝ<strong>×</strong>ÛÛÐÐ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÐØÖº ´½ºµ<br />

µ (σ,<br />

ØÓÖÝ<strong>×</strong>ØÓÒÓÑÐݹֺÁÒØÓÒÓÖÑÐÙ¸ØØÓÒÖÙ<strong>×</strong>ØÓØÖÐ ÏÒÛÕÙÒØÞØØÓÖݸÛÛÐÐÖÕÙÖØØØ<strong>×</strong><strong>×</strong>ÝÑÑØÖ<strong>×</strong>ÔÖ<strong>×</strong>ÖÚØ<br />

X ′µ (σ, τ) = X µ (σ, τ) .<br />

ØÓÒ Ì<strong>×</strong>ÓÓÙÛÐÐÚØÓØÖØÑÓÖÖÙÐÐÝÛÒÕÙÒØÞÒØØÓÖݺ ´½ºµ<br />

ØÓÚØÓÒºÀÚÒØØÓÒ¸ÛÒÖÚØÕÙØÓÒ<strong>×</strong>ÓÑÓØÓÒÓÑÒÖÓÑ ÓÒØÑÒ<strong>×</strong>ÓÒÓ<strong>×</strong>Ô¹ØÑÒÓÒØÑ<strong>×</strong><strong>×</strong>ÓØÖÓÙÒ<strong>×</strong>ØغÓÖÒÓÛÛÛÓÖÛØ ÌÖÕÙÖÑÒØÓØØÓÖÝØÓÒÓÑÐÝÖÛÐÐÑÔÓ<strong>×</strong>ÖØÒÓÒ<strong>×</strong><strong>×</strong>ØÒÝÓÒØÓÒ<strong>×</strong><br />

S<br />

ÑÓØÓÒÓÑÒÖÓÑØ<strong>×</strong>ØÓÒ<strong>×</strong>Ù<strong>×</strong>ØØØÛÓ¹ÑÒ<strong>×</strong>ÓÒÐÐÒÖÛÚÕÙØÓÒ<br />

σ)ºÌÙÐÖ¹ÄÖÒÕÙØÓÒ<strong>×</strong>Ó<br />

´½ºµ<br />

ØÒÒÒÖÐ<strong>×</strong>ÓÐÙØÓÒ<strong>×</strong>ØÓØÙÒØÓÒ<strong>×</strong>X<br />

abÚ<strong>×</strong>Ø´ØÛÓ¹ÑÒ<strong>×</strong>ÓÒеÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ Ì<strong>×</strong>ÑÙ<strong>×</strong>ظÓÓÙÖ<strong>×</strong>¸<strong>×</strong>ÙÔÔÐÑÒØÛØØÓÒ<strong>×</strong>ØÖÒØÕÙØÓÒ<strong>×</strong>ºÌÓÒ<strong>×</strong>ØÖÒØ<br />

µ (τ,<br />

0ºÌÚÖØÓÒÓØØÓÒÛØÖ<strong>×</strong>ÔØØÓØÑØÖ ÕÙØÓÒ<strong>×</strong>ÒØ<strong>×</strong><strong>×</strong>ÖδS =<br />

´½ºµ<br />

γ<br />

ÌÓÑÓÖÔ<strong>×</strong>ÑÒÚÖÒÒØÛÓ¹ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ÑÔÐ<strong>×</strong>ØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ<strong>×</strong> 0º<br />

= ∂ α X µ ∂ β X µ − 1 2 γ αβ ∂ γ X µ ∂ γ X µ .<br />

ÌÙ<strong>×</strong>¸ØÓÒ<strong>×</strong>ØÖÒØÕÙØÓÒ<strong>×</strong>ÑÔÐÝÑÒ<strong>×</strong>ØØØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖT αβ =<br />

<br />

′<br />

αβ (σ, τ) = e2ω (σ, τ)) γ αβ (σ, τ) ,<br />

= −κ<br />

∫<br />

δγ αβ<br />

dσdτ η µν η αβ ∂ α X µ ∂ β X ν .<br />

( ∂<br />

□X µ 2 )<br />

≡<br />

∂τ − ∂2<br />

X µ = 0 .<br />

2 ∂σ 2<br />

T αβ (σ, τ) = −(κ) −1 (−γ)<br />

−1/2 δS<br />

δγ αβ .

ÓÒ<strong>×</strong>ÖÚºÌÏÝÐÒÚÖÒÓØØÓÒS¸Û<strong>×</strong>Ù<strong>×</strong>ØØ<strong>×</strong>ØØÑÒØÓÓÒÓÖÑÐ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÏÛÐÐØØÓØ<strong>×</strong>ÖÑÖÒØÖÛÜØÓÙÒÖÝÓÒØÓÒ<strong>×</strong>ºÏÛÐÐÐ<strong>×</strong>ÓÒ ØØÓÖÝ<strong>×</strong><strong>×</strong>ÐÒÚÖÒغÁÒØÛÓ¹ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>¸ØÓÒÓÖÑÐÖÓÙÔ<strong>×</strong>ÒÒعÑÒ<strong>×</strong>ÓÒк ÒÚÖÒÓØØÓÖÝÑÔÐ<strong>×</strong>ØØØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ<strong>×</strong>ØÖÐ<strong>×</strong><strong>×</strong>ºÌ<strong>×</strong>ÑÒ<strong>×</strong><br />

ØÓÒºÁÛØØÓÓÖÒØÖÓÒØÓ ÁØÛÓÖй<strong>×</strong>Ø<strong>×</strong>ÓÙÒÖݸØÖ<strong>×</strong>Ð<strong>×</strong>Ó<strong>×</strong>ÙÖØÖÑÒØÚÖØÓÒÓØ<br />

αβÛÒÕÙÒØÞÒØØÓÖݺ ØÓÜÑÒØÔÓ<strong>×</strong><strong>×</strong>ÐØÝÓÒÒÓÑÐÝÒØØÖÓT<br />

Ý µÛÐÐÐ<strong>×</strong>ÓÚÓÙÒÖÝØÖÑÚÒ<br />

−∞ < τ < ∞, 0 ≤ σ < l .<br />

ÌÒ¸ØÚÖØÓÒÓØØÓÒÛØÖ<strong>×</strong>ÔØØÓX ÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>¸ÓÒÑÔÓ<strong>×</strong><strong>×</strong>ÔÖÓØÝÓÒØÓÒÓÒØÐ<strong>×</strong> ÏÒØ<strong>×</strong>ØÖÑØÓÚÒ<strong>×</strong>ÒØÖÖÖÒØÛÝ<strong>×</strong>ÒÛØØÒÔÔÒºÓÖ ´½º½¼µ<br />

− .<br />

´½º½½µ<br />

ÒÚÒ<strong>×</strong>ºÇÒÒÖÕÙÖØØØÓÑÔÓÒÒØÓØÑÓÑÒØÙÑÒÓÖÑÐØÓØÓÙÒÖÝ ÓØÛÓÖÐ<strong>×</strong>ØÚÒ<strong>×</strong>¸ØØ<strong>×</strong>¸ ÓÖØ<strong>×</strong>ÓÒÓÔÒ<strong>×</strong>ØÖÒØÖÖØÛÓÔÓ<strong>×</strong><strong>×</strong>ÐÛÝ<strong>×</strong>ÒÛØÓÙÒÖÝØÖÑ<strong>×</strong><br />

X µ (τ, l) = X µ (τ, 0), ∂ σ X µ (τ, l) = ∂ σ X µ (τ, 0), γ αβ (τ, l) = γ αβ (τ, 0) .<br />

´½º½¾µ<br />

ÑÓÑÒØÙÑ<strong>×</strong>ÓÛÒØÖÓÙØÒ<strong>×</strong>ÓØ<strong>×</strong>ØÖÒÒÒØÖ<strong>×</strong>ÔØ<strong>×</strong>D+1¹ÑÒ<strong>×</strong>ÓÒÐ Ø<strong>×</strong>ØÖÒÑÓÚÖÐÝÒ<strong>×</strong>Ô¹ØѺÌ<strong>×</strong>ÓÓÓÙÒÖÝÓÒØÓÒ<strong>×</strong>ÑÒ<strong>×</strong>ØØÒÓ ÈÓÒÖÒÚÖÒº<br />

µºÌÒ<strong>×</strong>Ó<br />

∂ σ X µ (τ, 0) = ∂ σ X µ (τ, l) = 0 .<br />

Ì<strong>×</strong>ÖÐÐØÆÙÑÒÒÓÙÒÖÝÓÒØÓÒ<strong>×</strong>ÓÒØÙÒØÓÒ<strong>×</strong>X Ò 0¸Ò ´½º½¿µ<br />

ÓÒØÓÒºÖÐØÓÙÒÖÝÓÒØÓÒ<strong>×</strong>ÖÈÓÒÖÒÚÖÒÒÒÛÛÐÐÒÓØ .,DºÌ<strong>×</strong><strong>×</strong>ÒÓÛÒ<strong>×</strong>ØÖÐØÓÙÒÖÝ<br />

ÐØÖÒØÚÐÝÓÒÒÜØØÛÓÒ<strong>×</strong>ÓØ<strong>×</strong>ØÖÒ<strong>×</strong>ÓØØδX µ =<br />

X µ∣ ∣<br />

σ=0<br />

= X µ µ∣ ∣<br />

0 σ=l<br />

= X µ l<br />

,<br />

ÛÖX 0ÒX µ lÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong>Òµ=0, .<br />

<br />

µ<br />

1 ∫ ∞<br />

∣<br />

dτ(−γ) 1/2 δX µ ∂ σ ∣∣<br />

σ=l<br />

X<br />

2πα ′ µ<br />

−∞<br />

X<br />

.<br />

σ=0

ÓÒ<strong>×</strong>ÖØÑÖºÌݸÓÛÚÖ¸ÔÐÝÚÖÝÑÔÓÖØÒØÖÓÐÒ<strong>×</strong>ØÖÒØÓÖÝÒØ<strong>×</strong>ØÙÝ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÕÙØÓÒ<strong>×</strong>ÓÑÓØÓÒº Ó¹ÖÒ<strong>×</strong>ºÇÒÛÚÓ<strong>×</strong>ÒÓÙÒÖÝÓÒØÓÒ¸ÛÒÐÓÓÓÖ<strong>×</strong>ÓÐÙØÓÒ<strong>×</strong>ØÓØ ÏÛÐÐ<strong>×</strong>ØØÓØÐعÓÒÓÓÖÒØ<strong>×</strong>ÓÒØÛÓÖÐ<strong>×</strong>ØÛÖÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

ÏÐ<strong>×</strong>ÓÒ ´½º½µ<br />

ÏØØ<strong>×</strong>ÒØÓÒ<strong>×</strong>¸ØÛÚÕÙØÓÒÓÑ<strong>×</strong> ´½º½µ<br />

σ ± = τ ± σ .<br />

∂ ± = 1 2 (∂ τ ± ∂ σ<br />

´½º½µ<br />

) .<br />

ÒØÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÒÚÓÐÚÒØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖÓÑ ´½º½µ<br />

Ì<strong>×</strong>ÖØÎÖ<strong>×</strong>ÓÖÓÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ºÌÓÒ<strong>×</strong>ÖÚØÓÒÓØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ¹ ´½º½µ<br />

ÝÏÝÐÒÚÖÒ¸ØÒÖݹÑÓÑÒØÙÑÓÒ<strong>×</strong>ÖÚØÓÒÕÙØÓÒÖÙ<strong>×</strong>ØÓ ÓÑ<strong>×</strong>∂ − T ++ + ∂ + T −+<br />

´½º½µ<br />

= 0ÛØ<strong>×</strong>ÑÐÖÖÐØÓÒÓÖ−↔+ºÆÓÛ¸<strong>×</strong>ÒT 0º<strong>×</strong>f<strong>×</strong>ÖØÖÖݸ<br />

)ØÓÚÕÙØÓÒ<br />

ÓÙØØ<strong>×</strong>ÐØÖº ØÓÖ<strong>×</strong>ÙÐ<strong>×</strong>ÝÑÑØÖ<strong>×</strong>ÐØÓÚÖØÖÛÓÓ<strong>×</strong>ØÓÚÖÒØÙºÏÛÐÐ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÑÓÖ Ø<strong>×</strong>ÑÔÐ<strong>×</strong>ÒÒÒØ<strong>×</strong>ØÓÓÒ<strong>×</strong>ÖÚÕÙÒØØ<strong>×</strong>ºÌ<strong>×</strong>ÓÒ<strong>×</strong>ÖÚÕÙÒØØ<strong>×</strong>ÓÖÖ<strong>×</strong>ÔÓÒ ÌÑÔÐØÓÒ<strong>×</strong>ÓØ<strong>×</strong><strong>×</strong>ØØÑÒØÖÚÖÝÔºÓÖÒÝÙÒØÓÒf(X<br />

ÌÒÖÐ<strong>×</strong>ÓÐÙØÓÒÓØÛÚÕÙØÓÒ<strong>×</strong><strong>×</strong><br />

+<br />

ÑÔÐ<strong>×</strong>ØØØÙÖÖÒØfT ++<strong>×</strong>ÓÒ<strong>×</strong>ÖÚ<strong>×</strong>ÛÐи<strong>×</strong>Ò∂ − (fT ++ ) =<br />

´½º¾¼µ X µ (σ, τ) = X µ R (τ − σ) + Xµ L<br />

(τ + σ)<br />

<br />

∂ + ∂ − X µ = 0 ,<br />

T ++ = ∂ + X µ ∂ + X µ = 0 ,<br />

T −− = ∂ − X µ ∂ − X µ = 0 .<br />

−+<br />

∂ − T ++ = 0 .<br />

= T +− = 0

ÓÖÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>Ø<strong>×</strong>ÝÒÔÖÓÓÙÒÖÝÓÒØÓÒ<strong>×</strong>ØÒÖÐ<strong>×</strong>ÓÐÙØÓÒ<strong>×</strong>ÚÒÝ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

´½º¾½µ<br />

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)<br />

,<br />

´½º¾¾µ<br />

)<br />

.<br />

´½º¾¿µ<br />

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µ<strong>×</strong>ØØÓØÐ<strong>×</strong>ØÖÒÑÓÑÒØÙÑ<strong>×</strong>ÖÒØ<br />

nÖÓÙÖÖÓÑÔÓÒÒØ<strong>×</strong>¸ÛÛÐÐ<br />

ÓÒØÓÒ<strong>×</strong>ÓÖØÐØÒÖØÑÓÚÒÑÓ<strong>×</strong>ØÓÓÑÒÒØÓ<strong>×</strong>ØÒÒÛÚ<strong>×</strong>ºÌÖØ<br />

ÛÖx µ<strong>×</strong>ØÒØÖ¹Ó¹Ñ<strong>×</strong><strong>×</strong>ÔÓ<strong>×</strong>ØÓÒÒp ÖÑÓØÓÒÓØ<strong>×</strong>ØÖÒÒØÖÓÑ<strong>×</strong><strong>×</strong>ÒØα µ<br />

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nµº 0ØÓÒ µ ÒÒØ<strong>×</strong>ØÖÒØÒ<strong>×</strong>ÓÒκ<strong>×</strong>l = (2α ′ ) 1/2 = (1/2πκ)<br />

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0ºÐ<strong>×</strong>ÐÐݸØÚÒ<strong>×</strong>ÒÓ mÒ ÓÖÓÔÒ<strong>×</strong>ØÖÒ<strong>×</strong>H = L 0ÒÓÖÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>H L 0 + ˜L<br />

ØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖØÖÒ<strong>×</strong>ÐØ<strong>×</strong>ÒØÓØÚÒ<strong>×</strong>ÒÓÐÐÓÙÖÖÓÒØ<strong>×</strong>L ÓÖØÓÔÒ<strong>×</strong>ØÖÒ¸Ò µÚ<strong>×</strong><br />

˜L mºÁÑÔÓ<strong>×</strong>ÒØ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒØÓÒ<strong>×</strong>ØØ<strong>×</strong>Ð<strong>×</strong>ØÓØÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÐÐÓÒØÓÒM ´½º¾µ<br />

= −p µ p<br />

´½º¾µ M (α −n · α n + ˜α −n · ˜α n )<br />

<br />

X µ R = 1 2 xµ + 1 2 l2 p µ (τ − σ) + i 2 l ∑ 1<br />

n αµ n exp ( −2πin(τ − σ)<br />

l<br />

n≠0<br />

X µ L = 1 2 xµ + 1 2 l2 p µ (τ + σ) + i 2 l ∑ 1<br />

n ˜αµ n exp ( −2πin(τ + σ)<br />

l<br />

n≠0<br />

X µ = x µ + l 2 p µ τ + il<br />

∞∑<br />

n=−∞,n≠0<br />

1<br />

n αµ n e−inτÓ<strong>×</strong>(nσ) ,<br />

µ µ µ<br />

m = κ<br />

∫ l<br />

0<br />

∫ l<br />

e −2imσ T −− dσ ,<br />

˜Lm = κ e −2imσ T ++ dσ .<br />

0<br />

=<br />

2<br />

∑ ∞<br />

2 = 2 α ′<br />

M 2 = 1 α ′<br />

∞<br />

∑<br />

n=1<br />

n=1<br />

α −n · α n<br />

=

ÓÖÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>ºÌ<strong>×</strong>ØÖÑÒØÑ<strong>×</strong><strong>×</strong>ÓÚÒ<strong>×</strong>ØÖÒ<strong>×</strong>ØØÒØÕÙÒØÙÑØÓÖݺ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

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mÒ˜Lm<strong>×</strong>Ø<strong>×</strong>ÝÒÐÖÑÓÒ<strong>×</strong>ØØÑ<strong>×</strong>ÐÚ<strong>×</strong>ÐÐØÎÖ<strong>×</strong>ÓÖÓÐÖ<br />

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[L m , L n ] = (m − n)L m+n .<br />

ÒÒØ<strong>×</strong><strong>×</strong>ÔØÖÙÑÒ<strong>×</strong>ÛØÔÝ<strong>×</strong>Ð<strong>×</strong>ØØ<strong>×</strong>ØÚ<strong>×</strong>ºÏÖ<strong>×</strong><strong>×</strong>ÒØÐÐÝÐÓÓÒÓÖ<br />

ØÐعÓÒÓÓÖÒØ<strong>×</strong>Ò<strong>×</strong>Ô¹ØÑ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>º ÓÒÙ<strong>×</strong>ÒØ<strong>×</strong>ÑÒ<strong>×</strong>ØÐÝÓ<strong>×</strong>ØÖÒ<strong>×</strong>ÑÔÐÖØÓØØÓØ<strong>×</strong>ÔØÖÙѺÏÒ ÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓØÈÓÒÖÖÓÙÔÛÖÙÒØÖݺÏÛÐÐÛÓÖØ<strong>×</strong>ÓÙØÒØÐع<br />

ÌÐعÓÒÙ<strong>×</strong>ÓØÒÝ<strong>×</strong>ØØÒ ´½º¾µ<br />

X ± = (X 0 ± X D )/ √ 2 .<br />

ÓÓÖÒØ<strong>×</strong>ºÁØ<strong>×</strong>ÓÙиÓÛÚÖ¸ÄÓÖÒØÞÓÚÖÒØ<strong>×</strong>ÒØÙÒÖÐÝÒØÓÖÝØ<strong>×</strong><br />

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±ºÌÐع<br />

X + (σ, τ) = x + + p + τ .<br />

ÁÒØ<strong>×</strong>Ù¸X ÑÒ<strong>×</strong>ÓÒÓØ<strong>×</strong>Ô¹ØÑÒØÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÐÐÓÒØÓÒÓÖØØÓÖÝØÓÒÓÑÐݹֺ ÓØÒÝÙÜÒÖÓÑ<strong>×</strong>ÄÓÖÒØÞÒÚÖÒغÌÓÒØÓÒ<strong>×</strong>ÓÖØØÓÖÝØÓÔÖ<strong>×</strong>ÖÚ ÄÓÖÒØÞÒÚÖÒØÙÖÒÓÙØØÓÒØÐØÓØÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>´Û<strong>×</strong>ÔÓÓÖÐÖµÓÒØ<br />

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+<br />

ÓØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖºÌ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÔÝ<strong>×</strong>ÐÐÝÑÒØÚÖØÓÒ<strong>×</strong>ÓØ ÐÓÒØÙÒÐÖ<strong>×</strong>ÓÖÓѸÖÐÑÒظÐÚÒÓÒÐÝØ(D−1)ØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>ÖØÓÒ<strong>×</strong>º ÁÒÓÓ<strong>×</strong>ÒØÐعÓÒÙ¸ÛÖ¸ÒظÐÑÒØÒØØÛÓÐÓÒØÙÒÐÖ<strong>×</strong> ÑÒÓØÛÓÖÐ<strong>×</strong>ØÒØØÖØ<strong>×</strong>Ô¹ØÑØÒÒØØÓØ<strong>×</strong>ÙÖ¸ººØ<br />

ÓÖÓÑÒÕÙÒØÞÒØÖÑÒÒØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>Ö<strong>×</strong>ÓÖÓѺ µ<strong>×</strong>ÕÙÒØÙÑÓÔÖØÓÖ<strong>×</strong> Ì<strong>×</strong>ØÒÖÛÝØÓÕÙÒØÞØØÓÖÝ<strong>×</strong>ØÓÒØÖÔÖØØX <br />

1)

ÒÖÔÐØÈÓ<strong>×</strong><strong>×</strong>ÓÒÖØ<strong>×</strong>ÝÓÑÑÙØØÓÖ<strong>×</strong>ºÌÕÙйØÑÒÓÒÐÓÑÑÙØØÓÒ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÖÐØÓÒ<strong>×</strong>ÖØÒÚÒÝ<br />

Ì<strong>×</strong>Ú¸ÓÖØÓÑÑÙØØÓÒÖÐØÓÒ<strong>×</strong>ÓØÓ<strong>×</strong>ÐÐØÓÖÑÓ<strong>×</strong>ØÓÐÐÓÛÒÓÑÑÙØØÓÒ ´½º¾µ<br />

ÖÐØÓÒ<strong>×</strong><br />

[P<br />

[X µ (σ, τ), X ν (σ ′ , τ)] = [P τ µ (σ, τ), P τ ν (σ′ , τ)] = 0 .<br />

´½º¾µ<br />

[α<br />

ÛØ<strong>×</strong>ØصÒØÓÒÒ<strong>×</strong>ØØÓØÖÒØÖ¹Ó¹Ñ<strong>×</strong><strong>×</strong>ÑÓÑÒظ<br />

k〉¸<strong>×</strong>ÒØÓÒÒÐØÝØÐÓÛÖÒÓÔÖØÓÖ<strong>×</strong>´<strong>×</strong><strong>×</strong>Ø<br />

[α m µ , ˜αµ n ] = 0<br />

ÌÖÓÙÒ<strong>×</strong>ØØ|0;<br />

ÓÔÖØÓÖ<strong>×</strong>º k〉ÛØØÖ<strong>×</strong>Ò ÒÖÐ<strong>×</strong>ØØÒØÓ<strong>×</strong>ÔF kÒÙÐØÝØÒÓÒ|0;<br />

)¸n∈NÒÐÐÔÓ<strong>×</strong><strong>×</strong>Ð ´½º¿½µ<br />

Ó<strong>×</strong>ÐÐØÓÖ<strong>×</strong>ÖÔÖ<strong>×</strong>ÒØÒÒÒØÒÙÑÖÓÒØÖÒÐÖ<strong>×</strong>ÓÖÓѺÌÓÚÕÙØÓÒ ÌÒØÖ¹Ó¹Ñ<strong>×</strong><strong>×</strong>ÑÓÑÒØÖÙ<strong>×</strong>ØØÖ<strong>×</strong>ÓÖÓÑÓÔÓÒØÔÖØиÛÐØ<br />

Nº<br />

|ǫ;<br />

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m 1 , m 2 , . . .,m n<br />

m i<br />

0〉<strong>×</strong>ØÖÓÙÒ<strong>×</strong>ØØÓ<br />

∈<br />

ÐÓ<strong>×</strong><strong>×</strong>ØÖÒØ<strong>×</strong>ØÒ<strong>×</strong>ÓÖÔÖÓÙØÓØÐØÒÖعÑÓÚÒÓ<strong>×</strong>Ô<strong>×</strong>ºÚÖÝÑÔÓÖØÒØ ÔÔÖÒÓÚÐÐØÛØÒØ<strong>×</strong>ÔÓ<strong>×</strong>ØØ<strong>×</strong>Ó<strong>×</strong>ÒÐ<strong>×</strong>ØÖÒº ÌÓÔÒ<strong>×</strong>ØÖÒÓ<strong>×</strong>Ô<strong>×</strong><strong>×</strong>ÙÑÓÚÖØÓ<strong>×</strong>Ô<strong>×</strong>ÓÚÖÐÐÑÓÑÒØkºÓÖØ<br />

ÓÖÑ<strong>×</strong>ØÀÐÖØ<strong>×</strong>ÔÓ<strong>×</strong>ÒÐÓÔÒ<strong>×</strong>ØÖÒºÌ<strong>×</strong>ØØ|0;<br />

ÒØÚÒÓÖѺÌÔÝ<strong>×</strong>ÐÓ<strong>×</strong>ÔÛÐÐ<strong>×</strong>Ù<strong>×</strong>ÔÓØÙÐÐÓ<strong>×</strong>ÔºÏÒ ÑÒÙ<strong>×</strong><strong>×</strong>ÒÒØÖÓÑÑÙØØÓÒÖÐØÓÒ<strong>×</strong>ÒØÖÓÖØÓ<strong>×</strong>ÔÓÒØÒ<strong>×</strong><strong>×</strong>ØØ<strong>×</strong>ÛØ ÔÓÒØØÓÓ<strong>×</strong>ÖÚ<strong>×</strong>ØØØ<strong>×</strong>Ó<strong>×</strong>Ô<strong>×</strong>ÒÓØÔÓ<strong>×</strong>ØÚÒغÌØÑÓÑÔÓÒÒØ<strong>×</strong>Ú ½¼<br />

µ<br />

τ (σ, τ), Xν (σ ′ , τ)] = −iδ(σ − σ ′ )η µν ,<br />

µ<br />

m, α ν n] = mδ m+n η µν<br />

[˜α µ m , ˜αν n ] = mδ m+nη µν<br />

P µ |0; k〉 = k µ |0; k〉 ,<br />

α µ m<br />

|0; k〉 = 0 m > 0 . ´½º¿¼µ<br />

k〉 = ǫ(k, m 1 , m 2 , . . .,m n ) α µ 1<br />

−m 1 · · ·α µn<br />

−m n<br />

|0; k〉,

ØÓÙ<strong>×</strong>ØÎÖ<strong>×</strong>ÓÖÓÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ØÓÜÒÒÚÖÒØ<strong>×</strong>Ù<strong>×</strong>ÔÖÓÑØÙÐÐÓ<strong>×</strong>ÔºÌ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÛÐÝÓÒØÕÙÒØÙÑÓ<strong>×</strong>ÔºÌÓÙÖÖÓÒØ<strong>×</strong>ÓØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ ÎÖ<strong>×</strong>ÓÖÓÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÒØÐ<strong>×</strong><strong>×</strong>ÐØÓÖÝÑÓÙÒØØÓØÖÕÙÖÑÒØØØØÓÑÔÓÒÒØ<strong>×</strong><br />

ÛÖÚÒÝ −−¸ÚÒ<strong>×</strong>ºÏÒØÓÑÔÓ<strong>×</strong><strong>×</strong>ÑÐÖÓÒØÓÒ<strong>×</strong> ÓØÒÖݹÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖ¸T<br />

Ò<strong>×</strong>ÑÐÖÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÓÖ˜LmÒØ<strong>×</strong>ÓÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>¸ÛÛÖÕÙÖÒÒÐØ<br />

++ÒT ´½º¿¾µ<br />

L<br />

0ÓÔÖØÓÖ<strong>×</strong> mÖÓÔÖØÓÖ<strong>×</strong>¸<strong>×</strong>ÓÓÒÑÙ<strong>×</strong>ØÖ<strong>×</strong>ÓÐÚ<br />

0¸ÛÒÓÒÐÝÛÓÖÖÝ ÐÐØÔÝ<strong>×</strong>Ð<strong>×</strong>ØØ<strong>×</strong>ºÁÒØÕÙÒØÙÑØÓÖÝØα ØÓÖÖÒÑÙØ<strong>×</strong>ºËÒα m−nÓÑÑÙØ<strong>×</strong>ÛØα nÙÒÐ<strong>×</strong><strong>×</strong>m =<br />

ÓÙØØÓÔÖØÓÖL 0ºÏÒØL 0Ò˜L0ÙÔØÓÒÙÒØÖÑÒ ´½º¿¿µ<br />

´½º¿µ<br />

ÒÒØÔÝ<strong>×</strong>Ð<strong>×</strong>ØØÓÒØÓÒ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓL ÓÒ<strong>×</strong>ØÒØ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>|φ〉<br />

ÔÝ<strong>×</strong>ÐÓ<strong>×</strong>ÔÓÔÓ<strong>×</strong>ØÚÒØÒÓÖѺÌÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÐÐÓÒØÓÒÛÐÐÐ<strong>×</strong>ÓÙÒÖÓ ÑÓØÓÒÙØÓØÓÒ<strong>×</strong>ØÒØa<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ÌÓÒ<strong>×</strong>ØÒØa<strong>×</strong>ÙÒØÖÑÒÓÖÒÓÛ¸ÒÛÐÐÜÙ<strong>×</strong>ÒØÓÒØÓÒØØØ<br />

∈ F phy m − aδ m,0 )|φ〉 = 0 m ∈ N .<br />

ÓÖÓÔÒ<strong>×</strong>ØÖÒ<strong>×</strong>¸<strong>×</strong>ÓØØØÓ<strong>×</strong>ÐÐØÓÖÖÓÙÒ<strong>×</strong>ØØ<strong>×</strong>Ñ<strong>×</strong><strong>×</strong><strong>×</strong>ÕÙÖ−2a¸ÒÜØØÓÒ<strong>×</strong> ´½º¿µ<br />

ÚÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÕÙÖÐÖÖØÒØ<strong>×</strong>ÝÒÝÑÙÐØÔÐÓ2ºÓÖÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>ØÓÒØÓÒ<br />

´½º¿µ ÓÑ<strong>×</strong>´ÛØα ′ ½½<br />

|φ〉 ∈ <br />

= 1/2µ<br />

˜F<br />

phy<br />

m = 1 2<br />

∞∑<br />

α m−n · α n ,<br />

−∞<br />

∞∑<br />

L 0 = 1 2 α2 0 + α −n · α n ,<br />

n=1<br />

(L m − aδ m,0 )|φ〉 = 0 m ∈ N ,<br />

(˜L<br />

M 2 = −2a + 2<br />

∞∑<br />

α −n · α n ,<br />

n=1<br />

∞∑<br />

∞∑<br />

M 2 = −8a + 8 α −n · α n = −8a + 8 ˜α −n · ˜α n .<br />

n=1<br />

n=1

0ÛØ ÔØÖ½ºËØÖÒÌÓÖÝ<br />

Ì<strong>×</strong><strong>×</strong>ØÓÒÐÝÓÒ<strong>×</strong>ØÖÒØÕÙØÓÒØØÓÙÔÐ<strong>×</strong>ØÐØÒÖØÑÓÚÒÑÓ<strong>×</strong>ºÈÝ<strong>×</strong>Ð ´½º¿µ ÁÑÔÓ<strong>×</strong>ÒØÓÒØÓÒ(L 0 − ˜L 0 )|φ〉 =<br />

<strong>×</strong>ØØ<strong>×</strong>ÖÓÙÒÝÓÓ<strong>×</strong>ÒÒÔÒÒØÐÝØÐعÑÓÚÒÒÖعÑÓÚÒ<strong>×</strong>ØØ<strong>×</strong>ÓÓ<strong>×</strong>й<br />

−−ºÌÔÝ<strong>×</strong>Ð<strong>×</strong>ØØ<strong>×</strong>ÖÖÕÙÖØÓÒÒÐØÝ<br />

mÒ˜LmÓÖÖ<strong>×</strong>ÔÓÒØÓØÖÑ<strong>×</strong>ÓÒØ ÐØÓÒ¸<strong>×</strong>ÙØØÓØÓÚÓÒ<strong>×</strong>ØÖÒغÌÓØÖL ÒÓÒÞÖÓÖÕÙÒÝÒT ÏÒØÒÙÑÖÓÔÖØÓÖ<strong>×</strong>´ÓÖÓ<strong>×</strong>ÐÐØÓÖÐÚе<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

++ÒT ´½º¿µ ØÔÓ<strong>×</strong>ØÚÖÕÙÒÝÓÑÔÓÒÒØ<strong>×</strong>L m<br />

´½º¿µ<br />

|φ〉 = 0 m = 1, 2, . . .<br />

ÏÖØÒÒØÖÑ<strong>×</strong>ÓØÒÙÑÖÓÔÖØÓÖ<strong>×</strong>ØÎÖ<strong>×</strong>ÓÖÓÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÓÑ<br />

1ÛØÛØn¸ÔÔÐØÓØ ÌÝÓÙÒØØÒÙÑÖÓÓÔÖØÓÖ<strong>×</strong>α −nÒ˜α −n , n ≥<br />

k〉º<br />

ÐÓ<strong>×</strong><br />

ÖÓÙÒ<strong>×</strong>ØØ|0;<br />

ÁØØÙÖÒ<strong>×</strong>ÓÙØ´Ù<strong>×</strong>ÒØÒÓ¹Ó<strong>×</strong>ØØÓÖѵØØØØÓÖÝ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÒÓÒÐÝØ ÓÔÒ ´½º¼µ<br />

<strong>×</strong>Ô¹ØÑÑÒ<strong>×</strong>ÓÒ<strong>×</strong>26ÒØÚÐÙÓØÓÒ<strong>×</strong>ØÒØa=1ºÏÛÐÐÒÓØÔÖÓÚÓÖ ÑÓØÚØØÛÝØ<strong>×</strong>Ò<strong>×</strong>ÓÛÒºÀÓÛÚÖÖÐÝ<strong>×</strong>ÝÓÑÔÙØØÓÒØÓØÒÓÖÑ<br />

(k 2 + M 2 )|φ〉 = 0 M 2 = 8N − 8a ÒN Ó<strong>×</strong>ØØ<strong>×</strong>ÓÖÐÓÛÑ<strong>×</strong><strong>×</strong>ÐÚÐ<strong>×</strong><strong>×</strong>ÓÛ<strong>×</strong>ØÒÓÖØ<strong>×</strong>ØÛÓÓÒØÓÒ<strong>×</strong>ºÌÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒÒ<br />

(k<br />

ÖØÐÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒº ÇÔÒËØÖÒ 26Òa=1<strong>×</strong>ÐÐØÖØÐÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒºÐÓÛÛÚØ<strong>×</strong>ÔØÖÙÑÓØ<br />

D =<br />

ØÝÓÒ<strong>×</strong>´ÔÖØÐ<strong>×</strong>ØÖÚÐÐÒ<strong>×</strong>ØÖØÒÐص¸ÒÄÓÖÒØÞ<strong>×</strong>ÐÖ<strong>×</strong> 0ºÌ<strong>×</strong><strong>×</strong>ØØ<strong>×</strong>Ú −2¸ÒØÝÖ ´µÓÖN = 0¸ÓÖÖ<strong>×</strong>ÔÓÒÒØÓ<strong>×</strong>ØØ<strong>×</strong>ÓØÓÖÑ|0; 2 =<br />

½¾ ´µÓÖN = 1¸ÓÖÖ<strong>×</strong>ÔÓÒÒØÓ<strong>×</strong>ØØ<strong>×</strong>ÓØÓÖÑǫ −1 |0; k〉¸M2 =<br />

N ≡<br />

∞∑<br />

α −n · α n =<br />

n=1<br />

∞∑<br />

˜α −n · ˜α n .<br />

n=1<br />

∞∑<br />

α −n · α n , Ñ ≡<br />

n=1<br />

∞∑<br />

˜α −n · ˜α n .<br />

n=1<br />

2 + M 2 )|φ〉 = 0 M 2 = 2N − 2a<br />

k〉¸M ·α<br />

=<br />

Ñ|φ〉 = 0

ØÖ<strong>×</strong>ÓÖÓÑÓÑ<strong>×</strong><strong>×</strong>Ð<strong>×</strong><strong>×</strong>ÚØÓÖÔÖØк ÔØÖ½ºËØÖÒÌÓÖÝ<br />

Ò 2ÓÙÖºÌÝÖ ´½º½µ ´µÓÖN<br />

ØÒ<strong>×</strong>ÓÖº 2ºÌ<strong>×</strong><strong>×</strong>ØØ<strong>×</strong>ÚØÖ<strong>×</strong>ÓÖÓÑÓÑ<strong>×</strong><strong>×</strong>Ú<strong>×</strong>ÓÒ¹ÖÒ<br />

= 2ØÖ<strong>×</strong>Ø<strong>×</strong>ØØ<strong>×</strong>ÛØÔÓ<strong>×</strong>ØÚ(Ñ<strong>×</strong><strong>×</strong>) α −2|0; µ k〉 −1α µ −1 ν |0; k〉 , 0ÛØÖÒ<strong>×</strong>ÓÖÑÙÒÖ ÛØM 2 =<br />

ÒÖعÑÓÚÖ<strong>×</strong>ÒØÖ<strong>×</strong>ØÐÚÐÑØÒÓÒØÓÒÖÐØÒØØÛÓº ÐÓ<strong>×</strong>ËØÖÒÓÖÐÓ<strong>×</strong><strong>×</strong>ØÖÒ<strong>×</strong>ØÖÖØÛÓ<strong>×</strong>Ø<strong>×</strong>ÓÑÓ<strong>×</strong>ÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÐع ØÚÖÓÙ<strong>×</strong>ØÒ<strong>×</strong>ÓÖÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓØÄÓÖÒØÞÖÓÙÔº ´ÚµÓÖÖÚÐÙ<strong>×</strong>ÓNØÖÓÙÖÚÖÓÙ<strong>×</strong><strong>×</strong>ØØ<strong>×</strong>ÛØM2 ><br />

−2¸<strong>×</strong>ÓØÝÖØÝÓÒ<strong>×</strong>¸<br />

ØÓØØÒ<strong>×</strong>ÓÖÔÖÓÙØÓÓÒÐعÑÓÚÒÒÓÒÖعÑÓÚÒÑ<strong>×</strong><strong>×</strong>Ð<strong>×</strong><strong>×</strong>ÚØÓÖºÓÖ¹ ÒÄÓÖÒØÞ<strong>×</strong>ÐÖ<strong>×</strong> ´µËØØ<strong>×</strong>ÓÖÖ<strong>×</strong>ÔÓÒÒØÓ|0; k〉 = |0; k〉 L ⊗ |0; k〉 RÚM 0ÓÖÖ<strong>×</strong>ÔÓÒÒ<br />

=<br />

ÒÓÖѺÌ<strong>×</strong>ÝÑÑØÖØÖÐ<strong>×</strong><strong>×</strong>ÔÖØÓǫÚ<strong>×</strong>ØÖÚØÓÒºÌÒØ<strong>×</strong>ÝÑÑØÖÔÖØ Ö<strong>×</strong>ÔÓÒÒØÓØØÖÔÖØÓǫØÖ<strong>×</strong>ÄÓÖÒØÞ<strong>×</strong>ÐÖ¸ØÐØÓÒ¸ÛØÔÓ<strong>×</strong>ØÚ ´µÓÖN = 1¸ØÖÓÙÖ<strong>×</strong>ØØ<strong>×</strong>ÓØÓÖÑǫ α −1˜αν µ −1 |0; k〉ÚM =<br />

0ÛØÖÒ<strong>×</strong>ÓÖÑÙÒÖ<br />

µνº ÓǫÚ<strong>×</strong>ÖÒØÛÓÒØ<strong>×</strong>ÝÑÑØÖØÒ<strong>×</strong>ÓÖÙ<strong>×</strong>ÙÐÐÝÒÓØÝB ½º¾ØÚÖÓÙ<strong>×</strong>ØÒ<strong>×</strong>ÓÖÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓØÄÓÖÒØÞÖÓÙÔº<br />

η¹ÔÖÓÙØ<strong>×</strong>ÖÓÑÓÙÒØÒÓ<strong>×</strong>ÐÐØÓÖÜØØÓÒ<strong>×</strong><br />

´µÓÖÖÚÐÙ<strong>×</strong>ÓNØÖÓÙÖÚÖÓÙ<strong>×</strong><strong>×</strong>ØØ<strong>×</strong>ÛØM2 ><br />

ÁÒØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>¸ÛÛÐÐÚÓ<strong>×</strong>ÓÒØÓÓÒ<strong>×</strong>ÖØÓÐÐÓÛÒØÖÓÚÖØÓÔÒ<strong>×</strong>ØÖÒ´ÓÖ<br />

kÒØÖÓÙÖÐÖ<br />

ÁÒØÐعÓÒÙ¸ØÙÐÐÓ<strong>×</strong>Ô<strong>×</strong>ÒÖØÝØØÓÒÓÐÐÓÑÒØÓÒ ´½º¾µ ÐعÑÓÚÒ<strong>×</strong>ØÓÖÓØÐÓ<strong>×</strong>Ó<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒµÔÝ<strong>×</strong>ÐÓ<strong>×</strong>ÔF ÌÖF k<br />

(N)ÒÓØØÒÙÑÖÓ ÓØÓ<strong>×</strong>ÐÐØÓÖÑÓ<strong>×</strong>ÓؾØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ºÄØP 24<br />

½¿<br />

α<br />

µν<br />

(<br />

q<br />

L 0 −1 ) .<br />

2 2

Ó<strong>×</strong>ÐÐØÓÖÜØØÓÒ<strong>×</strong>ØÐÚÐNÖ<strong>×</strong>ÒÖÓÑØ24ØÖÒ<strong>×</strong>ÚÖ<strong>×</strong><strong>×</strong>ÐÖ<strong>×</strong>ºÌÒ¸ÓÒ<strong>×</strong> ÔØÖ½ºËØÖÒÌÓÖÝ<br />

k〉ºÇÒÒ<strong>×</strong>ÓÛØØ ´½º¿µ ÌÖF<br />

ÏØÙ<strong>×</strong><strong>×</strong>ØØÑÓÙÐÓØÖÓÙÒ<strong>×</strong>ØØÒÖݸØÒÚÖ<strong>×</strong>ÓØÔÖÓÙØÓÒ<br />

k<br />

ÛÖE 0<strong>×</strong>ØL 0ÒÚÐÙÓØÖÓÙÒ<strong>×</strong>Øظ|0; ´½ºµ ÌÖF<br />

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24<strong>×</strong>ØÒÖØÒÙÒØÓÒÓØÓ<strong>×</strong>ÐÐØÓÖÒÖÝØÚÖÓÙ<strong>×</strong>ÐÚÐ<strong>×</strong>º<br />

k<br />

24ºÆÓÛÓÒ<strong>×</strong>ÖØØÛ<strong>×</strong>ØØÖ<br />

ØÙÒØÓÒ<strong>×</strong>¸η(τ)<br />

´½ºµ<br />

ÖÐØÔ<strong>×</strong><strong>×</strong>µ<strong>×</strong>ÚÒÝ <strong>×</strong>ÑÔÐÓÑÔÙØØÓÒ<strong>×</strong>ÓÛ<strong>×</strong>ØØØØÛ<strong>×</strong>ØØÖ´ÒÓÖÒØÖÓÙÒ<strong>×</strong>ØØÒÖÝÒ<br />

γ = 1 a 1<br />

2 a2 · · ·m amÛØ∑ i=1 ia i =<br />

ÌÖF k<br />

´½ºµ<br />

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½º¿ 24ºÁØØÙÖÒ<strong>×</strong>ÓÙØØØÔÖ<strong>×</strong>ÐÝ<strong>×</strong>ÙÓÙÒØÒÔÖÓÐÑ<strong>×</strong>Ö<strong>×</strong>ÒØÓÙÒØÒÓ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒÖØÒÑÓÐ<strong>×</strong>º ÌÖF k<br />

<strong>×</strong>Ôγ= 1<br />

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ÒÑÐݸØÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒÓÒÖ<strong>×</strong>ÓÈË<strong>×</strong>ØØ<strong>×</strong>ÒØÛÓÑÐ<strong>×</strong>Ó<strong>×</strong>ØÖÒ ØÒØÖÓÙØÓÒ¸ÒÔØÖ2ÛÖÝÖÚÛØÔÖÓÐÑÛÛ<strong>×</strong>ØÓ<strong>×</strong>ØÙÝÒØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>¸<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØ<strong>×</strong>ØÓÖ<strong>×</strong>ÒÖÚÛØÜÔÐØÓÙÒØÒÖÖÓÙØÝÚ 2¹ÈË ØÓÖÝØÀÄÑÓÐ<strong>×</strong>ÒØØÝÔÁÁÑÓÐ<strong>×</strong>ºÏ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÓÙÒØÒÓØ1 ½ Ò1<br />

m<br />

(<br />

q<br />

L 0 −1 ) = q E 0−1<br />

∞∑<br />

P 24 (N) q N ,<br />

N=0<br />

( )<br />

(<br />

q<br />

L 0<br />

) 24<br />

= q<br />

E 0<br />

1<br />

∏ ∞<br />

n=1 (1 − = q 1 E0<br />

qn ) η(τ) . 24<br />

(<br />

g q<br />

L 0 −1 ) .<br />

(<br />

g q<br />

L 0 −1 1<br />

) ∼ ∏ m<br />

i=1 η(a iτ) ≡ 1<br />

g γ (τ) .

ÔØÖ½ºËØÖÒÌÓÖÝ<br />

ÑÓÐ<strong>×</strong>º<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖ<strong>×</strong>ØÖÒØÓÖ<strong>×</strong>ºÏÒØÔØÖÛØ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØÀÄ<br />

ÒÐÐÝØØÓÖÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÏ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ØÖÙØÙÖÒÖÔÖ<strong>×</strong>ÒØØÓÒ ÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸ÛÖÙÐÐÝÒØÖÓÙÒÄÐÖ<strong>×</strong>Ò ÁÒÔØÖ3¸<strong>×</strong><strong>×</strong>йÓÒØÒÖÚÛÓØ<strong>×</strong>ÙØÓÄÐÖ<strong>×</strong>ºËØÖØÒÛØ<br />

ÒËÒ½℄ÒÐ<strong>×</strong><strong>×</strong>ÓN =<br />

ÖÖÚÛÓËÒ³<strong>×</strong><strong>×</strong>ØÙÝÓØÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓ1<br />

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2ÓÖÖØÓÒ<strong>×</strong>ØÓØ ÓØ1 2¹Ò1<br />

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ÁÒÔØÖ6¸ÛÑØÓÒÒØÓÒØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÚÒØÑÓÙ¹<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>º<br />

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S BH = A/(4G N ),<br />

ØÞÖÓÀÛÒØÑÔÖØÙÖ<strong>×</strong>ÚÒÝ ØÒÖÝÓØ<strong>×</strong>ØØ<strong>×</strong>ÖÖÝÒØ<strong>×</strong>ÖÓÒÙÖØÓÒ¸ØÒØ<strong>×</strong>ØØ<strong>×</strong>ØÐÒØÖÓÔÝ<br />

ÛÖG ´¾º¾µ<br />

ÑÖÓ<strong>×</strong>ØØØÞÖÓØÑÔÖØÙÖºÁQÐÐ<strong>×</strong>Ø<strong>×</strong>ØÓÖ<strong>×</strong>ÖÖÝ<strong>×</strong>ØظÒd(Q)<br />

ØÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒÒÒÓØÖÖÓÙØÓÖÐÐÐÓÐ<strong>×</strong>ºÇÒÒ<strong>×</strong>ØÓÛÓÖÒ<br />

stat¸<strong>×</strong>ÒÓÚ¸ÛÓÙÐÖÕÙÖÓÒØÓÚØÐÓÐØÞÖÓÀÛÒ<br />

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S (Q) =ÐÒd(Q).<br />

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´¾º¿µ<br />

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ØÖÑ<strong>×</strong>ÙÔØÓØÛÓÖÚØÚ<strong>×</strong>S<br />

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=<br />

.<br />

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∫<br />

d 4√ [<br />

1<br />

]<br />

−g R − 1 4<br />

16πG F µνF µν<br />

N<br />

ds 2 = −f(ρ)dτ 2 + f −1 (ρ)dρ 2 + ρ 2 dΩ 2 ,

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)Ý ´¾ºµ<br />

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t = λτ/a 2 , r = λ −1 (ρ − a),<br />

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4π<strong>×</strong>Òθ .<br />

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θdφ<br />

F ρτ = q e<br />

4πρ 2,<br />

F θφ = q m<br />

f(ρ) = 1 − 2G NM<br />

+ G N<br />

ρ 4πρ 2(q2 e + qm) 2 .<br />

= 1<br />

S BH = 1 4 (q2 e + q 2 m) .<br />

=<br />

√<br />

GN<br />

( )<br />

2 = a 2 − r 2 dt 2 + dr2<br />

r 2<br />

F r,t = q e<br />

4π ,<br />

4π (q2 e + q 2 m) .<br />

F θφ = q m<br />

θdφ 2 ),

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0ÐÑØÒ´¾º½µ ÒÐ<strong>×</strong>ÓÔÙØØÕÙÐØÓÜÚÐÙ¸Ðν= 0ºÁÒØν= ÓÖÑÙÐÛØÐØÝØÖ(−1) m¸ÛÛÐÐÔÖÓØÓÙØØ1 ÓÒÐÝØ1 2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÛÓ<strong>×</strong>ÒÖÝ<strong>×</strong>ÚÒÝ´¾º½¾µºÌÒØν →<br />

¾<br />

Φ 10 (ρ, σ, ν) = e 2πi(ρ+σ+ν)<br />

l<br />

χ K3 (τ, z) =<br />

< −1º<br />

64<br />

∏<br />

(k,l,m)>0<br />

∈<br />

∑<br />

h≥0,m∈Z<br />

Φ 10 (Z) = ∑ k,l,m<br />

l,<br />

(<br />

l<br />

c(4h − m 2 ) e 2πi(hτ+mz) ,<br />

D(k, l, m)e −2πi(kρ+lσ+mν) ,<br />

e ,q m ) = D( 1 2 q2 e, 1 2 q2 m,q e · q m ) .

ÛØ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ÌÓÚÓÖÑÙÐ<strong>×</strong>ÓÛ<strong>×</strong>ØØØÒÙ<strong>×</strong>¹ØÛÓ<strong>×</strong>ÙÖÔÖÑØÖÞÝZØÓÖ<strong>×</strong>ÒØÓØÛÓ ´¾º¾½µ<br />

<strong>×</strong>ÒÓØÖÒÓÒ¹ØÖÚÐÓÖØÒÖÝÓÖÑÙиÛÒÓÑÔÖØÑÖÓ<strong>×</strong>ÓÔ<br />

2¹ÈËÑÓÙк<br />

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ÛÛÐÐ<strong>×</strong>ØÙÝÓÖØÖ<strong>×</strong>ØÓØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÀÄÒØÝÔÁÁÑÓÐ<strong>×</strong>ºÏÛÐÐÖÚ ÏØØ<strong>×</strong>ÖÒØÖÓÙØÓÒÒØÓØÎÎÔÖÓÔÓ<strong>×</strong>иÛÒÓÛØÙÖÒØÓØØÛÓÑÓÐ<strong>×</strong>ØØ<br />

Ø<strong>×</strong>ÓÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>¸ÒÓÐÐÓÛÒÚ¸ÂØÖÒËÒ<strong>×</strong>ÓÛÓÛØÑÓÙÐÖÓÖÑ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÓÖ<br />

ÑÓÐ<strong>×</strong>ºÏÖ<strong>×</strong>Ø<strong>×</strong>ØÖØÝ<strong>×</strong>ÖÒ¸ÖݸØÀÄÒØÝÔÁÁÑÓÐ<strong>×</strong>ÐÓÛ¸ÓÖ ÏÛÐÐÐ<strong>×</strong>ÓÐÓÓØØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒØÝÓÒÒÖ<strong>×</strong>ÒØØÝÔÁÁ ÒÕÙ<strong>×</strong>ØÓÒ<strong>×</strong>ÒÖØÝÒÜÔÐØÓÙÒØÒÓØÐÓÐÑÖÓ<strong>×</strong>ØØ<strong>×</strong>ÒØÑÓк ÓÖÑÙÐ<strong>×</strong>ÑÐÖØÓØÓÚÓÒÓÖØÒÖ<strong>×</strong>ÓØ1 2¹ÈËÒ1<br />

ÑÓÚÒÓÒØÓÙÒÖ<strong>×</strong>ØÒÒØÜÔÐØÓÙÒØÒÓØÝÓÒÒÖ<strong>×</strong>ÒD=4¸<br />

¾º4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÑÓк<br />

ÌÀÄÅÓÐ<strong>×</strong><br />

N =<br />

NÓÖÓÐÚÒÝØØÖÒ<strong>×</strong>ÓÖÑØÓÒÓÖÖ<strong>×</strong>ÔÓÒÒØÓ1/NÙÒØÓ<strong>×</strong>Ø<br />

NÓÖÓÐ<strong>×</strong>ÔÖÓÚÙ<strong>×</strong>ÛØ<br />

<strong>×</strong>ØÙݾ℄ºËØÖØÒÖÓÑ<strong>×</strong>ܹÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ØÖÒ¹<strong>×</strong>ØÖÒÙÐØݸÓÒ<strong>×</strong><strong>×</strong>ØØØØÖÓØ 20)<strong>×</strong><strong>×</strong>ÓØ<br />

ÌØÖÓØ<strong>×</strong>ØÖÒÓÑÔØÓÒT 6ÒØ<strong>×</strong><strong>×</strong>ÝÑÑØÖZ ÓÙÖ¹ÑÒ<strong>×</strong>ÓÒÐÓÑÔØØÓÒ<strong>×</strong>ÛØN = 4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺÏÖØÒT 6<strong>×</strong>T 4<br />

ÓÒ<strong>×</strong>ÖØZ ÒŜ <strong>×</strong>ÆÙÐÒÒÚÓÐÙØÓÒÓÑÒÛØØ1/N<strong>×</strong>ØÓ˜S1ºÌÖ<strong>×</strong>ØÖ<strong>×</strong>ÖÔØÓÒØØ<br />

4ºÌ<strong>×</strong>Ð<strong>×</strong>ØÓØÀÄÑÓÐ<strong>×</strong>ØØÛÛÐÐ<br />

1Ò<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>Z NÒÚÓÐÙØÓÒÓØÆÖÒÐØØÓ<strong>×</strong>ÒØÙÖ(4, ÛØØØÖÓØ<strong>×</strong>ØÖÒÓÑÔØÓÒT<br />

ÖÖÝÓÙØØÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒºÙÖ¾º½<strong>×</strong>ÙÑÑÖÞ<strong>×</strong>ØÒÓÙÐØ<strong>×</strong>º<br />

<strong>×</strong>ØÖÒÓÑÔØÓÒT ÌÐÓÛ¹ÒÖÝØÓÖÝÓÒ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÓØÓÐÐÓÛÒÓ<strong>×</strong>ÓÒÐ<strong>×</strong><br />

1ØÓ˜S1ÒÓÐÐÓÛÒØÝÒ˹ÙÐØÝØ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>ØÓ<br />

4 <strong>×</strong>˜S 1 <strong>×</strong>S 1<strong>×</strong>ÙÐØÓØØÝÔÁÁ<strong>×</strong>ØÖÒÓÑÔØÓÒK3<strong>×</strong>˜S<br />

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lim<br />

ν→0<br />

e iπq·Z·q<br />

Φ 10 (Z) −→ 1 e iπρq2 e<br />

e iπσq2 m<br />

ν 2 η(ρ) 24 η(σ) . 24<br />

1¸ <strong>×</strong>Ŝ1 <strong>×</strong>S 1º <strong>×</strong>S<br />

1<br />

∗ N

ÌÝÔÁÁÓÒ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

<strong>×</strong>ÖÔØÓÒ¿ ÙÐÌÝÔÁÁÓÒ Ë<br />

1ÌÙÐÞ 1ÌÝÔÁÁÓÒ <strong>×</strong>ÖÔØÓÒ½ ÙÐØÝÀØÖÓØÓÒ <strong>×</strong>ØÖ¹<strong>×</strong>ØÖ <strong>×</strong>ÖÔØÓÒ¾<br />

´<strong>×</strong>ÝÑÑØÖÓÖÓÐ<strong>×</strong>ÓØØÖÓØ<strong>×</strong>ØÖÒºº¸ÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>µÛÐÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒ<strong>×</strong> ÙÖ¾º½ÌÒÓÙÐØ<strong>×</strong>ÒØÀÄÑÓкÌÓÚÒ<strong>×</strong>ÜÔØØÓÓÐ ÖÖÓÙØÒ<strong>×</strong>ÖÔØÓÒ¿º<br />

K3<strong>×</strong>S 1 <strong>×</strong> ˜S 1<br />

˜S eS 1ØÓb<br />

ÖÚÔÓØÓÒ<strong>×</strong>Ò<br />

1ºÌÕÙÒØÞØÓÒÓÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÔÒ<strong>×</strong>ÖÔØÓÒ¾ ØÖZ N¹ÓÖÓÐÒÓK3 <strong>×</strong> S<br />

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m)¸M<strong>×</strong>(6+m)<strong>×</strong>(6+m)ÑØÖÜÚÐÙ ,´¾º¾¾µ<br />

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<strong>×</strong>ÑÒ<strong>×</strong>ØÒØÕÙØÓÒ<strong>×</strong>ÓÑÓØÓÒÒ<strong>×</strong>ÓÑÔØÐÛØØÖÕÙÒØÞØÓÒ¿℄º ´¾º¾¿µ Ð<strong>×</strong>ØÖÒØÓØ(6 +<br />

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Z)<strong>×</strong>Ø˹ÙÐØÝ<strong>×</strong>ÝÑÑØÖÝØØ<br />

)<br />

(<br />

ÑØÖ<strong>×</strong>º<br />

SO(6,<br />

½ºÌØÖÓØÐØÓÒÓÑÒ<strong>×</strong>ÛØØÜÓÒ´ÓØÒÝÙÐÞÒØÒØ<strong>×</strong>ÝÑÑع<br />

m; Z)<strong>×</strong>Ø̹ÙÐØÝ<strong>×</strong>ÝÑÑØÖÝÒΓ (N) ⊂ SL(2,<br />

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Z)ÚØÓÖºÌÙ<strong>×</strong>¸ØÐØÖÖ<strong>×</strong><br />

HºÍÒÖS¹ÙÐØݸS H<br />

¾ºÌ(6+m)ÚØÓÖÐ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong>SO(6, m;<br />

q e´Ö<strong>×</strong>ÔºÑÒØÖ<strong>×</strong>q ¾ ÏÒN =<br />

∫<br />

S =<br />

−−−→<br />

K3<strong>×</strong>S 1 <strong>×</strong><br />

S<br />

−−−−−→<br />

K3<strong>×</strong>S 1 <strong>×</strong>Ŝ1<br />

d 4 x √ [<br />

−g R − ∂ µS H ∂ µ ¯SH<br />

2ÁÑ(S H ) + 1 8ÌÖ(∂ 2 µ ML ∂ µ ML)<br />

−−−→<br />

T 4 <strong>×</strong>S 1 <strong>×</strong>Ŝ1<br />

− 1 4ÁÑ(S H ) F µν LML F µν + 1 4Ê(S H ) F µν L ˜F µν ]<br />

T<br />

Γ1 (N) <strong>×</strong> SO(6, m; Z) )∖( SL(2)<br />

U(1) <strong>×</strong> SO(6, m)<br />

SO(6) <strong>×</strong> SO(m)<br />

1<br />

1, 2, 3, 5, 7¸m=([48/(N + 1)] − 2)<br />

→ (aS H +b)/(cS H +d)º

Z)ºÙÖØÖ¸ØÐØÖÒÑÒØÖ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong> ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

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ÓÙÐØÙÒÖØS¹ÙÐØÝÖÓÙÔ¸Γ 1 (N)¸ÛÖΓ (N)<strong>×</strong>Ø<strong>×</strong>Ù¹ÖÓÙÔÓSL(2, mÖÓÑØÖÚØÓÖ<strong>×</strong>º ÑØÖ<strong>×</strong>( a<br />

c d )ÛØa ´¾º¾µ<br />

ÇÒÒÓÖÑØÖ̹ÙÐØÝÒÚÖÒØ<strong>×</strong>ÐÖ<strong>×</strong>¸q2 e¸q2 mÒq ZºÌÖÜ<strong>×</strong>ØÑÒÝÑÓÖÒÚÖÒØ<strong>×</strong><br />

)<br />

1ÒÑÓÖÔÔÖÛÒ<br />

1(N)ºÌÖ<strong>×</strong><br />

Q<br />

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c d ) ∈ Γ<br />

ÖÕÙÒØÞ<strong>×</strong>ÙØØNq 2 e , q2 m ∈ 2ZÒq · q m ∈<br />

ÙØÓØ<strong>×</strong>ÖØÒØÙÖÓØ̹ÙÐØÝÖÓÙÔ¼℄ÓÖN =<br />

2¹ÈËÑÙÐØÔÐØ<strong>×</strong>ØØÔÖ<strong>×</strong>ÖÚØ<strong>×</strong>ÙÔÖÖ<strong>×</strong>´ÛØ16<strong>×</strong>ØØ<strong>×</strong>ÒÑÙÐØÔÐص<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÑØØÛÓÒ<strong>×</strong>ÓÈË<br />

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ÓÙÖ¹ÑÒ<strong>×</strong>ÓÒÐÓÑÔØØÓÒ<strong>×</strong>ÛØN =<br />

<strong>×</strong>ØØ<strong>×</strong>´µ1 Ò´µ1 Ñ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØ1<br />

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−ÈË=<br />

4<br />

mºÌÈËÑ<strong>×</strong><strong>×</strong><br />

2¹ÈËÓÖÑÙÐ<br />

4¹ÈËÑ<strong>×</strong><strong>×</strong>ÓÖÑÙÐ<br />

Ì<strong>×</strong>ÕÙÖÓØÑ<strong>×</strong><strong>×</strong>Ó1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÑÜ(M+ 2 , M2 −<br />

ÐØÖÒÑÒØÖ<strong>×</strong>ÖÔÖÐÐдÓÖÒعÔÖÐÐеºº¸q e ∝ q<br />

ÓÖÑÙÐÓÖ1 2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒÓØÒ<strong>×</strong><strong>×</strong>ÔÐÞØÓÒÓØ1 ÚÒÓÚºÏÒq e ∝ q m¸ØØÖÑ<strong>×</strong>Ò<strong>×</strong>Ø<strong>×</strong>ÕÙÖÖÓÓØÔÔÖÒÒØ1 ÓÖÑÙÐÚÒ<strong>×</strong>ÐÒØÓØ1<br />

¾<br />

N > 1º<br />

(<br />

M<br />

2<br />

±<br />

)<br />

1<br />

b<br />

=<br />

Nµº<br />

1<br />

d = 1 modNÒc = 0 mod<br />

≡<br />

(<br />

q 2 e<br />

−q e · q m<br />

e<br />

−q e · q m<br />

Z)<br />

e·q<br />

q 2 m<br />

a b<br />

1<br />

[<br />

S H − ¯S (q e + S H q m ) T (M + L)(q e + ¯S H q m )<br />

H<br />

± 1 2<br />

√<br />

(q<br />

T e (M + L)q e )(q T m(M + L)q m ) − (q T e (M + L)q m ) 2 ]<br />

( ) M<br />

2<br />

1<br />

−ÈË=<br />

2<br />

1<br />

[<br />

S H − ¯S H<br />

´¾º¾µ (q e + S H q m ) T (M + L)(q e + ¯S<br />

]<br />

H q m ) .

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ØÖÐÐÝÖ<strong>×</strong>ØØ<strong>×</strong>ÖÒÓÒØÓÓÒÓÖÖ<strong>×</strong>ÔÓÒÒÛØØ<strong>×</strong>ØØ<strong>×</strong>ÓØÀÄÓÖÓÐ<br />

2¹ÈËËØØ<strong>×</strong><br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ºËÙй ÓÙÒØÒ1<br />

2¹ÈËÑÙÐØÔÐØ»ØÖÓØ<strong>×</strong>ØÖÒ<strong>×</strong>ØØ<strong>×</strong><br />

2nØÖØÓÒÐÖ<strong>×</strong>Ö<strong>×</strong>ÖÓÑØ<br />

Ŝ1¿℄ºÄØd(n)ÒÓØØÒÖÝ ÏÛÐÐÒÓÛÓÒ<strong>×</strong>ÖØÓÙÒØÒÓÔÙÖÐÝÐØÖÐÐÝÖ1 ÓØØÖÓØ<strong>×</strong>ØÖÒÓÑÔØÓÒT 4 <strong>×</strong> S 1 <strong>×</strong><br />

ÓØÖÓØ<strong>×</strong>ØÖÒ<strong>×</strong>ØØ<strong>×</strong>ÖÖÝÒÖNq 2 e =<br />

ØÛ<strong>×</strong>Ø<strong>×</strong>ØÓÖ<strong>×</strong>ÒØÀÄÓÖÓÐÒºÚÖÝ1 ÒÖÝ16 = 2 8/2ºÌÒØÒÖØÒÙÒØÓÒÓd(n)<strong>×</strong>¿¾¸¾¸¿℄´ÓÖN ÛÖ ´¾º¾µ Òk +<br />

(i √ N) −k−2 f (k) (τ/N) ,<br />

(τ)ºÌ<strong>×</strong>ÖÐÚйNÒÙ<strong>×</strong>¹ÓÒÑÓÙÐÖ<br />

2m<strong>×</strong>ÚÒÝ ´¾º¾µ<br />

f (k) (τ) ≡ η(Nτ) k+2 η(τ) k+2 .<br />

ÌÒÖÝÓÔÙÖÐÝÑÒØÐÐÝÖ<strong>×</strong>ØØ<strong>×</strong>ÛØÖq m =<br />

<strong>×</strong>ÑÐÖÓÖÑÙÐÛØf (k) (τ/N)ÖÔÐÝf ÒØÖÚØÓÖ<strong>×</strong>ÒÐÒÖÐÝÒÔÒÒغÂØÖÒËÒÒÖÐÞØÎÎÔÖÓ¹ ¾ºº¿ 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÖÒ<strong>×</strong><strong>×</strong>ÖÐÝÝÓÒÒÖØÖÛØØÐØÖÒѹ<br />

4¹ÈËËØØ<strong>×</strong> ÓÖÑ<strong>×</strong>ÛØÛØ(k + 2)ºÓÖ(N, ÓÙÒØÒ1<br />

4¹ÈËÝÓÒ<strong>×</strong><strong>×</strong>ÒÖØÝËÐÑÓÙÐÖÓÖÑÓ<br />

<strong>×</strong>Û<strong>×</strong>ÛÖÐÖ¸1<br />

<strong>×</strong>ØÓÐÐÓÛÒÔÖÓÔÖØ<strong>×</strong> ÑÓÙÐÖÓÖÑÙ<strong>×</strong>ÒØØÚÐØÓÛÂÓÓÖѺÌÓÒ<strong>×</strong>ØÖÙØÑÓÙÐÖÓÖÑ<br />

(Z)ºÌÝÐ<strong>×</strong>ÓÔÖÓÚÒÜÔÐØÓÒ<strong>×</strong>ØÖÙØÓÒÓØ ÔÓ<strong>×</strong>ÐØÓØ<strong>×</strong>Ó<strong>×</strong>ÝÑÑØÖZ N¹ÓÖÓÐ<strong>×</strong>ÓØØÖÓØ<strong>×</strong>ØÖÒÓÒT 6ÓÖN =<br />

ÌÝÔÖÓÔÓ<strong>×</strong>ØØØÒÖÝÓ1<br />

Z)º´ËÔÔÒÜÓÖÒÓØØÓÒµ<br />

ÛØk= 24 − N+1 2ÒÐÚÐN¸˜Φk 0¸Ø<strong>×</strong>ØÖØØÓÖÞØÓÒÔÖÓÔÖØÝ ´µÁØ<strong>×</strong>ÒÚÖÒØÙÒÖØ˹ÙÐØÝÖÓÙÔΓ 1 (N)<strong>×</strong>ÙØÐÝÑÒØÖÓÙÔG Sp(2,<br />

´µÁÒØÐÑØz 2<br />

10)¸Ø<strong>×</strong>ÑØ<strong>×</strong>ØÎÎÓÖÑÙдպ´¾º¾½µµº ´¾º¾µ<br />

→<br />

¾ ÆÓØØØÓÖ(N, k) = (1,<br />

2 = 24/(N + 1)µ∞ ∑<br />

=<br />

n=0<br />

d(n) exp( 2πinτ<br />

N ) = 16<br />

24º<br />

(k)<br />

k) = (1, 10)¸f (10) (τ) = η(τ)<br />

lim<br />

z 2 →0<br />

1<br />

˜Φ k (Z) = (i √ N) −k−2 (2πz 2 ) 2 f (k) (z 1 /N) f (k) (z 3 )<br />

1, 2, 3, 5, 7<br />

2, 3, 5, 7¾℄º<br />

(N) ⊂

´µÁØÖÔÖÓÙ<strong>×</strong>ØÒØÖÓÔÝÓÖÐÖÐÓÐ<strong>×</strong>¾℄º ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ÓÖÖØÓÒ<strong>×</strong>ØÓØÐÓÛ¹ÒÖÝØÚØÓÒÚÒÒÕº´¾º¾¾µºËÙÓÖÖØÓÒ<strong>×</strong><br />

2´ÖÖÚØÚµ ´ÚµÓÖ1<br />

ÏØØ<strong>×</strong>ÖÒØÖÓÙØÓÒ¸ÛÑÓÚÓÒØÓØÓØÖÑÓÐÛÛÐÐÓÒ<strong>×</strong>ÖÒØ<strong>×</strong>ÛÓÖ ÐØÓÒÓÒ¹ÞÖÓÒØÖÓÔÝÙ<strong>×</strong>ÒÏг<strong>×</strong>ÒÖÐÞØÓÒÓØÀÒØÖÓÔÝÓÖÑÙÐÓÖ Ò<strong>×</strong>ØÒÖÚØÝØØÖ<strong>×</strong>ÛØØÔÖØÓÒÖÓÑØÑÓÙÐÖÓÖѸ℄º<br />

2¹ÈËÐÓÐ<strong>×</strong>¸ØÓÖÑÙÐÐ<strong>×</strong>ØÓÔÖØÓÒÓÖR 4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺÌ<strong>×</strong>ÑÓÐ<strong>×</strong>Ö<strong>×</strong>ÑÐÖØÓØ<br />

¾º ÌÌÝÔÁÁÅÓÐ<strong>×</strong><br />

4º ØØÝÔÁÁÓÑÔØØÓÒ<strong>×</strong>ÛØN =<br />

8<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÒÓÙÖ¹<br />

ÀÄÑÓÐ<strong>×</strong>ÛØØK3¸ÔÔÖÒÒØØÝÔÁÁ»<strong>×</strong>ÖÔØÓÒ¸ÒÖÔÐÝT 5µÓÖÓÐ<strong>×</strong>ÓØ<strong>×</strong>ܹ ÌÝÔÁÁ<strong>×</strong>ØÖÒØÓÖÝÓÑÔØÓÒ<strong>×</strong>ܹØÓÖÙ<strong>×</strong><strong>×</strong>N =<br />

ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ºÏÛÐÐÓÒ<strong>×</strong>ÖܹÔÓÒØÖZ N´N = 1, 2, 3, 4,<br />

ÛÐÐÒØÓ<strong>×</strong>ÔÝØÙÐØÝÖѺ<strong>×</strong>ÖÔØÓÒÓÒÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓØÝÔÁÁ<strong>×</strong>ØÖÒØÓÖÝ <strong>×</strong>ÛÛÐÐÑÓÚÒØÛÒ<strong>×</strong>ÚÖÐ<strong>×</strong>ÖÔØÓÒ<strong>×</strong>ÓØÓÖÓÐÖÐØÝÙÐØݸÛ<br />

1ºÌØÓØÐØÓÒÓ 4¸ÙØØ<strong>×</strong> 4º ØÓÖÙ<strong>×</strong>ØØÔÖ<strong>×</strong>ÖÚN = 4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺÌÓÖÓÐÔÖÓÙÖÒÚÓÐÚ<strong>×</strong><strong>×</strong>ÔÐØØÒT T 4 <strong>×</strong> S 1 <strong>×</strong> ˜S 1ÒÓÓ<strong>×</strong>ÒØØÓÒÓZ N<strong>×</strong>ÙØØØ<strong>×</strong>ÜÔÓÒØ<strong>×</strong>ÓÒT ØÓÒ<strong>×</strong>ÓÑÔÒÝ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>1/N<strong>×</strong>ØÐÓÒØÖÐS ØÓÖÓÐ<strong>×</strong>Ö¸ºº¸Ø<strong>×</strong>ÒÓÜÔÓÒØ<strong>×</strong>ºÁØØÙ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>ØÓ<strong>×</strong>ÔÝØØÓÒÓT NØÓÒº )º<br />

NØÓÒ<strong>×</strong> ÓÒ<strong>×</strong>ܹØÓÖÙ<strong>×</strong>ÛØØÓÐÐÓÛÒZ N ≠ 5ÄØω exp(2πi/N)Ò(z 1 , z 2 )ÓÑÔÐÜÓÓÖÒØ<strong>×</strong>ÓÒT 4ºÌZ ÒÖØÝ(z 1 , z 2 ) → (ωz 1 , ω −1 z 2<br />

mod5ºÌ<strong>×</strong>ÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓ 4º<br />

N<br />

ÙÐØ<strong>×</strong>´<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÐØÖÒØ<strong>×</strong>ÔØÖµÖÐØ<strong>×</strong>Ø<strong>×</strong>ØÓÓØÖØÝÔÁÁ»ÓÑÔØØÓÒ<strong>×</strong>º ÕÙ<strong>×</strong>¹ÖÝ<strong>×</strong>ØÐÐÒÓÑÔØØÓÒº<br />

= 5ÄØω= 4 = R 4 /Γ<br />

ÁÒÔÖØÙÐÖ¸ØÒÐÓÓØÀÄ<strong>×</strong>ØÖÒØÙÖÒ<strong>×</strong>ÓÙØØÓØØÝÔÁÁ<strong>×</strong>ØÖÒ<strong>×</strong>ÙÖ<br />

A4¸ÛÖΓ A4<strong>×</strong>ØÖÓÓØÐØØÓÄÐÖA ÌZ 5ÒÖØÓÖ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ω 3ÓÖÓÐ<strong>×</strong>ÓÒ<strong>×</strong>ÖÒ¿½℄ºÒÒÓ<br />

rÛØr= 1, 2, 3, 4<br />

¾º¾ºÌØÛÓ¹ÖÚØÚÐÓÛ¹ÒÖÝØÚØÓÒ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒÝ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÒØ ÒÙÑÖÓÚØÓÖÑÙÐØÔÐØ<strong>×</strong>Ò<strong>×</strong>ÒØÐØÓØÓÒ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÖØÀÄÑÓд<strong>×</strong>Õº<br />

ÇÙÖÓÒ<strong>×</strong>ÖØÓÒ<strong>×</strong>ÒÖÐÞØN = 2,<br />

¾<br />

=<br />

exp(2πi/5)ÒT<br />

6<br />

=

ÌÝÔÁÁÓÒ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

<strong>×</strong>ÖÔØÓÒ¿ ÙÐÌÝÔÁÁÓÒ Ë<br />

1ÌÙÐÞ 1ÌÝÔÁÁÓÒ <strong>×</strong>ÖÔØÓÒ½ ÙÐØÝÌÝÔÁÁÓÒ <strong>×</strong>ØÖ¹<strong>×</strong>ØÖ<br />

ÙÖ¾º¾ÌÒÓÙÐØ<strong>×</strong>ÒØØÝÔÁÁÑÓÐ<strong>×</strong>ºÌÓÚÒ<strong>×</strong>ÜÔØØÓ <strong>×</strong>ÖÔØÓÒ¾<br />

¾´<strong>×</strong>ÝÑÑØÖÓÖÓÐÓØØÝÔÁÁ<strong>×</strong>ØÖÒµÛÐÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒ<strong>×</strong>ÖÖÓÙØÒ <strong>×</strong>ÖÔØÓÒ¿º<br />

1ºÌÕÙÒØÞØÓÒÓÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÔÒ<strong>×</strong>ÖÔØÓÒ<br />

T 4 <strong>×</strong>S 1 <strong>×</strong> ˜S 1<br />

˜S eS 1ØÓb ÓÐØÖZ N¹ÓÖÓÐÒÓT 2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÚÒÒØÀÄÑÓÐ<br />

4 <strong>×</strong> S<br />

¾ºº½ HÒÒØÛØØÐØÓÒÒ<strong>×</strong>ÖÔØÓÒ¾º ´¾º¾¾µµºËÑÐÖÐݸØÑ<strong>×</strong><strong>×</strong>ÓÖÑÙÐÓÖ1 4¹ÈËÒ1 Ð<strong>×</strong>ÓÓÐÖÛØS N¹ØÓÒÖÓÑØÆË5¹ÖÒ<br />

4´<strong>×</strong>ÖÔØÓÒ¾µºÌÙÐØÝÔÁÁ<strong>×</strong>ØÖÒ<strong>×</strong><strong>×</strong>ÓÐØÓÒ<br />

4ºÏÖÒØÖ<strong>×</strong>ØÒØ<strong>×</strong>ØÙØÓÒÛÖØ<strong>×</strong><br />

4´<strong>×</strong>ÖÔØÓÒ½µ<strong>×</strong>ÙÐØÓ<br />

Z<br />

ÍÒÖ<strong>×</strong>ܹÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ØÖÒ¹<strong>×</strong>ØÖÒÙÐØݸØÝÔÁÁ<strong>×</strong>ØÖÒÓÒT NÓÖÓÐØÓÒ<strong>×</strong>ÑÒØÓÒÓÚº ØÝÔÁÁ<strong>×</strong>ØÖÒÓÒØ̹ÙÐØÓÖÙ<strong>×</strong>̂T ÓØÒÝÛÖÔÔÒØÆË5¹ÖÒÓÒT <strong>×</strong>ÓÑÔØØÓÓÙÖ¹ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÖ<strong>×</strong>Z NØÓÒÒØÚ1+1¹ÑÒ<strong>×</strong>ÓÒÐÛÓÖÐÚÓÐÙÑØÓÖÝÓØÆË5¹ÖÒ 3µÒØÙÐ 5ÓÖÓÐ<strong>×</strong> ÎÒËÒÚÓØÒØÓÖÖ<strong>×</strong>ÔÓÒÒÓÖÓÐØÓÒ´ÓÖN<br />

<strong>×</strong>ÓÒ¹ÖÒÒØ<strong>×</strong>ÝÑÑØÖØÒ<strong>×</strong>ÓÖ´ÛØ<strong>×</strong>йÙÐÐ<strong>×</strong>ØÖÒصÒØÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÓÖÒ ÌÐ<strong>×</strong>ÒØÛÓÖÐÚÓÐÙÑØÓÖÝÓ<strong>×</strong>ÒÐÆË5¹ÖÒÓÒ<strong>×</strong><strong>×</strong>ØÓÚ<strong>×</strong>ÐÖ<strong>×</strong>¸<br />

4´<strong>×</strong>¸℄ÓÖÖÐØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒµº<br />

= 2,<br />

<strong>×</strong>ÖÔØÓÒ℄ºÏÛÐÐÓØÒØÖÖ<strong>×</strong>ÙÐØÒØ<strong>×</strong>ÒÖÐÞØÓÒÓÖØN = 4, Ý<strong>×</strong>ØÙÝÒØZ ÓÒT ÖÒ¹ËÛÖÞ<strong>×</strong>ØÖÒÒØÐعÓÒÙ¸℄º<br />

1¹ÑÒ<strong>×</strong>ÓÒÐØÓÖݺÍ<strong>×</strong>Ò<strong>×</strong>ØÖÒ¹<strong>×</strong>ØÖÒÙÐØݸØ<strong>×</strong>ØÓÖÝÛÐÐØØÓØÝÔÁÁ<br />

4ØÓÓØÒØÐ<strong>×</strong>ÓÒÒØÚ ÓÙÖÖÐÖÑÓÒ<strong>×</strong>ºÌ<strong>×</strong>ÖØÓÑÔÓÒÒØ<strong>×</strong>Ó<strong>×</strong>ÒÐ(2, 0)ØÒ<strong>×</strong>ÓÖÑÙÐØÔÐØÒ5 +<br />

ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ºÏÒÑÒ<strong>×</strong>ÓÒÐÐÝÖÙØÐ<strong>×</strong>ÓÒT<br />

ØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>ÖØÓÒ<strong>×</strong>ØÓØÆË5¹ÖÒº<br />

4ÒØʹ<strong>×</strong>ÝÑÑØÖÝÒØÒØÓÖÓØØÓÒ<strong>×</strong>ÓÙØØÓÙÖ<br />

1 +<br />

ÄØÙ<strong>×</strong>ÓÖÒÞØÐ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓSO(4) <strong>×</strong>SO(4) RÛÖØÖ<strong>×</strong>ØSO(4)<br />

SU(2)<br />

¾ºÌØ<strong>×</strong>ÐÖÒØØÛÓ¹ÓÖÑÒØ<strong>×</strong>ÝÑÑØÖÙÐÒÓÑÒÒ ÓÒÑÒ<strong>×</strong>ÓÒÐÖÙØÓÒÓÒØÓÙÖ¹ØÓÖÙ<strong>×</strong>º<br />

R<strong>×</strong>ÖÓÑØT )ºÌ<strong>×</strong>ÓÑÓÙÖÒÓÒ¹ÖÐ<strong>×</strong>ÐÖ<strong>×</strong> ½ºÓÙÖ<strong>×</strong>ÐÖ<strong>×</strong>¸x m¸ÖÒØÖÔÖ<strong>×</strong>ÒØØÓÒ(1, 4 v<br />

R<strong>×</strong>ÔÒ¹ ¿¼ ÛÖØØÒ<strong>×</strong>Y αβÒY ˙α ˙βÛÖα<strong>×</strong>SU(2) L<strong>×</strong>ÔÒ¹ÐÒÜÒ˙β<strong>×</strong>SU(2)<br />

−−−→<br />

T 4 <strong>×</strong>S 1 <strong>×</strong><br />

S<br />

−−−−−→<br />

T 4 <strong>×</strong>S 1 <strong>×</strong>Ŝ1<br />

−−−→<br />

̂T 4 <strong>×</strong>S 1 <strong>×</strong>Ŝ1<br />

= SU(2) L <strong>×</strong>

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ αβÓÑØÓÙÖ<br />

ØÐعÓÒÙÓØØÝÔÁÁ<strong>×</strong>ØÖÒº ÓÑÒÛØØÓÙÖÒÓÒ¹ÖÐÓ<strong>×</strong>ÓÒ<strong>×</strong>¸ØÝÓÑØÖÒ¹ËÛÖÞÓ<strong>×</strong>ÓÒ<strong>×</strong>Ò<br />

˙βÓÑÓÙÖÖعÑÓÚÒÖÐÓ<strong>×</strong>ÓÒ<strong>×</strong>ºÏÒ ÐÒܺÇÒÑÒ<strong>×</strong>ÓÒÐÖÙØÓÒÓÒØÓÙÖ¹ØÓÖÙ<strong>×</strong>¸ØY ÐعÑÓÚÒÖÐÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØY<br />

ÖØÖÒ¹ËÛÖÞÖÑÓÒ<strong>×</strong>ÒØÐعÓÒÙÓØØÝÔÁÁ<strong>×</strong>ØÖÒº<br />

˙α<br />

1¹ÑÒ<strong>×</strong>ÓÒÐØÓÖÝØ<strong>×</strong><br />

RºÌ<strong>×</strong>ÓÑ ¿ºÌÖÑÓÒ<strong>×</strong>Öψ AβÒψ ˙βÛÖA<strong>×</strong><strong>×</strong>ÔÒÓÖÒÜÓSO(4) A<br />

ØÐعÒÖعÑÓÚÒÖÑÓÒ<strong>×</strong>ÒØØÚ1 +<br />

L<strong>×</strong>ÚÒÝ ÁÒØÓÚ<strong>×</strong>ØÙÔ¸ØØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÙÒÖØØØZ N<strong>×</strong>ÙÖÓÙÔÓSU(2) ´¾º¿¼µ<br />

ÒÜαºÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØØÖÐÖÑÓÒ<strong>×</strong>ÐÐØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong> ÇÒÒ<strong>×</strong>ØØØÓÒÐÝÐ<strong>×</strong>ØØØÖÒ<strong>×</strong>ÓÖÑÙÒÖØ<strong>×</strong>ØÓÒÖØÓ<strong>×</strong>ØØÖÖÝØ<br />

,<br />

ÌÙ<strong>×</strong>Û<strong>×</strong>ØØÓØÖÑÓÒ<strong>×</strong>ÔÙÔØÔ<strong>×</strong>ωÒØÓØÖÓÙÖÔÙÔØÔ<strong>×</strong> ´¾º¿½µ<br />

αβØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong><strong>×</strong> ´¾º¿¾µ 2Ò<br />

ω −1ºÌÐY Y<br />

ÌÙ<strong>×</strong>¸ØÛÓÐ<strong>×</strong>ÖÒÚÖÒØÙÒÖØZ NÒØÓØÖØÛÓØÖÒ<strong>×</strong>ÓÖÑÛØÔ<strong>×</strong><strong>×</strong>ω ω −2 RÐ<strong>×</strong>ØÑÔÔØÓ´<strong>×</strong>ݵÖعÑÓÚÖ<strong>×</strong>ºÌÙ<strong>×</strong>¸Û<strong>×</strong><br />

Nº<br />

LÐ<strong>×</strong>ØÑÔÔ<br />

ÖÑÓÒ<strong>×</strong>º<br />

.ÐÐÓØÖÐ<strong>×</strong>ÖÒÚÖÒØÙÒÖØZ ÁÒØÑÒ<strong>×</strong>ÓÒÐÖÙØÓÒÓØØ(2, 0)ØÓÖÝÓÒT AαÚÖ<strong>×</strong>ØÓÓÙÖÐعÑÓÚÒÖÐ<br />

4¸ØSU(2)<br />

αβÚ ØÓ´<strong>×</strong>ݵÐعÑÓÚÖ<strong>×</strong>ÒØSU(2) ØØØÓÖÓÐ<strong>×</strong>ÖÐØÓÒºÁÒÔÖØÙÐÖ¸ØÓÙÖÓ<strong>×</strong>ÓÒ<strong>×</strong>ØØÖ<strong>×</strong>ÖÓÑY Ö<strong>×</strong>ØÓÓÙÖÐعÑÓÚÒÖÐÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØψ ¿½<br />

4º<br />

( )<br />

g β ω 0<br />

α ≡<br />

0 ω −1<br />

ÛÖω= exp(2πi/N)ÓÖN = 2, 3,<br />

ψ Aα → g α β ψ Aβ .<br />

αβ → g α γ g α δ Y γδ .

¾ºº¾ NØÓÒÖÓÑØÈÓÒÖÈÓÐÝÒÓÑÐ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ N¹ØÓÒÓÖ<br />

Z<br />

ÓÒ<strong>×</strong>ÖØÈÓÒÖÔÓÐÝÒÓÑÐÓÖT 4ÛØÝØÔ<strong>×</strong><strong>×</strong>ÙÒÖØZ<br />

ØÓÒºÌÙ<strong>×</strong><strong>×</strong>ÜÓØÓ<strong>×</strong>ÓÒ<strong>×</strong>ÖÐÛÝ<strong>×</strong>ÔÖÓÒØÓØÖØÛÓÚÖØÓÒÐÑÓÒ ØÓÖÝÒÓÔÓÛÖ<strong>×</strong>ÛØÖÑÓÒ<strong>×</strong>ºÌÓÒØÑÙÐØÔÐÝÒØØÖÑÚ<strong>×</strong>ØÓÖÓÐ 1¹ÑÒ<strong>×</strong>ÓÒÐ<br />

ØÖÑÒÝØÔ<strong>×</strong>º<br />

1´¾º¿¿µ<br />

ω + 2 4x2 − 2xω − 2x ω +<br />

ÁÒØÓÚÜÔÒ<strong>×</strong>ÓÒ¸ÛÒØÝÚÒÔÓÛÖ<strong>×</strong>ÓxÛØÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØ1 5ØÓÒÓÒØÙÐ<br />

+<br />

ÓÖÖ<strong>×</strong>ÔÓÒÒÈÓÒÖÔÓÐÝÒÓÑÐ<strong>×</strong><br />

4ØÓÚÒÝ ÁØÔÔÖ<strong>×</strong>ØØÓÒÒÙ<strong>×</strong>ØÈÓÒÖÔÓÐÝÒÓÑÐØÓÓØÒØZ ØÝÔÁÁ<strong>×</strong>ØÖÒºÀÓÛÚÖ¸ØØ<strong>×</strong>ØÓÛÖØØÒ<strong>×</strong>SO(4)ØÓÒÖØÖØÒSU(2) L<br />

<strong>×</strong>ÙÖÓÙÔºÏÒN = 5¸ØZ 5ØÓÒ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÒÝÓÓ<strong>×</strong>ÒØT R 4 /Γ A4ºÌÒÚÐÙ<strong>×</strong>ÓØÒÖØÓÖÓØZ 5ÖÚÒÝωr¸´r =<br />

ËÛÖÞØÝÔÁÁ¹<strong>×</strong>ÙÔÖ<strong>×</strong>ØÖÒØØÛÙ<strong>×</strong>ØÖÚº ÏÒÓÛÔÖ<strong>×</strong>ÒØØØÐ<strong>×</strong>ÓØÓÖÓÐØÓÒ´ÓÒØÐعÑÓÚÖ<strong>×</strong>µÓÖØÖÒ¹ 2´¾º¿µ<br />

− x/ω − x/ω 2 − ωx − ω 2 x − ω 2 x 3 − ωx 3 − x 3 /ω − x 3 /ω<br />

ÔÖÓÖÑÓÒ<strong>×</strong>º 1ºÌÙ<strong>×</strong>¸ÓÒ<strong>×</strong>ØÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÒع ƾ℄ω=−1ÑÔÐ<strong>×</strong>ØØω ØÑ<strong>×</strong>ÐÚ<strong>×</strong>º<br />

−1ØÑ<strong>×</strong><br />

2 =<br />

Æ¿℄ω = exp(2πi/3)ÇÒ<strong>×</strong><strong>×</strong>ÜÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÛÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÛÔÙÔÔ<strong>×</strong><strong>×</strong>ω Òω 2ºÓÙÖÖÑÓÒ<strong>×</strong>ÓØÓωØÑ<strong>×</strong>ØÑ<strong>×</strong>ÐÚ<strong>×</strong>ÒØÓØÖÓÙÖÓØÓω Æ℄ω=exp(2πi/5)Ì<strong>×</strong><strong>×</strong>ÖÒØÖÓÑØÓØÖØÖÜÑÔÐ<strong>×</strong>ºÇÒÒ<strong>×</strong>ÙÔÛØ<br />

exp(πi/2)ÇÒ<strong>×</strong><strong>×</strong>ÜÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÛÓÒعÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ºÓÙÖÖÑÓÒ<strong>×</strong><br />

−1ØÑ<strong>×</strong>ØÑ<strong>×</strong>ÐÚ<strong>×</strong>º 4µºÌØ<br />

Æ℄ω =<br />

ÓØÓωØÑ<strong>×</strong>ØÑ<strong>×</strong>ÐÚ<strong>×</strong>ÒØÓØÖÓÙÖÓØÓω ÓÙÖÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÓØÖÓÙÖÒÝÔ<strong>×</strong>ωr´r = 1, 2, 3,<br />

¿¾<br />

N = 1, 2, 3, 4º<br />

(1 − ωx) 2 (1 − ω −1 x) 2 = x 4 − 2x 3 ω − 2x3<br />

ω + x2 (ω 2 + ω −2 ) + x2<br />

(1 − ωx)(1 − ω 2 x)(1 − ω −1 x)(1 − ω −2 x)<br />

= 1 + 2x 2 + x 4 + (ω 3 + ω + 1/ω + 1/ω 3 )x 2<br />

1, 2, 3, 4µºÌ

ÖÑÓÒ<strong>×</strong>ÖÙÔÒØÓØÛÓ<strong>×</strong>Ø<strong>×</strong>ÓÓÙÖÖÑÓÒ<strong>×</strong>ºÏØÒ<strong>×</strong>ظÓÒÖÑÓÒÔ<strong>×</strong> ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ÌÙ<strong>×</strong>¸Ø<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒÚ<strong>×</strong>Ö<strong>×</strong>ØÓÒ<strong>×</strong>ÝÑÑØÖÓÖÓÐÓØØÝÔÁÁ<strong>×</strong>ØÖÒ<br />

ÓÒ<strong>×</strong>Öº 6ÒØÙ<strong>×</strong><strong>×</strong>ÒÐÓÓÙ<strong>×</strong>ØÓÀÄÓÑÔØØÓÒ<strong>×</strong>ÓØØÖÓØ<strong>×</strong>ØÖÒºÊÐÐØØ 4ØØÛ ÙÔÔ<strong>×</strong>ω r´r<br />

ÓÒT <strong>×</strong>ÑÒØÓÒÒÔÖÚÓÙ<strong>×</strong><strong>×</strong>ØÓÒ¸ÓÑÔÙØÒØÝÓÒ<strong>×</strong>ÔØÖÙÑ<strong>×</strong>ÒÓÒ¹ØÖÚÐÙ<strong>×</strong>ÝÓÒ<strong>×</strong> ¾ºº¿ ÌÝÔÁÁÝÓÒÒÖÝÖÓÑÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ØØÖÓØ<strong>×</strong>ØÖÒÖ<strong>×</strong><strong>×</strong><strong>×</strong>ØØÝÔÁÁÆË5¹ÖÒÛÖÔ<strong>×</strong>K3ÒØÔÐÓT ÓÒÓØÔÔÖÒØÔÖØÙÖØÚ<strong>×</strong>ÔØÖÙÑÓ<strong>×</strong>ØÖÒØÓÖݺÁÒظÝÓÒÓÙÒØÒÒ¹ <strong>×</strong><strong>×</strong>ÖÐÝÖÕÙÖ<strong>×</strong>ÓÑÔÙØÒØÖ<strong>×</strong>ÓÖÓÑÓÑÒÖÓÑØ<strong>×</strong>ÓÐØÓÒ<strong>×</strong>ØÓÖÓØ ØÓÖݺÌÝÓÒÒÖÝÓÖÑÙÐÒÓØÒÒØÛÓÖÒØÛÝ<strong>×</strong>¸ÚÒÖ<strong>×</strong>ØÓ ØÖØÚÓÖÑÙÐÓÖÑÙÐØÔÐØÚÓÒº <strong>×</strong><strong>×</strong>ÓÛÒÒ¾℄¸ÓÖØÀÄÑÓÐ<strong>×</strong>¸ØÖÖØÛÓÑÓÙÐÖÓÖÑ<strong>×</strong>ØØÓÒÓÒ<strong>×</strong>ØÖÙØ<strong>×</strong><br />

(Z)µÒÒÓØÖ<br />

ÓØËÐÑÓÙÐÖÓÖÑÓÖØØÝÔÁÁÑÓÐ<strong>×</strong><strong>×</strong>ÚÒÝ<br />

2¹ÓÖÖØÓÒ<strong>×</strong>ÒØÀÄ<strong>×</strong>ØÖÒºÄØÙ<strong>×</strong>ÐÐØ ÓÒ<strong>×</strong>ØÒÖØÒÙÒØÓÒÓØÝÓÒÒÖ<strong>×</strong>´ÒÓØݘΦk ´ÒÓØÝΦ k (Z)µ<strong>×</strong>ØÓÒÖÐØØÓR (Z)ÒΨ ÓÖÖ<strong>×</strong>ÔÓÒÒÑÓÙÐÖÓÖÑ<strong>×</strong>ÒØØÝÔÁÁÑÓÐ<strong>×</strong>ØÓ˜Ψk<br />

4¸ÓÒ<strong>×</strong>k=1º<br />

k<br />

´¾º¿µ<br />

(Z)ºÌÛØk<br />

ØÀÄÒØÝÔÁÁÑÓÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÛÒÝÚ¸ÂØÖÒËÒ½¸¿½℄º ÛÖÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÓÒ<strong>×</strong>ØÖÙØÓÒÓÐÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÔÔÖÒÒØØ<strong>×</strong>ÛÓÖºÆÓÛÛ ØÙÖÒØÓØÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒÓÝÓÒ<strong>×</strong>ØØ<strong>×</strong>Ò<strong>×</strong>ØØÓÑÔÙØØÓÒÒØ<strong>×</strong>Ó<br />

ÛÒN+ 1|12ºº¸N = 2, 3, 5ºÓÖN (Z)ÒΨ ÏÛÐÐ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Ψk k (Z)ÓØØÝÔÁÁÓÖÓÐ<strong>×</strong>ÒÔØÖ6<br />

¾º ÁÒØÖ<strong>×</strong>ØÓØ<strong>×</strong>ÔØÖ¸ÛÛÐÐ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐÝ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓØÑÓÐ<strong>×</strong>ØÀÄÒØÝÔ<br />

4ËÙÔÖ<strong>×</strong>ÝÑÑØÖËØÖÒ<strong>×</strong><br />

ÌØÓÒÓØ<strong>×</strong>ÝÑÑØÖÝÖÓÙÔ<strong>×</strong>ÒÖØÝØÖÒ<strong>×</strong>ÓÖÑØÓÒgÛÒÚÓÐÚ<strong>×</strong><br />

ÓÙÒØÒÝÓÒ<strong>×</strong>ÒN =<br />

N<strong>×</strong>ÝÑÑØÖݺ ÁÁºÓÒ<strong>×</strong>Ö<strong>×</strong>ÖÔØÓÒ¿ÛÖÓÒ<strong>×</strong>ØÝÔÁÁ<strong>×</strong>ØÖÒØÓÖÝÓÑÔØÓÒM<strong>×</strong> ˜S<br />

ÛÖM<strong>×</strong>ØÖK3ÓÖT 4ºÏØÒØÒÓÖÓÐÓØ<strong>×</strong>ØÓÖÝÝZ ¿¿<br />

= 1, 2, 3, 4µº<br />

k + 2 = 12<br />

N + 1 ,<br />

=<br />

1 <strong>×</strong>S 1

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ÐÓ<strong>×</strong>ÐÝÓÐÐÓÛ<strong>×</strong>ØÖÚÛÓËÒ¾¼℄º<br />

1ØÓØÖÛØÒÓÖÖNØÖÒ<strong>×</strong>ÓÖÑØÓÒ˜gÒMºÌ<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺÇÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÖ<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÒÖØÓÖ<strong>×</strong><br />

1/NÙÒØ<strong>×</strong>ØÐÓÒØÖÐS ØÖÒ<strong>×</strong>ÓÖÑØÓÒ˜g<strong>×</strong>Ó<strong>×</strong>Ò<strong>×</strong>ÙØØØÓÑÑÙØ<strong>×</strong>ÛØØN ÖÓÑÓÒØÛÓÖй<strong>×</strong>ØÒ<strong>×</strong><strong>×</strong>ØÔÐÙ<strong>×</strong>ÖÓØØÓÒÓÒØÐعÑÓÚÒÖ<strong>×</strong><strong>×</strong>ÓÖÓѺ <strong>×</strong>ÖÔØÓÒ¾¸ØÓØÖÒ<strong>×</strong>ÓÖÑØÓÒĝØØØ<strong>×</strong>ÓÒÐÝ<strong>×</strong><strong>×</strong>ØÓÒØÖعÑÓÚÒÖ<strong>×</strong>Ó ÓÐÐÓÛÒØÒÓÙÐØ<strong>×</strong>¸ÛÚ<strong>×</strong>ÒØØØØÖÒ<strong>×</strong>ÓÖÑØÓÒ˜gØ<strong>×</strong>ÑÔÔ¸Ò<br />

=<br />

ÓØÔÖÒØØÓÖÝÒÒÔÖ<strong>×</strong>ÖÚ<strong>×</strong>ØN <strong>×</strong>ÖÔØÓÒ¾<strong>×</strong>ÓØÒÝØÒÒ<strong>×</strong>ÝÑÑØÖÓÖÓÐÓØÖÓØÓÖØÝÔÁÁ<strong>×</strong>ØÖÒ<br />

=<br />

<strong>×</strong>ØÜÓÒ¹ÐØÓÒÒØ<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒÒØÓØÓÑÔÐÜ<strong>×</strong>ØÖÙØÙÖÑÓÙÐÙ<strong>×</strong>ÓØ<br />

1ØÓØÖÛØØØÖÒ<strong>×</strong>ÓÖÑØÓÒĝº ØÓÖÝÓÒT 4 S1Ý1/NÙÒØÓ<strong>×</strong>ØÐÓÒS ÖÓÑØÑÒ<strong>×</strong>ÓÒÐÖÙØÓÒÓØØÒ¹ÑÒ<strong>×</strong>ÓÒÐÑØÖ¸ÆËÆËÒع<strong>×</strong>ÝÑÑØÖØÒ<strong>×</strong>ÓÖ<br />

ÐÐØ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÓÑ<strong>×</strong>ÖÓÑØÖعÑÓÚÒ<strong>×</strong>ØÓÖÓØÛÓÖй<strong>×</strong>غÌÐS<br />

ÐÒÙÐ<strong>×</strong>¸ÛØÓÙØÒÝÙÖØÖÐØÖ¹ÑÒØÙÐØÝØÖÒ<strong>×</strong>ÓÖÑØÓÒºÌ<br />

1ÒØÖ<strong>×</strong>Ø<strong>×</strong>ÖÔØÓÒººÌÑØÖÜÚÐÙ<strong>×</strong>ÐÖÐMÒÓ<strong>×</strong>ÒÓÖÑØÓÒ<br />

µÖÖÐØØÓØÓÒ<strong>×</strong>ÓÑÒ<br />

S1ºÒØÓÑÔÓÒÒØ<strong>×</strong><br />

H<br />

ØÓÖÙ<strong>×</strong>˜S1 <strong>×</strong>S<br />

ÓÙØØ<strong>×</strong>ÔÒ<strong>×</strong>ÞÓØÓÑÔØØÓÒ<strong>×</strong>ÔM ′ <strong>×</strong> Ŝ <strong>×</strong><br />

ÓØÆËÆË<strong>×</strong>ØÓÖ2¹ÓÖÑÐÓÒغÌÙÐ<strong>×</strong>A e¸ÒÚÖÓÙ<strong>×</strong><strong>×</strong>ÓÐØÓÒ<strong>×</strong>ÖÖÝÑÒØÖ<br />

¾ºº½ mº ÌÖÒÝÓÒ<strong>×</strong>ØÖÓÙÙÐØ<strong>×</strong><br />

ÐÑÒØÖÝ<strong>×</strong>ØÖÒ<strong>×</strong>ØØ<strong>×</strong>ÖÖÝÐØÖÖq 4¹ÈËÝÓÒ<strong>×</strong>ÔÓ<strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong>Ö<strong>×</strong>ÛÖÑÙØÙÐÐÝÒÓÒ¹ÐÓÐÒØÖÓÖØÝ<br />

q<br />

ÊÐиØ1<br />

mÓ<strong>×</strong>ØØÖÒÒØ<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒºÏØØ eÒ<br />

1<strong>×</strong>ØÙ<strong>×</strong>ÕÙÒØÞÒÑÙÐØÔÐ<strong>×</strong>Ó1/Nº<br />

1ÓÖÓÖÓÐÒ<strong>×</strong>ØÒØÓ ÓÒÓØÔÔÖÒØÔÖØÙÖØÚ<strong>×</strong>ÔØÖÙÑÓØØÓÖݺÌÐØÖÖÚØÓÖq ØÑÒØÖÚØÓÖq ÓÓÖÒØÖÓS1 /Z NÒ˜S1ØÓ1ºÌÖÙ<strong>×</strong>ÓS 5¹ÖÒ<strong>×</strong>ÛÖÔÔ<br />

1ºÌÑÓÑÒØÙÑ NÒØZ ÙÒØ<strong>×</strong>ÓÒÙ½¹ÖÒÖ¸ÛÖβ<strong>×</strong>ÚÒÝØÙÐÖÖØÖÓMÚÝ<br />

NÓÖÓÐÒØÓÒÒÚÓÐÚ<strong>×</strong>2π/NØÖÒ<strong>×</strong>ÐØÓÒÐÓÒS 1¸<strong>×</strong>ÒÐÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐ<strong>×</strong><strong>×</strong>ÓØÛØ ÐÓÒS ÏÓÒ<strong>×</strong>ÖØÓÐÐÓÛÒÝÓÒÓÒÙÖØÓÒÒ<strong>×</strong>ÖÔØÓÒ¿Q ÓÒM <strong>×</strong> S 1¸Q 1½¹ÖÒ<strong>×</strong>ÛÖÔÔÓÒS ØÖИS1ÛØÒØÚÑÒØÖ¸ÑÓÑÒØÙÑ−k/NÐÓÒS 4º 1ÒÑÓÑÒØÙÑJ ′¸<strong>×</strong><br />

ÐÓÒ˜S1ºÐ<strong>×</strong>Ó¸<strong>×</strong>Ò¹ÖÒÛÖÔÔÓÒMÖÖ<strong>×</strong>¸<strong>×</strong><strong>×</strong>عÖÒÖ¸−β<br />

24¸ØÒؽ¹ÖÒÖÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>(Q 1 − βQ 5<br />

¿<br />

)º´β<strong>×</strong>ÞÖÓÛÒM=T 4Ò1 ÁÒÓØØÀÄÒØÝÔÁÁÑÓÐ<strong>×</strong>¸M ′<strong>×</strong>ÓÙÖ¹ØÓÖÙ<strong>×</strong>ºÁÒØØÝÔÁÁÑÓÐ<strong>×</strong>¸ØÓÙÖ¹ØÓÖÙ<strong>×</strong>¸M ÓØÒÝ̹ÙÐÞÒÐÐÖÐ<strong>×</strong>ÓÒM = T 4ÒÛÛÐÐÒÓØØÝ̂T<br />

<strong>×</strong> Ŝ <strong>×</strong>

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ØÓÚÓÒÙÖØÓÒÒ<strong>×</strong>ÖÔØÓÒ¿Ð<strong>×</strong>ØÓÖÒØÓÒÙÖØÓÒÒ<strong>×</strong>ÖÔØÓÒ½º<br />

K3ºµ<br />

ÄØÙ<strong>×</strong>ÖÔд¹½µ¹ÖÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐÝ<strong>×</strong>ÒÐÆË5¹ÖÒÛÖÔÔÓÒ ÓÐÐÓÛÒØÙÐØÝÒÒÙ<strong>×</strong>ÒØ<strong>×</strong>ÒÓÒÚÒØÓÒ<strong>×</strong>Ù<strong>×</strong>Ò¾¼℄¸ÓÒ<strong>×</strong><strong>×</strong>ØØ ÛÒM =<br />

1<strong>×</strong>ØÖÐ̹ÙÐØÓ˜S1º<br />

ËÒØÓÖÓÐÖÐ<strong>×</strong>ÒÓØÔÖØÔØÒÒØ̹ÙÐØÝØÖÒ<strong>×</strong>ÓÖÑØÓÒ¸ØÓÖÓÐ<br />

M<br />

ØÓÒÓÑÑÙØ<strong>×</strong>ÛØØ̹ÙÐØÝØÖÒ<strong>×</strong>ÓÖÑØÓÒº<br />

1JÙÒØ<strong>×</strong>ÓÑÓÑÒØÐÓÒ 1ºÌ<br />

′ <strong>×</strong>S 1¸Q 5ÆË5¹ÖÒ<strong>×</strong>ÝQ 5ÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐ<strong>×</strong>ÐÓÒŜ 1<strong>×</strong>ÖÔÐÝ−JÙÒÑÒØÐ<strong>×</strong>ØÖÒ<strong>×</strong>ÛÒÒŜ 1¸ÛÖŜ ÙÖØָؽÖÓÑ<strong>×</strong>(−Q<br />

Ø<strong>×</strong>ØÓÒ¸ÐÐÙÒÑÒØÐ<strong>×</strong>ØÖÒ<strong>×</strong>ÖÖÔÐÝÆË5¹ÖÒ<strong>×</strong>ÒÚÚÖ<strong>×</strong>ºÌÙ<strong>×</strong>¸ÒØ<br />

1 + βQ 5 )ÙÒÑÒØÐ<strong>×</strong>ØÖÒ<strong>×</strong>ÛÖÔÔÒÓÒS Z NÓÖÓÐØÓÒÒÚÓÐÚ<strong>×</strong>Z ÒÐÐݸÓÒÖÖ<strong>×</strong>ÓÙØ<strong>×</strong>ØÖÒ¹<strong>×</strong>ØÖÒÙÐØÝØÓÖÖÚØØ<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒºÍÒÖ<br />

NÓÖÓÐÓM ′Ò<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>1/NÙÒØÓ<strong>×</strong>ØÐÓÒS ÒÙÖ¾º¿º<br />

1ºÌÖ<strong>×</strong>ÙÐØ<strong>×</strong><strong>×</strong>ÙÑÑÖÞ<br />

1ÆË5¹ÖÒ<strong>×</strong> ÒÛÚQ 1ÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐ<strong>×</strong>ÐÓÒŜ1¸(−Q 1 +βQ 5 )ÆËÛÖÔÔÒM −k/NÙÒØ<strong>×</strong>ÓÑÓÑÒØÙÑÐÓÒS1¸−JÆË5¹ÖÒ<strong>×</strong>ÛÖÔÔÒM ′ <strong>×</strong> Ŝ1¸Q<br />

ÛÖÔÔÒM ′ 1¸Ò<strong>×</strong>ÒÐÙÒÑÒØÐ<strong>×</strong>ØÖÒÛÖÔÔÒS Q 5³<strong>×</strong>ÛÖÔÔÒM Q 1½³<strong>×</strong>ÛÖÔÔÒS <strong>×</strong>ÖÔØÓÒ¿ ÒÓÙÐØ<strong>×</strong><br />

Q 5ÃÃÑÓÒÓÔÓÐÓÖŜ (−Q 1 +βQ 5 )ÆËÛÖÔÔÒM ÑÓÑÒØÙÑ−k/NÐÓÒS1<br />

ÑÓÑÒØÙÑ−k/NÐÓÒS 0º <strong>×</strong>ÖÔØÓÒ¾<br />

1<br />

1ºÏÒ<br />

ÑÓÑÒØÙÑJÐÓÒ˜S1<br />

−JÆËÛÖÔÔÒM ′<br />

−1ÃÃÑÓÒÓÔÓÐÓÖ˜S1 ÓÒÙÒº<strong>×</strong>ØÖÒÛÖÔÔÒS1<br />

⃗n¸ÙÒÑÒØÐ<strong>×</strong>ØÖÒÛÒÒÖ<strong>×</strong>ÐÓÒØÑÝ⃗wÒÆË5¹ÖÒ¸ÒÃÐÙÞ¹ÃÐÒ Ì<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒÜÐÙ<strong>×</strong>ÚÐÝÓÒØÒ<strong>×</strong><strong>×</strong>ÖÔØÓÒÒØÖÑ<strong>×</strong>ÓÙÒÑÒØÐ<strong>×</strong>ØÖÒ<strong>×</strong>¸ <strong>×</strong>Ŝ1Ý<br />

ÙÖ¾º¿ÌÖÒÝÓÒÓÒÙÖØÓÒ<strong>×</strong>ºÏÒM=K3¸M ′ = T 4Òβ=<br />

M = T 4¸ØÒM = ̂T 4Òβ=<br />

Ø<strong>×</strong>ÐØÖÒÑÒØÖ<strong>×</strong>Ö WÖ<strong>×</strong>ÔØÚÐÝØÒØ̹ÙÐØÝÒÚÖÒØ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÖÓÑ ÆË5¹ÖÒ<strong>×</strong>¸ÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐ<strong>×</strong>ÒÑÓÑÒغÁÛÒÓØÑÓÑÒØÐÓÒS1 ÏÓÐÐÓÛØÓÒÚÒØÓÒ<strong>×</strong>ÓÐÐÓÛÒ¾¼℄<br />

ÑÓÒÓÔÓÐÖ<strong>×</strong>Ý⃗ NÒ⃗ ´¾º¿µ<br />

q 2 e = 2⃗n · ⃗w , q2 m = 2 N ⃗ · ⃗W , q e · q m = ⃗n · ⃗N + ⃗w · ⃗W<br />

¿<br />

.<br />

˜S<br />

<strong>×</strong> S<br />

<strong>×</strong><br />

1<br />

′<br />

S 1<br />

−−−−−−−−−→<br />

′<br />

1<br />

′ <strong>×</strong> Ŝ1<br />

<strong>×</strong>S 1<br />

<strong>×</strong> S 1¸

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ØÓÒ¸<br />

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NÓÖÓÐØÓÒÓÑÑÙØ<strong>×</strong>ÛØØÒØÖÙÐØÝÒÒ<strong>×</strong>ØÖÓÖÛÐÐÒÒ<br />

1¸ÛÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÒÖÙÒØÖÐÖÙ<strong>×</strong>ÝØÓÖÓNºÌÙ<strong>×</strong>ÙÒÑÒØÐ ´¾º¿µ<br />

q 2 e = 2k , q2 m = 2Q 5(Q 1 − βQ 5 ) , q e · q m = J .<br />

1ÒØÓÖÓÐØÓÖÝ<br />

ÌZ <strong>×</strong>ÓØØÛÒ<strong>×</strong>ÐÝÖÓÙØØ<strong>×</strong>ØÓÒÝÓÒÖ<strong>×</strong>ºÌZ NÓÖÓÐØ<strong>×</strong>Ý1/N <strong>×</strong>ØÐÓÒS ÙÒØÓÑÓÑÒØÙÑÐÓÒS 1<strong>×</strong>NÒÒÑÓÑÒØÙÑÐÓÒS ÒØÓÖÓÐØÓÖݺ ÓÒÙÖØÓÒ<strong>×</strong> 1ØÖØÓÖÓÐÛÒØÓ<strong>×</strong>ØÖØ<br />

1ÓÖÓÖÓкÌÖ<strong>×</strong>ÙÐØÒ ÓÑ<strong>×</strong>n/NºÌÓÑÒØÒJÆË5¹ÖÒ<strong>×</strong>ØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>ØÓS ÛØNÓÔ<strong>×</strong>ÓJÆË5¹ÖÒ<strong>×</strong><strong>×</strong>ÝÑÑØÖÐÐÝÖÖÒÓÒS Ì˹ÙÐØÝ<strong>×</strong>ÝÑÑØÖÝÓØ<strong>×</strong>ØÓÖÝÒØ<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒ<strong>×</strong>ÖÐØØÓØ̹<br />

1<br />

2 q2 e = 2k/N ,<br />

2 q2 m = Q 1 Q 5 , q e · q m 1Ö<strong>×</strong>Ø ´¾º¿µ<br />

= J ,<br />

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1<br />

ÓÒÙÖØÓÒØÓØÓÙÒØÒÓÝÓÒÒÖ<strong>×</strong>ÒØÀÄ<strong>×</strong>ØÖÒº<br />

)ÒØÓÒÙÖØÓÒ<strong>×</strong>ÖÒØÔÖÚÓÙ<strong>×</strong><strong>×</strong>ØÓÒº<br />

1ÖÚÒÝ 4¹ÈË<strong>×</strong>ÙÔÖÑÙÐØÔÐØ<strong>×</strong>Ö¹ ÄØd(q e ,q )ÒÓØØÒÙÑÖÓÓ<strong>×</strong>ÓÒÑÒÙ<strong>×</strong>ÖÑÓÒ1<br />

ÌÕÙÒØÙÑÒÙÑÖ<strong>×</strong>kÒJÒÖ<strong>×</strong>ÖÓÑØÖÖÒØ<strong>×</strong>ÓÙÖ<strong>×</strong><br />

m<br />

ÖÝÒÚÒ<strong>×</strong>ØÓÖ<strong>×</strong>(q e ,q m<br />

´¾º¿µ ÌÝÓÒÖ<strong>×</strong>ÓØÓÒÙÖØÓÒÛÒQ 5 =<br />

1º<br />

¿<br />

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1<br />

(N)º<br />

q 2 e = 2k/N , q2 m = 2(Q 1 − β) , q e · q m = J .<br />

0

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0ÐÓÒ˜S1º<br />

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l 0 ′ + l 0 + L = k , j 0 + J ′ = J .<br />

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ÄØf(˜ρ, ˜σ,<br />

.<br />

).´¾º¾µ<br />

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d KK (l 0) ′ e 2πil′ 0 ρ<br />

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d KK<br />

´¾º¿µ<br />

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ÌØÓÖÓ1/64ÖÑÓÚ<strong>×</strong>ØÒÖÝÓ<strong>×</strong>ÒÐ1<br />

f(ρ, σ, ν) = ∑<br />

k,Q 1 ,J<br />

[ ]<br />

d(q e ,q m ) e 2πi σ(Q 1 −1)/N+ρk+νJ<br />

f (˜ρ, ˜σ, ˜ν) = 1 ( ∑<br />

)<br />

64 e−2πiσ/N (−1) J ′ d D1 (Q 1 , L, J ′ ) e 2πi(σQ 1/N+ρL+νJ ′ )<br />

Q 1 ,L,J<br />

( ∑<br />

′ )(∑<br />

(−1) j 0<br />

d CM (l 0 , j 0 ) e 2πil 0ρ+2πij 0 ν<br />

l 0 ,j 0<br />

ν)<strong>×</strong> f(ρ, σ,<br />

f(ρ,<br />

σ, ν) = [ Ê S ∗ (K3/Z N )(ρ, σ, ν) <strong>×</strong> EÌÆ(ρ, ν) <strong>×</strong> g(ρ) ] −1 .<br />

l ′ 0<br />

¿

ÛÖ<br />

[<br />

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−1<br />

≡ (−1) J ′ d D1 (Q 1 , L, J ′ ) e 2πi(σQ 1/N+ρL+νJ ′) ,<br />

Q 1 ,L,J ′<br />

[<br />

EÌÆ(ρ, ν) ] −1 ≡ 1 ∑<br />

(−1) j 0<br />

d CM (l 0 , j 0 ) e 2πil 0ρ+2πij 0 ν ,<br />

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4<br />

l 0 ,j 0<br />

[ ] −1<br />

g(ρ) ≡ 1 ∑<br />

d KK (l 0 ′ 16<br />

) e2πil′ 0 ρ .<br />

l 0<br />

′<br />

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Z)Û<strong>×</strong>Ø<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong><strong>×</strong> N¸ÓÛÚÖ¸ØÝÓÙÒ<br />

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˜Φ(˜ρ, ˜σ, ˜ν) = exp(2πi(˜α˜ρ + ˜γ˜σ + ˜ν))<br />

1∏<br />

N−1<br />

∏ ∏<br />

ÐÓÛºÏÛÐÐ<strong>×</strong>ÓÖØÐÝÓÑÔÙØØÓÚÕÙØÓÒÜÔÐØÐÝÝÓÑÔÙØÒÓØ<br />

<strong>×</strong><br />

[1 − exp{2πi(˜σk + ˜ρl + ˜νj)}] P N−1<br />

s=0 e−2πils/N c (r,s)<br />

b (4kl−j 2)<br />

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b=0 r=0 k∈Z+ r N ,<br />

l∈Z,j∈2Z+b<br />

k,l≥0,j

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N¹ÓÖÓÐÓØØÖÓØ ÑÙÐØÔÐØÚØÓÖÓ16 = 2<br />

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2¹ÈË<strong>×</strong>ØØ<strong>×</strong>º<br />

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k/N(n, k, ∈ Z)ÐÓÒS

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′<br />

∑<br />

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¿<br />

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LÒÓÒ¹ÒÚÖÒØÖÑÓÒ<strong>×</strong>ºÌ<br />

+<br />

<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ØÓÒÐÝÓÒØ<strong>×</strong>ÐÖ<strong>×</strong>ÒØÖعÑÓÚÒÖÑÓÒ<strong>×</strong>ºÐÐ<br />

<strong>×</strong><strong>×</strong>ÖÝ<strong>×</strong>ØÓÓÙÖÖÐعÑÓÚÒU(1) 0ÕÙÒØÙÑÒÙÑÖÓ−1ºÌÙÒÖÓÒ<br />

0ÕÙÒØÙÑÒÙÑÖ1¸ÛÐØ ÒØÖØÒØÓÖÝÓÓÙÖÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒÓÙÖÖعÑÓÚÒU(1)<br />

ÌÓÓÑÔÙØØÔÖØØÓÒÙÒØÓÒ¸Ö<strong>×</strong>ØÓÒ<strong>×</strong>ÖØÖÐعÑÓÚÒÖÑÓÒ<strong>×</strong>Û<br />

1º ØÛÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒØÛÓÓØÖعÑÓÚÒÖÑÓÒ<strong>×</strong>ÖÖÝj ÓØÖØÛÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒÖعÑÓÚÒÖÑÓÒ<strong>×</strong>ÖÖÝj 0ÕÙÒØÙÑÒÙÑÖ<strong>×</strong>ºÌÖÓÒØÖÙØÓÒ<strong>×</strong>ÚÒÝ ØÐ<strong>×</strong>ÖÖÝÒØÖÐÑÓÑÒØÐÓÒS<br />

ÛÖF<strong>×</strong>ØØÓØÐÓÒØÖÙØÓÒØÓØ<strong>×</strong>Ô¹ØÑÖÑÓÒÒÙÑÖ¸ÜÔØÖÓÑØÖÑÓÒ<br />

ÖÖÝÓÒÐÝl 0ÕÙÒØÙÑÒÙÑÖ<strong>×</strong>ÙØÒÓj ´¾ºµ<br />

ÓÒ<strong>×</strong>ÝÑÔØÓØÓÙÖ¹ÑÒ<strong>×</strong>ÓÒÐÓ<strong>×</strong>ÖÚÖºÌØÓÖÓ4ÓÑ<strong>×</strong>ÖÓÑØÕÙÒØÞØÓÒÓ ÞÖÓ¹ÑÓ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØØÖÓÒ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÒÖØÓÖ<strong>×</strong>¸ÖÓÑØÔÓÒØÓÚÛ<br />

ZÖ(˜ρ) ≡ÌÖÖÐعÑÓÚÒÖÑÓÒ<strong>×</strong>((−1) (1 − e 2πinN ˜ρ ) 4 ,<br />

ØÖÖÑÓÒÞÖÓÑÓ<strong>×</strong>º ÆÜØÛÓÑÔÙØØÔÖØØÓÒÙÒØÓÒÓÖØÔÖØØØ<strong>×</strong>ÒØÖØÒºÌÖÖØÛÓ 0Ó ÔÖØ<strong>×</strong>ØÓØ<strong>×</strong>¸ØÞÖÓÑÓÓ<strong>×</strong>ÐÐØÓÖ<strong>×</strong>ÒØÒÓÒ¹ÞÖÓÑÓ<strong>×</strong>ºÝØÒØ<strong>×</strong>ÞR<br />

F<br />

(−1) j 0<br />

e 2πi˜ρl 0<br />

e 2πi˜νj 0<br />

) = 4<br />

∞∏<br />

n=1<br />

1)¹

ØÌÙ¹ÆÍÌ<strong>×</strong>ÔØÓÐÖ<strong>×</strong>ÓØØØÑØÖ<strong>×</strong>ÐÑÓ<strong>×</strong>ØظÒÒÐÓÐÖÓÒÓ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

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ZÓ<strong>×</strong>(˜ρ, ˜ν) ≡ÌÖÓ<strong>×</strong>ÐÐØÓÖ<strong>×</strong>((−1)<br />

0ÖÓÑØÓÙÖ¹ÑÒ<strong>×</strong>ÓÒÐÔÓÒØÓÚÛº<br />

(1 − e 2πinN ˜ρ+2πi˜ν ) 2 (1 − e 2πinN ˜ρ−2πi˜ν ) 2,<br />

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ÓÚÛ¸ØÝÚ<strong>×</strong>ØØ<strong>×</strong>Ø<strong>×</strong>(−1)<br />

<strong>×</strong>ÔºÌÔÖØØÓÒÙÒØÓÒÓÖØ<strong>×</strong>ÑÓ<strong>×</strong><strong>×</strong>ÚÝ<br />

F = (−1) j<br />

ÈÙØØÒØÓØÖÐÐØÓÒ<strong>×</strong>ØØÙÒØÔÖØØÓÒÙÒØÓÒ<strong>×</strong>¸ØÔÖØØÓÒÙÒØÓÒ<strong>×</strong><strong>×</strong>ÓØ<strong>×</strong> ´¾º½µ<br />

ÛØØÒØÖÓÑ<strong>×</strong><strong>×</strong>ÑÓØÓÒÓØ1¹5<strong>×</strong>Ý<strong>×</strong>ØÑÒØÌÙ¹ÆÍÌ<strong>×</strong>Ô<strong>×</strong>ÚÒÝ<br />

ZÞÖÓ(˜ν) ≡ÌÖÞÖÓÑÓ<strong>×</strong>((−1)<br />

(1 − e 2πi˜ν ) . 2 },´¾º¾µ {(1 − e 2πinN ˜ρ ) 4 (1 − e 2πinN ˜ρ+2πi˜ν ) −2 (1 − e 2πinN ˜ρ−2πi˜ν ) −2 <br />

∑<br />

l 0 ,j 0<br />

F ∞∏<br />

=<br />

F<br />

n=1<br />

(−1) j 0<br />

e 2πi˜νj 0<br />

= −<br />

(−1) j 0<br />

e 2πi˜ρl 0<br />

e 2πi˜νj 0<br />

)<br />

1<br />

∞∑<br />

j 0 e 2πi˜νj 0<br />

= −<br />

j 0 =1<br />

dØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>(l 0 , j 0 )(−1) j 0e 2πil 0˜ρ+2πij 0˜ν = ZÖ(˜ρ)ZÓ<strong>×</strong>(˜ρ˜ν)ZÞÖÓ(˜ν)<br />

= −4e 2πinu ˜ (1 − e 2πi˜ν ) −2<br />

<strong>×</strong><br />

∞∏<br />

n=1<br />

e2πi˜ν

¾ºº ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

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ÃÐÙÞ¹ÃÐÒÑÓÒÓÔÓÐÛÐÐÙ<strong>×</strong>ÒØÒØØÓÒØÛÒØÒÙÐÖÑÓÑÒØÙÑ<br />

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ÐÒ<strong>×</strong>ÐÓÒM=T 1¹ÖÒ<strong>×</strong>ÒÓÒ<strong>×</strong>ÖØÝÒÑ<strong>×</strong>ÓØ5¹ÖÒÛÖÔÔÓÒT ØÛÓÖйÚÓÐÙÑØÓÖÝÓØ5¹ÖÒ<strong>×</strong>(1+1)¹ÑÒ<strong>×</strong>ÓÒÐÛÓÒØÒ<strong>×</strong>Ø<strong>×</strong>ÐÖ<strong>×</strong><br />

1ÖÓÑØÔÓÒØÓÚÛÓ<br />

1ºÀÓÛÚÖ¸Ø<br />

4 <strong>×</strong> S<br />

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NÛÐ<br />

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NºÆØÖ<strong>×</strong>ÒÝÑÓÑÒØÙÑÐÓÒ˜S1ºËÑÐÖÐݸØÛÓ<br />

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0<strong>×</strong>ÛÓÑÓÛÒÖÓÑÓÙÖØÓÚ¹ÑÒ<strong>×</strong>ÓÒ<strong>×</strong>º<br />

1ÓØÓÖÑk + 1<br />

ØÓØÖØÛÓÓØÓÖÑk − 1<br />

ÓØÖÑÓÒÑÓ<strong>×</strong>ÖÖÝÑÓÑÒØÙÑk + NÐÓÒS 1 1ÛÐØÓØÖØÛÓÖÖÝk − 1<br />

ØÓ<strong>×</strong>ÐÐØÓÖ<strong>×</strong>ÖÐØÖÝØÓÖÓ(−1) j<br />

ÌÙ<strong>×</strong>¸dÏÐ<strong>×</strong>ÓÒ(l 0 , j 0

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

0ÐÓÒ˜S1¸ØÒ Ø<strong>×</strong>ÑÓ<strong>×</strong>ÖÖÝÒØÓØÐÑÓÑÒØÙÑl 0 /NÐÓÒS1Òj dÏÐ<strong>×</strong>ÓÒ(l<br />

l 0 ,j<br />

ÌÙÐÐÔÖØØÓÒÙÒØÓÒÓØÓÚÖÐÐÝÒÑ<strong>×</strong>ÓØ1¹5<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÚÒÝØ<br />

0<br />

.´¾º¿µ<br />

ÔÖÓÙØÓØÔÖØØÓÒÙÒØÓÒ<strong>×</strong>´¾º¾µÒ´¾º¿µÓØÝÒÑ<strong>×</strong>ÓØØÖÒ<strong>×</strong>ÚÖ<strong>×</strong>ÑÓ<strong>×</strong><br />

(1 − e 2πil˜ρ−2πi˜ν )<br />

(u)¸ÒÒÓØÒØØ4ºÌÒÐÖ<strong>×</strong>ÙÐØÒÛÖØØÒÓÑÔØÐÝ<br />

Ò´ÓÖM=T 4µÓØÏÐ<strong>×</strong>ÓÒÐÒ<strong>×</strong>ÐÓÒT Ù<strong>×</strong>ÒØÓÒØ<strong>×</strong>c (r,s)<br />

{ 2 4µ´¾º¿µÒÛÖØØÒ<strong>×</strong> ´¾ºµ ÓÖM N<br />

1<br />

(2 − N e2πis/N − e −2πis/N )ÓÖM ÌÔÖÓÙØÓ´¾º¾µÒ´ÓÖM = T<br />

´¾ºµ<br />

1<br />

Í<strong>×</strong>Ò´¾º¾µ¸´¾º¼µ¸´¾ºµÒ´¾ºµÛÔÙØØÓØÖØÙÐÐÔÖØØÓÒÙÒØÓÒ ¾ºº ÌÙÐÐÈÖØØÓÒÙÒØÓÒ<br />

4º ÓÖÓØM = K3ÒM T −2πi(eαeρ+eν)ÒØk=0ØÖÑÒØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÓÑÖÓÑØ ÌÑÙÐØÔÐØÚØÓÖe ØÖÑ<strong>×</strong>ÒÚÓÐÚÒd CM (l 0 , j 0 )Òd (l ′ 0 )ºÓÑÔÖÒÛØØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÓÖ˜Φ<br />

∑<br />

=<br />

0 , j 0 )(−1) j 0<br />

e 2πil 0˜ρ+2πij 0˜ν<br />

∏<br />

∏<br />

(1 − e 2πil˜ρ ) −2<br />

l∈NZ+1,l>0<br />

∏<br />

l∈NZ+1,l>0<br />

∑<br />

(1 − e 2πil˜ρ−2πi˜ν )<br />

b<br />

c (0,s)<br />

1 (−1) =<br />

l∈NZ−1,l>0<br />

∏<br />

l∈NZ−1,l>0<br />

(1 − e 2πil˜ρ ) −2 ∏<br />

(1 − e 2πil˜ρ+2πi˜ν )<br />

∏ ∞<br />

dÅ(l 0 , j 0 )(−1) j 0<br />

e 2πil 0˜ρ+2πij 0˜ν = −4e −2πi˜ν<br />

l 0 ,j 0 l=1<br />

∞∏<br />

(1 − e 2πil˜ρ+2πi˜ν ) − P N−1<br />

∞∏<br />

s=0 e−2πils/N c (0,s)<br />

1<br />

l=1<br />

=<br />

1∏<br />

N−1<br />

∏<br />

f (˜ρ, ˜σ, ˜ν) = e −2πi(eαeρ+eν)<br />

b=0 r=0<br />

KK<br />

∏<br />

k∈Z+ r N ,l∈Z,<br />

j∈2Z+b<br />

k,l≥0,<br />

j0<br />

∏<br />

l∈NZ−1,l>0<br />

= K3<br />

= T 4 .<br />

(1 − e 2πil˜ρ+2πi˜ν )<br />

(1 − e 2πil˜ρ ) 2 P N−1<br />

s=0 e−2πils/N c (0,s)<br />

1<br />

(1 − e 2πil˜ρ−2πi˜ν ) − P N−1<br />

s=0 e−2πils/N c (0,s)<br />

(1 − e 2πi(eσk+eρl+eνj) ) − P N−1<br />

.´¾ºµ<br />

s=0 e−2πils/N c (r,s)<br />

b (4kl−j 2)<br />

(˜ρ, ˜σ, ˜ν)Ò

´¾ºµÛÒÖÛÖØ´¾ºµ<strong>×</strong> ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

)<strong>×</strong>ÚÒÝ ´¾ºµ<br />

f<br />

ÛÖC<strong>×</strong>ØÖÖÐÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>Ù<strong>×</strong>ÔÓØØÖÓÑÔÐÜÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ÔÐÐÐ ´¾ºµ ÌÒÖÝd(q e ,q<br />

˜ν)¸ÚÒÝ<br />

m<br />

Ý(˜ρ, ˜σ,<br />

ÒØÖØÓÒºÌÙ<strong>×</strong>¸ÛÚÓÑÔÙØØÒÖÝÓÖÑÙÐÝÜÔÐØÐÝÓÙÒØÒØÐ ØÖÑÒÖÓÑØÖÕÙÖÑÒØØØØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓÒÚÖÒØÒØÖÓÒÓ i³<strong>×</strong>Ö ÀÖM 1 , M 2ÒM 3ÖÐÖÙØÜÔÓ<strong>×</strong>ØÚÒÙÑÖ<strong>×</strong>ÛØM 3

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>¿¿℄´<strong>×</strong>Ð<strong>×</strong>Ó¼¸½℄µºÓÒ<strong>×</strong>ÖØÓÐÐÓÛÒÝÓØÓÖ<strong>×</strong>ÓÒ<br />

2¹ÈËÝÓÒ<strong>×</strong><br />

4¹ÈË<strong>×</strong>ØØÝ<strong>×</strong> ÛØÓÓÖÒØλµ<strong>×</strong>ØÖÐÓÑÒ<strong>×</strong>ÓÒÓÒ<strong>×</strong>Ù<strong>×</strong>ÔÖÓ<strong>×</strong><strong>×</strong>ÛÓÒ1 ÒØÓÔÖÓ1 ÓÒ1 4¹ÈËÝÓÒÒØÓØÛÓ1<br />

dÖ<strong>×</strong>ÙØØ¿¿℄´¾ºµ<br />

,<br />

ÛÖØÒÑØ<strong>×</strong>ÓØÝÑÔÐÝØØØÒØÖ<strong>×</strong>a,<br />

¿ºÜÒÒØØÛÓÝÔÖÓÙØ<strong>×</strong>ÑÔÐ<strong>×</strong>ØÕÙÚÐÒÙÒÖ0º<br />

b, c,<br />

¾ºÌÕÙÚÐÒÖÐØÓÒ(a, b, c, d) ∼ (aσ −1 , bσ −1 , cσ, dσ)ÛØσ≠<br />

ÇÒÒ<strong>×</strong>ÓÛØØÝ<strong>×</strong>ÙØÐÙ<strong>×</strong>ÓØÕÙÚÐÒ<strong>×</strong>ÚÒÓÚ¸ÓÒÒÐÛÝ<strong>×</strong>ÓÓ<strong>×</strong><br />

ØÖÑÒÝØÕÙØÓÒ¿¿¸¼℄ 4ºÁÒØÙÔÔÖ¹ÐÔÐÒ¸Ø<strong>×</strong>ÛÐÐ<strong>×</strong>ÖÖÙÐÖÖ<strong>×</strong><br />

ºÖÕÙÒØÞØÓÒÖÕÙÖ<strong>×</strong>ad, bd, bc ∈ ZÒac<br />

∈ Γ 1<br />

´¾º¼µ<br />

(N)ÓÖN 2, 3,<br />

ÛÖE<strong>×</strong>ÖÐÙÒØÓÒÓÐÐÓØÖÑÓÙÐMºÁØ<strong>×</strong><strong>×</strong>ÝØÓ<strong>×</strong>ØØØÖ<strong>×</strong>ÒØÖ<strong>×</strong>ØØ<br />

ÒØÖÓÖÓØÙÔÔÖÐÔÐÒÛØÖÙ<strong>×</strong>Ð<strong>×</strong>ÓÒÖ<strong>×</strong>ÒÐÐØ<strong>×</strong>ÛØØÒØÖÔØ<strong>×</strong>ÓÒ<br />

0¸ØÒØÖÓØÖÐÑÓÚ<strong>×</strong>ÒØÓØ<br />

0¸ØÖ<strong>×</strong>Ö<strong>×</strong>ѹÖÐ<strong>×</strong>ÒØÖ ÖÐλÜ<strong>×</strong>ØØÔÓÒØ<strong>×</strong>b aÒd<br />

<strong>×</strong>ÓÐØÓÒÒÓÒ¹ÞÖÓE<strong>×</strong>ØÓÓÖѳØ<strong>×</strong>ѹÖÐ<strong>×</strong>ÒØÓÖÙÐÖÖ<strong>×</strong>¸<strong>×</strong>ÓÛÖ<strong>×</strong>ØÖØØ<br />

0¸ØÖÐ<strong>×</strong>ÓÑ<strong>×</strong>ØÖØ<br />

cÓÖÒÝEºÏÒE =<br />

ÓÒØÖÐλ¹Ü<strong>×</strong>ÛØÖÙ<strong>×</strong>1 2acºÏÒE 0º<br />

0ºÌ<br />

≠<br />

ØÖÐÜ<strong>×</strong>ÖÑÒÒÙÒÒºÏÒØÖa = 0ÓÖc ÓÒÐÓÓ<strong>×</strong>ÓÖØÐÖ<strong>×</strong>Ø<strong>×</strong>ѹÖÐÛØÓÒÒØλ=0ÓÒØÖÐÜ<strong>×</strong>ØØ<strong>×</strong>ÓÑÔØÐ ÙÒÑÒØÐÓÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÝÖ<strong>×</strong>ØÖ<strong>×</strong>ØÖØÒØÚÐÙÓÊ(λ)ØÓØÒØÖÚÐ<br />

ÐÒ<strong>×</strong>ÔÖÔÒÙÐÖØÓØÖÐÜ<strong>×</strong>ÓÖE = 0ÒÑÒ<strong>×</strong>ÙØÐÒÐÓÖE ÛØØÕÙÒØÞØÓÒÓÖ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>ѹÖÐÒØÖ<strong>×</strong>Ø<strong>×</strong>ØÖÐÜ<strong>×</strong>Ø<strong>×</strong>ÓÑÔÓÒØÒ 1ÓÖÖ<strong>×</strong>ÔÓÒØÓØÛÓÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØݺÆÜظ <strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒØÓØ<strong>×</strong>ÛÒE=<br />

[0, 1]ºÌ<strong>×</strong>ØÖØÐÒ<strong>×</strong>Ê(λ) 0,<br />

<br />

½ºad − bc = 1º<br />

( )<br />

a b<br />

c d<br />

(<br />

=<br />

)<br />

q e<br />

−→<br />

q m<br />

=<br />

(<br />

)<br />

ad q e − bd q m<br />

⊕<br />

ca q e − cb q m<br />

2ac<br />

(<br />

)<br />

−bc q e + bd q m<br />

−ac q e + ad q m<br />

(a, b, c, d) → (c, d, −a, −b)º<br />

∈ NZº<br />

] [Ê(λ) − ad+bc 2 ]<br />

+<br />

[ÁÑ(λ) + E 2<br />

= 1+E2<br />

2ac<br />

4a 2 c 2 ,<br />

=<br />

≠

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

ÝÐÓÓÒÓÖÒÓØÖ<strong>×</strong>ѹÖÐÛØÓÒÒØ1/NØÐÐÓÒØ<strong>×</strong>ØѹÛÝÔÓÒØ1/2º <strong>×</strong>ÑÐÖÔÖÓÙÖ<strong>×</strong>ÓÒ<strong>×</strong>ØÖØÒÛØØÐÖ<strong>×</strong>Ø<strong>×</strong>ѹÖÐÛØÓÒÒÓÒØÔÓÒØ<br />

1]Ø<strong>×</strong>ØÙÖÒ<strong>×</strong>ÓÙØØÓØ1/NºÌÔÖÓÙÖ<strong>×</strong>ØÒ´ÖÙÖ<strong>×</strong>ÚÐݵÖÔØ ØÒØÖÚÐ[0,<br />

ÙÒÑÒØÐÓÑÒ<strong>×</strong>ØÒÚÒÝÖ<strong>×</strong>ØÖØÒØÓØÖÓÒÓÙÒÝØ<strong>×</strong><strong>×</strong>ѹ ´¾º½µ<br />

λ = 1ÓÒØÖÐÜ<strong>×</strong>ºÇÒÓØÒ<strong>×</strong>ØÓÐÐÓÛÒ<strong>×</strong>ØÓÔÓÒØ<strong>×</strong>ÓÖN 0ºÌÙÒÑÒØÐÓÑÒ<strong>×</strong>ÖÚÒÒÙÖ¾ºº<br />

1ØÓÒÒØݺÌØÛÓ<strong>×</strong>ØÖØÐÒ<strong>×</strong>ÑÝ<br />

( 0, 1) , 1 1 (0, 1, 1) , 1 2 1 (0, 1, 1, 2, 1) . 1 3 2 3 1<br />

ÖÐ<strong>×</strong>ÒØØÛÓÛÐÐ<strong>×</strong>ÓÒÒØÒλ=0, ÒÐÙÝÒØÔÓÒØ<strong>×</strong>³−1 0Ò1<br />

=<br />

1, 2, 3<br />

N=1<br />

N=2<br />

Ø<strong>×</strong>ÑÖÓÒÔÔÖ<strong>×</strong><strong>×</strong>ØÏÝÐÑÖÓÃÅÄ<strong>×</strong>ÙÔÖÐÖÒ<strong>×</strong>º<br />

3ÀÄÑÓÐ<strong>×</strong>ºÏÛÐÐÐØÖ<strong>×</strong>ØØ<br />

N=3<br />

0 1/3 1/2 2/3 1<br />

ÑØÓµ 3¸Ø<strong>×</strong>ÔØÙÖÓ<strong>×</strong>ÒÓØØÖÑÒØÓÒÒ<strong>×</strong>ÒÒÒØÒÙÑÖÓ<strong>×</strong>ѹÖÐ<strong>×</strong><br />

4¸ØÓÐÐÓÛÒ<strong>×</strong>ÕÙÒ<strong>×</strong>ÓØÒÓÒ´Ù<strong>×</strong>ÒËÒ³<strong>×</strong><br />

ÙÖ¾ºÙÒÑÒØÐÓÑÒ<strong>×</strong>ÓÖØN = 1, 2,<br />

ÓÖN ><br />

ØÓÓØÒÐÓ<strong>×</strong>ÓÑÒºÓÖN =<br />

ÔØÒÙÖ¾ººÁØÑÝØÓÙØÓ<strong>×</strong>ÖÙÐÖÔÓÐÝÓÒÛØÒÒØ<strong>×</strong>ÛØØ<br />

( 0, 1, 1, 3, 2 −2n+1<br />

, . . ., , −n<br />

, . . ., 1 n+1<br />

, . . ., , 2n+1<br />

1 4 3 8 5 −4n −2n−1 2 2n+1<br />

nÒÓØØ<strong>×</strong>ѹÖÐÛØÒØÖÔØ<strong>×</strong>( 1Ö<strong>×</strong>ÔØÚÐݺÌÙÒÑÒØÐÓÑÒÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÓÚ<strong>×</strong>ÕÙÒ<strong>×</strong><br />

0ÖÔÖ<strong>×</strong>ÒØØØÛÓ<strong>×</strong>ØÖØÐÒ<strong>×</strong>Ø ÄØα 2n−1<br />

, n<br />

ÔØ<strong>×</strong>( 4n<br />

n+1<br />

, )ÓÖÐÐn 2n+1 ∈ ZºÆÓØØØα 0Òβ 2n+1 4n<br />

Ê(λ) = 0,<br />

ÒÒعÑÒ<strong>×</strong>ÓÒÐÖÐÖÓÙÔ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ÝÑÑØÖÝÖÓÙÔ¸D ∞ = Z ⋊ Z 2ºD (1)<br />

nØ<strong>×</strong>ѹÖÐÛØÒØÖ¹ ´¾º¾µ<br />

. . . , 3, 5, 2, 3, 1) . 4n 5 8 3 4 1<br />

∞<strong>×</strong>ÒÖØ<br />

2n+1)Òβ ¼

ÝØÛÓÒÖØÓÖ<strong>×</strong>ÖØÓÒyÒ<strong>×</strong>ØγÚÒÝ ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

´¾º¿µ<br />

<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> 2ÒÖØÝδÒ<br />

y : α n → α −n , β n → β −n−1Òγ : α n → α n+1 , β n → β n−1 ,<br />

∞º ´¾ºµ <strong>×</strong>Ø<strong>×</strong>ÝÒØÖÐØÓÒ<strong>×</strong>y2 = 1Òy ·γ ·y = γ −1ºÌÖ<strong>×</strong><strong>×</strong>ÓÒZ<br />

δ : α n ←→ β n .<br />

ÌØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>(γ, δ)ÒÖØÒÓØÖÖÐ<strong>×</strong>ÝÑÑØÖÝD (2)<br />

α 0<br />

Weyl chamber for N = 4<br />

α<br />

β<br />

1<br />

<strong>×</strong>ÓØÒÓÑÒØÓÖÓÖÑÙкÆÓØØØØÑØÖÓØ<strong>×</strong>ѹÖÐ<strong>×</strong>ÖÙÒ<strong>×</strong> ÖÐ<strong>×</strong>ÒØÖÔÖ<strong>×</strong>ÒØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ØØÔÔÖÛØÑÙÐØÔÐØÝÓÒÒØ<strong>×</strong>ÙÑ Ó<strong>×</strong>ѹÖÐ<strong>×</strong><strong>×</strong>ØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒÒØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÓØ<strong>×</strong>ѹ<br />

4ÀÄÑÓÐ<strong>×</strong>ÓÙÒÝÒÒÒØÒÙÑÖ<br />

1<br />

0 1/4 1/3 1/2 2/3 3/4<br />

3/8 5/8<br />

1<br />

Ä<strong>×</strong>ÙÔÖÐÖØØÛÓÒ<strong>×</strong>ØÖÙغ <strong>×</strong>ѹÖÐ<strong>×</strong>ºÏÛÐÐÐØÖ<strong>×</strong>ØØØ<strong>×</strong>ÑÖÓÒÔÔÖ<strong>×</strong><strong>×</strong>ØÏÝÐÑÖÓÃÅ<br />

2<strong>×</strong>ÔÔÖÓ<strong>×</strong>ÐÑØÔÓÒØÓØÒÒØ<strong>×</strong>ÕÙÒÓ<br />

ÙÖ¾ºÌÙÒÑÒØÐÓÑÒÓÖN =<br />

ÄØÖ¸ÛÒÛ<strong>×</strong>ØÙÝÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÐØØÓØÀÄÑÓÐ<strong>×</strong>¸ÛÛÐÐ<strong>×</strong>ØØØ Ì<strong>×</strong>ÓÑÔÐØ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓØÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓØÀÄÓÖÓÐ<strong>×</strong>º<br />

ÓÒØ<strong>×</strong>ÐÓ<strong>×</strong>ÖØÓ1 2ºÌÔÓÒØ1<br />

ÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÖÖÐØØÓØÛÐÐ<strong>×</strong>ÓØÏÝÐÑÖÓØÓÖÖ<strong>×</strong>ÔÓÒÒ<br />

β −1<br />

α −1<br />

β 0<br />

½

ÔØÖ¾ºÓÙÒØÒÝÓÒ<strong>×</strong>ÒËØÖÒÌÓÖÝ<br />

¾º½¼ ÓÒÐÙ<strong>×</strong>ÓÒÒÊÑÖ<strong>×</strong><br />

ÛÐÐÜÔÐÓÖØ<strong>×</strong>ØÖÙØÙÖÓØ<strong>×</strong>ÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÐØÖÔØÖ<strong>×</strong>ºÏÛÐÐÓÒ<strong>×</strong>ØÖÙØØ<strong>×</strong> <strong>×</strong>ØÖÒØÓÖ<strong>×</strong>ºÌÒÖÝÓØÝÓÒ<strong>×</strong>ØØ<strong>×</strong>ÖÒÖØÝÑÓÙÐÖÓÖÑ<strong>×</strong>ºÏ<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖ<br />

Ø<strong>×</strong>ÖÖÐØØÓØÐÖ<strong>×</strong>ØÖÙØÙÖÓÑÒÖÓÑØÑÓÙÐÖÓÖÑ<strong>×</strong>º ÑÓÙÐÖÓÖÑ<strong>×</strong>ÝÖÒØÑØÓ<strong>×</strong>ÒÔØÖ5Ò<strong>×</strong>ØÙÝØÖÐÖ<strong>×</strong>ÒÔØÖ6º<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ºÏÛÐÐÐØÖ<strong>×</strong>ÓÛ<br />

ÁÒØ<strong>×</strong>ÔØÖÛÚÐÓÓØØÔÖÓÐÑÓÓÙÒØÒÝÓÒ<strong>×</strong>ÒN =<br />

ÏÐ<strong>×</strong>Ó<strong>×</strong>ØÙØÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓÖØ1<br />

¾

3<br />

ÃÅÄÐÖ<strong>×</strong><br />

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4¹ÈËÝÓÒ<strong>×</strong>ÒØÀÄÑÓÐ<strong>×</strong>º<br />

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<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÛÓÙÖÒØÒÖÝÓÖÑÙÐÓ1<br />

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ØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÒÒÛØÒÜÑÔÐÓÃÅÄ <strong>×</strong>ÙÔÖÐÖºÌ<strong>×</strong>ÔØÖ<strong>×</strong><strong>×</strong>ÑÓ<strong>×</strong>ØÐÝÓÒ¾¸¿¸¸¸¿¸¸℄ºÌÖÖ<strong>×</strong>Ð<strong>×</strong>Ó ÒÓÙÖØÓ<strong>×</strong>¸¸¸¼¸½¸¾¸¿¸℄ÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒÖÐØÓÒØÓ <strong>×</strong>ØÖÒØÓÖݺ

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÐÖÓÐعÒÚÖÒØ<strong>×</strong>ÑÓÓØÚØÓÖÐ<strong>×</strong>ÓÒØÖÓÙÔµ¸ÓÖÙ<strong>×</strong>Ø<strong>×</strong>ÒÐÖÓÚÖÐ ¿º¾ ÄÐÖÒÙÒÖ<strong>×</strong>ØÓÓÒÖÐØÓÒØÓÄÖÓÙÔ¸ÛÓ<strong>×</strong>ÐÖØ<strong>×</strong>´<strong>×</strong>Ø ÒØÓÒÒÈÖÓÔÖØ<strong>×</strong><br />

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ØÓÐÐÓÛÒÔÖÓÔÖØ<strong>×</strong> ÛØ<strong>×</strong>ÐÖÑÙÐØÔÐØÓÒÝÐÑÒØ<strong>×</strong>ÓKºÁØ<strong>×</strong>ÄÐÖ¸ÒØÓÒ¸ØÐ<strong>×</strong>Ó<strong>×</strong><br />

gØØ<strong>×</strong><strong>×</strong>ØÖÙØÚÓÚÖØÓÒÒÓÑÔØÐ ÔÓ<strong>×</strong><strong>×</strong>µÒÓÛÛØÔÖÓÙØ[., .] : g<strong>×</strong>g →<br />

´µÒØ<strong>×</strong>ÝÑÑØÖÝ<br />

.]<strong>×</strong>ÐÒÖ¸ ´µ[.,<br />

[x, x] = 0, ∀x ∈ g (ÒÒ[x,<br />

ÒÙÐÖÑÓÑÒØÙѺÐÐØÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ÛÛÐÐ<strong>×</strong>ØÙÝÛÐÐÑØÖÜÄ ÜÑÔÐ<strong>×</strong>ÓÄÐÖ<strong>×</strong>ÒÔÝ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÙÐÑÐÖÖÓÑØ<strong>×</strong>ØÙÝÓØØÓÖÝÓ<br />

´µÂÓÒØØÝ[[x,<br />

C)¸Û<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ÓØÚ<br />

y],<br />

ÓØÓÚÒØÓÒÐÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÛÓ<strong>×</strong>ÒØÓÒÛÒÜØÑÓØÚØ ÌÑÓÖÒØÖ<strong>×</strong>ØÒÄÐÖ<strong>×</strong>¸ÒØÓÒ<strong>×</strong>ÛÛÐÐÐÒÛظÖ<strong>×</strong>Ù¹Ð<strong>×</strong><strong>×</strong><br />

n)ÑØÖ<strong>×</strong>ÓÚÖCº ÐÖ<strong>×</strong>¸ØØ<strong>×</strong>¸ØÝÒÙÒÖ<strong>×</strong>ØÓÓ<strong>×</strong><strong>×</strong>ÙÐÖÓgl(n,<br />

aº<br />

ÐÖÓÐÐ(n <strong>×</strong><br />

<strong>×</strong>ØÖÙØÙÖÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>º <strong>×</strong>ÙÐÖ¸ÐÐØÖØÒ<strong>×</strong>ÙÐÖ¸ÛÐÐÔÐÝÒÑÔÓÖØÒØÖÓÐÒÙÒÖ<strong>×</strong>ØÒÒØ 0ºÇÒ<strong>×</strong>Ù ÒØÓÒ¿º¾º¾Ä<strong>×</strong>ÙÐÖaÓg<strong>×</strong><strong>×</strong>Ù<strong>×</strong>Ô<strong>×</strong>Ø<strong>×</strong>ÝÒ[a, a] ⊆<br />

<br />

ÁØ<strong>×</strong>Ð<strong>×</strong>ÓÄÐÖºÄ<strong>×</strong>ÙÐÖa<strong>×</strong>ÐÐÐÒ[a, a] =<br />

´¿º½µ<br />

´¿º¾µ<br />

y] = −[y, x], ∀x, y ∈ g),<br />

z] + [[y, z], x] + [[z, x], y] = 0, ∀x, y, z ∈ g .

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÒиÝØ<strong>×</strong>ÒØÓÒ¸<strong>×</strong>Ð<strong>×</strong>ÓÄ<strong>×</strong>ÙÐÖºÁaÒbÖÐ<strong>×</strong>ÒÄÐÖ¸ aº<br />

ÒØÓÒ¿º¾ºÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖg<strong>×</strong><strong>×</strong>ØÓ<strong>×</strong>ÑÔÐØ<strong>×</strong>ÒÓÒ¹ÐÒ<br />

ÒØÓÒ¿º¾º¿<strong>×</strong>Ù<strong>×</strong>ÔaÓg<strong>×</strong>ÐÐÒÐØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>[a, g] ⊆<br />

ÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖg<strong>×</strong><strong>×</strong>ØÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐg<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓÖØ<strong>×</strong>ÙÑ<br />

0µÒg<strong>×</strong>ÒÓÔÖÓÔÖÒÓÒ¹ÞÖÓÐ<strong>×</strong>´ººØ<strong>×</strong>ÓÒÐÝÐ<strong>×</strong>ÖgÒ0µº ØÒ<strong>×</strong>ÓÖa +<br />

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´ºº[g, g] ≠<br />

ÒØÓÒØØØ<strong>×</strong><strong>×</strong>ÑÔÐ<strong>×</strong>ØØÓÔ<strong>×</strong><strong>×</strong>ØÓØÒÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ÖÓÑÒعÑÒ<strong>×</strong>ÓÒÐ ÒØÓÒÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ºÁØ<strong>×</strong>ÚØ<strong>×</strong> <strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÏÛÐÐÒØÖÓÙØÚÐÐÝÒÖØÓÖ<strong>×</strong>¸ÒØÚÐÐݹ <strong>×</strong>ѹ<strong>×</strong>ÑÔÐØÝØÖÓÙØÃÐÐÒÓÖѺ ËÖÖÖÐØÓÒ<strong>×</strong>ØÝ<strong>×</strong>Ø<strong>×</strong>ݸØØÔÔÖÓÔÖØÙÒØÙÖºÀÖÛÚØÒØÓÒÓ<br />

ÑÔÔÒ¸ÚÒÝ →ÒCg¸ÐÐØÓÒØ ÓÖÒÝÄÐÖÛÒÒÐÒÖÑÔ: g<br />

ÒÐÒÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ´xyµÖÓÑgØÓØ<strong>×</strong>кÌÃÐÐÒÓÖÑÓg<strong>×</strong>ÚÒÝ ÛÖÒCg<strong>×</strong>Ø<strong>×</strong>ÔÓÐÐC¹ÐÒÖÑÔ<strong>×</strong>ÖÓÑgØÓgº ÏÒÒÓÛÒØÃÐÐÒÓÖÑÓÒgºÚÒØÛÓÐÑÒØ<strong>×</strong>xÒyÒg¸ÛÒ<br />

x(y)<br />

ÌÃÐÐÒÓÖÑ<strong>×</strong>ÒÚÖÒØÒØ<strong>×</strong>Ò<strong>×</strong>ØØ ´¿ºµ<br />

B(x, y) =ÌÖ(xy)<br />

ÐÐÖØÒ³<strong>×</strong>ÖØÖÓÒÓÖ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐØݺ ÌÓÖÑ¿º¾ºÄÐÖg<strong>×</strong><strong>×</strong>ѹ<strong>×</strong>ÑÔÐÒÓÒÐÝØÃÐÐÒÓÖÑÓÒg<strong>×</strong>ÒÓÒ¹ ÏÒÒÓÛÒÒÕÙÚÐÒØÒØÓÒÓÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<br />

<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ØÓÑÓÖÒÖÐÄÐÖ<strong>×</strong>¸<strong>×</strong>ÓÑÓØÒØÓÒ<strong>×</strong>ÖÑÓÖ<br />

ÒÖغ ÌÖ<strong>×</strong>ÓÒÓÖØÑÙÐØÔÐÒØÓÒ<strong>×</strong><strong>×</strong>ØØÛÒÛÔ<strong>×</strong><strong>×</strong>ÖÓÑÒعÑÒ<strong>×</strong>ÓÒÐ<br />

b¸a ∩ bÒ[a, b]º<br />

= [x, y] , ´¿º¿µ<br />

.<br />

B([x, y], z) = B(x, [y, z]) .

<strong>×</strong>ÙØÐÒÖÐÞØÓÒ<strong>×</strong>ØÒÓØÖ<strong>×</strong>¸ÒÒØ<strong>×</strong>ÐÔÙÐØÓÙÒÖ<strong>×</strong>ØÒØÒØÓÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ØÓÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>º ÖÓÑÖÒØÔÓÒØ<strong>×</strong>ÓÚÛºÏÚ<strong>×</strong>ÓÑÜÑÔÐ<strong>×</strong>ÓÄÐÖ<strong>×</strong>¸ÓÖÑÓÚÒÓÒØÓ<br />

¿º¾º½ ÌÑÓ<strong>×</strong>ØÑÐÖÜÑÔÐÓÖÓÙÔÖÓÑÔÝ<strong>×</strong><strong>×</strong><strong>×</strong>ØØÖ¹ÑÒ<strong>×</strong>ÓÒÐÖÓØØÓÒÖÓÙÔ<br />

SO(3)ºÁØ<strong>×</strong>ØÖÓÙÔÓÐÐÖÓØØÓÒ<strong>×</strong>ÓÙØØÓÖÒÓÒØØÖ¹ÑÒ<strong>×</strong>ÓÒÐÙÐÒ ÜÑÔÐ<strong>×</strong><br />

ÌÄÐÖ<strong>×</strong><strong>×</strong>ÓØØÓØ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÔÓ3<strong>×</strong>3ÓÑÔÐÜÑØÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÝÒXÌ=−X¸ ÒÓØÝso(3)¸ÛÖØ<strong>×</strong>ÙÔÖ<strong>×</strong>ÖÔØÌÒÓØ<strong>×</strong>ØØÖÒ<strong>×</strong>ÔÓ<strong>×</strong>ÓÑØÖܺ<br />

3¸ÛØÓÑÔÓ<strong>×</strong>ØÓÒ<strong>×</strong>ØÖÓÙÔÓÔÖØÓÒºÁØÖÔÖ<strong>×</strong>ÒØ<strong>×</strong>Ø<strong>×</strong>ÝÑÑØÖ<strong>×</strong>Ó<strong>×</strong>ÔÖº 1º <strong>×</strong>Ô¸R nÓÑÔÐÜÑØÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÝÒ<br />

C)¸ÒÓÛÒ<strong>×</strong>Ø<br />

<strong>×</strong>ÑØÖÜÖÓÙÔ¸Ø<strong>×</strong>ØÖÓÙÔÓ3<strong>×</strong>3ÖÐÑØÖ<strong>×</strong>A<strong>×</strong>ÙØØAÌA = I¸ÒØA ÁØ<strong>×</strong><strong>×</strong>ÔÜÑÔÐÓÑÓÖÒÖÐÐ<strong>×</strong><strong>×</strong>ÓÄÐÖso(n,<br />

2ØÖÐ<strong>×</strong><strong>×</strong>ÑØÖ<strong>×</strong>ÓÚÖCºÏÖÐÖÑÒØÓÒØÎÖ<strong>×</strong>ÓÖÓÐÖ<strong>×</strong>Ø<strong>×</strong>Ý C)ºÁØ<strong>×</strong>Ø <strong>×</strong>ÔÐÓÖØÓÓÒÐÄÐÖ¸Û<strong>×</strong>Ø<strong>×</strong>ÔÓÐÐn <strong>×</strong><br />

ÒÓØÖÑÐÖÜÑÔÐ<strong>×</strong>Ø<strong>×</strong>ÔÐÐÒÖÄÐÖ¸ÒÓØsl(n, αβºÏÛÐÐ<strong>×</strong>ÒØÓÙÖ<strong>×</strong>ÓØ<strong>×</strong><br />

C)<strong>×</strong>ØÐÖÓ <strong>×</strong>ÔÓÐÐn <strong>×</strong> nÓÑÔÐÜØÖÐ<strong>×</strong><strong>×</strong>ÑØÖ<strong>×</strong>ÓÚÖCºÌÐÖsl(2,<br />

2 <strong>×</strong><br />

ØÓÙÖÖÑÓ<strong>×</strong>L mÓØÒÖÝÑÓÑÒØÙÑØÒ<strong>×</strong>ÓÖT )ºÏÛÐÐ<strong>×</strong>ØØØ<strong>×</strong>ÝÑÔÐØ<br />

C)<strong>×</strong>Ø<strong>×</strong>ÐÓÓÔÐÖº 2nÓÑÔÐÜ<br />

ÝÓÒÒÖÝÓÖÑÙк ÖÓÙÔÔÐÝ<strong>×</strong>ÚÖÝÑÔÓÖØÒØÖÓÐÒØØÓÖÝÓËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ØØÓÙÖÒØ<br />

ÔØÖØØØÎÖ<strong>×</strong>ÓÖÓÐÖ<strong>×</strong>ÖÐØØÓØÐÖsl(2, ÒÓØÖÜÑÔÐ<strong>×</strong>Ø<strong>×</strong>ÝÑÔÐØÐÖsp(n)ºÁØ<strong>×</strong>Ø<strong>×</strong>ÔÓ2n <strong>×</strong><br />

ÑØÖ<strong>×</strong>X<strong>×</strong>ÙØØJXÌJ = X¸ÛÖJ ÄÐÖ<strong>×</strong><strong>×</strong>ÛÐк<br />

nÓÑÔÐÜÑØÖ<strong>×</strong>ºÌÝÖÐÐ<strong>×</strong>ÑÔÐ<br />

C)¸ÒÓÛÒ<strong>×</strong>Ø<br />

¿º¿<br />

ÐÐØÓÚÑØÖÜÐÖ<strong>×</strong>Ö<strong>×</strong>ÙÐÖ<strong>×</strong>ÓØÄÐÖgl(n,<br />

ÊÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong><br />

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0 I n<br />

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ρ : g<br />

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:g→ÒCg .<br />

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ρ([x, y]) = ρ(x)ρ(y) − ρ(y)ρ(x) ÓÖÐÐx, y<br />

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= [x, y] .<br />

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.<br />

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1 z2 2 + . . . + a mz2 m ,<br />

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¼<br />

ρ(h)<br />

( ) ( ) ( )<br />

1 0 0 1 0 0<br />

h = , e = , f =<br />

0 −1 0 0 1 0<br />

[h, e] = 2e , [h, f] = −2f , [e, f] = h .<br />

B] = 2B , [A, C] = −2C , [B, C] = A ,<br />

= A , ρ(e) = B , ρ(f) = C<br />

f(z 1 , z 2 ) = a 0 z1 m + a 1z1 m−1 z 2 + a 2 z m−2<br />

ρ m (x)f = −(x 11 z 1 + x 12 z 2 ) ∂f − (x 21 z 1 − x 11 ) ∂f ,<br />

∂z 1 ∂z 2<br />

sl(2, C) →

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ÌØÓÒÓρ m (h)ÓÒ<strong>×</strong><strong>×</strong>ÐÑÒØz <strong>×</strong>ρ 2 2<br />

<strong>×</strong>ØØzk <strong>×</strong>ÒÒÚØÓÖÓÖρ (h)ÛØÒÚÐÙ(m 2k)ºÁÒÔÖØÙÐÖ¸ρ 2<br />

<strong>×</strong>ÓØØ<br />

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2 .<br />

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xÒy<strong>×</strong>ÚÒÓÚ¸Ø<strong>×</strong><br />

C)ÒØÖ<strong>×</strong>ÓÒ<strong>×</strong>ÙÓÖÒØÖm ≥<br />

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´¿º½µ<br />

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0µÓÖρ(x)u´Ö<strong>×</strong>Ôºρ(y)uµ<strong>×</strong>Ò ÒÒÚØÓÖuÓρ(h)¸ØÖρ(x)u = 0´Ö<strong>×</strong>Ôºρ(y)u ½<br />

(ρ m (h)f)(z) = −z 1<br />

∂f<br />

∂z 1<br />

+ z 2<br />

∂f<br />

∂z 2<br />

.<br />

∂ ∂<br />

ρ m (h) = −z 1 + z 2 .<br />

∂z 1 ∂z 2<br />

1 kzm−k<br />

m<br />

1 zm−k<br />

m<br />

ρ m (x) = −z 2<br />

∂<br />

∂z 1<br />

,<br />

−<br />

ρ m (x)z1 k zm−k 2 = −k z k−1<br />

m (h)<br />

(h) z k 1 zm−k 2 = (m−2k)z k 1 zm−k<br />

ρ m (y) = −z 1<br />

∂<br />

∂z 2<br />

,<br />

1 z2 m−k+1 ,<br />

ρ m (y)z1 k zm−k 2 = (k − m) z k+1<br />

1 zm−k<br />

1 z m−k−1<br />

C)<br />

ρ(h)ρ(x)u = (α + 2)ρ(x)u ,<br />

ρ(h)ρ(y)u = (α − 2)ρ(y)u .<br />

=

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

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ÕÙÚÐÒØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÚØ<strong>×</strong>ÑÛØ<strong>×</strong>ÒÑÙÐØÔÐØ<strong>×</strong>ºÏÛÐÐÓÑØÓ<br />

)³<strong>×</strong>ÛÖØ<strong>×</strong>ØÓÑÜÑÐÐÝ<br />

ØÒØÓÒÓÛØ<strong>×</strong>ÓÖÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖÐØÖÒØ<strong>×</strong><strong>×</strong>ØÓÒºÓÖÒÓÛ¸<br />

<strong>×</strong>ÓÐÐØÓÒÓ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ÓØρ(h i<br />

C)º<br />

<strong>×</strong>Ø<strong>×</strong>ÝÒ 2ÒÒØÖ<strong>×</strong>ºÌ ÛÓÒØÒÙÛØsl(3,<br />

ÓÖsl(3, C)¸ÐÐØÛØ<strong>×</strong>ÖÓØÓÖѵ=(m , m 2 )ÛØm , m<br />

ÛØÚØÓÖ<strong>×</strong>ÓØÓÒØÖÔÖ<strong>×</strong>ÒØØÓÒÖÐÐÖÓÓØÚØÓÖ<strong>×</strong>ºÌØ<strong>×</strong>¸ÓÖÚØÓÖz<br />

ÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÖÓÓØαº 2<strong>×</strong>ÐÐÖÓÓØÒØÐÑÒØz<strong>×</strong>ÐÐØÖÓÓØÚØÓÖ )µ¸ÛÖ<br />

[h<br />

<strong>×</strong>ÙÐÖÓØÄÐÖºÁØ<strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ÒØÖÐÖÓÐÒØ<strong>×</strong>ØÙÝÓØ<strong>×</strong>ØÖÙØÙÖÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÁØ<strong>×</strong>ÐÐØÖØÒ<br />

ØÔÖα iÖØ<strong>×</strong>ØÓÑÜÑÐÐÝÓÑÑÙØÒÐÑÒØ<strong>×</strong>ÒØÄÐÖºÌ<strong>×</strong><strong>×</strong>ØÔÐÝ<strong>×</strong><br />

≡ (a 1 , a 2 ) ∈ C<br />

ÌÖÓÓØ<strong>×</strong>´ÒÛØ<strong>×</strong>µÖÒ<strong>×</strong>Ø<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>Óh i´ρ(h i<br />

Øh <br />

ρ(h 1 )v = m 1 v ,<br />

ρ(h 2 )v = m 2 v .<br />

1<br />

1 , z] = a 1 z , [h 2 , z] = a 2 z ,<br />

= (m 1 , m 2 ) ∈<br />

1

ÒØÓÒ¿º¿º¿ÌÖØÒ<strong>×</strong>ÙÐÖÓÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg<strong>×</strong>Ø ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÓÑÔÐÜ<strong>×</strong>Ù<strong>×</strong>ÔhÓgÛØØÓÐÐÓÛÒÔÖÓÔÖØ<strong>×</strong> 0¸ h¸<br />

ÓÒØÓÒ(i)<strong>×</strong>Ý<strong>×</strong>ØØh<strong>×</strong>ÓÑÑÙØØÚ<strong>×</strong>ÙÐÖÓg¸ÒÓÒØÓÒ(ii)<strong>×</strong>Ý<strong>×</strong>ØØØ<br />

´µÓÖÐÐh 1Òh h¸h<strong>×</strong>ÓÒÐÞк<br />

2Òh¸[h 1 , h 2 ] =<br />

´µÓÖÐÐx ∈ h¸ØÒx ÓÒØÓÒ(iii)<strong>×</strong>Ý<strong>×</strong>h<strong>×</strong>ÓÒÐÞиÒ<strong>×</strong>ÒÐÐØh∈hÓÑÑÙظØh³<strong>×</strong><br />

´µÓÖÐÐh ∈<br />

ØÖÛ<strong>×</strong>ÓÒÐÝÓÒÐÑÒظhº<br />

<strong>×</strong>ÑÜÑÐÐÝÓÑÑÙØØÚºÁØ<strong>×</strong>ØÙ<strong>×</strong>¸ØÒÓÖÑÐÞÖN ÒÖÓÓØ<strong>×</strong>Ô<strong>×</strong>ÒÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ÖØÒ<strong>×</strong>ÙÐÖºÏØØÓÚÒÖÐÒØÓÒÓØÖØÒ<strong>×</strong>ÙÐÖ¸ØÖÓÓØ<strong>×</strong> ÌÖÒÓÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg<strong>×</strong>ÒØÓØÑÒ<strong>×</strong>ÓÒÓØ<strong>×</strong><br />

g<br />

Ð<strong>×</strong>ÓÓÑÑÙظÒØÙ<strong>×</strong>ØÝÖÐ<strong>×</strong>ÓÓÒÐÞÐ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐݺÓÖØ<strong>×</strong>Ósl(2, C)<br />

ÒØÓÒ¿º¿ºÖÓÓØÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg´ÛØÖ<strong>×</strong>ÔØØÓØÖØÒ<strong>×</strong>Ù¹<br />

gÛØ ∗<strong>×</strong>ÙØØØÖÜ<strong>×</strong>Ø<strong>×</strong>ÒÓÒ¹ÞÖÓÐÑÒØ ÐÖhµ<strong>×</strong>ÒÓÒ¹ÞÖÓÐÒÖÙÒØÓÒÐα∈h<br />

ËÓ¸ÖÓÓØ<strong>×</strong>Ù<strong>×</strong>Ø´ÒÓÒ¹ÞÖÓµÓÐÐØÓÒÓ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ÓÖØh³<strong>×</strong>ºÌ<strong>×</strong>ØÓ<br />

hº<br />

x ∈<br />

ÓÖÐÐh ∈<br />

ÖÐØÓÒ<strong>×</strong>ØÛÒØÐÑÒØ<strong>×</strong>¸ÛÒÒÓÛÜÔÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÑÒÓÖÑØÓÒ¸Ù<strong>×</strong>ÒØÓÒÔØ<br />

α<strong>×</strong>ÐÐÖÓÓØÚØÓÖ´ÓÖØÖÓÓØαµº<br />

C)ÒÛÓÖÒÓÙØØÚÖÓÙ<strong>×</strong>ÓÑÑÙØØÓÒ ÐÐÖÓÓØ<strong>×</strong><strong>×</strong>ÒÓØLºÌÖÓÓØ<strong>×</strong>Ôg α<strong>×</strong>Ø<strong>×</strong>ÔÓÐÐx∈gÓÖÛ[h, x]<br />

ÓÖÐÐh∈hºÒÐÑÒØÓg 2¸ÒØ<br />

ZØÓÖÖ<strong>×</strong>ÔÓÒÒÖÓÓØÚØÓÖº <strong>×</strong>ÔÝØÚÖÓÙ<strong>×</strong>ÓÑÑÙØØÓÒÖÐØÓÒ<strong>×</strong>¸ÛÛÓÐÓÛºÀÖ¸αÒÓØ<strong>×</strong>ØÖÓÓØÒ<br />

C)ºÚÒØÚÖÓÙ<strong>×</strong>ÖÓÓØ<strong>×</strong><strong>×</strong>ÒÓÙØÓ<br />

ÓÒØÓØ<strong>×</strong><strong>×</strong>ÐÑÒØ<strong>×</strong>Ósl(3, ÓÖÓÓØÚØÓÖ<strong>×</strong>¸<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÌÚØÓÖ<strong>×</strong>x iÒy iÖÒÚØÓÖ<strong>×</strong>ÓÖh 1Òh ÓÐÐØÓÒÓØÒÚÐÙ<strong>×</strong>ÖØÖÓÓØ<strong>×</strong>ÓÖsl(3,<br />

<br />

[h,<br />

g¸[h, x] = 0ÓÖÐÐh ∈<br />

∈<br />

x] = α(h)x ,<br />

(h) = {x ∈ g|[x, h] ⊆ h}ÓhÒgº<br />

= α(h)x

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

α<br />

Z<br />

(2, −1) x 1<br />

(−1, 2) x 2<br />

(1, 1) x 3 ´¿º¾¾µ<br />

(−2, 1) y<br />

ÚÖÓÙ<strong>×</strong>ÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ÛÒÛÚ<strong>×</strong>ØÙØÖØÓÖÝÐØØÐÑÓÖºÌÓÖÖÝÒÓÖÑØÓÒ ÓÙØÐÐØÖÓÓØ<strong>×</strong><strong>×</strong>ÖÙÒÒظÒØ<strong>×</strong><strong>×</strong>ÙÒØØÓÓÒÐÝÛÓÖÛØØÛÓÖÓÓØ<strong>×</strong>ÖÓÑØ<br />

1<br />

(1, −2) y 2<br />

2ºÏÛÐÐÐØÖÓÑØÓØ<br />

ÓÚ<strong>×</strong>ܸ<strong>×</strong>ØÓØÖ<strong>×</strong>ÒÛÖØÒÒØÖÑ<strong>×</strong>ÓØ<strong>×</strong>ØÛÓºÏ<strong>×</strong>ÒÐÓÙØØÖÓÓØ<strong>×</strong><br />

(−1, −1) y 3 .<br />

Ì<strong>×</strong><strong>×</strong>ÜÖÓÓØ<strong>×</strong>ÓÖÑÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÒÚÒØÓÒÐÐÝÐÐA ÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒÙ<strong>×</strong>ÙÐÐÝÒÓØΠºÌÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÚØÔÖÓÔÖØÝ ÒÜÔÖ<strong>×</strong><strong>×</strong>ØÓØÖÓÙÖÖÓÓØ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØѺÌÓÓØÐÐ<strong>×</strong>1Ò2<strong>×</strong> ÖØÖÖݸÒ<strong>×</strong>ÕÙÚÐÒØØÓÐÐÐÒØÑØÓØÖÛݺÌØÛÓÖÓÓØ<strong>×</strong>ÖÐÐØ ØØÐÐØÖÓÓØ<strong>×</strong>ÒÜÔÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÐÒÖÓÑÒØÓÒ<strong>×</strong>ÓØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛØ ÒØÖÓÒØ<strong>×</strong>¸<strong>×</strong>ÙØØØÓÒØ<strong>×</strong>ÖÐÐÔÓ<strong>×</strong>ØÚÓÖÐÐÒØÚºÅÓÖÒÖÐÐݸ <strong>×</strong>Ñ<strong>×</strong>ÑÔÐÄÐÖÓÖÒrÛÐÐÚrÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐ<br />

´¿º¾µ<br />

ÕÙÐØÓÞÖÓ´ÙØÒÓØÐÐÞÖÓ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐݵºÇÒÛÜ<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Øα³<strong>×</strong>ÓÖ<br />

ÖÓÓØ<strong>×</strong>Ö<strong>×</strong>ÙØØÓÖÒÝα j³<strong>×</strong>ÖÒØÖ<strong>×</strong>ÒØÖÐÐÖØÖØÒÓÖÕÙÐØÓÞÖÓÓÖÐÐÐ<strong>×</strong><strong>×</strong>ØÒÓÖ<br />

∈<br />

ÛÖØn +¸ÒÖØÙ<strong>×</strong>ÐÐØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>º −ºÆÓØØØÐÐØÐÑÒØ<strong>×</strong>ÓΠÖ<br />

+¸ÒØα³<strong>×</strong> Ûn j ≥ 0ÖÐÐØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>´ÛºÖºØØÓ<strong>×</strong>ÒΠµ¸ÒÓØL ÛØn j ≤ 0ÖÐÐØÒØÚÖÓÓØ<strong>×</strong>¸ÒÓØL 2<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

ÒL ÓÖØ<strong>×</strong>Ósl(3, C)ÐÐØÖÓÓØ<strong>×</strong>ÒÜÔÖ<strong>×</strong><strong>×</strong>ÒØÖÑ<strong>×</strong>Óα 1Òα<br />

α 1 = (2, −1) ´¿º¾¿µ<br />

α 2 = (−1, 2) ,<br />

L¸ÛÚ<br />

α = n 1 α 1 + n 2 α 2 + · · · + n r α r ,

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

(2, −1) = α 1<br />

(−1, 2) = α 2<br />

(1, 1) = α 1 + α 2 ´¿º¾µ<br />

(−2, 1) = −α<br />

ØÓÚÖØÖ<strong>×</strong>ÕÙÐÐÝ<strong>×</strong>ÙØк<br />

1<br />

(1,<br />

2<strong>×</strong>¸ÓÓÙÖ<strong>×</strong>¸ÖØÖÖݺÒÝÓØÖÔÖÛ<strong>×</strong>Ø<strong>×</strong><strong>×</strong><br />

−2) = −α 2<br />

(−1, −1) = −α 1 − α 2 .<br />

0¸Û<strong>×</strong>ØØØ ÌÓÓÔÒÓÙØα ÐÖgÒÓÑÔÓ<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÑ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

hÚ<strong>×</strong>ÓÑÔÓ<strong>×</strong>ØÓÒÓØÄÐÖg<strong>×</strong>ÖØ<strong>×</strong>ÙÑÓ<br />

1Òα iÖ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐÝÓÒÐÞкÌÄ ÓÒ<strong>×</strong>ÖÒØÐÑÒØ<strong>×</strong>Óh<strong>×</strong>ØÓ<strong>×</strong>ÐÓÒÒØÓØÖÓÓØ<strong>×</strong>Ôg ÓÒØØÓÒÓØh i ∈<br />

ÖÓÓØ<strong>×</strong>Ô<strong>×</strong>´ÓÑÔÖÛØ´µµ¸<strong>×</strong>ÒÐÐØh Ì<strong>×</strong>ÑÒ<strong>×</strong>ØØÚÖÝÐÑÒØÓgÒÛÖØØÒÙÒÕÙÐÝ<strong>×</strong><strong>×</strong>ÙÑÓÒÐÑÒØÓhÒ ´¿º¾µ<br />

g = ⊕ g α = ⊕ ⊕ ⊕<br />

g α h g α .<br />

<strong>×</strong>ÒÚØÓÖ<strong>×</strong>ÓØh³<strong>×</strong>Ú<strong>×</strong>Ù<strong>×</strong>ÓÒØÖÓÐÓÚÖØ<strong>×</strong>ØÖÙØÙÖÓØÄÐÖº<strong>×</strong>ÚØÓÖ ÛØÛÚÒÓÒºÆÓÛ¸ÛÒØÓ<strong>×</strong>ÓÛØÓÑÔÓ<strong>×</strong>ØÓÒÓØÄÐÖ<br />

α∈L<br />

αºÌ<strong>×</strong><strong>×</strong>ØÖ<strong>×</strong>Ø<strong>×</strong>ÒÓÑÖÒÓ<strong>×</strong>ÓÑÑØÓØÓ<br />

α∈L − α∈L +<br />

ÓÒÐÑÒØÖÓÑÖÓÓØ<strong>×</strong>Ôg −ÖØÚØÓÖ<strong>×</strong>Ô<strong>×</strong>ÒÖØÝØÐÑÒØ<strong>×</strong><br />

ظÒ<strong>×</strong>ÛÛÐÐ<strong>×</strong>¸ÛÒÖÙØ<strong>×</strong><strong>×</strong>ÒÓØÔÖÓÐÑÚÒÙÖØÖÛÒÛÖÐØ ÛØÔÓ<strong>×</strong>ØÚÒÒØÚÒÚÐÙ<strong>×</strong>ÛºÖºØØÖØÒ<strong>×</strong>ÙÐÖhÖ<strong>×</strong>ÔØÚÐݺ Ì<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ΠÓØÄÐÖ<strong>×</strong>ÐÐØÒÓÖÑØÓÒÓØÄÐÖÒ<br />

ÕÙÚÐÒØÖѺÌ<strong>×</strong><strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ÒÓØÙ<strong>×</strong>Ø<strong>×</strong>ÖÓÖÙ<strong>×</strong>ØÄÐÖ ØÑØÓ<strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÛÖØÛÓÐÄÐÖ<strong>×</strong>ÔØÙÖÝÑØÖܸÓÖÒ<br />

<strong>×</strong>Ô<strong>×</strong>g=N + ⊕ h ⊕ N −¸ÛÖN + /N<br />

<strong>×</strong>Ñ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒØÓÒØÒÙÑÖÓÐ<strong>×</strong><strong>×</strong><strong>×</strong>º ÛÓ<strong>×</strong>ÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ØÝÖ<strong>×</strong>ÓÑÓÖÔØÓ¸ÙØÒØÛÐÐÐÐÓÛÙ<strong>×</strong>ØÓÐ<strong>×</strong><strong>×</strong>ÝÐÐØÔÓ<strong>×</strong><strong>×</strong>Ð<br />

ÓÛÚÖ¸ÛÖÓÖ<strong>×</strong>ÓÑÓØÔÖÓÔÖØ<strong>×</strong>ÓØÖÓÓØ<strong>×</strong>¸ÛØÓÙØÔÖÓÓ¸ÐÓÛº ÏÛÐÐÐÓÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÖÝØÓÓÑÔÐØØ<strong>×</strong>ØÙÝÛÚ<strong>×</strong>ØÖØÛØØÜÑÔÐ<br />

C)¸ØÖÛ<strong>×</strong>ØÙÝØÏÝÐÖÓÙÔÒÏÝÐÖØÓÒ<strong>×</strong>ÓØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѺÖ<strong>×</strong>ØÐݸ<br />

<br />

Ósl(3,

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´ÚµÁα<strong>×</strong>ÖÓÓØÓg¸ØÒØÓÒÐÝÑÙÐØÔÐ<strong>×</strong>ÓαØØÖÖÓÓØ<strong>×</strong>ÖαÒ−α´ÓÑÔÖ<br />

∗<strong>×</strong>ÖÓÓظØÒ<strong>×</strong>Ó<strong>×</strong>−α´ÓÑÔÖÛØ´µµ¸ ´µÓÖÒÝαÒβÒh∗¸[g ÛØ´µÒ´µµ¸<br />

∗¸ ´µÁα∈h ´µÌÖÓÓØ<strong>×</strong><strong>×</strong>ÔÒh<br />

Ò<strong>×</strong>Ò´¿º¾µº ´ÚµÁαÒβÖÖÓÓØ<strong>×</strong>¸ØÕÙÒØØÝ2 αÖÓÒ¹ÑÒ<strong>×</strong>ÓÒи<br />

.〉<strong>×</strong><br />

h<strong>×</strong>Ù<br />

〈α,α〉<strong>×</strong>ÒÒØÖ¸ÛÖØÒÒÖÔÖÓÙØ〈., 〈α,β〉<br />

´ÚµÓÖÐÐÖÓÓØ<strong>×</strong>α¸ØÖÓÓØ<strong>×</strong>Ô<strong>×</strong>g ´ÚµÓÖÖÓÓØα¸ÛÒÒÒÓÒ¹ÞÖÓÐÑÒØ<strong>×</strong>x α ∈ g α¸y α ∈ g −aÒh ∈<br />

ØØx α , y C)º<br />

ÒÔÒÒØÓØÖÓÖÒ<strong>×</strong>Ò<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÌ<strong>×</strong>Ñ<strong>×</strong><strong>×</strong>Ò<strong>×</strong>Ù<strong>×</strong>¸ÑÒÝÓ ÀÚÒ<strong>×</strong>ØÙØÖÓÓØ<strong>×</strong>ÒØÖÔÖÓÔÖØ<strong>×</strong>¸ÓÒÒÓÒÓÒ<strong>×</strong>ÖÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong><br />

ÐÖÝÕÙÓØØÑÔÓÖØÒØÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÓÙØÖÓÓØ<strong>×</strong>¸ÒÖ<strong>×</strong><strong>×</strong>ÒÒØÓ<strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong> ØÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÓÙØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÒÚÓÐÚÓÒÐÝØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÒÒÓØØÄÐÖ<strong>×</strong>ÖÓÑ<br />

ÛÐÐÒÓØ<strong>×</strong>ÖÚÙ<strong>×</strong>ÒÝÒÛÔÙÖÔÓ<strong>×</strong>ØÓÚÓØ<strong>×</strong>ÔØÓ<strong>×</strong>ØÙÝÒØѺÀÓÛÚÖ¸ÛÑÒØÓÒ ÛØÝÓѺÌÖÓÖ¸ÓÒÒ<strong>×</strong>ØÙÝØØÓÖÝÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÓÒØÖÓÛÒºÏÚ<br />

−αÒh α<strong>×</strong>ÔÒ<strong>×</strong>ÙÐÖÓg<strong>×</strong>ÓÑÓÖÔØÓsl(2,<br />

ÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ¸ÓÒÒ<strong>×</strong><strong>×</strong>ÓØØÓØÒ<strong>×</strong>ØÖØÖÙÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ ÒÚ¹ÚÖ<strong>×</strong>ºÇÒÒÙ<strong>×</strong>ØÐ<strong>×</strong><strong>×</strong>ØÓÒÓØ<strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÒØÖÒ<strong>×</strong>ÐØ <strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>Ù<strong>×</strong>ØÖ<strong>×</strong>ÔÓÒØØÓØÛÝÖÓÑØÓÚ<strong>×</strong>ØÙݺÚÒ<br />

ØÖØÒÑØÖ<strong>×</strong>ÓÖØ<strong>×</strong>ÖÙÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÛÛÐÐÒÔÖØÙÐÖÓÖÖÒÓ ØØÓÐ<strong>×</strong><strong>×</strong>ÝÒ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>ÓØ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÓÒ<strong>×</strong>ØÓÓØ ÖØÒ<strong>×</strong>ÙÐÖ¸ÛÓ<strong>×</strong><strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ØÖÓÓØ<strong>×</strong>ÖºÄØÖÛÒÛÓÒ<strong>×</strong>ØÖÙØ<br />

<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑØÛÒ<strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÒÖØÒÑØÖ<strong>×</strong>ºÏÚÐÖÝ<strong>×</strong>ÒØ<strong>×</strong><br />

2ÑÒØÓÒÓÚµÒ<strong>×</strong>ÙØØÓØظØÖ<strong>×</strong>Ò<br />

Ó<strong>×</strong>ÒÓØÑØØÖºÏÐ<strong>×</strong>ÓÑÒØÓÒØØØÓÓØØÛÓÖÓÓØ<strong>×</strong><strong>×</strong>ÖØÖÖݸÒÒÝ ÛÖØØÒØÓØÖÖÓÓØ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØÑÒ<strong>×</strong>ØÓÖÖÒÓØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><br />

C)¸ÛÖÛ<strong>×</strong>ÒÐÓÙØØÛÓÖÓÓØ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Ò ØÖÓÓØ<strong>×</strong>´ÚÞºØÐÐÐÒÓα ØÛÒÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÖØÒÑØÖ<strong>×</strong>¸ÚÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸Ù<strong>×</strong>ÙÐÛ ÓØÖ<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ØØ<strong>×</strong>Ø<strong>×</strong>ÝØÔÖÓÔÖÖØÖÖÕÙÐÐÝÓÓÒØ<strong>×</strong>ºÐ<strong>×</strong>Ó¸ ÒÝØÛÓÖØÒ<strong>×</strong>ÙÐÖ<strong>×</strong>ÓgÖÓÒÙØØÓÓØÖºÌÙ<strong>×</strong>¸ØÓÑØ<strong>×</strong><strong>×</strong>ÓØÓÒ<br />

1Òα ÔÔÒÒØÜÑÔÐÓsl(3,<br />

¼<br />

α , g β ] ⊂ g α+β¸<br />

α

ÒØÓÜÑÒØÒÔÒÒÓØÓÚ<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>Ñ<strong>×</strong>ÓØÓÓØÓÖÖÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÓØÖÓÓØ<strong>×</strong>º<br />

ØÓÚÕÙ<strong>×</strong>ØÓÒÓØÒÔÒÒÓØÓÓÓÖÖÒÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ò<strong>×</strong><strong>×</strong>ÓØÒ ¿º¿º ÏÛÐÐ<strong>×</strong>ØÙÝØÓØÏÝÐÖÓÙÔÓgÒÓÛ¸ÒÒÓÒ<strong>×</strong>Ó¸ØÖÝØÓÙ<strong>×</strong>ØØÓÖ<strong>×</strong><strong>×</strong> ÌÏÝÐÖÓÙÔ<br />

ÏÝÐÖÓÙÔ¸ÛÒÖÑØÒÒÒÖÔÖÓÙØÓÒgÛ<strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÓÖÑØÖ<strong>×</strong> ÖØÒÑØÖ<strong>×</strong>ØÓÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÓÖÛÑÓØÚØØÓØ<br />

xÒy¸ÛÒÒÒÒÖÔÖÓÙØÓÒg<strong>×</strong><br />

ÓÒ<strong>×</strong>ÖØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>LÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖgÒ<strong>×</strong>Ø¸Π¸Ó<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong> ´¿º¾µ<br />

ÚØÓÖ<strong>×</strong>ÔÓÐÒÖÓÑÒØÓÒÓÐÐÖÓÓØ<strong>×</strong>α∈LºÆÓØØÖÒØÛÒEÒ <strong>×</strong>ºÄØØÚØÓÖ<strong>×</strong>ÔÒÖØÝΠEºE<strong>×</strong>Ù<strong>×</strong>ØØr¹ÑÒ<strong>×</strong>ÓÒÐÙÐÒ ØØÒÖØLºÁÒØØÓÖÝÓ<strong>×</strong>ØÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸Ø<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><strong>×</strong>ÐÐ<br />

〈x, y〉 =ÌÖ(xy ∗ )<br />

ÐÑÒØ<strong>×</strong>ÓΠÛØÒØÖÓÒØ<strong>×</strong>ÒÒ<strong>×</strong>ÙÛÝØØØÓÒØ<strong>×</strong>ÖØÖÐÐ ÒÓÒ¹ÒØÚÓÖÐÐÒÓÒ¹ÔÓ<strong>×</strong>ØÚ¸ÛÖ<strong>×</strong>E<strong>×</strong>Ù<strong>×</strong>ØØÚØÓÖ<strong>×</strong>ÔÒÖØÝΠÛØÓÙØ<br />

LºÐÑÒØα∈L<strong>×</strong><strong>×</strong>ÙØØØÒÜÔÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÐÒÖÓÑÒØÓÒÓØ<br />

ÒÝ<strong>×</strong>ÙÖ<strong>×</strong>ØÖØÓÒ<strong>×</strong>º<br />

ÒÓÛÒ<strong>×</strong>ÖØÓÒ¸Ù<strong>×</strong>ÓÑØÖÐÐݸØ<strong>×</strong>ÓÒÒØ<strong>×</strong>ÔE<strong>×</strong>ÛÛÐÐÜÔÐÒ<br />

ÓÖÒÝØÛÓÖÓÓØ<strong>×</strong>α, β ∈ L¸ÓÒ<strong>×</strong>ÖØÓÐÐÓÛÒÐÒÖØÖÒ<strong>×</strong>ÓÖÑØÓÒÓE<br />

E¸ØÖØ<br />

ÒØÓÑÔÓ<strong>×</strong>ØÓÒÓÖÖØÓÒ<strong>×</strong><strong>×</strong>ÒÖØÓÒ´ÐÐÛÓÖ³µºÌÙ<strong>×</strong>¸Ø<strong>×</strong>ÙÖÓÙÔ ÓÒØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>ÒØÒLØÓØ<strong>×</strong>кÐ<strong>×</strong>Ó¸ÖØÓÒÔÓ<strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong>ÒÒÚÖ<strong>×</strong>´Ø<strong>×</strong>е¸<br />

β<strong>×</strong>Ð<strong>×</strong>ÓÖÓÓغÌ<strong>×</strong>ØÓÐÐ<strong>×</strong>ÙÖØÓÒ<strong>×</strong>ÛÐÐØÙ<strong>×</strong>Ø<strong>×</strong>ÔÖÑÙØØÓÒ ÐÓÛºÓÖÒÓÛ¸ÛÒÓØØÔÖÓÔÖØÝØØw ÐÐØÏÝÐÖÓÙÔÓLÒÖØÓÒ<strong>×</strong>ÒÓÛÒ<strong>×</strong>ÏÝÐÖØÓÒºËÒØ<br />

α 2 = 1¸∀α ∈ LºÓÖÐÐα, ∈<br />

ÐÑÒØw α ·<br />

αÓÖα∈LÓÖÑ<strong>×</strong>ÖÓÙÔ<br />

ÙÒÖÐÝÒÄÐÖºÁÒØ<strong>×</strong>ÛÒØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÖÒØ<strong>×</strong>ÖÓÑÄÐÖg½<br />

ÒØÖÓÙÔºÆÓØØØÙÔØÓØ<strong>×</strong>ÔÓÒØØÖ<strong>×</strong>ÒÒÓÑÒØÓÒÓØÖÐØÓÒØÓØ ÖÒrÓg<strong>×</strong>ÒظØÏÝÐÖÓÙÔÒÖØ<strong>×</strong>ÖØÓÒ<strong>×</strong>ÛºÖºØØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong><strong>×</strong> ÓØÓÖØÓÓÒÐÖÓÙÔÓÒEÒÖØÝÐÐØÖØÓÒ<strong>×</strong>w<br />

´¿º¾µ<br />

〈α, β〉<br />

w α · β = β − 2<br />

〈α, α〉 α, β ∈ E ,<br />

β

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÒØÖÙÑÒØ<strong>×</strong>ÖÙÒÖ<strong>×</strong>ØÓÓº ÆÜظÛÒÕÙÒØØÝÒÓÛÒ<strong>×</strong>ØÏÝÐÚØÓÖρÓL<strong>×</strong> ´¿º¾µ<br />

ÛØÖØÒ<strong>×</strong>ÙÐÖh¸ØÏÝÐÖÓÙÔ<strong>×</strong>ÒÓØÝW(g, h)ºÏÛÐÐ<strong>×</strong>ÑÔÐÝÐÐØW<br />

ÒÓÑÒØÓÖÓÖÑÙÐÓÄÐÖ<strong>×</strong>¸ÓØÒØÒÒÒعÑÒ<strong>×</strong>ÓÒкÆÓÛ¸ÐÓÓÒØ ØÓÚÕÙØÓÒ¸ÛÒÓØØØØÛÝØÏÝÐÚØÓÖ<strong>×</strong>Ò¸<strong>×</strong><strong>×</strong>ÙÑÓÚÖØ ÌÏÝÐÚØÓÖÛÐÐÔÐÝÚÖÝÑÔÓÖØÒØÖÓÐÐØÖÛÒÛ<strong>×</strong>ØÙÝØÖØÖÒ<br />

∑<br />

ρ = 1<br />

<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>¸ÑÝÒÓØ<strong>×</strong>ÙØÐÓÖÒÖÐÞØÓÒØÓØ<strong>×</strong>ÓÒÒعÑÒ<strong>×</strong>ÓÒÐÄ<br />

2<br />

ÐÖ<strong>×</strong><strong>×</strong>ÒØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong><strong>×</strong>ÒÓØÒغÌÖ<strong>×</strong>Ò ÐØÖÒØÒØÓÒÛÐÒ<strong>×</strong>Ø<strong>×</strong>ÐØÓÒÖÐÞØÓÒØÓØÒÒØ<strong>×</strong>ÛØÓÙØÒÚÓÐÚÒ<br />

<strong>×</strong>Ø<strong>×</strong><strong>×</strong> ÒÒØ<strong>×</strong>ÙÑ<strong>×</strong>ÒÛÒØÐÓÛº ÒØÓÒ¿º¿ºÌÏÝÐÚØÓÖ¸ρ¸ÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑL<strong>×</strong>ÒØÓØÚØÓÖÛ<br />

Õº´¿º¿¼µÒ<strong>×</strong>ØÏÝÐÚØÓÖÚÒÓÖÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ÒØ<strong>×</strong>Ø<strong>×</strong><br />

ÝÔÖÔÐÒØÖÓÙØÓÖÒÒE<strong>×</strong>ÙØØVÓ<strong>×</strong>ÒÓØÓÒØÒÒÝÖÓÓغÓÒ<strong>×</strong>ÖÒ ÒØÓÒØØÛÛÐÐÙ<strong>×</strong>ÖÓÑÖÓÒº ÏÒÐ<strong>×</strong>ÓÚÐÓÔÓÑØÖÐÔØÙÖÓØÓÚ<strong>×</strong>ÒØ<strong>×</strong>ÔEºÄØV<br />

〈ρ, α i 〉 = 1, ÓÖÐÐα ∈ Π<br />

0¸ÒØÖ<strong>×</strong>ÓVÛÐÐÐÑÒØ<strong>×</strong>ØØ<strong>×</strong>Ø<strong>×</strong>ÝØÒÕÙÐØÝ<br />

ÓÖÒÝÚØÓÖαÒØÓÒ¹ÑÒ<strong>×</strong>ÓÒÐÓÖØÓÓÒÐÓÑÔÐÑÒØÓV¸ÐØÙ<strong>×</strong>ÒÓØØ<strong>×</strong><strong>×</strong><br />

E)¸ØÝÔÖÔÐÒVÔÖØØÓÒ<strong>×</strong>Ø<strong>×</strong>ÔEÒØÓØÛÓ<strong>×</strong><strong>×</strong>º<br />

ÐÑÒØαÛ<strong>×</strong>ÔÖÔÒÙÐÖØÓØ<strong>×</strong>ÝÔÖÔÐÒºÌÙ<strong>×</strong>¸VÛÐÐØ<strong>×</strong>ØÓÐÑÒØ<strong>×</strong>µ ÒE<strong>×</strong>ÙØØ〈α, µ〉 = <strong>×</strong>ÓÖLºÆÓÛÛÒÑØÓÒÒØÓÒØÓÛØÛÐÖÒØÓÚ¸ÛÒØÝØ<br />

γºÒÐÑÒØÛ<strong>×</strong>ÒÓØ<br />

ÚÒØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ(L, +<strong>×</strong> <strong>×</strong>Ø<strong>×</strong>ÝÒ〈α, µ〉 +ÒL −Ö<strong>×</strong>ÔØÚÐݺÒÝÐÑÒØαÓL ÐÐÓÑÔÓ<strong>×</strong>ÐØÖÜ<strong>×</strong>ØβÒγ<strong>×</strong>ÙØØα=β+ ÓÑÔÓ<strong>×</strong>Ð<strong>×</strong>ÐÐÒÓÑÔÓ<strong>×</strong>кÌ<strong>×</strong>ØÓÐÐÒÓÑÔÓ<strong>×</strong>ÐÐÑÒØ<strong>×</strong>ÒL −¸<strong>×</strong>ÒÒØ<strong>×</strong>ÔE¸ÖÜØÐÝØ<strong>×</strong>Ø<strong>×</strong>ÓÔÓ<strong>×</strong>ØÚÒÒØÚÖÓÓØ<strong>×</strong>ÓØ <strong>×</strong>ÓÖÑLÛØØ<strong>×</strong>ΠÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØÄÐÖgºÌÒ¸Ø<strong>×</strong>Ø<strong>×</strong>L α¸ØØÒÖØØÏÝÐÖÓÙÔÖÖØÓÒ<strong>×</strong>ÛØ −µº<br />

+<br />

ÒL ÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓg´ÒØÒÓØØÓÒØÓÐÐØÑL +ÒL ¾ ÓÑØÖÐÐݸØÖØÓÒ<strong>×</strong>¸w<br />

〈α, µ〉 > 0ÓÖ〈α, µ〉 < 0º<br />

> 0Ò〈α, µ〉 < 0ÝL<br />

α∈L + α .<br />

i

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÓØÒÝÖØÒØÖÓÓØβ¸ÛØÖ<strong>×</strong>ÔØØÓØÝÔÖÔÐÒÔÖÔÒÙÐÖØÓα¸ÒEº<br />

(β)ÛÓÙÐØÚØÓÖ<br />

ÐÑÒØ<strong>×</strong>ÓΠºÌ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÐÐØÓÔÒÙÒÑÒØÐÏÝÐÑÖÒE´ÖÐØÚØÓ<br />

Ö<strong>×</strong>ÔØØÓØÝÔÖÔÐÒÒEÔÖÔÒÙÐÖØÓØÖÓÓØαºw α<br />

iÖÐÐØ 0ÓÖ<br />

ÒÓÔÒÏÝÐÑÖ<strong>×</strong><strong>×</strong>ÒÝÓÖØÄÐÖgÌ<strong>×</strong>ÐÑÒØ<strong>×</strong>ÖÑÔÓÖØÒØÒ<br />

ÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØαÛÐÐÔÖØØÓÒEÒØÓØÛÓÐÚ<strong>×</strong><strong>×</strong>ÙØØØÖ〈α, µ〉 <<br />

〈α,<br />

ØÖÔÖ<strong>×</strong>ÒØØÓÒØÓÖÝÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÖÐÐØÓÑÒÒØÒØÖÐ<br />

ΠºÇÒÑØÛÓÒÖÛØÓØÐÑÒØ<strong>×</strong>ÓØÐÓ<strong>×</strong><br />

µ〉 > 0ºÚÒΠ¸ÛÓÒ<strong>×</strong>ÖØÒØÖ<strong>×</strong>ØÓÒÓØ<strong>×</strong>Ø<strong>×</strong>〈α , H〉 > 0¸ÛÖα ΠµºÌÐÓ<strong>×</strong>ÙÒÑÒØÐÏÝÐÑÖÒE´ÖÐØÚØÓΠµ<strong>×</strong>Ø<strong>×</strong>ØÓÐÐH ÌÏÝÐÑÖÔÒ<strong>×</strong>ÓÒØ<strong>×</strong>ΠÒÖÒظÙØÕÙÚÐÒظ<strong>×</strong>ÛÐÐÚ<br />

∈<br />

<strong>×</strong>ÙØØ〈α i , H〉 ≥ 0ÓÖÐÐα ∈<br />

CÓÖ<br />

C)º<br />

ÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÏÝÐÖÓÙÔØ<strong>×</strong><strong>×</strong>ÑÔÐÝÒØÖÒ<strong>×</strong>ØÚÐÝÓÒØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<br />

ÐÑÒØ<strong>×</strong>ºÏÛÐÐØÐÓÙØØÑÛÒÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÖÔÖ<strong>×</strong>ÒØØÓÒØÓÖÝÓsl(3, CÒØÓØÖÛÝ<br />

<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒÐ<strong>×</strong>ÓÓÒØ<strong>×</strong>ØÓÏÝÐÑÖ<strong>×</strong>º ÖÓÙÒºËÓ¸ØÖ<strong>×</strong>ÓÒ¹ØÓ¹ÓÒÓÖÖ<strong>×</strong>ÔÓÒÒØÛÒÏÝÐÑÖ<strong>×</strong>Ò<strong>×</strong><strong>×</strong>¸ÓÖ<strong>×</strong>Ø ÖÒØÏÝÐÑÖºÓÖÓÔÒÏÝÐÑÖC¸ØÖÜ<strong>×</strong>Ø<strong>×</strong>ÙÒÕÙ<strong>×</strong>Π L<strong>×</strong>ÙØØC<strong>×</strong>ØÓÔÒÙÒÑÒØÐÏÝÐÑÖ<strong>×</strong><strong>×</strong>ÓØØÓΠ ¾ºËÐÖÔÖÓÙØ<strong>×</strong>ÖÒÚÖÒØÙÒÖWº ÓÒÝÖÔÖ<strong>×</strong>ÒØØÓÒÓgÒÚÖÒغ<br />

∗ØØÐÚØ<strong>×</strong>ØÓÛØ<strong>×</strong> ÏÓÒÐÙÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓØÏÝÐÖÓÙÔÛØÛÔÖÓÔÖØ<strong>×</strong>ÓW ½ºÌÏÝÐÖÓÙÔ<strong>×</strong>Ø<strong>×</strong>ØÓÐÒÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>Óh ´¿º¿½µ<br />

¿ºÌÏÝÐÖÓÙÔØ<strong>×</strong><strong>×</strong>ÑÔÐÝÒØÖÒ<strong>×</strong>ØÚÐÝÓÒØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ò Ð<strong>×</strong>ÓÓÒØ<strong>×</strong>ØÓÏÝÐÑÖ<strong>×</strong>ºÓÖÒÝ<strong>×</strong><strong>×</strong>ΠÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸ÒÓÖÒÝw∈W¸ ÓÖÒÝw∈Wº<br />

〈w(α), w(β)〉 = 〈α, β〉,<br />

ºÌ<strong>×</strong>ØΠÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒÖØ<strong>×</strong>ØÛÓÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>Ø<strong>×</strong>ÑÙÒÖØÏÝÐ ØÑw(Π)<strong>×</strong>Ò<strong>×</strong><strong>×</strong>Ó<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>º<br />

ºÌÖØÓÒÛØÖ<strong>×</strong>ÔØØÓ<strong>×</strong>ÑÔÐÖÓÓØαØ<strong>×</strong>ØØÓØ<strong>×</strong>ÒØÚ¸ÒÔÖÑÙØ<strong>×</strong> ÛÒÛÖÓÒ<strong>×</strong>ØÖÙØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑØÖÒÓÑÒØÓÖÒØØ<strong>×</strong>º ÖÓÙÔºÓÖÒÝÖÓÓØα¸W(α)<strong>×</strong>ÔÒ<strong>×</strong>ØÛÓÐÖÓÓØ<strong>×</strong>ÔºÌ<strong>×</strong>ÔÓÒØÛÐÐÙ<strong>×</strong>ÙÐÐØÖ<br />

ØÖ<strong>×</strong>ØÓØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>º ¿<br />

i<br />

i<br />

E

ºWÒÓØÓÒÐÝÔÖÑÙØ<strong>×</strong>ØÖÓÓØ<strong>×</strong>¸ÙØØÛØ<strong>×</strong>ÓÒÝÓØÖ<strong>×</strong>ØÛØÑÓÙк ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ºËÒØÏÝÐÖÓÙÔ<strong>×</strong>ÒÖØÝØÙÒÑÒØÐÖØÓÒ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓØ<br />

i¸ÒÝÐÑÒØw∈WÒÛÖØØÒ<strong>×</strong>ÛÓÖ³ÒØÙÒÑÒØÐ<br />

ÒØÑÒÑÙÑÒÙÑÖÓÖØÓÒ<strong>×</strong><strong>×</strong>ÐÐÖÙÜÔÖ<strong>×</strong><strong>×</strong>ÓÒº ÔÓ<strong>×</strong><strong>×</strong>Ð<strong>×</strong>ÙÖØÓÒ<strong>×</strong>ØØÒÖØw<strong>×</strong>ÐÐØÐÒØl(w)ÓwºÒÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ ÖØÓÒ<strong>×</strong>ºÚÒw∈WÑÝÜÔÖ<strong>×</strong><strong>×</strong>ÝÖÒØÛÓÖ<strong>×</strong>¸ÒØÑÒÑÙÑ <strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α<br />

ºÖÓÑØÒØÓÒÓØÏÝÐÚØÓÖ<strong>×</strong>Ø<strong>×</strong>ÙÑÓÐÐØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>¸ØÖØÓÒ ºÌÐÒØlÓÝ<strong>×</strong>l(w)<br />

ÏÝÐÚØÓÖº<br />

=<br />

iÛÐÔÖÑÙØÒÐÐØÓØÖ iÖÓÑØ ÓρÛØÖ<strong>×</strong>ÔØØÓ<strong>×</strong>ÑÔÐÖÓÓØα iÙ<strong>×</strong>ØØ<strong>×</strong>α iØÓ−α<br />

´¿º¿¾µ ÖÓÓØ<strong>×</strong>ÑÓÒØÑ<strong>×</strong>ÐÚ<strong>×</strong>ºÌÙ<strong>×</strong>¸ÖØÓÒÛØÖ<strong>×</strong>ÔØØÓα ½½ºÓÖØÓØÏÝÐÖÓÙÔÓÒØÒ<strong>×</strong>ÜØÐÝÓÒÔÓÒØÒØÐÓ<strong>×</strong>ÏÝÐÑÖº ½¼ºÌÏÝÐÚØÓÖρÐÛÝ<strong>×</strong>Ð<strong>×</strong>ÒØÓÔÒ´ÒÒÐÓ<strong>×</strong>µÏÝÐÑÖº<br />

½¾ºÌÏÝÐÖÓÙÔ<strong>×</strong>ÖÓÜØÖÖÓÙÔ<strong>×</strong>ºÇÒ<strong>×</strong><br />

iÙ<strong>×</strong>Ø<strong>×</strong>ÙØÖØ<strong>×</strong>α w αi (ρ) = ρ − α i .<br />

( ) 2+|aij |<br />

wαi w 2 ijÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong>ÖÐØ ´¿º¿¿µ<br />

αj = 1ÛÒ|a | = 0, 1Òi j .<br />

ÓÑÔÙØØÖØÖÓ<strong>×</strong>ØÛØÑÓÙÐ<strong>×</strong>ÓgÒØ<strong>×</strong>ÖØÖÒÒÓÑÒØÓÖ Ì<strong>×</strong>ÓÒÐÙ<strong>×</strong>ÓÙÖ<strong>×</strong>ØÙÝÓØÏÝÐÖÓÙÔÓÖÒÓÛºÏÛÐÐØØÓÙ<strong>×</strong>ÒØØÓ<br />

jº ÙÖØÖ¸ÛÒ|a ij | ≥ 2¸ØÖÖÒÓÖÐØÓÒ<strong>×</strong>ºÌÐÑÒØ<strong>×</strong>a ÖÓÑØÖÒÓÑÒØÓÖÒØØ<strong>×</strong>ºÆÜظÛÓÑØÓØÓÐ<strong>×</strong><strong>×</strong>ØÓÒÓÒع ÒØØ<strong>×</strong>ºÁØÐ<strong>×</strong>ÓÔÐÝ<strong>×</strong>ÚÖÝÑÔÓÖØÒØÖÓÐÒÓÒ<strong>×</strong>ØÖÙØÒØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

ØÓØÖÓÓØ<strong>×</strong>α<br />

ÝÒÒÖÑ<strong>×</strong>º ÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÒØÔÖÓ<strong>×</strong><strong>×</strong>ÐÖÒÒÓÙØÖØÒÑØÖ<strong>×</strong>Ò<br />

iÒα <br />

l(w −1 )<br />

ij<br />

≠

¿º¿º ÖØÒÅØÖ<strong>×</strong>¸ÝÒÒÖÑ<strong>×</strong>ÒÐ<strong>×</strong><strong>×</strong>ØÓÒÓÒع ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÔÖÓÙØ<strong>×</strong>ÓØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α∈Π¸ÐÐØÖØÒÑØÖܺÇÒÒÐ<strong>×</strong><strong>×</strong>Ý ÇÒÛÚÜ<strong>×</strong>Ø¸Π¸Ó<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Óg¸ÛÒ<strong>×</strong><strong>×</strong>ÓØØÓØÑØÖܸÓÒÒÖ ÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><br />

ØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÓÚÖCÙ<strong>×</strong>ÒØÖØÒÑØÖÜ<strong>×</strong><strong>×</strong>ÓØ ÒÒ¸ÓØÙÒÖÐÝÒ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg¸ØÖØÒÑØÖÜA(g)ÓØ<br />

rÑØÖÜÛØÐÑÒØ<strong>×</strong> }¸ÛÖr<strong>×</strong>ØÑÒ<strong>×</strong>ÓÒÓE¸ ØÓØ<strong>×</strong>ÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѺÒÙÑÖØÒΠ<strong>×</strong>Π={α 1 , α 2 , . . .,α r<br />

<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg<strong>×</strong>Ør ijÖØ<strong>×</strong>ÑÓÒ<strong>×</strong>ØØÔÔÖÒØÒØÓÒÓØÏÝÐÖÓÙÔ<strong>×</strong> ´¿º¿µ<br />

<strong>×</strong><br />

a = 2 〈α i, α j 〉<br />

ÓØÖØÒÑØÖÜÖÒØÖ<strong>×</strong>ºÌÖØÒÑØÖÜÔÒ<strong>×</strong>ÓÒØÒÙÑÖØÓÒÓΠÒ ÓÜØÖÖÓÙÔÓÚºÙ<strong>×</strong>ØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓÖÑ<strong>×</strong><strong>×</strong>ÓØÖÓÓØ<strong>×</strong>Ô¸Ø<br />

〈α j , α j 〉 .<br />

ÖÒØÒÙÑÖØÓÒ<strong>×</strong>ÐØÓÖÒØÖØÒÑØÖ<strong>×</strong>ØØÖÓÒÙØØÓÓÒÒÓØÖ<br />

〈α,α〉<strong>×</strong>ÒÒØÖ¸ÐÐØÐÑÒØ<strong>×</strong> ÌÐÑÒØ<strong>×</strong>a ÓÒÚÖ<strong>×</strong>ÐݸÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖÒÒØÖÓÙØ<strong>×</strong>ÖØÒ ÝÔÖÑÙØØÓÒÑØÖܺ ÌÓÚÖÝ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ¸ÛÒ<strong>×</strong><strong>×</strong>ÓØÖØÒÑØÖÜ<strong>×</strong>ÒÓÚº<br />

ÖØÒÑØÖÜ<strong>×</strong>ÒÓÒ¹ÒÖظÒ<strong>×</strong>ÒØÕÙÒØØÝ2 〈α,β〉<br />

r}ÓÖÒr<strong>×</strong>Ø<strong>×</strong>ÝÒØÓÐÐÓÛÒÓÒØÓÒ<strong>×</strong><br />

ZÓÖÐÐiÒj¸<br />

2ÓÖÐÐi¸<br />

ÑØÖܺÚÒÖиÒÓÑÔÓ<strong>×</strong>и(r <strong>×</strong> r)<strong>×</strong>ÝÑÑØÖÑØÖܾA=(a<br />

{1, 2, . . .,<br />

´µa ij<br />

0¸<br />

∈<br />

´µa ii<br />

j¸<br />

=<br />

´µa ij = 0 ⇔ a ji =<br />

ÑØÖܺ <strong>×</strong>ÝÑÑØÖÞÐØÖÜ<strong>×</strong>Ø<strong>×</strong>ÒÓÒ¹ÒÖØÓÒÐÑØÖÜD<strong>×</strong>ÙØØA=DBÛÖB<strong>×</strong><strong>×</strong>ÝÑÑØÖ ¾Ì<strong>×</strong>ÝÑÑØÖÓÒØÓÒÒÜØÒØÓÒÐÙ<strong>×</strong>ÝÑÑØÖÞÐÑØÖ<strong>×</strong>ºÑØÖÜA<strong>×</strong><strong>×</strong>ØÓ<br />

0¸ ´Úµa ij ∈ Z ≤0ÓÖi ´ÚµØA ><br />

<br />

≠<br />

ij<br />

)¸i, j ∈ I =

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

I ÓÒÒ<strong>×</strong>ÄÐÖg(A)ÒÖØÝØÒÖØÓÖ<strong>×</strong>e i¸f i¸Òh <strong>×</strong>Ø<strong>×</strong>ÝÒØÓÐÐÓÛÒÓÒØÓÒ<strong>×</strong>ÓÖi, j ∈<br />

´¿º¿µ<br />

ÒØÖÓÓØ<strong>×</strong>º(i)<strong>×</strong>Ý<strong>×</strong>ÐÐØÒØÖ<strong>×</strong>ÓØÖØÒÑØÖÜÖÒØÖ<strong>×</strong>¸Û<strong>×</strong><strong>×</strong>ÝØÓ ÑÓØÚØÓÒ¸<strong>×</strong>ÓÐØÙ<strong>×</strong>ÓÒ<strong>×</strong>ÖÛØÓØÕÙØÓÒ<strong>×</strong>ÓÚ<strong>×</strong>Ý<strong>×</strong>ÓÙØØÄÐÖ ÌÓÚÕÙØÓÒ<strong>×</strong>ÑÝ<strong>×</strong>ÑÐØØÐ<strong>×</strong>ØÖÒÛÒÔÖ<strong>×</strong>ÒØÛØÓÙØØÒ<strong>×</strong><strong>×</strong>ÖÝ<br />

[e i , e j ] = [f i , f j ] = 0A (e i ) 1−a ij<br />

e j = (f ) 1−a ij<br />

f j = 0 ÓÖi j .<br />

ÛÝØÓ´µº(ii)<strong>×</strong>Ù<strong>×</strong>ØØÓÓÓÒÚÒÒØÒÓÖÑÐÞØÓÒÓÒÓÓ<strong>×</strong><strong>×</strong>ÓÖØÒÒÖ<br />

〈α,α〉<strong>×</strong>ÒÒØÖ¸ÛÓ<strong>×</strong>Ò<strong>×</strong><strong>×</strong>Ó<strong>×</strong>ÐÐØ<br />

ÖØ<strong>×</strong>Ø<strong>×</strong>ÝÑÑØÖÝÓØ<strong>×</strong>ÐÖÔÖÓÙØÒÖÓÓØ<strong>×</strong>ÔºÓÖØÑÒÒÓÓÒØÓÒ(iv)¸ ÓÒ<strong>×</strong>ÖØÓÐÐÓÛÒÓÒØÓÒ<strong>×</strong>ØØÓÒÒ<strong>×</strong>ÓÛÓÒØÒÒÖÔÖÓÙØ<strong>×</strong>ÓÖÒÝØÛÓÖÓÓØ<strong>×</strong><br />

ÙÒÖ<strong>×</strong>ØÒÛÒÓØØØØÕÙÒØØÝ2 〈α,β〉<br />

ÔÖÓÙØÛÐ<strong>×</strong>ÓÓ<strong>×</strong>ÐÐØÛÝØÓØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ósl(2, C)ºÓÒØÓÒ(iii)<br />

αÒβ<br />

ÆÓÛ¸Ø<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛÖÒ<strong>×</strong>ØÓ<strong>×</strong>ÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÛÛÖÒÓÑÔÓ<strong>×</strong>Ð¸Ò 0¸<br />

´¿º¿µÖÒÓÛÒ<strong>×</strong>ØÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ºÏÖÐÖÒÒÖÐ<strong>×</strong>ѹ<br />

0º<br />

<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÛÖÓØÒ<strong>×</strong>ÖØ<strong>×</strong>ÙÑ<strong>×</strong>Ó<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸Ò ÌÖØÒÑØÖܸA¸ÙÒÕÙÐÝÒ<strong>×</strong>ØÄÐÖÛÛÐÐg(A)ºÌÖÐØÓÒ<strong>×</strong> ÒØÖÒÓÒÝØÛÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><strong>×</strong>ÒÚÖÖÓÓظÒ<strong>×</strong>ÓØÓÐÐÓÛ<strong>×</strong>ØØ(α<br />

Ð<strong>×</strong>ÓØÖÓÙØÖÃÐÐÒÓÖѺÌÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ÒÒ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄ<br />

i , α j ) ≤<br />

ÒÒØÑØÖÜÐÑÒØ<strong>×</strong>a ÐÖÖÚÖÝÙ<strong>×</strong>ÙÐÖÓÑØÔÓÒØÓÚÛÓÓÙÖÒÐÑÓÖÙØÒØÓÒÒع<br />

ij ≤<br />

Ä<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>ÛÛÐÐ<strong>×</strong>ÛÒÛÒÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>ÓÖØÐݺ ÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ºÁØ<strong>×</strong>Ø<strong>×</strong>ÓÒØÓÒÛÓ<strong>×</strong>ÒÖÐÞØÓÒ<strong>×</strong>Ø<strong>×</strong>ÑÔÐ<strong>×</strong>ØÛÝØÓ ÑÓÚÖÓÑÒعÑÒ<strong>×</strong>ÓÒÐÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ØÓÒÒعÑÒ<strong>×</strong>ÓÒÐÃÅ ÄØÙ<strong>×</strong>ÐÓÓØØÒØÖ<strong>×</strong>ÓØÖØÒÑØÖÜÑÓÖÐÓ<strong>×</strong>ÐݸÒ<strong>×</strong>ÛØÛÒ<strong>×</strong>Ý<br />

)¸ÒÙ<strong>×</strong>Ò(ii)Û ÓÙØØѺÖ<strong>×</strong>ظÝØØÖÒÐÒÕÙÐØÝ(α i , α j ) 2 ≤ (α i , α i )(α j , α<br />

<br />

j<br />

[h i , h j ] = 0 ; [e i , f j ] = δ ij h j ;<br />

[h i , e j ] = a ij e j ; [h i , f j ] = −a ij f j ;<br />

i<br />

ij<br />

= 0<br />

≠<br />

i¸ÓÖi={1, 2, . . ., r}¸<br />

´¿º¿µ<br />

(α, β) > 0 ⇒ α − β ∈ L<br />

(α, β) < 0 ⇒ α + β ∈ L .

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong> ØØÒÕÙÐØÝa ij , a ji<br />

ÓÖ ijØÓ ´¿º¿µ<br />

≤ 4ÛØÕÙÐØÝÓÐÒÓÖi = jÒa ÆÓÛ¸Ù<strong>×</strong>Ò(iii)Ñ(iv)Û<strong>×</strong>ØÔÓ<strong>×</strong><strong>×</strong>ÐØ<strong>×</strong>ÓÖa a ij = a ji = 0<br />

ØÐÑÒØ<strong>×</strong>ÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÖÑÙÐØÔÐÝÒÓÒ¹ÞÖÓÓÒ<strong>×</strong>ØÒظÓÒØ<strong>×</strong>ÒÓØÖÖÓÓØ<br />

ijÚØÒÐØÛÒØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѺÁÐÐ<br />

〉<strong>×</strong>ÙÒÒ<br />

ÒÐ<strong>×</strong>ØÛÒØÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒØÖÓÓØ<strong>×</strong>ÔE<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÓÖØÛÓÖÓÓØ<strong>×</strong>αÒ ÓØαÒβÖÑÙÐØÔÐÝØ<strong>×</strong>ÑÒÓÒ¹ÞÖÓÓÒ<strong>×</strong>ØÒغËÓ¸ØØÙÐÐÒØ<strong>×</strong>ÓÖÓÓØ<strong>×</strong><br />

ÌÐÑÒØ<strong>×</strong>α β〉¸ØÖÖØÓÐÐÓÛÒÔÓ<strong>×</strong><strong>×</strong>ÐØ<strong>×</strong><br />

ijÒÓÒÓÖÑØÓÒÓÙØØ <strong>×</strong>Ý<strong>×</strong>ØÑØØ<strong>×</strong>ÕÙÚÐÒØØÓØÓÖÒÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѺÌÕÙÒØØÝ2 〈α<br />

〈α j ,α j<br />

ÖÒÓØÑÔÓÖØÒظÙØÓÒÐÝØÖÖØÓ<strong>×</strong>ºÌÐÑÒØ<strong>×</strong>α β〉¸ÒØÒÐØÛÒαÒβ<strong>×</strong>60ÓÖ120¸<br />

β¸ÛÖα<strong>×</strong>ÒÓØÑÙÐØÔÐÓβ¸Ò〈α, α〉 ≥ 〈β,<br />

´Úµ β〉¸ÒØÒÐØÛÒαÒβ<strong>×</strong>45ÓÖ135¸ ´µ〈α,<br />

ÒØÒÐØÛÒαÒβ<strong>×</strong>30ÓÖ150º<br />

α〉 = 〈β,<br />

´µ〈α, α〉 = 2〈β,<br />

ØÒØÖÖÒÓÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÓÒØÖØÓ<strong>×</strong>ÓØÖÐÒØ<strong>×</strong>ºÁØÒÐØÛÒØÛÓÖÓÓØ<strong>×</strong> ËÓ¸ØØÛÓÖÓÓØ<strong>×</strong>ÖÒÓØÑÙÐØÔÐ<strong>×</strong>ÓÓØÖÒÖÒÓØÔÖÔÒÙÐÖØÓÓØÖ¸<br />

〈α,<br />

3ºÁØÛÓÖÓÓØ<strong>×</strong>ÖÔÖÔÒÙÐÖ<br />

ÆÓÛÛÓÑØÓØÓÝÒÒÖÑ<strong>×</strong>ÒÐ<strong>×</strong><strong>×</strong>ØÓÒ¸ÙÔØÓÕÙÚÐÒ¸Ó<br />

αÖÐ<strong>×</strong>ÓÖÓÓØ<strong>×</strong>ºÓÑÔÖØ<strong>×</strong>ÛØØÓÒØÓÒ´µÓ´¿º¿ºµº<br />

β<strong>×</strong>ÖÓÓظÒØÒÐØÛÒαÒβ<strong>×</strong><strong>×</strong>ØÖØÐÝ ØÒØÖØÓÓØÖÐÒØ<strong>×</strong>ÑÙ<strong>×</strong>ØØÖ1, √ 2,ÓÖ√ ÖѺ <strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖg¸ÒØÖÑ<strong>×</strong>ÓØÖØÒÑØÖ<strong>×</strong>¸ÓÖÒÓØÐÐØÝÒÒ <strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÇÒÒÐ<strong>×</strong><strong>×</strong>ÝØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ÒÒØÓÖÖ<strong>×</strong>ÔÓÒÒ<br />

αÒβ<strong>×</strong><strong>×</strong>ØÖØÐÝÓØÙ<strong>×</strong>¸ØÒα +<br />

ÙظØÒα ÌÓ<strong>×</strong>ØÑÓØÚØÓÒ¸ÓÒ<strong>×</strong>ÖØÓØÖÓÓØ<strong>×</strong>ÔÓÑÔÓ<strong>×</strong>ØÓÒÓØÄ<br />

− βÒβ ÐÖºÏÓÒ<strong>×</strong>ÖØÑÜÑÐÐÒ<strong>×</strong>ÙÐÖÓgÒØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ѹ<strong>×</strong>ÑÔÐÝÓÒ<br />

´µ〈α, β〉 = 0¸<br />

−<br />

ÓÖ ÓÖ<br />

a ij = a ji = −1<br />

a ij = −1, a ji = −2<br />

a ij = −1, a ji = −3 .<br />

ij<br />

∈ {0, 1, 2, 3}ÓÖi≠jº<br />

i,α j 〉<br />

α〉 = 3〈β, β〉, ´¿º¿µ

gÚÒØÄÐÖ<strong>×</strong>ÖØ<strong>×</strong>ÙÑÓÖÓÓØ<strong>×</strong>Ô<strong>×</strong>ºÌÄÐÖ<strong>×</strong>ÖÓÒÙÔÒØÓ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÒÜØÒÚØÖÓÓØ<strong>×</strong>ÔÓÑÔÓ<strong>×</strong>ØÓÒØÓØÐ<strong>×</strong><strong>×</strong>ØÓÒÓÄÐÖ<strong>×</strong>ºÌ ÒÚØÓÖ<strong>×</strong>ÓÖØÐÑÒØ<strong>×</strong>ÓØÖØÒ<strong>×</strong>ÙÐÖºÌÙ<strong>×</strong>¸Ð<strong>×</strong><strong>×</strong>ÝÒÖØÒ<strong>×</strong>ÙÐÖ<strong>×</strong><br />

<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Π¸ÒÒÒØÖØÒÑØÖÜÓÖÝÒÒÖÑÓÖÖ<strong>×</strong>ÔÓÒÒØÓΠº ÌÙ<strong>×</strong>¸ÒÐ<strong>×</strong><strong>×</strong>ÝÒØÚÖÓÙ<strong>×</strong>ÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÛÖÖÙ ÒÓÖÑØÓÒÓØÖØÒ<strong>×</strong>ÙÐÖ¸ÒÒÓg¸<strong>×</strong>ÓÒØÒÒØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<br />

ÚÖÓÙ<strong>×</strong>ÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ØÖÓÖÖ<strong>×</strong>ÔÓÒÒÝÒÒÖÑ<strong>×</strong>ÖÐ<strong>×</strong>ØÐÓÒÛØØ<strong>×</strong>ѹ ÓÖÑÓÐ<strong>×</strong><strong>×</strong>ØÓÒØÓÖÑ´ÛØÓÙØÔÖÓÓµØÓÛÖ<strong>×</strong>ØÒÓØ<strong>×</strong><strong>×</strong>ØÓÒÛÖØ ØÓ<strong>×</strong>ØÙÝÒØÚÖÓÙ<strong>×</strong>Ð<strong>×</strong><strong>×</strong><strong>×</strong>ÓÖØÒÑØÖ<strong>×</strong>ÓÖÝÒÒÖÑ<strong>×</strong>ºÏ<strong>×</strong>ØØØ<strong>×</strong>ÒØ<br />

ÚÒÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѺ <strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ØÝ<strong>×</strong>ÖºÐÓÛÛÚÓÛÓÒÓÒ<strong>×</strong>ØÖÙØ<strong>×</strong>ØÝÒÒÖÑ ÌÓØ<strong>×</strong>ØΠÓÖ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛÒÐ<strong>×</strong>Ó<strong>×</strong><strong>×</strong>ÓØÖÔ¸ÓÒ<strong>×</strong><strong>×</strong>ØÒÓÚÖØ<strong>×</strong>ÒÐÒ<strong>×</strong><br />

jÖÓÒÝ<strong>×</strong>ÔÒÒÙÔÓÒØÒÐØÛÒØ<br />

jÖÓÖØÓÓÒÐØÒÛÔÙØÒÓØÛÒ<br />

iÓΠÛ<strong>×</strong><strong>×</strong>ÓØ ÓÒÒØÒØѸÒÓÛÒ<strong>×</strong>ÝÒÒÖѺÌÓÐÑÒØα ÚÖØÜv iºÌÛÓÚÖØ<strong>×</strong>v iÒv 2ØÑ<strong>×</strong>Ø<strong>×</strong>ÓÖØÖ¸ÒØÖ jÚ<br />

ÔÓÒØÒÖÓÑØÚÖØÜ<strong>×</strong><strong>×</strong>ÓØØÓØÐÓÒÖÖÓÓØØÓÛÖØÚÖØ<strong>×</strong><strong>×</strong>ÓØØÓØ<br />

<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α iÒα jºÁØÛÓÖÓÓØ<strong>×</strong>α iÒα jÖÒÓØ<br />

<strong>×</strong>ÓÖØÖÖÓÓغÄÓÓÒØ´¿º¿µÛ<strong>×</strong>ØØØÖÖÓÒÐÝØÖÔÓ<strong>×</strong><strong>×</strong>ÐÐÒØÖØÓ<strong>×</strong>Ò<br />

ØÓÖÖ<strong>×</strong>ÔÓÒÒÚÖØ<strong>×</strong>v iÒv jºÏÔÙØÓÒØÛÒv iÒv jα iÒα j<strong>×</strong>√ jÛØÒÖÖÓÛ Ø<strong>×</strong>ÑÐÒظØÛÓ<strong>×</strong>ØÐÓÒÖÓα ØÖÔÓ<strong>×</strong><strong>×</strong>ÐÒÐ<strong>×</strong>ØÛÒØÖÓÓØ<strong>×</strong>º iÒα j<strong>×</strong>√ <strong>×</strong>ØÐÓÒÖÓα<br />

ÌÛÓÝÒÒÖÑ<strong>×</strong>Ö<strong>×</strong>ØÓÕÙÚÐÒØØÖ<strong>×</strong>ÓÒ¹ØÓ¹ÓÒ¸ÒÓÒØÓÑÔ<br />

iÒα 3ØÑ<strong>×</strong>Ø<strong>×</strong>ÓÖØÖºÁÒØÓÒ¸α iÒα ÓÖØÓÓÒÐÓÖÓØ<strong>×</strong>ÑÐÒظÛÓÖØØØÛÒv Ù<strong>×</strong>ÓØØÓÒÓØÏÝÐÖÓÙÔÓÒØѸØÕÙÚÐÒÐ<strong>×</strong><strong>×</strong>ÓÝÒÒÖÑ<strong>×</strong> ØÖØÓÒÓØÖÖÓÛ<strong>×</strong>ºËÒÒÝØÛÓ<strong>×</strong><strong>×</strong>ΠÓÖØ<strong>×</strong>ÑÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÖÕÙÚÐÒØ ÓØÚÖØ<strong>×</strong>ÓÓÒØÓØÚÖØ<strong>×</strong>ÓØÓØÖØØÔÖ<strong>×</strong>ÖÚ<strong>×</strong>ØÒÙÑÖÓÓÒ<strong>×</strong>Ò<br />

iÒv ÖÕÙÚÐÒغÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÖÖÙÐØ<strong>×</strong>ÝÒÒÖÑ<strong>×</strong>ÓÒÒغÏÒÓÛ Ð<strong>×</strong>ØÐÐØÝÒÒÖÑ<strong>×</strong>ÓØÐ<strong>×</strong><strong>×</strong>Ð<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>º ÒÔÒÒØÓØÓÓØ<strong>×</strong>ΠºÌÛÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÛØÕÙÚÐÒØÝÒÒÖÑ<strong>×</strong><br />

ÖÒnº<br />

C)ºÁØ<strong>×</strong>Ó<br />

ÖÒnº<br />

½ºA nÌÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑA n<strong>×</strong>ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓØÄÐÖsl(n C)ºÁØ<strong>×</strong>Ó<br />

+ 1,<br />

¾ºB nÌÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑB n<strong>×</strong>ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓØÄÐÖso(2n<br />

<br />

+ 1,

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

nº<br />

C)ºÁØ<strong>×</strong>ÓÖÒnº<br />

¿ºC ÁÒÖÒÓÒ¸ØÖ<strong>×</strong>ÓÒÐÝÓÒÔÓ<strong>×</strong><strong>×</strong>ÐÝÒÒÖѸÖØÒØØØÖ<strong>×</strong>ÓÒÐÝÓÒ ÏÒÓØÛÒØÖ<strong>×</strong>ØÒÔÓÒØ<strong>×</strong>ÓÙØØÓÚÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ØØÔÔÒÒÐÓÛÖÒº ÌÝÒÒÖÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØØÓÚÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÖÚÒÒº(3.1)º<br />

nÌÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑC n<strong>×</strong>ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓØÄÐÖsp(n, ºD nÌÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑD n<strong>×</strong>ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓØÄÐÖso(2n,<br />

2<strong>×</strong><strong>×</strong>ÓÒÒظÖØÒØØØØ <strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑÐ<strong>×</strong><strong>×</strong>ÓÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÖÒÓÒºÌÄÐÖso(2, 2Ö<strong>×</strong>ÓÑÓÖÔ¸ÖØÒ<br />

C)<br />

<strong>×</strong><strong>×</strong>ÓØØÓØÐ<strong>×</strong><strong>×</strong>ÐÄÐÖ<strong>×</strong>¸ØÖÖÚÜÔØÓÒÐÖÖÙÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸<br />

<strong>×</strong>ÒÓØ<strong>×</strong>ѹ<strong>×</strong>ÑÔиÒØÖÑÒÒØÖÄÐÖ<strong>×</strong>sl(2, C),<br />

3Ö <strong>×</strong>ÓÑÓÖÔºÁÒÖÒØÛÓ¸ØÝÒÒÖÑD so(4, C) ∼ = sl(2, C)⊕sl(2, C)ºÐ<strong>×</strong>Ó¸ØÝÒÒÖÑ<strong>×</strong>B 2ÒC ØØØØso(5, C) ∼ = sp(2,<br />

8º<br />

C)ºÁÒØÓÒØÓØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong><br />

C)ºÁÒÖÒØÖ¸ØÝÒÒÖÑ<strong>×</strong>A 3ÒD <strong>×</strong>ÓÑÓÖÔ¸ÖØÒØØØØsl(4, C) ∼ = so(6,<br />

ÖÓÑØÓÐÐÓÛÒÐ<strong>×</strong>غ ÌÓÖÑ¿º¿ºÚÖÝÖÖÙÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓÔÖ<strong>×</strong>ÐÝÓÒÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ ÆÓÛÛ<strong>×</strong>ØØØÐ<strong>×</strong><strong>×</strong>ØÓÒØÓÖѸÛØÓÙØÔÖÓÓÐÓÛº ÒÓØG 2 , F 4 , E 6 , E 7ÒE 1º ½ºÌÐ<strong>×</strong><strong>×</strong>ÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>A n¸n ≥<br />

so(3, C)Òsp(1, C)Ö<br />

¾ºÌÐ<strong>×</strong><strong>×</strong>ÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>B n¸n ≥ 2<br />

ËÒÚÖÝÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÒÙÒÕÙÐÝÓÑÔÓ<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÑÓÖÖÙÐÖÓÓØ<strong>×</strong><br />

¿ºÌÐ<strong>×</strong><strong>×</strong>ÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>C n¸n 8º ºÌÐ<strong>×</strong><strong>×</strong>ÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>D <strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>º<strong>×</strong>ÖÙÓÖ¸Ð<strong>×</strong><strong>×</strong>ØÓÒÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>Ð<strong>×</strong>ØÓØÐ<strong>×</strong><strong>×</strong>ØÓÒÓ<strong>×</strong>ѹ <strong>×</strong>ÝØÑ<strong>×</strong>¸ØÐ<strong>×</strong><strong>×</strong>ØÓÒÓÖÖÙÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>Ð<strong>×</strong>ØÓØÐ<strong>×</strong><strong>×</strong>ØÓÒÓÐÐÖÓÓØ<br />

n¸n ºÌÜÔØÓÒÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>G 2 , F 4 , E 6 , E<br />

ÌÓÖÑ¿º¿ºÚÖÝÓÑÔÐÜ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓÔÖ<strong>×</strong>ÐÝÓÒÐÖ <strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒÐ<strong>×</strong><strong>×</strong>ØÓÒÓØÖÖÙÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>Ð<strong>×</strong>ØÓØÐ<strong>×</strong><strong>×</strong>ØÓÒ ÓØ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÛÛ<strong>×</strong>ØØÒØÓÖÑÓØÓÖÑÐÓÛº<br />

7ÒE ÖÓÑØÓÐÐÓÛÒÐ<strong>×</strong>Ø <br />

≥ 3<br />

≥ 4

1¸ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

3¸ 2¸<br />

½ºsl(n + 1, C), n ≥<br />

4¸<br />

¾ºso(2n + 1, C), n ≥<br />

¿ºsp(n, C), n ≥<br />

8º<br />

ºso(2n, C), n ≥<br />

ÓÙÖ<strong>×</strong>ÒØÖØ<strong>×</strong>ÙѺ ØÓ<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑÝ<strong>×</strong>ÔÝÒÛ<strong>×</strong>ÑÔÐ<strong>×</strong>ÙÑÑÒ<strong>×</strong>ÓÙÖÒÓÛÑÒÝØÑ<strong>×</strong>ÓÒ <strong>×</strong>ѹ<strong>×</strong>ÑÔÐÐÖ<strong>×</strong>ÖØ<strong>×</strong>ÙÑÓ<strong>×</strong>ÑÔÐÐÖ<strong>×</strong>¸Ò<strong>×</strong>ÙÒÕÙÐÝØÖÑÒÙÔ ºÌÜÔØÓÒÐÄÐÖ<strong>×</strong>G 2 , F 4 , E 6 , E<br />

ÖÑÛÔØÓÖÐÐÝÔØÙÖ<strong>×</strong>ÐÐØÒÓÖÑØÓÒÓÙØØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÒÒØ ÏÚÐ<strong>×</strong>Ó<strong>×</strong>ÒØØØÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÒÒ<strong>×</strong><strong>×</strong>ÓØÖÔÐÐØÝÒÒ ÌÙ<strong>×</strong>¸ÛÚÐ<strong>×</strong><strong>×</strong>ØÚÖÓÙ<strong>×</strong><strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸ÒØÖÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>º<br />

7ÒE ÄÐÖØÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓº<br />

A<br />

n<br />

E 6<br />

B n<br />

E 7<br />

C<br />

ÙÖ¿º½ÌÝÒÒÖÑ<strong>×</strong>ÓÖØÐ<strong>×</strong><strong>×</strong>Ð<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><br />

n<br />

E 8<br />

Dn<br />

F 4<br />

G<br />

ØÖ<strong>×</strong>ØÙÝÒØÒÖÐ<strong>×</strong>ØÖÙØÙÖÓÄÐÖ<strong>×</strong>¸ÛÒÓÛÓÑØÓØÖÔÖ<strong>×</strong>ÒØØÓÒ ¿º¿º ÊÔÖ<strong>×</strong>ÒØØÓÒÌÓÖÝÓËѹËÑÔÐÄÐÖ<strong>×</strong><br />

2<br />

ÔÖØÙÐÖ<strong>×</strong>ÛÛÓÖØÓÙØÜÔÐØÐÝÓÖº<br />

C)<strong>×</strong>Ø ØÓÖÝÓÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÛØØÜÑÔÐÓsl(3, αØÓÖÖ<strong>×</strong>ÔÓÒÒÖÓÓØÚØÓÖºÏÚÐÖÝ<strong>×</strong>ÒØÓÒ<strong>×</strong>ØÖÙØÓÒÓÖÔÖ<strong>×</strong>ÒØØÓÒ C)Ò ÏÛÐÐ<strong>×</strong>ØÐÐÛÓÖÒÛØØ<strong>×</strong><strong>×</strong>´¿º¾¼µºÄØα = (a 1 , a 2 )ÖÓÓØÓsl(3,<br />

Z<br />

¼

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÄÐÖ<strong>×</strong>ÓÖÒr>1º Ì<strong>×</strong>ØÖØÒÔÓÒØÒÐÓÓÒÓÖ<strong>×</strong>ØÖÙØÙÖ<strong>×</strong>ØÓÒØÒÖÐÞØÓÒÓ´µºÏ<br />

C)ºÏÛÐÐÙ<strong>×</strong><strong>×</strong>ÓÑÓÛØÛÐÖÒØØÖÒ<strong>×</strong>ÛØÑÓØÓÒ<strong>×</strong>ÓÙÖÓÖ<br />

ÖÙÐݺÌ<strong>×</strong>ØÓÓÔÖØÓÖ<strong>×</strong>ρ(h)ÓÖÐÐh∈hÖ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐÝÓÒÐÞÐÒÚÖÝ ÚÒÖÓÓØ<strong>×</strong>ÒÛØ<strong>×</strong><strong>×</strong>ØÒÚÐÙ<strong>×</strong>Óρ(h)ºÌ<strong>×</strong>ÙØÐÒÖÐÞØÓÒ<strong>×</strong>ØÓ Ó<strong>×</strong>ÖÚØØÒÒÝÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒØÖØÒ<strong>×</strong>ÙÐÖhØ<strong>×</strong>ÓÑÔÐØÐÝ<br />

ÓÖsl(2,<br />

ØÖØ<strong>×</strong>ÙÑÓØ<strong>×</strong>ÛØ<strong>×</strong>Ô<strong>×</strong>ºÌ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>Óρ(h)ÖÐÐÛØ<strong>×</strong>º )<strong>×</strong><br />

ÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒρÓgÓÒÚØÓÖ<strong>×</strong>ÔVÛØÖØÒ<strong>×</strong>ÙÐÖh¸Ò<br />

2ÖÒØÖ<strong>×</strong>ºÓÖ ÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒ¸ÒÒ¸ÚÖÝÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒ(ρ, V<br />

ÓÖsl(3, C)¸ØÛØ<strong>×</strong>ÖÓØÓÖѵ=(m , m 2 )¸ÛÖm ÓÖÐÐh∈hºÒÓÒ¹ÞÖÓÚØÓÖv<strong>×</strong>Ø<strong>×</strong>ÝÒØÓÚÕÙØÓÒ<strong>×</strong>ÐÐÛØÚØÓÖ<br />

∗<strong>×</strong>ÐÐÛØÓÖρØÖÜ<strong>×</strong>Ø<strong>×</strong>ÒÓÒ¹ÞÖÓÚØÓÖvÒV<strong>×</strong>ÙØØ<br />

1Òm ´¿º¿µ<br />

ÓÖØÛص¸ÒØ<strong>×</strong>ØÓÐÐÚØÓÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÝÒ´¿º¿µ<strong>×</strong>ÐÐØÛØ<strong>×</strong>Ô<br />

ÐÑÒص ∈ h<br />

ÓÖÒÝÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒρÓg¸ØÛØ<strong>×</strong>ÓρÒØÖÑÙÐØÔÐØÝÖ ÒÚÖÒØÙÒÖØØÓÒÓØÏÝÐÖÓÙÔº ÛØÛصºÌÑÒ<strong>×</strong>ÓÒÓØÛØ<strong>×</strong>Ô<strong>×</strong>ÐÐØÑÙÐØÔÐØÝÓØÛغ<br />

ÌÒÖÐÞØÓÒÓ´µ<strong>×</strong>ØÓÓÑÒÒØÒØÖÐÐÑÒØ<strong>×</strong>ºÒÓÖÖÔÖ<br />

C)ºÂÙ<strong>×</strong>ØÐØÒØÖ<strong>×</strong>mÓÙÖ<strong>×</strong>Ø<strong>×</strong>ØÒÚÐÙ<strong>×</strong>ÓØÖÖÙÐ<br />

2ÒÒÓÒ¹ÒØÚÒØÖ<strong>×</strong><strong>×</strong>ÐÐÓÑÒÒØÒØÖÐй<br />

ÐÑÒØÓÙÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ØÛØÓ<strong>×</strong>ÓÑÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒºÅÓÖÒÖÐÐݸÓÖ<br />

(m 1 , m 2<br />

C)<strong>×</strong>ÓÑÒÒØÒØÖÐÐÑÒØÒ¸ÓÒÚÖ<strong>×</strong>ÐݸØØÚÖÝÓÑÒÒØÒØÖÐ<br />

C)¸ÛÛÐÐ<strong>×</strong>ØØØ<strong>×</strong>ØÛØÓÖÖÙÐÖÔÖ<strong>×</strong>Òع<br />

)ÛØm ÒÒØÖÓÖÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØαÒÓÑÒÒØÒØÖÐØ<strong>×</strong>ÒÓÒ¹ÒØÚº<br />

1Òm ÑÒØÓsl(3, ÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ósl(2, ØÓÒÓsl(3,<br />

ÑÖºÌ<strong>×</strong><strong>×</strong>ØÒÖÐÞØÓÒÓ´µÒ´µØÓÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>º ÏÝÐÖÓÙÔºÁØ<strong>×</strong>ÔÖ<strong>×</strong>ÐÝØ<strong>×</strong>ÐÑÒØ<strong>×</strong>ØØÖÓÒØÒÒØÐÓ<strong>×</strong>ÙÒÑÒØÐÏÝÐ ÛØ<strong>×</strong>ÒÒØÖÐÐÑÒغÌ<strong>×</strong>ØÓÒØÖÐÐÑÒØ<strong>×</strong><strong>×</strong>ÒÚÖÒØÙÒÖØØÓÒÓØ ÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ¸ÒÐÑÒص∈h<strong>×</strong>ÐÐÒÒØÖÐÐÑÒØ2 Ì<strong>×</strong>ÒÒÓØÖÓÓØ<strong>×</strong>ÓÖØÖÔÖ<strong>×</strong>ÒØØÓÒØÓÖÝÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><br />

〈µ,α〉<br />

C)Ö<strong>×</strong>ÒÐÓÛÖ¸ )ÛØÓÖ α Ð<strong>×</strong>ÒØÒÖÐÞØÓÒÓ´µºÌÓÔÖØÓÖ<strong>×</strong>ρ(x)Òρ(y)Ósl(2, Ö<strong>×</strong>ÔØÚÐݸØÒÚÐÙ<strong>×</strong>Óρ(h)ºÄØα=(a 1 , a 2 )ÖÓÓØÓsl(3, C)ÒÐØZ ÓÖÖ<strong>×</strong>ÔÓÒÒÖÓÓØÚØÓÖºÄØρÖÔÖ<strong>×</strong>ÒØØÓÒÓsl(3, C)¸Òµ(m 1 , m 2<br />

½<br />

1<br />

ρ(h)v µ = 〈µ, h〉v µ ,<br />

〈α,α〉<strong>×</strong>

ρÒvØÓÖÖ<strong>×</strong>ÔÓÒÒÛØÚØÓÖºÌÒ¸ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ρ(h 1<br />

)v<strong>×</strong>ÒÛÛØÚØÓÖÛØÛØ ´¿º¼µ<br />

)ρ(z α = (m 1 + a 1 )ρ(Z α ,<br />

ÓÖÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ¸ÐØvÛØÚØÓÖÛØÛصÒ<strong>×</strong>ÙÔÔÓ<strong>×</strong> ´¿º½µ ÌÙ<strong>×</strong>¸ØÖρ(Z α )v = 0ÓÖρ(Z α<br />

FhÛÚ<br />

µ + α = (m 1 + a 1 , m 2 + a 2 ) .<br />

´¿º¾µ<br />

x α<strong>×</strong>ÒÐÑÒØÓg αºÌÒ¸ÓÖÐÐh ∈<br />

ÛØ<strong>×</strong>Û<strong>×</strong>Ù<strong>×</strong>ØØÓÑÔÖ<strong>×</strong>ÓÒÓØÒØÖ<strong>×</strong>¸ÙØÓÖÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<br />

C)ØÛØ<strong>×</strong>ÛÖÒØÖ<strong>×</strong>mÒØÒÓØÓÒÓÓÑÔÖÒØÛÓ αº<br />

ÐÖÝÛØØÑÒ<strong>×</strong>ØÓ<strong>×</strong>ÝÛØ<strong>×</strong>ÖØÒÒÓØÖºÏÛÐÐÐÐÙ<strong>×</strong>ØÖØØÓÖØ<br />

ÌÓÚÕÙØÓÒ<strong>×</strong>Ý<strong>×</strong>ØØρ(x α<br />

)ÒÛÒØÓ<br />

)v<strong>×</strong>ØÖÞÖÓÓÖ<strong>×</strong>ÛØÚØÓÖÛØÛص ÓÖØ<strong>×</strong>Ósl(2,<br />

ØÓÖÑ 2´Õº´¿º¾¿µµ¸ÒØÛÓÛØ<strong>×</strong> ØÛØ<strong>×</strong>ÓÐÐØÓÒÓØ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ÓÐÐØρ(h 2ÒÛÖØØÒÒ<br />

i<br />

<strong>×</strong>Ósl(3, C)ºÚÒØØÛÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α 1Òα ´¿º¿µ<br />

µ 1Òµ 2¸Û<strong>×</strong>ÝØص 1<strong>×</strong>ÖØÒµ 2´ÒÓص 1 ≽ µ 2µµ − µ<br />

µºÌ<strong>×</strong><strong>×</strong>ÐÐØ<strong>×</strong>ØÛØÓρº<br />

µ 1 − µ 2 = aα 1 + bα 2<br />

µ¸ÓÖÐÐÛØ<strong>×</strong><br />

,<br />

C)¸<br />

ÒÒÝÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ¸ÒØÓÖÑÓØÓÖÑÐÓÛºÁÒØ<strong>×</strong>Ó<br />

ÛØa≥0Òb≥0ºÒÐÓÓÙ<strong>×</strong>ØÓØÐÖ<strong>×</strong>ØÒÚÐÙÒÖÔÖ<strong>×</strong>ÒØØÓÒÓsl(2, ØÖÜ<strong>×</strong>Ø<strong>×</strong>Ûص 0ÒÖÔÖ<strong>×</strong>ÒØØÓÒÓsl(3, C)<strong>×</strong>ÙØص 0 ≽<br />

C)¸<br />

ÚÐÙÓρ(h)ºÌ<strong>×</strong><strong>×</strong>ÒÓ´µ¸´µ¹´Áµ<strong>×</strong>ØØÒÝØÛÓÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ó<br />

C)ÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒρ(h)<strong>×</strong>ÓÒÐÞиÒØÖ<strong>×</strong>ÐÖ<strong>×</strong>ØÒ¹ ÆÓÛÛÚÒÓÙ<strong>×</strong>ØÓÔÙØØÓØÖØÒÖÐÞØÓÒ<strong>×</strong>Ó´µ¸´µ¹´ÁµØÓsl(3,<br />

ÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÛØØ<strong>×</strong>ØÒÚÐÙmºÆÓÛÛ<strong>×</strong>ØØØØÓÖÑÓ<strong>×</strong>Ø ÒÓÒ¹ÒØÚÒØÖ¸Ò¸ÓÒÚÖ<strong>×</strong>ÐݸÓÖÚÖÝÒÓÒ¹ÒØÚÒØÖm¸ØÖ<strong>×</strong>ÒÖÖ¹<br />

C)ÛØØ<strong>×</strong>ÑÐÖ<strong>×</strong>ØÒÚÐÙÖÕÙÚÐÒغÌ<strong>×</strong>ØÒÚÐÙ<strong>×</strong>ÐÛÝ<strong>×</strong><br />

sl(2,<br />

C)ÒÒÖÐÞØØÓÒÝÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖº<br />

sl(2,<br />

¾ ÛØÓÖsl(3,<br />

ρ(h 2 )ρ(z α )v = (m 2 + a 2 )ρ(Z α )v .<br />

ρ(h)ρ(x α )v = (〈µ, h〉 + 〈α, h〉)ρ(x α )v .<br />

+<br />

1

ÌÓÖÑ¿º¿º ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

)Ö<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÐÝÓÒÐÞÐÒÚÖÝ<br />

C)<strong>×</strong>ØÖØ<strong>×</strong>ÙÑÓ<br />

¾ºÐÐØÛØ<strong>×</strong>¸µ¸ÖÒØÖÐÐÑÒØ<strong>×</strong>º ÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒºÅÓÖÒÖÐÐݸÒÚÖÝÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒ<br />

)<strong>×</strong>ØÖØ<strong>×</strong>ÙÑÓØ<strong>×</strong>ÛØ<strong>×</strong>Ô<strong>×</strong>º<br />

½ºÚÖÝÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒρÓsl(3, Ø<strong>×</strong>ÛØ<strong>×</strong>Ô<strong>×</strong>ØØ<strong>×</strong>¸ρ(h 1 )Òρ(h<br />

ÕÙÚÐÒØÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÚØ<strong>×</strong>Ñ<strong>×</strong>ØÛغÒÒÝØÛÓÖÖ¹<br />

2<br />

ÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ(ρ, V<br />

0¸ÒØÛÓ ¿ºÚÖÝÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÓsl(3,<br />

ºÌÛÓÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÛØØ<strong>×</strong>Ñ<strong>×</strong>ØÛØÖÕÙÚÐÒغ <strong>×</strong>ØÖÙÓÖÒÝÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖº<br />

C)ÛØØ<strong>×</strong>Ñ<strong>×</strong>ØÛØÖÕÙÚÐÒغÌ<strong>×</strong>Ñ<br />

C)<strong>×</strong>ÙÒÕÙ<strong>×</strong>ØÛص ÙÐÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ósl(3,<br />

ØÓÖÑ 0Óπ<strong>×</strong>Ó ºÁπ<strong>×</strong>ÒÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÓsl(3, C)¸ØÒØ<strong>×</strong>ØÛص ÐÑÒغ ÑÒØØØØ<strong>×</strong>ØÛØÓÚÖÝÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓÑÒÒØÒØÖÐ<br />

2ÒÒÓÒ¹ÒØÚÒØÖ<strong>×</strong>ºÌ<strong>×</strong>ÙØÐÒÖÐÞØÓÒ<strong>×</strong>Ø<strong>×</strong>Øع<br />

µ<br />

ÛØm 1Òm 2ÖÒÓÒ¹ÒØÚÒØÖ<strong>×</strong>¸ØÒØÖÜ<strong>×</strong>Ø<strong>×</strong>ÒÖÖÙÐÖÔÖ<strong>×</strong>Òع<br />

ÖÔÖ<strong>×</strong>ÒØØÓÒº ÐÖ¸ÚÖÝÓÑÒÒØÒØÖÐÐÑÒØÓÙÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ØÛØÓÒÖÖÙÐ<br />

)ºÓÖÒÖÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄ ºÁm 1Òm ØÓÒρÓsl(3, C)ÛØ<strong>×</strong>ØÛص = (m 1 , m 2<br />

1)ºÓÖ 0)ºÓÖ )<strong>×</strong> ÌØÖÚÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÒÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÛØ<strong>×</strong>ØÛØ(0,<br />

sl(2, C)ÒÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÛØ<strong>×</strong>ØÛØmÛ<strong>×</strong>ÓÑÒ<strong>×</strong>ÓÒ(m<br />

ØÓÛÖ<strong>×</strong>ØÒÓØÔØÖÛÒÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÒÓÑÒØÓÖÒØØÝÓÄÐÖ<strong>×</strong>ºÓÖ ÏÛÐÐÓÑØÓØÖÔÖ<strong>×</strong>ÒØØÓÒØÓÖÝÓÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ÖÝ<br />

sl(3, C)¸ØÑÒ<strong>×</strong>ÓÒÓØÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÛØ<strong>×</strong>ØÛØ(m , m<br />

ÒÓÛ¸ÛÙ<strong>×</strong>ØÖÔØÙÐØÛØÛÚÐÖÒØÓÙØÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸ÓÖ ÛÑÓÚÓÒØÓØØÓÔÓÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>º<br />

2<br />

¿<br />

0<br />

0 = (m 1 , m 2 )<br />

+<br />

1<br />

1<br />

2 (m 1 + 1)(m 2 + 1)(m 1 + m 2 + 2) .

´µÒÖÐÞÒ´µÛ<strong>×</strong>ØØØÖÜ<strong>×</strong>Ø<strong>×</strong>ÑÜÑÐÐÒ<strong>×</strong>ÙÐÖ¸ÐÐØ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÖØÒ<strong>×</strong>ÙÐÖ¸ÓgÛØ<strong>×</strong><strong>×</strong>ѹ<strong>×</strong>ÑÔÐÝÓÒgÒÚÖÝÖÖÙÐÖÔÖ<strong>×</strong>Òع<br />

´µÌÑÙÐØÔÐØÝÓÚÖÝÖÓÓØ<strong>×</strong>ÓÒº ØÓÒ<strong>×</strong>ÚÒ<strong>×</strong>ØÖØ<strong>×</strong>ÙÑÓÛØ<strong>×</strong>Ô<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓØÖØÒ<strong>×</strong>ÙÐÖº ÌÒÚÐÙ<strong>×</strong>ÖØÖÓÓØ<strong>×</strong>ÒÛØ<strong>×</strong>ÓØÄÐÖº<br />

´µÚÖÝÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒÓg<strong>×</strong>ÙÒÕÙ<strong>×</strong>ØÛØÛ<strong>×</strong>ÓÑÒÒØ ´µÒÖÐÞÒ´µ¸ÐÐØÛØ<strong>×</strong>ÖÒØÖÐÐÑÒØ<strong>×</strong>º<br />

´µÚÖÝÓÑÒÒØÒØÖÐÐÑÒØ<strong>×</strong>Ø<strong>×</strong>ØÛØÓÒÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>º ÒØÖÐÐÑÒØÒØÛÓÕÙÚÐÒØÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÚØ<strong>×</strong>Ñ<strong>×</strong>Ø ÛغËÔÓÒØ´µ¸´Àµ¸´Áµ<br />

´µÌ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>ÒÚÒØÓÔÓ<strong>×</strong>ØÚÒÒØÚÖÓÓØ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓ<strong>×</strong><strong>×</strong> ÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÓÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><strong>×</strong>ÒÓØÙÒÕÙ¸ÒÓÖ<strong>×</strong>ØÖÓÖÖÒº Ë´µ<br />

´µÌÄÐÖ<strong>×</strong>ÔÐØ<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÑ ´¿ºµ<br />

´µÌÖ<strong>×</strong>ÖÓÙÔÓÔÖÑÙØØÓÒ<strong>×</strong>ÓØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Û<strong>×</strong>ÒÖØ ÒÐÐØÖÓÓØ<strong>×</strong>Ô<strong>×</strong>ÖÓÒ¹ÑÒ<strong>×</strong>ÓÒк<br />

g = ⊕ g α = ⊕ ⊕ ⊕<br />

g α h g α ,<br />

α∈L α∈L − α∈L<br />

´µÌÏÝÐÖÓÙÔ<strong>×</strong>ÒعÑÒ<strong>×</strong>ÓÒкÁØÖ<strong>×</strong>ÙÔØÖÓÓØ<strong>×</strong>ÔÒØÓÑÖ<strong>×</strong>ÒÓÛÒ ÝÖØÓÒ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÒØÖÓÓØ<strong>×</strong>Ô¸ÒÓÛÒ<strong>×</strong>Ø<br />

<strong>×</strong>ØÏÝÐÑÖ<strong>×</strong>º ÏÝÐÖÓÙÔºÌ<strong>×</strong>ØÓÙÒÑÒØÐÖÐØÓÒ<strong>×</strong>ÒÖØØÏÝÐÖÓÙÔº<br />

+<br />

´µÌ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÔØÙÖÐÐØÒÓÖÑØÓÒÓØÄÐÖgºÌ ´µÌÖÜ<strong>×</strong>Ø<strong>×</strong>ÚØÓÖρÐÐØÏÝÐÚØÓÖÛÐÛÝ<strong>×</strong>Ð<strong>×</strong>ÒØÐÓ<strong>×</strong>ÏÝÐ ÑÖº<br />

ÖÓÓØ<strong>×</strong>ÒØÖÓÓØ<strong>×</strong>Ô<strong>×</strong>ÐÐØÖØÒÑØÖܺÁØÓÒØÒ<strong>×</strong>ÐÐØÒÓÖÑØÓÒÓÙØ ÒÒÖÔÖÓÙØÑØÖÜÓÒ<strong>×</strong>ØÖÙØÖÓÑØÒÒÖÔÖÓÙØ<strong>×</strong>ÓØÚÖÓÙ<strong>×</strong>ÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐ ØÄÐÖgº

´ÐµÒÓØÖÕÙÚÐÒØ<strong>×</strong>ÖÔØÓÒÓØÄÐÖg<strong>×</strong>ØÖÓÙØ<strong>×</strong>ÝÒÒÖÑÛ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´ÑµÌ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖgÒ<strong>×</strong>ÖÕÙÚÐÒØÐÝØÖÓÙØ<strong>×</strong>ÐÒÖÓÖÑ ÓÒØÒ<strong>×</strong>Ø<strong>×</strong>ÑÒÓÖÑØÓÒ<strong>×</strong>ØÖØÒÑØÖܺ ÓÖØÖÓÙØ<strong>×</strong>ÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>º<br />

ÚÖÝÖÖÙÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong><strong>×</strong>ÓÑÓÖÔÔÖ<strong>×</strong>ÐÝÓÒÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÖÓÑØÓÚ Ð<strong>×</strong>غ<br />

8º ´ÒµÌÖÖÓÙÖÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÐ<strong>×</strong><strong>×</strong>ÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ÒÑÐݸA n<br />

3)¸ÒD n (n ≥ 4)ÒÚÜÔØÓÒÐÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>¸ÒÑÐݸG , F 4 , E 6 , E<br />

´ÔµÌÖ<strong>×</strong>ÓÒ¹ØÓ¹ÓÒÓÖÖ<strong>×</strong>ÔÓÒÒØÛÒØÐ<strong>×</strong><strong>×</strong><strong>×</strong>Ó<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>Ò<br />

8º<br />

7ÒE ´ÓµÚÖÝ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong><strong>×</strong>ÓÑÓÖÔÔÖ<strong>×</strong>ÐÝØÓÓÒÐÖÖÓÑÑÓÒ<strong>×</strong>Øsl(n +<br />

ÜÔØÓÒÐÄÐÖ<strong>×</strong>G 2 , F 4 , E 6 , E<br />

¿ºÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>º<br />

Ï<strong>×</strong>ØÖØÛØØØÓÖÝÓÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ÒÓÛºÇÒØÒÛÓÖÒÓÖÙ<strong>×</strong> ÁÒÒØÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong><br />

7ÒE ØÒÖÐÞØÓÒÓØØÓÙÐÒØÒÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>µÒÒ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong> <strong>×</strong>ÛÒÓÛÛÖØÓÒ<strong>×</strong>ØÓÐÓÓÒÒÛØÖÓÙÐÝØÓÜÔØÒÔÙÖ<strong>×</strong>ÙÒØѺÌÓ<br />

ÒÚØÓÖ<strong>×</strong>ÒØÓÖÖ<strong>×</strong>ÔÓÒÒÒÚÐÙ<strong>×</strong>ÓÐÐØ<strong>×</strong>ÐÑÒØ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>ÓÙÐÚÙ<strong>×</strong>Ø ÜÔÐÓÖØ<strong>×</strong>ØÖÙØÙÖ¸Û<strong>×</strong>ÓÙÐ<strong>×</strong>ØÖØÝØÖÝÒØÓÒÖØÒ<strong>×</strong>ÙÐÖ¸´ÓÖÛØÚÖ<br />

Ü<strong>×</strong>Ø<strong>×</strong>¸ØÝÒÒÖÑÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÄÐÖºÌÖÛÐÐØÒÖÐÞØÓÒ ÖÓÓØ<strong>×</strong>ØÖÙØÙÖ¸ÒÖÓÓØ<strong>×</strong>ÔÓÑÔÓ<strong>×</strong>ØÓÒÓØÄÐÖºÌÒ¸ÛÒ<strong>×</strong>Ó<br />

ÓØÏÝÐÖÓÙÔ¸ÖØÖÒÒÓÑÒØÓÖÓÖÑÙÐÐ<strong>×</strong>Ó<strong>×</strong>ÛÛÐÐ<strong>×</strong>º ÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓÖØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>¸ÒÖÓÑØ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØØÖØÒÑØÖÜÒ¸Ø<br />

<strong>×</strong>ÑÔÐ<strong>×</strong>ØÛÝÓ<strong>×</strong>ØÙÝÒØÑ<strong>×</strong>Ù<strong>×</strong>ÙÐÓÒÐÝØÓÓÚÖÓÑØÒØÐÖÒÒØÙØÓÒ ÐØÓÙÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÒÖÐÞØÓÒ<strong>×</strong>ÓÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸Ø<strong>×</strong> ÓÖÛÔÖÓÛØØ<strong>×</strong>ØÙÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>¸Û<strong>×</strong>ÓÙÐÑÒØÓÒØØ<br />

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(n ≥ 1), B n (n ≥ 2), C n (n ≥<br />

2<br />

4¸ÒØ<br />

1, C), n ≥ 1¸so(2n + 1, C), n ≥ 2¸sp(n, C), n ≥ 3¸so(2n, C), n ≥

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ÒÑÔÓ<strong>×</strong>Ò ÄÐÖ<strong>×</strong>ºÚÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖg¸ÛÒØÖÝÒÓÒ<strong>×</strong>ØÖÙØÒ´l¹<br />

i¸ØÓØÐÖ<br />

<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> Ì<strong>×</strong>ÛÐÐÑÓÝØÖØ<strong>×</strong>ØÛÒØÓÖÒÐÒÖØÓÖ<strong>×</strong>ØÓÒÐÙØÒØÖÐÒÖØÓÖ<strong>×</strong><br />

ÑÒ<strong>×</strong>ÓÒеÒØÖÐÜØÒ<strong>×</strong>ÓÒØÓØÝ<strong>×</strong>ÑÔÐÝÒlÒØÖÐÒÖØÓÖ<strong>×</strong>¸k [t α , k i ] = 0ÓÖi<br />

ØÂÓÒØØÝÒØÙ<strong>×</strong>ÒÒÓØÓÑÔÐØÐÝÖØÖÖݺÌÒÙÑÖÓ<strong>×</strong>ÓÐÙØÓÒ<strong>×</strong>ØÓ ´¿ºµ<br />

ØÓÚÕÙØÓÒ<strong>×</strong>ÙØØÓØÂÓÒØØÝÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ØÒÙÑÖÓÒÔÒÒØ<br />

iÚØÓ<strong>×</strong>Ø<strong>×</strong>Ý<br />

[t α , t β ] = f αβ γ tγ + f αβ i ki ,<br />

ÛÖf<br />

<strong>×</strong>ÓÐÙØÓÒºÀÒ¸ØÒØÖÓÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ØÖÚкÌÙ<strong>×</strong>¸<br />

iÖÓÑØÓÚÕÙØÓÒ<strong>×</strong>ÓÛ<strong>×</strong>¸<br />

αβ<br />

γÖØ<strong>×</strong>ØÖÙØÙÖÓÒ<strong>×</strong>ØÒ<strong>×</strong>ÓgºÌÒÛ<strong>×</strong>ØÖÙØÙÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong>f αβ<br />

ØÓÒØÖÐÐÝÜØÒØÄÐÖg¸ÛÒØÓÐØÖØ<strong>×</strong><strong>×</strong>ØÖÙØÙÖØÓÐÐÓÛÓÖØÜØÒ<strong>×</strong>ÓÒº<br />

0<strong>×</strong>ØÓÒÐÝÔÓ<strong>×</strong><strong>×</strong>Ð<br />

Ì<strong>×</strong>Ð<strong>×</strong>ØÓØÓØÐÓÓÔÐÖÓÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖº<br />

ÒØÖÐÜØÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓÒÒÛÖØÓÛÒÓÖgºÒÒØk ÄØg<strong>×</strong>ÑÔÐÄÐÖ¸ÒÓÒ<strong>×</strong>ÖØ<strong>×</strong>ÔÓÒÐÝØÑÔ<strong>×</strong>ÖÓÑØÖÐ<br />

ÓÖÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸ØØØØÖÚÐ<strong>×</strong>ÓÐÙØÓÒf αβ<br />

i =<br />

ÓÑÔÐÜÔÐÒÛØÓÓÖÒØzºÌÒ<strong>×</strong><strong>×</strong>ÓÖØÓÚÚØÓÖ<strong>×</strong>ÔÓÒÐÝØÑÔ<strong>×</strong> ÒÖØ<strong>×</strong>ÒØÙÖÐÖØÓÔÖØÓÒÖÓÑØÄÐÖg<strong>×</strong><br />

1ØÙÒØÖÐÒØ<br />

S 1ØÓgº<strong>×</strong>ÓÖ¸ÐØ{t |α = 1, . . .,r}<strong>×</strong><strong>×</strong>Óg¸ÒS ÖÓÑS ÒØÙ<strong>×</strong>¸ nºÌ<strong>×</strong><strong>×</strong>Ô ´¿ºµ<br />

1ØÓgÛÐÐÓØÓÖÑ{tα n Z}¸ÛÖt n = tα ⊗ z<br />

γÖ<strong>×</strong>ØÖÙØÙÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong>ÓgºÏØØÓÚÖØØ<strong>×</strong><strong>×</strong>ÔÓÑ<strong>×</strong>Ä ´¿ºµ<br />

[t α m , tβ n ] ≡ [tα ⊗ z m , t β ⊗ z n ] = [t α , t β ] ⊗ (z m · z n ),<br />

ÒÙÑÖÓÒÖØÓÖ<strong>×</strong>º<br />

[t α m, t β n] = f αβ γ t γ ⊗ z m+n = f αβ γ t γ m+n,<br />

ÛÖf αβ loop<strong>×</strong>ÒÒÒØ<br />

loopÒÖØ ÐÖÐÐØÐÓÓÔÐÖ¸ÒÓØg loopºÆÓØØØØ<strong>×</strong>ÙÐÖÓg ÝØÒÖØÓÖ<strong>×</strong>tα loopÒØ<strong>×</strong>ÑÛÝ<strong>×</strong>ÛÓÖgºÏ 0<strong>×</strong>Ù<strong>×</strong>ØØ<strong>×</strong>ÙÐÖgºÆÓØØظØÐÖg ÆÓÛ¸ÛÒÐÓÓÓÖÒØÖÐÜØÒ<strong>×</strong>ÓÒØÓg <br />

α<br />

´¿ºµ<br />

= 1, . . ., l, α = 1, . . .,r .<br />

|α = 1, . . ., r; n ∈<br />

α

ØÖÝØÑÓ<strong>×</strong>ØÒÖÐÒ<strong>×</strong>ØÞÓÖØÖØÓØÒÖØÓÖ<strong>×</strong><strong>×</strong> ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿ºµ<br />

m0ÒÔÙØ<br />

[t α m, t β n] = f αβ γ t γ ⊗ z m+n + (f αβ i ) mnk i .<br />

ÙÒÕÙ<strong>×</strong>ÙØÒ<strong>×</strong>ÓÖÓÖg¸ÒØØ<strong>×</strong>ØÃÐÐÒÓÖÑÓgºÌÙ<strong>×</strong>¸ØÒØÖÐÜØÒ<strong>×</strong>ÓÒ<strong>×</strong> ´ØØ<strong>×</strong>¸ÒØÓÒصÒÒØ<strong>×</strong>ØÖÙØÙÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong><strong>×</strong>ÓÙÐÖÓÑÒÒÚÖÒØØÒ<strong>×</strong>ÓÖ<br />

ÏÑÔÓ<strong>×</strong>ØÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÓÑÒÖÓÑØÂÓÒØØÝÒØØØØØÐÖg<br />

ÓÒÐÝÓÒ¹ÑÒ<strong>×</strong>ÓÒиÒ<strong>×</strong>ÔÖÓÔÓÖØÓÒÐØÓØÃÐÐÒÓÖÑBÓgºÓÖÓÒÚÒÒ¸Û<br />

<strong>×</strong><strong>×</strong>ÙÐÖÓg loop¸ÒÒØ<strong>×</strong>ØÖÙØÙÖÓÒ<strong>×</strong>ØÒØ<strong>×</strong>(f<br />

βºÁØØÙÖÒ<strong>×</strong>ÓÙØØÖ<strong>×</strong><br />

αβ<br />

i ) 00Ò(f i )<br />

ØÓÞÖÓºÆÓÛ¸ÓÖÜÚÐÙÓn¸ØÒÖØÓÖ<strong>×</strong>tα nØÖÒ<strong>×</strong>ÓÖÑÙ<strong>×</strong>ØÐØÒÖØÓÖ<strong>×</strong>t αβºÌ<strong>×</strong>Ú<strong>×</strong>Ø ÓØÓÒØÖÔÖ<strong>×</strong>ÒØØÓÒÓgÛØÖ<strong>×</strong>ÔØØÓØÒ<strong>×</strong>α,<br />

ÒÓÓ<strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong>ÙØØØÃÐÐÒÓÖÑÒØØ<strong>×</strong><strong>×</strong><strong>×</strong>ÕÙÐØÓδ ÓÐÐÓÛÒÖØ<strong>×</strong>ÓÖg ]ØÐÖÓÄÙÖÒØÔÓÐÝÒÓÑÐ<strong>×</strong>ÒzºÌÒØÐÓÓÔÐÖ<strong>×</strong>ØÒÚÒÝ<br />

loop<br />

´¿º¼µ<br />

´¿º½µ<br />

ÌÒÒعÑÒ<strong>×</strong>ÓÒÐÐÓÓÔÐÖ¸<strong>×</strong>Ù<strong>×</strong>ÙÐÐÝÛÖØØÒÒØÓÐÐÓÛÒÛݺÄØL=<br />

ÒÓÛÒ<strong>×</strong>ØÖÚØÓÒÛ<strong>×</strong>ØÓÐÐÓÛÒÖØ<strong>×</strong>ÛØØÓØÖÒÖØÓÖ<strong>×</strong>¸℄ ÏÒØÓÓÒÑÓÖÒÖØÓÖ¸d¸ØÓØ<strong>×</strong>ÒØÖÐÐÝÜØÒÒÒعÑÒ<strong>×</strong>ÓÒÐÐÖ¸<br />

C[z, z −1 ´¿º¾µ<br />

´¿º¿µ<br />

[d, t α m ] = −[tα , d] = mt α m ; [d, k] = 0 .<br />

ÝØÑ<strong>×</strong>Ù<strong>×</strong>ØØÄÐÖgÒÓÛÒ<strong>×</strong>ØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖÓØÒÄ<br />

ÌÓÚÓÒ<strong>×</strong>ØÖÙØÓÒ<strong>×</strong>ØÒĝ<br />

ÐÖĝº 0ÚÚÒ<strong>×</strong>ÒÖØ<strong>×</strong>ÛØØÖÚØÓÒ¸ÒØ<strong>×</strong>ÙÐÖÒÖØ<br />

=<br />

ÌÒÖØÓÖ<strong>×</strong>t α <br />

[t α m, t β n] = f αβ γ t γ ⊗ z m+n − mkδ αβ δ m+n,0 ,<br />

[k, t α n] = 0 .<br />

g loop = L ⊗ C g = C[z, z −1 ] ⊗ C g.<br />

C[z, z −1 ] ⊗ C g. ⊕ Ck ⊕ Cd .<br />

αβ α

¿ºº¾ ÌÊÓÓØËÝ<strong>×</strong>ØÑ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÓÒØÒØÒØÖÐÐÑÒØkÒdºÌÙ<strong>×</strong>¸ØÖØÒ<strong>×</strong>ÙÐÖ<strong>×</strong> ÏÖ<strong>×</strong>ØØÖÑÒØÑÜÑÐÐÒ<strong>×</strong>ÙÐÖºÌ<strong>×</strong>ÛÐÐÖØÒÐÝÓÒØÒØÖØÒ rºÁØÛÐÐÐ<strong>×</strong>Ó <strong>×</strong>ÙÐÖÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖg¸ÒÖØÝh ÌÖÓÓØ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓĤÒÓÙÒÝÓ<strong>×</strong>ÖÚÒØÓÐÐÓÛÒÖÐØÓÒ<strong>×</strong><br />

i ∈ H i = 1, . . .,<br />

Ĥ =<strong>×</strong>ÔÒ{k,<br />

Ò ´¿ºµ<br />

[h i<br />

Ø<strong>×</strong>ØÓÖÓÓØ<strong>×</strong><strong>×</strong> ´¿ºµ<br />

0 , ej n ] = (hi 0 , h0 j )ej n , [k, ej n ] = 0, [d, ej n ] = nej n<br />

Ò ´¿ºµ<br />

[h i 0, h j n] = [k, h j n] = 0, [d, h j n] = nh j n .<br />

ÏÖØÒØÖÓÓØ<strong>×</strong>ˆα<strong>×</strong>Ù<strong>×</strong>ØÚÐÝ<strong>×</strong>ØÖÔÐØÓÒÚÐÙ<strong>×</strong>ÙÒÖØÒÖØÓÖ<strong>×</strong>(h i 0<br />

ˆα n i = (αi , 0, n),<br />

nÓ´¿ºµ<strong>×</strong>ÑÙÐØÔÐØÝÓÒ¸ÛÐØÖÓÓØ<br />

α ∈ L(g), n ∈ Z,<br />

0ÒÖØÓÖ<strong>×</strong>Ò<strong>×</strong>Ò<br />

i<strong>×</strong>ÖÓÓØÓØ<br />

ˆα<br />

ÏØÓÙØØÒÖØÓÖdØÓ<strong>×</strong>ØÒÙ<strong>×</strong>ØÐÚÐnÓØÖÓÓظÐÐØÖÓÓØ<strong>×</strong>ÖÒÒØÐÝ<br />

0ÛØØÒÚÐÙ0ÒØÖÖr<strong>×</strong>ÙÒÖØÓÖ<strong>×</strong>º<br />

ÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÒÖØÓÖ<strong>×</strong>e i nÒh 0¸Ö<strong>×</strong>ÔØÚÐݺÌÖÓÓØα ÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖgºÓØÖÓÓØ<strong>×</strong>ˆα<br />

ÒÖغÌÒÖØÓÖdØÙ<strong>×</strong>Ö<strong>×</strong>³ØÐÖĝÓÖÒØÓØÐÚÐnÒÓÒÐÝØÒ ÀÖÛÒÙÒÖ<strong>×</strong>ØÒØÒ<strong>×</strong><strong>×</strong>ØÝØÓÒÐÙØÒÖØÓÖdÒØÖØÒ<strong>×</strong>ÙÐÖº<br />

i<br />

̂α n<strong>×</strong>ÑÙÐØÔÐØÝr¸<strong>×</strong>ÒØÓ<strong>×</strong>ÒÓØÔÒÓÒØÐÐjÓØh 0<br />

ÒÚÐÙÓh j Ì<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>Óĝ<strong>×</strong>ÒÓØÝ̂L¸ÒØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖ<br />

nÒ´¿ºµº nÓ´¿ºµÖÒÓÒ¹ÒÖغÏÒÒÓظÓÛÚÖ¸ÖÑÓÚØÒÖÝ<br />

ÐÒ<strong>×</strong>ºÌÖÓÓØ<strong>×</strong>ÖÙ<strong>×</strong>ØØÓÐÐØÓÒÓ<strong>×</strong>ÑÙÐØÒÓÙ<strong>×</strong>ÒÚÐÙ<strong>×</strong>ÓØÐÑÒØ<strong>×</strong>ÓØ<br />

0)¸ÒÓØL<strong>×</strong>ÓÖº<br />

ÖØÖÓÓØ<strong>×</strong>̂α i<br />

ÖØÒ<strong>×</strong>ÙÐÖºÀÓÛÚÖ¸ØÖØÒ<strong>×</strong>ÙÐÖÒØ<strong>×</strong><strong>×</strong><strong>×</strong>ÒØÖÐÐÝÜØÒØÓ ËÓÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÓgÒĝÖÚÖÝÑÙ<strong>×</strong>ÑÐÖÒÓÒ<strong>×</strong>ØÖÙØÐÓÒØ<strong>×</strong>Ñ<br />

ÓØÖÓÓØ̂α 0<br />

ÒÐÙØÛÓÑÓÖÒÖØÓÖ<strong>×</strong>ÒØÖÓÓØ<strong>×</strong>Ð<strong>×</strong>ÓÓÒØÒØÒÚÐÙ<strong>×</strong>ÓØ<strong>×</strong>ÒÖØÓÖ<strong>×</strong>º<br />

g<strong>×</strong>Ù<strong>×</strong>ØØ<strong>×</strong>Ù<strong>×</strong>Ø̂α 0 i = (αi , 0,<br />

<br />

d,<br />

h i 0<br />

| i = 1, . . .,r} ´¿ºµ<br />

, k, d)¸<br />

´¿ºµ<br />

n 0 = (0, 0, n), n ∈ Z\{0},<br />

j<br />

n , n ≠ j

ÐÐØ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÚÑÙÐØÔÐØÝÓÒ<strong>×</strong>ÓÖºÌÖ<strong>×</strong>¸ÓÛÚÖ¸ÓÒÑÓÖÖÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÓÑÒÖÓÑØÒÒعÑÒ<strong>×</strong>ÓÒÐÒØÙÖÓØÐÖºÁØ<strong>×</strong>ØÔÔÖÒÓÖÓÓØ<br />

nÛØÑÙÐØÔÐØÝrº<br />

<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÏÒØÝØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><strong>×</strong> ÓÑÔÓ<strong>×</strong>Ø<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>ÒØÓÔÓ<strong>×</strong>ØÚÒÒØÚÖÓÓØ<strong>×</strong>¸ÛØÖ<strong>×</strong>ÔØØÓØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ ÄÛØØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐ<strong>×</strong>¸ÛÒ<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Ò<br />

̂α 0<br />

Ò ÓÖ<br />

´¿º¼µ<br />

̂α 0 i = (α i , 0, 0) = α i<br />

<strong>×</strong>Ò<strong>×</strong>ØØØ<strong>×</strong>ÓÑÔÓ<strong>×</strong>кÄØÖÛÛÐÐ<strong>×</strong>¸ÒØÓÒØÜØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

1<strong>×</strong>ÐÐÒÑÒÖÝÖÓÓغÁØ<strong>×</strong>ÒÓØ<strong>×</strong>ÑÔÐÖÓÓØÒØ<br />

1)ºÏØØ<strong>×</strong>¸ØÒÖØÖÓÓØ<strong>×</strong>´¿ºµÖ<br />

̂α 1 0 = (−µ, 0, i) = δ − µ .<br />

ÏØØ<strong>×</strong>ÒØØÓÒÓÔÓ<strong>×</strong>ØÚ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Ø<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong><strong>×</strong><br />

nÒÓÙÖ<strong>×</strong><strong>×</strong>Ó<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>º<br />

ÀÖµ<strong>×</strong>Ø<strong>×</strong>ØÖÓÓØÓĝÒδ= (0, 0,<br />

Ù<strong>×</strong>Ø̂α n 0 = n·δ, n ≠ 0ºÌÖÓÓØ̂α<br />

´¿º½µ<br />

ÑÒÖÝÖÓÓØ<strong>×</strong>ØØÖÐ<strong>×</strong>Ó<strong>×</strong>ÑÔкÀÓÛÚÖ¸ÛÒÐÙ̂α 0<br />

ÓØÄÐÖĝ<strong>×</strong> −Ö<strong>×</strong>ÔØÚÐݸÛÒÚØÖÒÙÐÖÓÑÔÓ<strong>×</strong>ØÓÒ<br />

+ºÒÓØÒØ<strong>×</strong>ÙÐÖ<strong>×</strong>ÒÖØÝØÔÓ<strong>×</strong>¹<br />

´¿º¾µ<br />

= ÒØ<strong>×</strong>ØÓÒØÚÖÓÓØ<strong>×</strong><strong>×</strong>̂L− ̂L\̂L<br />

ØÚÒÒØÚÖÓÓØ<strong>×</strong>Ýĝ ÐØØÓÒÒÄÐÖºÖ<strong>×</strong>ظÛÒÖØÓÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>º ÁÒÒÐÓÝÛØ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸ÓÒÒ<strong>×</strong>ØÏÝÐÖÓÙÔÓÖØÓÒ<strong>×</strong>ÓØÛØ ¿ºº¿ ÏÝÐÖÓÙÔ +Òĝ ĝ<br />

ÆÓØØØÐÐØαÓÚÖØÖÐÖÓÓØ<strong>×</strong>¸Ù<strong>×</strong>ØÒÓÑÒØÓÖÓØÊÀËÛÓÙÐ ´¿º¿µ<br />

ÒÓØÑ<strong>×</strong>Ò<strong>×</strong>ÓÖÒÑÒÖÝÖÓÓغÙ<strong>×</strong>ÓØÓÚÓÖÑÓØÖØÓÒ¸ÑÒÝ ÔÖÓÔÖØ<strong>×</strong>ÓØÒÏÝÐÖÓÙÔÖÒÐÓÓÙ<strong>×</strong>ØÓØÓ<strong>×</strong>ÓØÏÝÐÖÓÙÔÓ<strong>×</strong>ÑÔÐ ÄÐÖ<strong>×</strong>ºÌÖÖ¸ÓÛÚÖ¸Ð<strong>×</strong>ÓÒÛØÙÖ<strong>×</strong>ÛÖÖÐØØÓØÜ<strong>×</strong>ØÒÓ ¼<br />

0<br />

i<br />

̂L + = {̂α = (α, 0, n) ∈ ̂L| n > 0ÓÖ(n = 0, α ∈ L)},<br />

= ĝ + ⊕ Ĥ ⊕ ĝ − .<br />

〈α, β〉<br />

w α · β = β − 2<br />

〈α, α〉 α .<br />

= 1, . . .,r, ´¿ºµ

0ÓÖÒÝÖÐÖÓÓØα¸ÒÒÓÒ<strong>×</strong> ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿ºµ ÑÒÖÝÖÓÓØ<strong>×</strong>ºÁÒÔÖØÙÐÖ¸ÒÓØØØ〈α,<br />

ÖÓÓØ<strong>×</strong>¸ 0}ÓÑÒÖÝ<br />

δ〉 =<br />

ËÒÒÝÖØÓÒ<strong>×</strong>ÒÙØÓÑÓÖÔ<strong>×</strong>ÑÓØÖÓÓØÐØظØ<strong>×</strong>Ð<strong>×</strong>ÓÑÒ<strong>×</strong>ØØØÏÝÐ ´¿ºµ<br />

= {nδ| n ≠ ÒÒÒÝÏÝÐÖØÓÒØ<strong>×</strong><strong>×</strong>ØÒØØÝÓÒØ<strong>×</strong>Ø̂Lim<br />

ÓÖÓÓØÐØغ ÖÓÙÔÑÔ<strong>×</strong>Ø<strong>×</strong>Ø̂LrÓÖÐÖÓÓØ<strong>×</strong>ÓÒØÓØ<strong>×</strong>к<br />

Ŵ<strong>×</strong>Ø<strong>×</strong>ÑÖØÔÖÓÙØÓØÏÝÐÖÓÙÔÓgÒØÖÓÙÔÓØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>ÒØ<br />

w α | ̂Lim = id bLim .<br />

ÌÖÓÓØ<strong>×</strong>¸ÓÛÚÖ¸ÒÓÛÚØÓÒÐÒÚÐÙ<strong>×</strong>ÒØÑÒØ<strong>×</strong>ÛÓÙÐ<strong>×</strong>ÓÛÙÔÒ ´¿ºµ<br />

m<strong>×</strong>ÚÒÝ<br />

Ŵ = W ⋉ T .<br />

´¿ºµ<br />

ØÚÖÓÙ<strong>×</strong>ÓÑÔÙØØÓÒ<strong>×</strong>ÓØÏÝÐÖÓÙÔ¸ÒÛÛÐÐ<strong>×</strong>ØØÒÓÛºÄØ̂α i<br />

Ò̂βi m = (β i , 0, m)ØÛÓÖÐÖÓÓØ<strong>×</strong>ºÌÒ¸ØÖØÓÒw i n ·<br />

m)¸ÒÒÓØØÕÙÒØØÝ ̂β<br />

n<strong>×</strong>´ÖÛÓÒ<strong>×</strong>Ö<br />

i<br />

i∨µº .´¿ºµ<br />

ÌÖØÓÒÒÜÔÖ<strong>×</strong><strong>×</strong>¸Ò¸<strong>×</strong>ØÖÔÐØÐØÖÓÓØ<strong>×</strong>̂α i<br />

ÒÖÐÛØ̂µ j m = (µj , k, 〉Ýα ÓÑÔÙØØÓÒØØÐÐÓÛ<strong>×</strong>Ù<strong>×</strong>ØÓÖ<strong>×</strong>ØØÓÚÖØÓÒÒÚÖÝ<strong>×</strong>Ù<strong>×</strong>ØÚÓÖѺÒÒ ÒØÚÖÝÒØÙØÚÔØÙÖÓØ<strong>×</strong>ØÖÙØÙÖÓØÏÝÐÖÓÙÔÛÖÖÝÓÙØÓÒÑÓÖ nºÏ<br />

α<strong>×</strong><br />

Ì<strong>×</strong><strong>×</strong>ØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÓÖØÖØÓÒÓÛØ̂µ j mÛØÖ<strong>×</strong>ÔØØÓÖÐÖÓÓØ̂α<br />

Í<strong>×</strong>ÒØ<strong>×</strong>ÛÒÛÖØ´¿ºµ<strong>×</strong> ´¿ºµ ÓÖÒÝÖÓÓØβj ∈ L¸ØØÖÒ<strong>×</strong>ÐØÓÒT ´¿º¼µ<br />

.<br />

w<br />

½<br />

w α · β = δ,<br />

bα<br />

2<br />

〈bα i n ,bαi n<br />

w bα i n · ̂β j m = ̂β j m − 2<br />

〈̂α i n, ̂α i n〉 [〈αi , β j 〉 + 0 · m + n · 0]̂α i n .<br />

(<br />

w bα i n · ̂µ j m = w α i · (µ j + nkα i∨ ), k, m + 1<br />

)<br />

2k [〈µj , µ j 〉 − 〈µ j + nkα i∨ , µ j + nkα i∨ 〉]<br />

T i α : ̂µ j m = (µ j , k, m) ↦→<br />

i<br />

n = (αi , 0, n)<br />

i<br />

(<br />

µ j + kα i , k, m + 1<br />

2k [〈µj , µ j 〉 − 〈µ j + kα i , µ j + kα i 〉]<br />

bα i n<br />

= w i α ◦ (T α i∨) n ,<br />

)

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÖØÓÒÓØÒÏÝÐÖÓÙÔÒÛÖØØÒÓØÓÖÑ<br />

α<strong>×</strong>ØÓÖÒÖÝÏÝÐÖØÓÒÛØ<strong>×</strong>ÓÒØÖ<strong>×</strong>ØÓÑÔÓÒÒØÓØØÖÔÐØÓ<br />

mÒ<strong>×</strong>ØÒØØÝÓÒØÐ<strong>×</strong>ØØÛÓÓÑÔÓÒÒØ<strong>×</strong>ºÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØÒÝÏÝÐ<br />

´¿º½µ<br />

ÛÖwi ØÖÓÓص i<br />

ÔÖÓÙØÓØÏÝÐÖÓÙÔÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖÒÖÓÙÔÓØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>T¸<br />

−1ÓÖÐÐw∈ŴºÌÐÒÖÓÙÔÓØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong><strong>×</strong><br />

{}.ÌÙ<strong>×</strong>¸ØÒÏÝÐÖÓÙÔŴ<strong>×</strong><strong>×</strong>ÑÖØ<br />

w bα i n<br />

= wα i ◦ T j β<br />

ÓÖ<strong>×</strong>ÓÑβ jºÐ<strong>×</strong>ÓT w i α<br />

= w ◦ T w i α<br />

◦ w<br />

ÒÑÔÓÖØÒØÔÖÓÔÖØÝÓØØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>ÖÓÙÔ<strong>×</strong>ØØØ<strong>×</strong>ÒÖØÝØ<strong>×</strong>ØÖÓÓØ ´¿º¾µ<br />

ÒÓÖÑÐ<strong>×</strong>ÙÖÓÙÔÓŴÛØW∩ T =<br />

ÔÔÖÒÒØÒØÓÒÓØÞÖÓÖÓÓØÒ´¿ºµ<strong>×</strong><br />

<strong>×</strong>ÓÒ<strong>×</strong>ÕÙÒ¸ÐÐÔÓ<strong>×</strong><strong>×</strong>ÐØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>ÖÓØÒÝÖØÓÒÛØÖ<strong>×</strong>ÔØØÓØÖÓÓØ ´¿º¿µ<br />

ØÒÏÝÐÖØÓÒÒØÓÚÓÖÑØÏÝÐÖÓÙÔÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖ ÒÖØ<strong>×</strong>ØÖØÓÒ<strong>×</strong>¸ÛÐØÑÒÖÝÖÓÓØÒÖØ<strong>×</strong>ØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>ºÌÙ<strong>×</strong>¸ØÒ<br />

1¸ÒÓÑÒØÓÒ<strong>×</strong>ÓØ<strong>×</strong>ÖØÓÒÛØÐÑÒØ<strong>×</strong>ÓWºÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØØÖÖ<strong>×</strong>ØÒ<br />

ÏÝÐÖÓÙÔ<strong>×</strong>ÒÖØÝ<br />

α 0<br />

Û<strong>×</strong>ÒعÑÒ<strong>×</strong>ÓÒкÌÒÏÝÐÖÓÙÔÐ<strong>×</strong>ÓÔÖÑÙØ<strong>×</strong>ØÖÒ<strong>×</strong>ØÚÐÝÒÖÐÝØ ´¿ºµ<br />

w i ≡ w bα i, i = 1, . . .,r .<br />

ÛÐ<strong>×</strong>ÓÒØÓÑÒÒØÒÏÝÐÑÖ ÝÖÑÓÚÒÐÐØÝÔÖÔÐÒ<strong>×</strong>ÛÖÐØÒÚÖÒØÝ<strong>×</strong>ÓÑÏÝÐÖØÓÒºËÑÐÖÐݸ ÒÏÝÐÑÖ<strong>×</strong>ÛÖØÓ<strong>×</strong>ÓÔÒ<strong>×</strong>Ù<strong>×</strong>Ø<strong>×</strong>ÓØÛØ<strong>×</strong>ÔÛÖÓØÒ ÌÒÏÝÐÖÓÙÔ<strong>×</strong>ÒÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>Ò<strong>×</strong>ØØÒعÑÒ<strong>×</strong>ÓÒÐÏÝÐÖÓÙÔW<br />

ÌÐÖÐ<strong>×</strong>ÓÑØ<strong>×</strong>ÏÝÐÚØÓÖÛ<strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ´¿ºµ<br />

.<br />

¾<br />

Ŵ = W ⋉ T .<br />

w bα 0<br />

1<br />

· µ = (w θ · µ i + kθ ∨ , k, m + 〈µ i , θ ∨ 〉 − k ∨ ) .<br />

P + k<br />

{ r∑ }<br />

µ i µ i | µ i ≥ 0<br />

i=0

ÒØÓÒ¿ºº½ÏÝÐÚØÓÖ<strong>×</strong>ÒØÓÚØÓÖρ<strong>×</strong>Ø<strong>×</strong>ÝÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿ºµ<br />

ÒÖÐÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>º<br />

iºÏÛÐÐ<strong>×</strong>ØØØÓÚÒØÓÒÐ<strong>×</strong>ÓÜØÒ<strong>×</strong>ØÓØÑÓÖ<br />

(ρ, α i ) = 1 (α 2 i, α i ),<br />

ÂÙ<strong>×</strong>ØÐÓÖØ<strong>×</strong>ÓÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸ÓÒÒÐ<strong>×</strong>ÓÐ<strong>×</strong><strong>×</strong>ÝØ ¿ºº Ð<strong>×</strong><strong>×</strong>ØÓÒÓÒÄÐÖ<strong>×</strong><br />

ÓÖÐÐÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α ÚÖÓÙ<strong>×</strong>Ð<strong>×</strong><strong>×</strong><strong>×</strong>ÓÒÄÐÖ<strong>×</strong>ÚØÖÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÒÕÙÚÐÒØÐÝØÖÓÙØÖ<br />

ÐÓÛ¸ÛÐ<strong>×</strong>ØØÝÒÒÖÑ<strong>×</strong>ÓÖØ<strong>×</strong>Ð<strong>×</strong><strong>×</strong><strong>×</strong>ÓÒÄÐÖ<strong>×</strong>º<br />

rÒ 2º ÝÒÒÖÑ<strong>×</strong>¸℄ºÌÖÖÓÙÖÒÒØÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ<strong>×</strong>ÐÐA r , B r , C<br />

D rºÁÒØÓÒ¸ØÖÖÚÜÔØÓÒÐÒÄÐÖ<strong>×</strong>ÐÐE , E 7 , E 8 , F 4ÒG<br />

6<br />

A 1<br />

E 6<br />

A n<br />

E 7<br />

B<br />

ÙÖ¿º¾ÌÝÒÒÖÑ<strong>×</strong>ÓÖØÒÄÐÖ<strong>×</strong><br />

n<br />

E 8<br />

C n<br />

F<br />

ÛÒÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÏÝÐÒÓÑÒØÓÖÓÖÑÙÐÓÖÄÐÖ<strong>×</strong>ºÏÛÐÐÒÓÛÚÖ Ì<strong>×</strong>ÓÒÐÙ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓÒÄÐÖ<strong>×</strong>ºÏÛÐÐÓÑØÓØÑÐØÖ<br />

4<br />

G<br />

D 2<br />

n<br />

ÒØÖÓÙØÓÒØÓØØÓÖÝÓ<strong>×</strong>ÙÔÖ¹ÐÖ<strong>×</strong>ÓÖÓÒØÓ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º<br />

¿

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

¿ºº½ ¿º ÖÁÒØÖÓÙØÓÒØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

ÄØÙ<strong>×</strong><strong>×</strong>ØÖØÛØØÒÓØÓÒÓ<strong>×</strong>ÙÔÖÚØÓÖ<strong>×</strong>Ôº<strong>×</strong>ÙÔÖÚØÓÖ<strong>×</strong>Ô<strong>×</strong>ÚØÓÖ ËÙÔÖÐÖ<strong>×</strong> 2¹Ö¸ºº<strong>×</strong>ØÓÑÔÓ<strong>×</strong>ØÓÒ<br />

ÒÒÒÖÐÒM¹ÖÚØÓÖ<strong>×</strong>Ô<strong>×</strong>ØÓÑÔÓ<strong>×</strong>ØÓÒ ´¿ºµ <strong>×</strong>ÔÓÚÖÐØØ<strong>×</strong>Z α<strong>×</strong><strong>×</strong>ØÓÓÑÓÒÓÙ<strong>×</strong>ÓÖαºÓÖÚØÓÖ<strong>×</strong>ÔV¸Ø<strong>×</strong><br />

V = V 0 ⊕ V 1 , 0, 1 ∈ Z 2<br />

´¿ºµ<br />

= Z/2Z ,<br />

V = ⊕ V α .<br />

α∈M<br />

ÒÐÑÒØÓV α+βºÌÖÒÒÑÓÖÒÖÐ<strong>×</strong>ÓÚÓÖÚØÓÖ<strong>×</strong>Ô<strong>×</strong>¸ÙØÛÛÐÐ<br />

VÖÜÑÔÐ<strong>×</strong><br />

)¸ÒØÜØÖÓÖÐÖ∧ ØÒ<strong>×</strong>ÓÖÐÖT(V )¸<strong>×</strong>ÝÑÑØÖÐÖS(V 2¹ÖÒ<strong>×</strong>º<br />

1¸ÓÖÛ<br />

ÙØÒÓÛÓÒ<strong>×</strong>ØÓÔÒÑÒØÓÒ<strong>×</strong><strong>×</strong>ØÒÝÑÔÓ<strong>×</strong>ÝØÖÒº <strong>×</strong>Ù¹ÐÖÓ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Ð<strong>×</strong>Ó<strong>×</strong>ÙÔÖÐÖ¸Ò<strong>×</strong>Ù¹<strong>×</strong>ÙÔÖÐÖIÓg<strong>×</strong><br />

ÓÖÚØÓÖ<strong>×</strong>Ô<strong>×</strong>º<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Z 2¹ÖÐÖ¸A=A 0 ⊕ A<br />

A α A β<br />

IºÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Ò<strong>×</strong>ÑÐÖØÓÖÙÐÖÄÐÖ¸<br />

⊆ A<br />

ÑÒÐÝÓÒ<strong>×</strong>ÖÒZ <strong>×</strong>Ø<strong>×</strong>ÝÒ<br />

ÐÐÒÐ[I, g] ⊆<br />

1ÛØÄÖØ ÒØÓÒ¿ºº½Ä<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Z 2ÖÐÖg = g 0 ⊕ g<br />

´¿º¼µ<br />

[x,<br />

Ä<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒÓØÄÐÖØÛÝÓÒÙÒÖ<strong>×</strong>ØÒ<strong>×</strong><strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>º ÖÐÐØÚÒÒÓÔÖØ<strong>×</strong>Ógº<br />

z]] = [[x, y]z] + (−1) d(x)d(y) [[x, z]y] ,<br />

ÛÖÓÖÒÝÓÑÓÒÓÙ<strong>×</strong>ÐÑÒØg∈ g n , n<br />

0ÑÓÙкÓÒ<strong>×</strong>ÖØ<strong>×</strong><strong>×</strong>ÓØÚÐÖ<br />

= 0, 1,(g) = nºÌ<strong>×</strong>Ù<strong>×</strong>Ô<strong>×</strong>g 0Òg 1<br />

g 0<strong>×</strong>ÒÓÖÒÖÝÄÐÖ¸ÛÐg 1<strong>×</strong>g <br />

Ò<br />

[x[y,<br />

y] = −(−1)(x)(y) [y, x] ´¿ºµ

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

2ÖÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ÒÓÑÓÖÔ<strong>×</strong>Ñ<strong>×</strong>gl(V<br />

ÌÄÖØ<strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ´¿º½µ<br />

)ÓØ<strong>×</strong>ÙÔÖÚØÓÖ<strong>×</strong>ÔVºÁØ<strong>×</strong>ÒØÙÖÐZ gl(V ) 0 = {f ∈ gl(V ) : f(V n ) ⊆ V n , n ∈ Z 2 } ,<br />

gl(V ) 1 = {f ∈ gl(V ) : f(V n ) ⊆ V n+1 , n ∈ Z 2 } .<br />

ÓÓÖÖÒÓÖÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>¸¼℄ ´¿º¾µ<br />

¿ºº¾ ÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

xy y x, x, y ∈ gl(V ) 1 .<br />

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.,n}ÓÖÓÙÒØÐÝÒÒØ<strong>×</strong>ØÒÛ<strong>×</strong>Ø<br />

ÒÓØHº iÓÖØÒÖØÓÖ<strong>×</strong>ÓØÄÐÖºÌÖØÒ<strong>×</strong>ÙÐÖÓGÛÐÐ<br />

IØÓÒÜØÓÒÖØÓÖ<strong>×</strong>ºÏÓÒØÒÙ <strong>×</strong>ÙÔÖÐÖºÁØÛÐÐØÖØ<strong>×</strong>Ø{1,<br />

ÚÒØÓØ<strong>×</strong><strong>×</strong>ØØØÚ<strong>×</strong>Ù<strong>×</strong>ÒÙÒÖ<strong>×</strong>ØÒÒÓØ<strong>×</strong>ØÖÙØÙÖÓØÃÅÄ<strong>×</strong>Ù¹<br />

. .<br />

<strong>×</strong>ÒØÛØNºÏÛÐÐÙ<strong>×</strong>Ø<strong>×</strong>ØS ÔÖÐÖÒØÖÑ<strong>×</strong>ÓØÒÖØÓÖ<strong>×</strong>ÖØØØÚÖÝÒÒÒºÏÛÐÐÔÙÖ<strong>×</strong>ÙÐØÖÒØ ÏÛÐÐÖ<strong>×</strong>ØÒÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ØÖÓÙØÖÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ºÌ<br />

⊆<br />

Ù<strong>×</strong>Òe i , h i¸Òf <br />

[x, y] =<br />

{<br />

xy − yx,xÓÖy∈gl(V ) 0 ,

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RÖÐÚØÓÖ<strong>×</strong>ÔÛØÒÓÒ¹ÒÖØ<strong>×</strong>ÝÑÑØÖÖÐÚÐÙÐÒÖÓÖÑ<br />

j¸ Iº<strong>×</strong>ÙØØ<br />

I¸<br />

ÄØH (., .)ÒÐÑÒØ<strong>×</strong>h , i ∈<br />

´µ(h i , h j ) ≤ 0i ØÓÔظ<strong>×</strong>ÓÐØÙ<strong>×</strong>ØÖÝØÓÓÒÚÒÓÙÖ<strong>×</strong>ÐÚ<strong>×</strong>ØØØÒ<strong>×</strong>ØÔÖÓÔÖØ<strong>×</strong>ÛÚÓÑ<br />

´µÁ(h i , h i<br />

CºÌÓÚÒØÓÒ<strong>×</strong>Ñ<strong>×</strong>ØÓÓÓ¸ÙÒÑÓØÚØÒÙÒÓÑÓÖØÐ<br />

Iº<br />

) > 0¸ØÒ2(h<br />

(h i ,h i<br />

∈ ZÓÖÐÐj )<br />

´µÁ(h i , h i ) > 0Òı S¸ØÒ(h ∈ ZÓÖÐÐj∈ (h i ,h i )<br />

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ÄØH=H R ⊗<br />

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R<br />

Ø<strong>×</strong>ØÓÒÓÒغÌ<strong>×</strong>ÛÐÐØÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ÓÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖº ÖÐØÓÒ<strong>×</strong>´¿ºµÒ´¿º¿µºÏØØÓ<strong>×</strong>ÒÑÒÛÒÓÛÒØÓÐÐÓÛÒ ÓÖÐÓÓÒØØÚÐÐݹËÖÖÖÐØÓÒ<strong>×</strong>ÓÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖ¸ÖÐÐØ<br />

S)<strong>×</strong><strong>×</strong>ÓØØÓØ<br />

ÖÐØÓÒ<strong>×</strong> ÖØÒÑØÖÜA¸ÛØØÐÒÄÐÖH<strong>×</strong>Ø<strong>×</strong>ÖØÒ<strong>×</strong>ÙÐÖ¸<strong>×</strong>ØÄ<strong>×</strong>ÙÔÖ¹<br />

iÛØi∈I<strong>×</strong>Ø<strong>×</strong>ÝÒØÓÐÐÓÛÒÒÒ ÒØÓÒ¿ºº¾ÓÖÖ<strong>×</strong>¹Ã¹ÅÓÓÝÄ<strong>×</strong>ÙÔÖÐÖG= (A, H,<br />

ÐÖÒÖØÝh i ∈ HÒÐÑÒØ<strong>×</strong>e , f<br />

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´µ[e i<br />

´µ[h, e<br />

i<br />

≠<br />

, f j ] = δ ij h i<br />

∈<br />

i ,h j )<br />

∈<br />

i ,h j )<br />

i<br />

i ] = (h, h i )e i¸[h, f j ] = −(h, h j )f j¸

´µe i = 0 =f ii /∈ S¸e i = 1 =f ii ∈<br />

´Úµ(e i ) 1−2a ij<br />

ÄØÙ<strong>×</strong>ÙÒÖ<strong>×</strong>ØÒØÓÚÒØÓÒºÄÓÓÒØ´¿º¿µÛÒÙÒÖ<strong>×</strong>ØÒØÓÖÒ<br />

a ii e j = (f a ii f j = 0 a > 0Òi ´Úµ(e i ) 1− a ij<br />

a ii e j = (f a ii f j = 0 i S, a ii > 0Òi ´Úµ[e i , e j ] = 0 = [f i , f j<br />

(iv)Ò(v)ÓÚºÌÓÒØÓÒ(iii)<strong>×</strong>ÜÔØÓ<strong>×</strong>ÙÔÖÄ<br />

0º ]a =<br />

i ) 1−2a ij<br />

ii ≠ j .<br />

i ) 1− a ij<br />

∈ ≠ j .<br />

ÐÖÛÖÛÛÐÐÒØÓ<strong>×</strong>ØÒÙ<strong>×</strong>ØÛÒØÚÒÒÓÐÑÒØ<strong>×</strong>ÓØÄ<br />

ij<br />

∅¸ÛÚÄÐÖÒÓÒØÓÒ<br />

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(iii)<strong>×</strong>ÖÙÒÒغÌÒØÖÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÓÒØÒÒØÖØÒ <strong>×</strong>ÙÐÖHºÊÐÐØØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÚØÖÚÐÒØÖ¸<br />

ÓØÖÐØÓÒ<strong>×</strong>(i), (ii),<br />

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ÐÖ´<strong>×</strong>Ð<strong>×</strong>Ó´¿º½µµº<strong>×</strong>ÜÔظÛÒS =<br />

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ØÖØÒÑØÖÜ<strong>×</strong>ÒÓØÖ<strong>×</strong>ØÖØØÓÔÓ<strong>×</strong>ØÚÓÖÔÓ<strong>×</strong>ØÚ<strong>×</strong>ѹÒغ ÔÓ<strong>×</strong>ØÚ<strong>×</strong>ѹÒظØÒØ<strong>×</strong>ÃÅÓÓÝÄ<strong>×</strong>ÙÔÖÐÖºÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<br />

0ÓÖÐÐı∈I¸ÙØØÖØÒÑØÖÜ<strong>×</strong> <strong>×</strong>ÙÔÖÐÖGºÌ<strong>×</strong>ÙÐ<strong>×</strong><strong>×</strong>ÓÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ÖØÓ<strong>×</strong>ÛÚS= Òa ii > 0ÓÖÐÐi∈IºÌÖØÒÑØÖÜ<strong>×</strong>ÔÓ<strong>×</strong>ØÚ¹ÒظººØ(A) ><br />

ÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÁa ØÒ<strong>×</strong>ÙÐÖ¸ÒÒÒÓÒØÝÒÒÖѸÒÓÒÐÒØÖÝÓØ<br />

ii<br />

}¸Û<strong>×</strong>ÛÒØ<strong>×</strong>ÓÒعÑÒ<strong>×</strong>ÓÒÐ<br />

><br />

iÓØÖ¹ Ì<strong>×</strong>ÔÒÓØÖÔÐØÓØÓÖÑ{e i , h i , f<br />

ÈÖÓÔÓ<strong>×</strong>ØÓÒ½ ÓÖØ<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ÖØÒÑØÖÜÓÖÖ<strong>×</strong>ÔÓÒØÓÓÒ<strong>×</strong>Ù<strong>×</strong>Ù¹ÐÖºÏÛÐÐÒÓÛÚ<strong>×</strong>ÑÐÖÓÑÔÓ<strong>×</strong>ØÓÒ<br />

i<br />

<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>¸Û<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓÒsl(2, C)ÐÖºÐÑÒØh ´µÁi∈I S¸Òa ii ≠ 0¸ØÒØÄ<strong>×</strong>ÙÔÖÐÖS i = Cf i ⊕Ch i ⊕Ce<br />

C)º<br />

i<strong>×</strong><br />

i<br />

ÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖG<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓsl(2, ´µÁi∈S¸ØÒØÄ<strong>×</strong>Ù¹<strong>×</strong>ÙÔÖÐÖS i = C[f i , f i ] ⊕ Cf i ⊕ Ch i ⊕ C[e i , e i ] ⊕ Ce<br />

1)º<br />

<br />

<strong>×</strong>ÓÑÓÖÔØÓsl(0,<br />

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

S¸

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

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i<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓØ ´µÁa ii = 0¸ØÒØÄ<strong>×</strong>Ù¹´<strong>×</strong>ÙÔÖµÐÖS i = Cf i ⊕ Ch i<br />

C)¸ÓÖ<br />

⊕ Ce<br />

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1)¸ÓÖÓ<strong>×</strong>ÑÔÐ <strong>×</strong>ØÙÒØÔÖÚÓÙ<strong>×</strong><strong>×</strong>ØÓÒ¸ÝÓÔ<strong>×</strong>ÓØ3¹ÑÒ<strong>×</strong>ÓÒÐÄÐÖsl(2, ÚÒ<strong>×</strong>ÑÔÐÖÓÓظÒÓØ5¹ÑÒ<strong>×</strong>ÓÒÐÄ<strong>×</strong>ÙÔÖÐÖsl(0, −ÖØ<strong>×</strong>Ù¹<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒÖØÝØ<br />

1)ÓÒGÓÑÔÓ<strong>×</strong><strong>×</strong> ÖÓÓغ<strong>×</strong>ÓÖ¸ØÓÒØØÓÒÓÓØ<strong>×</strong>sl(2, C)Òsl(0,<br />

¿ºº¿ iÖ<strong>×</strong>ÔØÚÐݺ ÖØ<strong>×</strong>ÙÑG = N + ⊕ H ⊕ N −¸ÛÖN /N<br />

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ÒØÓÒ¿ºº¿ÌÓÖÑÐÖÓÓØÐØØQ<strong>×</strong>ÒØÓØÖÐÒÖÓÙÔÒÖØ<br />

ÆÓ<strong>×</strong>ÙÖÔÖ<strong>×</strong><strong>×</strong>ØÖºÌ<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÖÒÒÑÒÒÖÚÖÝ<strong>×</strong>ÑÐÖØÓØÒع<br />

IÖÐÐØ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>º<br />

ijºÌ ÝØÐÑÒØ<strong>×</strong>α i¸i∈IÛØÖÐÚÐÙÐÒÖÓÖÑÚÒÝ(α i , α j ) = a<br />

<strong>×</strong>ÔØÛÑ<strong>×</strong>ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÃÅÄ<strong>×</strong>ÙÔÖÐÖÚÖÝÖÒØÖÓÑØØÓ ÒØ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐ<strong>×</strong>ÛÖØÝÛÖÐÛÝ<strong>×</strong>ÕÙÐØÓ2ºÌÖ<strong>×</strong>ÓÒÑÓÖÑÔÓÖØÒØ Z¸ÙÒÐ ÐÑÒØ<strong>×</strong>α i , i ∈<br />

ØÓØÖÄÐÖ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>ØÒÓØÓÒÓÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>½℄ºÄØÙ<strong>×</strong>ÙÒÖ<strong>×</strong>ØÒ Ø<strong>×</strong>ÖÙÐÐݺ<br />

ÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>ºÇÒÐݸØÓÙÒÓØÔÔÖÒظÒØ<strong>×</strong><strong>×</strong>ØÐÑÒØ<strong>×</strong>a ÓÖØ<strong>×</strong>ÓÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸Ø<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>Û<strong>×</strong>ÒØÒÐÐ<br />

ij ∈<br />

<strong>×</strong>ÑÒÖÝÖÓÓØ<strong>×</strong>ºÀÓÛÚÖ¸Ø<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛÖ<strong>×</strong>ØÐÐÐÐÖкÓÖØ<strong>×</strong>ÓÃÅÄ ØÖÓÓØ<strong>×</strong>ÛÖÖдÔÓ<strong>×</strong>ØÚÒØÒÓÖÑÛÖغÒÒÒÖÔÖÓÙØÒÒØÖÓÓØ<strong>×</strong>Ôµº<br />

<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÓÖÖ<strong>×</strong>ÓÙÒØØÓÒÒ<strong>×</strong>ØÓÚÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌ<strong>×</strong>Ñ<strong>×</strong> ÓÖÒÒعÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸Û<strong>×</strong>ÛØØØÖÔÔÖÒÛÒÓÖÓÓØ<strong>×</strong>ÒÓÛÒ<br />

ÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑÓÃÅÄ<strong>×</strong>ÙÔÖÐÖÑÖÐÝÖÒØÖÓÑØÓØÖÐ<strong>×</strong><strong>×</strong>ÓÄ ÐÖ<strong>×</strong>¸ÒØÓÖÒÒغÏÛÐÐ<strong>×</strong>ÓÛØ<strong>×</strong>ÔÖÓÔÖØÝÐØÖ<strong>×</strong>ØÒÓÑÒØÓÖÒØØÝÓ<br />

+<br />

∈ S

Q¸ÒØÒÖÐ<strong>×</strong>¸ÑÝÒÓØÒÒØÖÐÐØظ<strong>×</strong>ÒÒÒÖÐØÒÜÒ<strong>×</strong>Ø ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

<strong>×</strong>Ô<strong>×</strong>ÒÝØÓÚÖÓÓØÚØÓÖ<strong>×</strong>º<br />

I<strong>×</strong>ÓÙÒØÐÝÒÒØÒÛ<strong>×</strong>ØÖÒÓQ<strong>×</strong>ÒÓØÒغÆÓÛÛÓÑØÓØÖÓÓØ<br />

−αµ<strong>×</strong>Ø<strong>×</strong>Ù<strong>×</strong>ÔÓ ÒØÓÒ¿ººÓÖα =<br />

α<strong>×</strong>ÐÐØÑÙÐØÔÐØÝÓØÖÓÓØαº α<strong>×</strong>ÒÓÒ¹ØÖÚкÌ<br />

k=1i k ∈ Q¸ØÖÓÓØ<strong>×</strong>ÔG α´Ö<strong>×</strong>ÔºG GÒÖØÝØÐÑÒØ<strong>×</strong>[e ij [. . .[e i2 , e ]]]´Ö<strong>×</strong>Ôº[f i1 [. . .[f i2 , f ]]]µºÒÓÒ¹ÞÖÓÐÑÒØα<br />

i1<br />

ÓØÓÖÑÐÖÓÓØÐØØQ<strong>×</strong><strong>×</strong>ØÓÖÓÓØÓGØ<strong>×</strong>Ù<strong>×</strong>ÔG ÑÒ<strong>×</strong>ÓÒÓØÖÓÓØ<strong>×</strong>ÔG −αiÓÖØ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÖÓÒ¹ÑÒ<strong>×</strong>ÓÒкÐ<strong>×</strong>Ó¸<strong>×</strong>ÓÖÐÐØÖÓÓØ<strong>×</strong>α∈QÒ αiÒ ÌÓÙÒÖ<strong>×</strong>ØÒØ<strong>×</strong>¸ÓÑÔÖÛØØÐ<strong>×</strong>ØÖÐØÓÒÒ´¿º¿µºÐÐØÖÓÓØ<strong>×</strong>Ô<strong>×</strong>ÓÖi∈I ÖÚÒ<strong>×</strong>G ÒØÓÒ¿ºº = Ce<br />

1ÓGºÌÖ<strong>×</strong>Ð<strong>×</strong>ÓØÓÒÔØÓÔÓ<strong>×</strong>ØÚÒÒØÚÖÓÓغ<br />

α<strong>×</strong>ØÖÓÒØÒÒØÚÒÔÖØ<br />

iÒG = Cf iºÁÒÔÖØÙÐÖ¸Ò<strong>×</strong>ÓÖ¸ØÖÓÓØ<strong>×</strong>Ô<strong>×</strong>G GÜÔÖ<strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong>ÙÑÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÖÓÓØ<strong>×</strong>ÔG G 0ÓÖØÓÔÖØG <strong>×</strong><strong>×</strong>ÙÑÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>º a<strong>×</strong><strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ´Ö<strong>×</strong>ÔºÒØÚµα´Ö<strong>×</strong>Ôº−αµ<br />

1µº<br />

½ºÖÓÓØ= 0}Ò<strong>×</strong>ÛÖØØÒ<strong>×</strong>ÙÔÔ(α)º<br />

¾ºÖÓÓØα<strong>×</strong><strong>×</strong>ØÓÚÒ´Ö<strong>×</strong>ÔºÓµG α ≤ G 0´Ö<strong>×</strong>ÔºG kiÒ<strong>×</strong>ÛÖØØÒØ(α)º<br />

1µºÏØÒÛÖØd(α) =<br />

´Ö<strong>×</strong>Ôºd(α) =<br />

¿ºÌØÓÖÓÓØα = ∑ i<strong>×</strong>ÒØÓ∑ k i<br />

α<br />

ÏÖÓÒÞØÓÚ<strong>×</strong>ØØÑÒØ<strong>×</strong>ÒØÓÒØÜØÓÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄй Ö<strong>×</strong>¸ÙØÔÒÒÑÒØØÒÓÛÛÐ<strong>×</strong>ÓÚÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒØÐÖº<br />

ºÌ<strong>×</strong>ÙÔÔÓÖØÓα<strong>×</strong>Ø<strong>×</strong>Ø{i ∈ I : k i ≠<br />

ÓÖÒÝÖÓÓØα∈Lº¸ÑÙÐØ(α)ÑÙÐØ(−α)¸ÒÖÓÓØα<strong>×</strong>ÔÓ<strong>×</strong>ØÚÒÓÒÐÝØÖÓÓØ<br />

º<strong>×</strong>ÓØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>L<strong>×</strong>ÐÒÖÐÝÒÔÒÒØ<strong>×</strong>Ù<strong>×</strong>ØΠ<strong>×</strong>ÙØØÓÖÒÝα∈<br />

L, α = ∑ β∈Π k ββ¸ÛÖÓÖÐÐβ∈ Π¸ØÖÐÐØ<strong>×</strong>ÐÖ<strong>×</strong>k β ∈ Z<br />

ØÖØÒÓÑÔÓ<strong>×</strong>ÓÒØÃÅÄ<strong>×</strong>ÙÔÖÐÖG<strong>×</strong>ÖØ<strong>×</strong>ÙÑÓÖÓÓØ<strong>×</strong>Ô<strong>×</strong><strong>×</strong><br />

−α<strong>×</strong>ÒØÚºÌ<strong>×</strong>Ú<strong>×</strong>Ù<strong>×</strong>ÓÑÔÓ<strong>×</strong>ØÓÒÓØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>ÒØÓÔÓ<strong>×</strong>ØÚÒÒØÚ<br />

−º<strong>×</strong>ÓÖ¸Ø<strong>×</strong>ÐÐÓÛ<strong>×</strong>Ù<strong>×</strong>ØÓÖÐÞ<br />

+ÓÖÐÐk ÓÒ<strong>×</strong>ºÌ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>LÓÑÔÓ<strong>×</strong><strong>×</strong>ÒØÓL = L +<br />

´¿º¿µ<br />

∪ L<br />

h)xºÌÒ¸ÓÖ ÄØα ∈ LÒh α ∈ H<strong>×</strong>ÙØØÓÖÐÐx G αÒh Mh¸[h, x] = (h α<br />

<br />

,<br />

ÐÐy∈ G<br />

∑ j<br />

−αi<br />

ij<br />

αº<br />

G = (⊕ α∈L+ )G α ⊕ H ⊕ (⊕ α∈L− G α ) .<br />

∈ ∈<br />

−α¸[x, y] = (x, y)h<br />

β<br />

−º<br />

0<br />

∈ Z

ÓÖÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÏÃÒÓÑÒØÓÖÓÖÑÙÐÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÛÛÐиÓÖ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

Ø<strong>×</strong>ÓÓÑÔÐØÒ<strong>×</strong><strong>×</strong>ÚÒÓØÖÖØÖÞØÓÒÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÚÒÝ ÓÖÖ<strong>×</strong>ºÁØ<strong>×</strong>Ù<strong>×</strong>ÙÐÐÝÚÖÝÖØÓÔÔÐÝØÒØÓÒ¿ºº¾ÒØÖÑ<strong>×</strong>ÓØÒÖØÓÖ<strong>×</strong>Ò ÖÐØÓÒ<strong>×</strong>ØÓÚÒÄÐÖØÓÒÛØÖØ<strong>×</strong>ÃÅÄ<strong>×</strong>ÙÔÖÐÖÓÖÒÓغÀÒØ <strong>×</strong>Ù<strong>×</strong>ÙÐØÓÚÖÒØÖØÖÞØÓÒ<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÌÒØÓÒÐÓÛ <strong>×</strong>ÑÒÐÝÔÖ<strong>×</strong>ÒØÓÖÓÑÔÐØÒ<strong>×</strong><strong>×</strong>ÓÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒØ ÒØÓÓÒ<strong>×</strong>ØÖÙØ<strong>×</strong>ÙÖØÖÞØÓÒ<strong>×</strong>ÑÝÒÓØÑÑØÐÝÔÔÖÒØÙÖкÀÓÛÚÖ¸ <strong>×</strong>Ý<strong>×</strong>ØÑØÐÐÝÓÐÐÓÛÒØÚÐÓÔÑÒØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÛÐÐÑØÖÖ ÔÔÖØØÒÓÖ<strong>×</strong>ÙÖØÖÞØÓÒºÁØÛÓÙÐÐ<strong>×</strong>ÓÒÓÑÔÐظÓÛÚÖÖ Ä<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÐÓÛÛÚÖØÖÞØÓÒÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ÖÚÛÓÒÓÒ<strong>×</strong>ØÖÙØ<strong>×</strong>¸ØÓÓÑØ<strong>×</strong>ÓÑÓØÖ<strong>×</strong>ÙÐØ<strong>×</strong>ØØÐÔ<strong>×</strong>ÔØ<strong>×</strong>ØÙÝÓÃÅ<br />

ÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÒØÓÒÐÔÖÓÔÖØÝÓÖH<strong>×</strong>ØÜ<strong>×</strong>ØÒÓ <strong>×</strong>ÓÙÐÒÓØÔÔÖÚÖÝ<strong>×</strong>ÙÖÔÖ<strong>×</strong>ÒÖÓÑÓÙÖÓÒ<strong>×</strong>ØÖÙØÓÒÓØÖØÒ<strong>×</strong>ÙÐÖÓÖØ ÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖØÖØÒ<strong>×</strong>ÙÐÖH<strong>×</strong><strong>×</strong>йÒØÖÐÞÒºÌ<strong>×</strong>ÔÖÓÔÖØÝ<br />

ÖÙÐÖÐÑÒغÌ<strong>×</strong><strong>×</strong>ÒÐÑÒØhÒH<strong>×</strong>ÙØØØÒØÖÐÞÖÓhÒG<strong>×</strong>Hººº<br />

HºÌÜ<strong>×</strong>ØÒÓÖÙÐÖÐÑÒØÒÙ<strong>×</strong>ØÓÓØÒÓÙÒÓÒØÒÓÖÑ<strong>×</strong><br />

ÒØÓÒ¿ººÒÝÄ<strong>×</strong>ÙÔÖÐÖG<strong>×</strong>Ø<strong>×</strong>ÝÒØÓÐÐÓÛÒÓÒØÓÒ<strong>×</strong><strong>×</strong>ÃÅÄ <strong>×</strong>ÝÑÑØÖÐÒÖÓÖÑ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>¾¸¿℄ ÓØÖÓÓØ<strong>×</strong>ÓGºÆÓÛ¸ÛÒÃÅÄ<strong>×</strong>ÙÔÖÐÖÒØÖÑ<strong>×</strong>ÓØÒÓÒ¹ÒÖØ<br />

C<br />

<strong>×</strong>ÙÔÖÐÖº<br />

G (h) =<br />

½ºG<strong>×</strong><strong>×</strong>ÐÒØÖÐÞÒÚÒ<strong>×</strong>ÙÐÖHÛØØÔÖÓÔÖØÝØØG<strong>×</strong>ØÖØ<strong>×</strong>ÙÑ ÓÒ<strong>×</strong>Ô<strong>×</strong>ÓH¸ÒÐÐØÒ<strong>×</strong>Ô<strong>×</strong>ÖÒعÑÒ<strong>×</strong>ÓÒк<br />

HºÁØÖÖÓÒÐÝÒØÐÝÑÒÝ ¾ºÌÖ<strong>×</strong>ÒÓÒ¹ÒÖØÒÚÖÒØ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÐÒÖÓÖÑ(.,<br />

0µ¸α<strong>×</strong>ÐÐÔÓ<strong>×</strong>ØÚ´Ö<strong>×</strong>ÔºÒØÚµÖÓÓغ<br />

0¸ØÒØÒÓÖÑ<strong>×</strong>ÓØÖÓÓØ<strong>×</strong>ÓGÖÓÙÒÖÓÑrº<br />

.)ÒÓÒG ¿ºÌÖ<strong>×</strong>ÒÐÑÒØh∈Mh<strong>×</strong>ÙØØC G (h) =<br />

Ò<strong>×</strong>i∈I<strong>×</strong>ÙØØa ii ><br />

ÓÚºÓÖÚÒr∈R¸ØÖÜ<strong>×</strong>ØÓÒÐÝÒØÐÝÑÒÝÖÓÓØ<strong>×</strong>αÓGÛØ|α(h)|<<br />

(a)<strong>×</strong>Ø<strong>×</strong>ØÓÐÑÒØ<strong>×</strong>ÓGÛÓÑÑÙØ<br />

0º<br />

Áα(h) > 0´Ö<strong>×</strong>Ôºα(h) ºÄØαÒβÓØÔÓ<strong>×</strong>ØÚÓÖÓØÒØÚÖÓÓØ<strong>×</strong>ÓÒÓÒ¹ÔÓ<strong>×</strong>ØÚÒÓÖѺÌÒ(α, β)<br />

0ºÅÓÖÓÚÖ¸(α, β) = 0Òa H αÒ[x, G β ] =<br />

ÌÒØÖÐÞÖÓÒÐÑÒØaÓÖÓÙÔG¸ÒÓØC G<br />

ÛØaºC G<br />

½¼¼<br />

<<br />

(a) = {x ∈ G| xa = ax}<br />

∈<br />

≤

Ì<strong>×</strong>ÓÑÔÐØ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓÒØÒØÖÓÙØÓÒØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÏ<strong>×</strong>ØÐÐ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÒÓÑÒØÓÖÓÖÑÙÐÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÓÒ<strong>×</strong>ÖÒØÑÔÓÖØÒÓØ<strong>×</strong>ØÓ Ø<strong>×</strong>Ø<strong>×</strong><strong>×</strong>¸ÛÚ<strong>×</strong>ÚØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓÒØÒÓÑÒØÓÖÓÖÑÙÐØÐÐØÖÛÚÐÐØ ÚÓÒÑÔÓÖØÒØØÓ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>¸ØÓÙºÏÛÐÐÒÓÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÏÝйùÓÖÖ<strong>×</strong><br />

ÓÒØÖØÖØÓÖÝÓÄÐÖ<strong>×</strong>ØÓÑÓØÚØØÖØÖÒÒÓÑÒØÓÖÓÖÑÙÐ<strong>×</strong>º <strong>×</strong>ÖÕÙÖØÓÓÒ<strong>×</strong>ØÖÙØغÌÒÓÑÒØÓÖÒØØÝÓÙÖ<strong>×</strong>ÒØÖÔÖ<strong>×</strong>ÒØØÓÒØÓÖÝ ÓÄÐÖ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÔÐ<strong>×</strong>ÓØÏÝÐÖØÖÓÖÑÙкÏ<strong>×</strong>ØÖØÛØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒ<br />

¿º ¿ºº½ÒÓÑÒØÓÖÁÒØØ<strong>×</strong><br />

ÒÒØÑÒ<strong>×</strong>ÓÒÐÓÒ<strong>×</strong>ØÓÚØÖÖØØÖÙÒÖ<strong>×</strong>ØÒÒÓØÚÖÓÙ<strong>×</strong><strong>×</strong>ÔØ<strong>×</strong>Ó Ï<strong>×</strong>ØÖØÛØØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ÒØÒÖÙØØÓØ ÖØÖ<strong>×</strong>ÇÁÖÖÙÐÊÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong><br />

ØÒÓÑÒØÓÖÒØØÝÒÓÛØÝØÑÓ<strong>×</strong>ÓÒÓÒ<strong>×</strong>Ö<strong>×</strong>ØÑÓÖÒÓÒ¹ØÖÚÐ C)ÒØØ<br />

ÈË<strong>×</strong>ØØ<strong>×</strong>Ò<strong>×</strong>ØÖÒØÓÖݺÌÑÓØÚØÓÒÓÖÖØÖØÓÖÝ<strong>×</strong><strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÚÒØÛÓ ÓØ<strong>×</strong><strong>×</strong>ØÒÓÑÒØÓÖÒØØݸÛÛÐÐÚÖÝÖÙÐØÓÓÙÖÔÖÓÐÑÓÓÙÒØÒ Û<strong>×</strong>ØÙÖÐÖØÓ<strong>×</strong>ØÙÝØÓÒÔØÖØÖ<strong>×</strong>ÓÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ºÇÒÓØÓ<strong>×</strong>ÓÓØ<strong>×</strong> Ð<strong>×</strong><strong>×</strong>ÓÄÐÖ<strong>×</strong>ºÏÓØÓØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ósl(2, C)Òsl(3, ′Û<strong>×</strong>ÓÑÔØÐÛØØÓÔÖØÓÒÓg<br />

′¸ØÖ<strong>×</strong>Ò<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑÓÚØÓÖ<strong>×</strong>Ô<strong>×</strong><br />

)µÓÄÐÖg¸ ÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>¸VÒV ′´ØØ<strong>×</strong>¸ρ:g→gl(V )Òρ : g → gl(V ′<br />

Û<strong>×</strong>ÝØØV<strong>×</strong><strong>×</strong>ÓÑÓÖÔÓÖÕÙÚÐÒØØÓV gº ´¿ºµ<br />

T : V → V<br />

ÓÖÒÓغÁØÛÓÙÐÙ<strong>×</strong>ÙÐØÓÚÛÝÓØÖÑÒÒ<strong>×</strong>ÙÖÐØÓÒ<strong>×</strong>Ô<strong>×</strong>ØÛÒÖÔÖ¹ ÚÒÖÔÖ<strong>×</strong>ÒØØÓÒÙ<strong>×</strong>ÙÐÐÝ<strong>×</strong>ÓÑÔÐØ<strong>×</strong>ÖÔØÓÒÒØÖÑ<strong>×</strong>ÓÑØÖ<strong>×</strong>¸Ò<br />

<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÛØÓÙØÚÒØÓÓÒØÓØØÐ<strong>×</strong>ÓØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ºËÙÔÔÓ<strong>×</strong>ÛÓÙÐ Ø<strong>×</strong>ÒÓØÐÛÝ<strong>×</strong>ÔÔÖÒØÐÝÓÚÓÙ<strong>×</strong>ØÛÓÚÒÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ö<strong>×</strong>ÓÑÓÖÔØÓÓØÖ<br />

ÓÒ<strong>×</strong>ØÖÙØÕÙÒØØݸ<strong>×</strong>ÝÙÒØÓÒ¸ØØÔØÙÖ<strong>×</strong><strong>×</strong>ÓÑÒØÖÒ<strong>×</strong>ÕÙÐØÝÓØÖÔÖ<strong>×</strong>Ò¹<br />

ÓÖÐÐv∈ VÒÐÐg∈<br />

ØØÓÒÒ<strong>×</strong><strong>×</strong>ÙÒØØÓØÖÑÒÛØÖÓÖÒÓØØÛÓÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>Ö<strong>×</strong>ÓÑÓÖÔØÓ ÑØÑØÐÐݸÛÒÐ<strong>×</strong><strong>×</strong>ÙÒØÓÒ´ÙÒØÓÒØØ<strong>×</strong>ÒÚÖÒØÓÚÖÓÒÙÝÐ<strong>×</strong><strong>×</strong>¸½¼½<br />

ÓØÖÙ<strong>×</strong>ØÝÓÑÔÖÒØÚÐÙÓØÙÒØÓÒÓÒØÚÒÖÔÖ<strong>×</strong>ÒØØÓÒºËÔÒ<br />

′<br />

ρ ′ gT(v) = T(ρ g (v)),

ÛÒÓÙÖ<strong>×</strong>¸ÖØ<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>µØØÖ¹ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÖÔÖ<strong>×</strong>ÒØØÓÒÓ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄÐÖgÛÛ<strong>×</strong>ØÙÝÒÓÛº ØÖÞ<strong>×</strong><strong>×</strong>ÓÑÓÖÔÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ºÌ<strong>×</strong>Ð<strong>×</strong>ØÓØÓØÖØÖÓÒÖÖÙÐ<br />

µÓÒعÑÒ<strong>×</strong>ÓÒÐÖÖÙÐÖÔÖ<strong>×</strong>ÒØØÓÒ¸ÛØ<br />

CÚÒÝ ÒØÓÒ¿ºº½ÌÖØÖχ ÌÖØÖÓÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒØÖÑÒ<strong>×</strong>ØÖÔÖ<strong>×</strong>ÒØÓÒÙÔØÓ ´¿ºµ <strong>×</strong>ØÛص¸<strong>×</strong>Ò<strong>×</strong>ØÑÔÖÓÑh ÕÙÚÐÒºÌÙÒØÓÒÔÒ<strong>×</strong>ÓÒÐÝÓÒØÕÙÚÐÒÐ<strong>×</strong><strong>×</strong>ÓρÒ<strong>×</strong>Ø<strong>×</strong><strong>×</strong><br />

→<br />

ÀÖ¸ÛÚÒØÖØÖØÓÑÔÖÓÑhØÓCºÏÓÙÐÐ<strong>×</strong>ÓÓÒ<strong>×</strong>ÖÛØ<strong>×</strong>¸ ´¿ºµ<br />

µÓÖÛص<strong>×</strong> µÓÖ<br />

µ¸ÒÔÐÓh ∈ h<strong>×</strong>ÖÙÑÒØ<strong>×</strong>Óχ µºÖÓÑÕ´¿º¿µÛ<strong>×</strong>ØØρ(h) · v µ = 〈µ, h〉v<br />

ÛÖØ<strong>×</strong>ÙÑ<strong>×</strong>ÓÚÖØ<strong>×</strong>ØÓÐÐÛØ<strong>×</strong>ÒVº ´¿ºµ ÐÐÛØÚØÓÖ<strong>×</strong>v µ ∈ VÒÛÒÖÛÖØØÖØÖχ χ µ (h) = ∑ µÑÙÐØ(µ) h〉),<br />

ØÑÒ<strong>×</strong>ÓÒÓÒØÖÔÖ<strong>×</strong>ÒØØÓÒ (0)ÚÐÙØÓÒØÞÖÓÛØÚ<strong>×</strong> V<strong>×</strong>Ø ÌÓÔÖØÓÖρ(0)<strong>×</strong>d V <strong>×</strong> d VÑØÖÜÛØÐÐÒØÖ<strong>×</strong>ÕÙÐØÓÞÖÓ¸ÛÖd ÑÒ<strong>×</strong>ÓÒÓØÖÔÖ<strong>×</strong>ÒØØÓÒVºÌÖØÖχ ÌÖØÖÓÖØÖØ<strong>×</strong>ÙÑÓØÛÓÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong><strong>×</strong>ÕÙÐØÓØ<strong>×</strong>ÙÑÓØÖØÖ<strong>×</strong> <strong>×</strong>ÓØÒÝ<strong>×</strong>ÙØÖØÒØÖØÖÓØ<strong>×</strong>ÙÑÓÙÐÛ<strong>×</strong>ÕÙÓØÒØÓÙØÖÓÑØ ÖØÖÓØÓÖÒÐÖÔÖ<strong>×</strong>ÒØØÓÒº ÓØÓÒ<strong>×</strong>ØØÙÒØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ºËÑÐÖÐݸØÖØÖÓÕÙÓØÒØÓÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong><br />

µ<br />

χ<br />

<strong>×</strong>ØÛØÑÓÙдÖÔÖ<strong>×</strong>ÒØØÓÒµÓgºÁØ<strong>×</strong>ÐÐØÏÝÐÖØÖÓÖÑÙк ÄØVÒÖÖÙÐÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>ÒØØÓÒÓØÓÑÔÐÜ<strong>×</strong>ѹ<strong>×</strong>ÑÔÐÄ ÏÒÙ<strong>×</strong>ØØÓÒÓØÏÝÐÖÓÙÔÓÒØ<strong>×</strong>ØÓÛØ<strong>×</strong>ØÓÜÔÖ<strong>×</strong><strong>×</strong>ØÖØÖÓ<br />

½¼¾<br />

χ µ (x) : h ↦→ χ µ (h) =ÌÖexp(ρ(x)) .<br />

χ µ (gxg −1 ) = χ µ<br />

exp(〈µ,<br />

µ (0) = d V .

ÐÖgÛØ<strong>×</strong>ØÛصºÌÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿ºµ<br />

ÖÓÙÔºÌÙ<strong>×</strong>¸ÓÒÒÓÑÔÙØØÖØÖÓÒÖÖÙÐÒعÑÒ<strong>×</strong>ÓÒÐÖÔÖ<strong>×</strong>Òع ØÓÒÖÓÑØÒÓÛÐÓØØÓÒÓØÏÝÐÖÓÙÔÓÒØÐÑÒØ<strong>×</strong>ºÆÓÛÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong> ÛÖρ<strong>×</strong>ØÏÝÐÚØÓÖ<strong>×</strong>ÒÒ´¿º¾µÓÖ´¿º¿¼µ¸ÒØ<strong>×</strong>ÙÑ<strong>×</strong>ÓÚÖØÙÐÐÏÝÐ<br />

,<br />

ØÒÓÑÒØÓÖÒØØݺ ¿ºº¾ ÓÒ<strong>×</strong>ÖØÒÓÑÒØÓÖÓÕº´¿ºµ ÌÒÓÑÒØÓÖÁÒØØÝ<br />

Ì<strong>×</strong><strong>×</strong>ÒÓÛÒ<strong>×</strong>ØÒÓÑÒØÓÖÒØØݺÌÏÝÐÒÓÑÒØÓÖÓÖÑÙÐ<strong>×</strong><strong>×</strong>Ô¹ ´¿ºµ<br />

[ÜÔ( 1 〈α, h〉)<br />

Ð<strong>×</strong>ØÓÒÓØÏÝÐÖØÖÓÖÑÙÐØÓØØÖÚÐÖÔÖ<strong>×</strong>ÒØØÓÒºÓÒÚÒØÓÒÐÐݸØ<br />

−ÜÔ(− 2<br />

α∈L<br />

ÒÓÑÒØÓÖÓÖÑÙÐ<strong>×</strong>ÛÖØØÒ<strong>×</strong>´¿ºµ¸ÙØÓÖÓÙÖÔÙÖÔÓ<strong>×</strong>ÓÒÖÐÞÒØØÓÒÒع<br />

+<br />

.<br />

ÑÒ<strong>×</strong>ÓÒÐÄÐÖ<strong>×</strong>¸ÛÖ<strong>×</strong>ØØÒØÓÓÖÑØØÖ<strong>×</strong>ÙØÓÖØÒÖÐÞØÓÒº<br />

ÛÖw(ρ)<strong>×</strong>ØÑÓρÙÒÖØØÓÒÓØÐÑÒØwÓØÏÝÐÖÓÙÔºÌ ´¿º¼µ<br />

ÑÔÓÖØÒÓØÓÚÓÖÑÙиÖÓÑÓØØÒÖÐÒØÔÓÒØÓÚÛÓÓÙÖÔÖÓÐѸ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong><br />

ÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÓgÒØÖÖ<strong>×</strong>ÔØÚÑÙÐØÔÐØ<strong>×</strong>ÒÚÒØÄÀËÛÚÒÓÛÐ ÒÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÄØÙ<strong>×</strong>ÐÓÓØÕº´¿ºµÑÓÖÐÓ<strong>×</strong>ÐÝÒ<strong>×</strong>ÛØØÓÒØÒ<strong>×</strong><br />

ÓØÏÝÐÖÓÙÔÓgÒØ<strong>×</strong>ØÓÒÓÒÐÐØÖÓÓØ<strong>×</strong>ºÌÙ<strong>×</strong>¸ØÒÓÑÒØÓÖÒØØÝ ØØÑ<strong>×</strong>Ø<strong>×</strong>ÓÑÔÓÖØÒغÚÒØÊÀËÓÕº´¿ºµ¸ÛÚÒÓÛÐÓÐÐØ ÒÒÓØÓÚÖ<strong>×</strong>ØغÁØ<strong>×</strong>ØØÖØÓØÖÐØÓÒØÛÒØ<strong>×</strong>ÔØÖÙÑÓ1<br />

ÓÒØÒ<strong>×</strong>ÐÐØ<strong>×</strong><strong>×</strong>ÒØÐÒÓÖÑØÓÒÓÙØØÄÐÖgÒÚÒØÒÓÑÒØÓÖ ÒØØݸÓÒÒÓÒ<strong>×</strong>ØÖÙØgÓÑÔÐØÐÝÖÓÑغÁÒØØÓÖÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Ø ÔÐÝ<strong>×</strong>ÒØÖÐÖÓÐÒÓØÓÒÐÝÙ<strong>×</strong>ØÓÒØÒ<strong>×</strong>ØÒÓÖÑØÓÒÓGÒظÙØÐ<strong>×</strong>ÓÙ<strong>×</strong> ½¼¿<br />

∑<br />

w∈W<br />

χ µ (h) =<br />

(−1)l(w)ÜÔ[〈w(µ + ρ), h〉]<br />

∑<br />

w∈W (−1)l(w)ÜÔ[〈w(ρ), h〉]<br />

∑<br />

(−1) l(w)ÜÔ[〈w(ρ), h〉] =<br />

w∈W<br />

∏<br />

α∈L +<br />

(1 −ÜÔ(−〈α, h〉)) = ∑ w∈W<br />

∏<br />

=ÜÔ(〈ρ, h〉) ∏<br />

α∈L +<br />

[<br />

1<br />

2 〈α, h〉) ]<br />

]<br />

1 −ÜÔ(−〈ρ, h〉)<br />

det(w)ÜÔ(w(〈ρ, h〉) − 〈ρ, h〉) ,

ØÔÖÓÚ<strong>×</strong>ØÐÒÛØÙØÓÑÓÖÔÓÖÑ<strong>×</strong>º ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

C)º<strong>×</strong><strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÖ¸<br />

¸ÛØÖ<strong>×</strong>ÔØØÓØØÛÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌØÓÒÓØÖØÓÒ<strong>×</strong>ÓÒØÖÓÓØ<strong>×</strong><strong>×</strong>ÚÒÝ<br />

2ºÌÑÙÐØÔÐØÝÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÓÒºÌÏÝÐÖÓÙÔ<strong>×</strong>ÚÒ ÄØÙ<strong>×</strong><strong>×</strong>ØÖØÝÓÑÔÙØÒØÒÓÑÒØÓÖÒØØÝÓsl(3,<br />

sl(3, C)<strong>×</strong>ØÛÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α 1Òα 2ºÌ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>¸L +¸<strong>×</strong>ÚÒÝα 1¸α Òα 3 = α 1 + α<br />

ÝØÔÖÑÙØØÓÒÖÓÙÔÓØÖÐÑÒØ<strong>×</strong>¸S 3ºÌÐÑÒØ<strong>×</strong>ÖØÖØÓÒ<strong>×</strong>r α2ÖÚÒÝ<br />

1Òr ´¿º½µ<br />

2<br />

w αi (α j ) = (α i + α j<br />

´¿º¾µ<br />

)<br />

ÌÐÑÒØ<strong>×</strong>ÓS 3¸ÒÖØÝw α1Òw (1, w α1 , w α2 , w α1 · w α2 , w α2<br />

C)<strong>×</strong>ÚÒÝ )ºÈÙØØÒØÐÐØÓØÖÒØÓ<br />

· w α1 , w α1 · w α2 · w α1 )<br />

ÌØÓÒÓØ<strong>×</strong>ÜÐÑÒØ<strong>×</strong>ÓÒρ<strong>×</strong>(ρ, −α 1 , −α 2 , −ρ, α 1 , α 2<br />

´¿º¿µ Õ´¿ºµØÏÝÐÒÓÑÒØÓÖÓÖÑÙÐÓÖsl(3,<br />

ÛÖρ<strong>×</strong>ØÏÝÐÚØÓÖ¸Òw<strong>×</strong>ÒÐÑÒØÓØÏÝÐÖÓÙÔWºÒÓØÒu=e(−α<br />

ÆÓÛÓÒ<strong>×</strong>ÖØÊÀ˺ ´¿ºµ<br />

1 )<br />

]<br />

´¿ºµ<br />

1 − u − v + u 2 · v + u · v 2 − u 2 · v 2<br />

Ì<strong>×</strong>ÓÑÔÐØ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓØÒÓÑÒØÓÖÓÖÑÙÐÓÖÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ѹ ÊÀË= 1 + u 2 · v + u · v 2 − u 2 · v 2 − u − v]<br />

ÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>¸ÓÒÓØÑÒÔÔÐØÓÒ<strong>×</strong>ÓØÏÝÐÖÓÙÔ<strong>×</strong>ØÐÙÐØÓÒÓÖ¹ <strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÏÝÐÖÓÙÔÓÒÄÐÖ<strong>×</strong>ÛÒÒËØÓÒ¿ºº¿º<strong>×</strong>ÒØÒع <strong>×</strong>ÑÔÐÄÐÖ<strong>×</strong>ºÏÒÓÛÐÓÓØØÒÓÑÒØÓÖÒØØÝÓÒÄÐÖ<strong>×</strong>ºÏ<br />

2Ò−1ÓØÖÛ<strong>×</strong>ºÌÕÙÐØÝ<strong>×</strong>ÓÚÓÙ<strong>×</strong>º ÛÖÐ(w)<br />

ØÖ<strong>×</strong>Ó<strong>×</strong>ØÛØÑÓÙÐ<strong>×</strong>¸ÒØÒÓÑÒØÓÖÓÖÑÙÐÛ<strong>×</strong>ÒÓ<strong>×</strong>ÓÓØÓØ<br />

= +1ÓÖw= x, x<br />

½¼<br />

w αi (α i ) = −α i ,<br />

)¸ÛØ<br />

∏<br />

(1 − e(−φ)) = ∑ (−1)Ð(w) e(w(ρ) − ρ).<br />

φ∈L + w∈W<br />

Òv=e(−α 2<br />

[<br />

ÄÀË=(1 − u)(1 − v)(1 − uv) =<br />

1,<br />

[<br />

2

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

µÖÒ<strong>×</strong> ´¿ºµ ÏÝÐÖØÖÓÖÑÙк<strong>×</strong>ÓÖ¸ØÖØÖ<strong>×</strong>χ ÓÖÑÙп℄<br />

χ µ = ∑ λÑÙÐØ(µ) λ ,<br />

ÛÖρ<strong>×</strong>ØÏÝÐÚØÓÖÒ<strong>×</strong>ÓÖº<br />

ÛÖÛÚÒØÜÔ〈λ,<br />

ÌÒÓÑÒØÓÖÒØØÝÓÖØ<strong>×</strong>ÓÒÄÐÖ<strong>×</strong>ÓÑ<strong>×</strong><br />

λ<strong>×</strong>ÓÖÑÐÜÔÓÒÒÐ<strong>×</strong>ºÌÏÝйÃÖØÖ ´¿ºµ<br />

h〉 = e<br />

χ ,<br />

ÌÑÙÐØÔÐØ<strong>×</strong>ÓÐÐØÖÓÓØ<strong>×</strong>ÒØÒعÑÒ<strong>×</strong>ÓÒÐ<strong>×</strong>ÛÖ1¸ÒØØÖÑÓÒØ ´¿ºµ<br />

ÖØÒ<strong>×</strong>¸ØÖÓÖ¸ÒÓØÚØÑÙÐØÔÐØÝØÓÖº <strong>×</strong>ÙÑ<strong>×</strong>ÓØÒÓÑÒØÓÖÓÖÑÙкÌÝÓ<strong>×</strong>ÖÚØÓÒ´ÙØÓÅÓÒе<strong>×</strong>ØØ ÒÐØÖÒØÒØÓÒ<strong>×</strong>ÚÒÝÄÔÓÛ<strong>×</strong>ÝÒÅÐÒÛ<strong>×</strong>ØÐÓÖØÓÛÖØÒØ<br />

ÓÒØÖÙØÓÒ¸ÓÒÐÝÛÒÔÓ<strong>×</strong>ØÚÖÓÓØØ<strong>×</strong>ÑÔÔØÓÒÓÒ¹ÔÓ<strong>×</strong>ØÚÖÓÓغËÓÓÒÒ<strong>×</strong><br />

ρ]Ú<strong>×</strong>ØØÖØÒØÖÓØØÖÑ<strong>×</strong>ºÊÐÐØØÒÐÑÒØÓØÏÝÐ<br />

[w(ρ) −<br />

wÓÖÐÐw∈Ŵ¸<br />

ρ]ÓØÒ<strong>×</strong> ÖÓÙÔØ<strong>×</strong><strong>×</strong>ÔÖÑÙØØÓÒÓÐÐÖÓÓØ<strong>×</strong>´ÒÓØÒ<strong>×</strong><strong>×</strong>ÖÐÝÔÓ<strong>×</strong>ØÚµºÌÙ<strong>×</strong>¸[w(ρ) −<br />

Í<strong>×</strong>ÒØ<strong>×</strong>ÒØÓÒ¸ÛÒ<strong>×</strong>ØØ ´¿ºµ Ø<strong>×</strong>ØΦ ̂Φ w = w(̂L − ) ∩ ̂L + =<br />

{ˆα ∈ ̂L<br />

∣ ∣∣<br />

+ w −1 (ˆα) ∈ ̂L<br />

}<br />

−<br />

´¿º½¼¼µ<br />

.<br />

ρ − w(ρ) = 1 2<br />

+¸ÛÜÔÐÒ<strong>×</strong>Ø<br />

ÒÐÓÙØÒØÓÚÕÙØÓÒº Ð<strong>×</strong>ÔÔÖÒÒØÊÀËÓØÓÚÓÖÑÙкÁÑÒÖÝÖÓÓØ<strong>×</strong>ÓÒÓØÔÔÖÒØ<strong>×</strong>Ø<br />

wÓÖÒÄÐÖ<strong>×</strong><strong>×</strong>ØÑÒÖÝÖÓÓØ<strong>×</strong>ØÙÖÒÓÙØØÓÏÝÐÒÚÖÒØÒÒ ÌÒÓÑÒØÓÖÓÖÑÙÐØØÛÓÖ<strong>×</strong>ÓÖÒùÅÓÓÝÐÖ<strong>×</strong>¸ØÖÒÐÙÒØ<br />

ÛÖ〈̂Φ w 〉<strong>×</strong>Ø<strong>×</strong>ÙÑÓÐÑÒØ<strong>×</strong>ÓØ<strong>×</strong>Ø̂ΦwºÆÓØØØ−̂L = ̂L<br />

̂Φ<br />

½¼<br />

µ (h) =<br />

e<br />

∑w∈ c W (−1)l(w)w(µ+ρ)<br />

∑<br />

∑<br />

(−1) ∏<br />

l(w)w(ρ) =ρ<br />

w∈cW<br />

∑<br />

w∈ c W (−1)l(w)w(ρ)<br />

bα∈bL +<br />

(1 −−bα )ÑÙÐØ(bα) .<br />

ˆα∈ b L +<br />

[ˆα − w(ˆα)] ∼ 〈̂Φ w 〉 ,<br />

−

ÑÒÖÝÖÓÓØ<strong>×</strong>Ò̂L+¸<strong>×</strong>ØÏÝйÃÒÓÑÒØÓÖÓÖÑÙÐ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿º½¼½µ<br />

l¸ØÒ<strong>×</strong>ÔÐÞØÑØÓØ<br />

C)¸ÚÒÝØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØÑ ÏÒÓÛÓÒ<strong>×</strong>ÖØÜÑÔÐÓÒÒÄÐÖ¸̂ sl(n,<br />

l<br />

, l = n − 1)ºÏÖÚØÒÖÐÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÓÖA <strong>×</strong>ÓA (1)<br />

1ÒA (1)<br />

l´<strong>×</strong>Õ´¿º¼µº lÖÚÒÝ µ¸ÛÖδ<strong>×</strong><br />

<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong><strong>×</strong>ØÙÒÓÒÓØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖ̂L¸<br />

2ÓÖØ<strong>×</strong>ÓÐÐÙ<strong>×</strong>ØÖØÓÒºÌ<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓA (1)<br />

Ø<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØÓÖÞÓÒØÐ<strong>×</strong>ÙÐÖ¸A ÛØØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>Ò̂LÛÚÔÓ<strong>×</strong>ØÚÒÚÐÙ<strong>×</strong>ÛºÖºØdºÌÙ<strong>×</strong>Ø<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<br />

l¸ØÓØÖÛØØÖÓÓØδ 0ºÏÒØ<br />

−<br />

Ø<strong>×</strong>ÑÐÐ<strong>×</strong>ØÑÒÖÝÖÓÓظҵ<strong>×</strong>Ø<strong>×</strong>ØÖÓÓØÓA lÖÚÒݸ<br />

̂LºÌ Ì<strong>×</strong>ØÓÖÓÓØ<strong>×</strong><strong>×</strong>ÚÒÝÙÒØÓÒÐ<strong>×</strong>ÓØÓÖÑjδ + µ¸ÛÖj∈ ZÒµ ∈<br />

ÑÒÖÝÖÓÓØ<strong>×</strong>ÖÚÒÝÙÒØÓÒÐ<strong>×</strong>ÓØÓÖÑjδ¸ÛÖj∈ Z, j ≠<br />

ÌÖÐÖÓÓØ<strong>×</strong>ÚÑÙÐØÔÐØݽÒØÑÒÖÝÖÓÓØ<strong>×</strong>ÚÑÙÐØÔÐØÝlºÌÒÓѹ<br />

−{0}}.´¿º½¼¾µ ÖÓÓØ<strong>×</strong>ÓA ÒØÓÖÓÖÑÙÐ<strong>×</strong>ÚÒÝ<br />

(1)<br />

̂L + = {(s−1)·δ+ ˆα i +...ˆα i+k−1 , s·δ−(ˆα i +...ˆα i+k−1 ), s·δ | 1 ≤ k ≤ l; s ∈ Z<br />

Ê<strong>×</strong>ØÒØ<strong>×</strong>Õº´¿º½¼½µÛÚ<br />

+<br />

´¿º½¼¿µ<br />

∏<br />

´¿º½¼µ<br />

ØÓÓÑÔÙØØÒÓÑÒØÓÖÓÖÑÙкÌÓØÖÑÒØ<strong>×</strong>Ø̂Φ¸ÛÖÐÐØØÓÒÓØ<br />

−¸Ò〈̂Φ〉<strong>×</strong>Ø<strong>×</strong>ÙÑÓÐÐÐÑÒØ<strong>×</strong>Ó̂ΦºÏÒÓÛØ<strong>×</strong>ÑÔÐÒ<br />

ÐÑÒØ<strong>×</strong>ÓØÏÝÐÖÓÙÔÓÒØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>º ÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÓØÄÐÖºÏÒÓÛÒØÓÚÐÙØØ<strong>×</strong>Ø<strong>×</strong>̂ΦwÒØÏÝÐÖÓÙÔ<br />

= ÛÖ̂Φw ˆL + ∩ ˆL<br />

´¿º½¼µ Ŵ(̂L R ) = ̂L R Ŵ(̂L I ) = ̂L I ,<br />

½¼<br />

(A (1)<br />

∏ (<br />

1 − e<br />

−ˆα )ÑÙÐØ(ˆα) ∑<br />

=<br />

ˆα∈L +<br />

w∈Ŵ<br />

det(w) e −〈ˆΦ w〉 ,<br />

(1)<br />

ˆα∈ˆL +<br />

(1 − e(−ˆα))ÑÙÐØ(ˆα) = ∑ w∈ c W<br />

∏<br />

ˆα∈ˆL +<br />

(1 − e(−ˆα))ÑÙÐØ(ˆα) = ∑ w∈ c W<br />

(−1) l(w) e(w(ρ) − ρ) .<br />

(−1) l(w) e(−〈̂Φ w 〉)

ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ØÓÖÑ }ºÌÙ<strong>×</strong>¸̂ΦwÓÒ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÓÐÑÒØ<strong>×</strong>Ó ´¿º½¼µ<br />

I ∩ ̂l + ) = (̂L I ∩ ̂L +<br />

´¿º½¼µ<br />

)<br />

ºÌ<strong>×</strong>Ø̂Φw<strong>×</strong>Ø<strong>×</strong>ØÓÐÐÖÓÓØ<strong>×</strong>{̂α ∈ ̂L + | w −1 α ∈ ̂L<br />

¿ºº¿<br />

−<br />

{β, β + n.δ},ÛÖ, ∈ ̂L +Òm ´¿º½¼µ<br />

1¸ÖÓÑØÓÚÒØÓÒÓØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>¸<br />

+ n.δ}ÛÖ, ∈ ̂L −Òn Z + − {0} .<br />

ÒÓÑÒØÓÖÁÒØØÝÓÖ̂<br />

´¿º½¼µ<br />

ÓÖØÒùÅÓÓÝÐָ (1)<br />

ÒØØÝÚ<strong>×</strong><br />

ZºÈÙØØÒØÐÐØÓØÖÒØÓØÒÓÑÒØÓÖ<br />

ÛÚ̂L+ =<br />

,<br />

ÒØÏÝÐÖÓÙÔ<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓZ 2 ⋉ −1¸ØÓÚÒØØÝ<strong>×</strong>ÕÙÚÐÒØØÓØÂÓØÖÔÐ<br />

1´¿º½½¼µ<br />

e −n(2n−1)α 0<br />

e −n(2n+1)α 1<br />

− ∑ e −(n+1)(2n+1)α 0<br />

e −n(2n+1)α<br />

n∈Z<br />

ËØØÒe −α 0<br />

= rÒe 1<br />

= qr<br />

ÒØØÝÒÚÓÐÚÒØØØÙÒØÓÒϑ 1<br />

´¿º½½½µ<br />

¿ºº<br />

2 r n−1/2 .<br />

ÒÓÑÒØÓÖÒØØÝÓÖظ¸¸℄º )<strong>×</strong>ÒÜÑÔÐÒÛÖØÓÛÒØ ÒÓÑÒØÓÖÓÖÑÙÐÓÖ̂<br />

½¼<br />

ÏÔÔÐÝØÓÚ<strong>×</strong>ØÓØ<strong>×</strong>Ó̂<br />

2<br />

Ò<br />

ÓÖ<br />

Ŵ(̂L<br />

{β<br />

β<br />

β<br />

∈<br />

∈<br />

sl(2, C)<br />

Z +<br />

(n(̂α 1 + ̂α 0 ), n̂α 1 + (n − 1)̂α 0 , (n − 1)̂α 1 + n̂α 0<br />

∣ ∣∣ n = 1, 2, 3, . . .<br />

)<br />

∏<br />

(1 − e −nα 0<br />

e −nα 1<br />

)(1 − e −(n−1)α 0<br />

e −nα 1<br />

)(1 − e −nα 0<br />

e −(n−1)α 1<br />

)<br />

n≥1<br />

= ∑ n∈Z<br />

−α (τ, z)<br />

−iϑ 1 (τ, z) = q 1/8 r −1/2 ∞<br />

∏<br />

= ∑ n∈Z<br />

n=1<br />

(−1) n q (n−1/2)2<br />

(1 − q n ) ( 1 − q n−1 r ) ( 1 − q n r −1)<br />

sl(3, C)º<br />

sl(3, C) (A<br />

(1)

C)ºÌÖÖØÛÓÐÑÒØ<strong>×</strong>ÒØÖØÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

3¸ÒÖØÝØÛÓÐÑÒØ<strong>×</strong>ºÁØ ÌÓÖÞÓÒØÐÐÖÓÖ̂<br />

2´ÐÐØ<br />

sl(3, C)<strong>×</strong>sl(3,<br />

<strong>×</strong>ÙÐÖ¸̂α 1 ,Ò̂α 2ºÌÏÝÐÖÓÙÔÓsl(3, C)<strong>×</strong>S <strong>×</strong>ÒÖØÝØÖØÓÒ<strong>×</strong>ÛØÖ<strong>×</strong>ÔØØÓØØÛÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>̂α 1¸Ò̂α ÖØÓÒ<strong>×</strong>w bα1Òw bα2Ö<strong>×</strong>ÔØÚÐݵºÌ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØÒÄÐÖ̂<br />

2º<br />

)¸ÛÖ ÚÒÝØ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØÓÖÞÓÒØÐÐÖ¸ØÓØÖÛØ̂α 0 = δ − (̂α 1 + ̂α 2<br />

δ<strong>×</strong>Ø<strong>×</strong>ÑÐÐ<strong>×</strong>ØÔÓ<strong>×</strong>ØÚÑÒÖÝÖÓÓØÓA (1)<br />

ÌÏÝÐÖÓÙÔÓ̂<br />

2<strong>×</strong>ÚÒÝC)Ò<br />

sl(3, C)<strong>×</strong>Ø<strong>×</strong>ÑÖØÔÖÓÙØÓØÏÝÐÖÓÙÔÓsl(3, ÒÐÒÖÓÙÔ¸T¸ÓØÖÒ<strong>×</strong>ÐØÓÒ<strong>×</strong>ÒÖØÝØÛÓÐÑÒØ<strong>×</strong>( ∼<br />

ÄØt(m, n) ∈ TÒÐÐÓÛØÖÒ<strong>×</strong>ÐØÓÒÛÓ<strong>×</strong>ØÓÒÓÒ̂α<br />

ÁØÓÐÐÓÛ<strong>×</strong>ØØ 1Ò̂α<br />

´¿º½½¾µ ´¿º½½¿µ<br />

ÓÖÖÚØÝÒÓÖصºÌÙ<strong>×</strong>¸<br />

.tÛÖ<br />

t̂α 0 = ̂α 0 + qδ,<br />

<strong>×</strong>ÙØØ(m+n+q) = 0ºÌÐÑÒØ<strong>×</strong>ÓØÏÝÐÖÓÙÔÖÓØÓÖÑw = w α<br />

w α ∈ S 3Òt(m, n) ∈ TºÄØŴ Tº tØ<strong>×</strong>ÙÖÓÙÔÓŴÒÖØÝt(m, n)´ÛÖØØÒ<strong>×</strong>t<br />

ÓÒØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>¸ÛÚ¸ ÆÓÛ¸ØÓÓÑÔÙØØÒÓÑÒØÓÖÓÖÑÙиÛÒØÓØÖÑÒØØÓÒÓØÏÝÐ<br />

〉ºÖÓÑØÒØÓÒÓ̂Φ¸ÒØØÓÒÓØÐÑÒØ<strong>×</strong>ÓØÏÝÐÖÓÙÔ ŴL,´¿º½½µ<br />

α1 ·ŵ ˆα2 ·ŵ ˆα1 cot<br />

ÓÖw bα1 , w bα2 ∈ S 3Òt ÖÓÙÔÓÒ〈̂Φ t<br />

.´¿º½½µ<br />

̂Φ t = {̂α 1 +iδ, ̂α 2 +jδ, ̂α 1 +̂α 2 +kδ | 0 ≤ i ≤ (m−1), 0 ≤ j ≤ (n−1), 0 ≤ k ≤ (q−1)}<br />

t̂α 1 = ̂α 1 + mδ<br />

t̂α 2 = ̂α 2 + nδ<br />

= Z 2 ).<br />

bα<br />

Ŵ = ŴL ∪w bα1 ·ŴL ∪w bα2 ·ŴL ∪w bα1 ·w bα2 ·ŴL ∪w bα2 ·w bα1 ·ŴL ∪ŵ ˆ<br />

∈<br />

sl(3, C)Ö<br />

ÌÙ<strong>×</strong>¸〈̂Φ t 〉 = (m + k)̂α 1 + (n + k)̂α 2 +<br />

[m(m − 1) + n(n − 1) + k(k − 1)]<br />

δ<br />

2<br />

´¿º½½µ ½¼

ÈÙØØÒÐÐØÓÚØÓØÖÒØÒÓÑÒØÓÖÓÖÑÙиÛÚ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

∏<br />

(1 − u s 0 .us 1 .us 2 )2 (1 − u s−1<br />

s≥1<br />

−<br />

=<br />

∑<br />

r j ≡0(ÑÓ3)<br />

∑<br />

r 1 =1,r 2 =0,r 3 =2(ÑÓ3)<br />

+ ∑<br />

0 .u s 1 .us−1<br />

2 )(1 − u s−1<br />

0 .u s−1<br />

1 .u s 2 )<br />

<strong>×</strong> (1 − u s−1<br />

0 .u s 1 .us 2 )(1 − us 0 .us−1 1 .u s 2 )(1 − us 0 .us 1 .us−1 2 )<br />

u 1 6 (r 1(r 1 −2)+r2 2+r 3(r 3 +2))<br />

0 u 1 6 (r2 1 +r 2(r 2 +2)+r 3 (r 3 −2))<br />

1 u 1 6 (r 1(r 1 +2)+r 2 (r 2 −2)+r3 2)<br />

2<br />

u 1 6 (r 1(r 1 +2)+r2 2+r 3(r 3 −2))<br />

0 u 1 6 (r 1(r 1 −2)+r 2 (r 2 +2)+r3 2)<br />

1 u 1 6 (r2 1 +r 2(r 2 −2)+r 3 (r 3 +2))<br />

2<br />

u 1 6 (r 1(r 1 +2)+r 2 (r 2 −2)+r3 2)<br />

0 u 1 6 (r 1(r 1 −2)+r2 2+r 3(r 3 +2))<br />

1 u 1 6 (r2 1 +r 2(r 2 +2)+r 3 (r 3 −2))<br />

2<br />

r j ≡1(ÑÓ3)<br />

∑<br />

r 1 =0,r 2 =2,r 3 =1(ÑÓ3)<br />

ÓÙÖ<strong>×</strong>ØÙÝÓØÒÓÑÒØÓÖÒØØÝÓÖÒÄÐÖ<strong>×</strong>ºÏ<strong>×</strong>ÓÛØÑÓØÓÒ<strong>×</strong><br />

r j ≡2(ÑÓ3)<br />

ØØÓÙÖÙØÓØÔÖ<strong>×</strong>ÒÓØÑÒÖÝÖÓÓغÏÒÜØ<strong>×</strong>ØÙÝØÒÓÑÒØÓÖ<br />

)ÌØÓÑÔÐØ<strong>×</strong><br />

ÒØØÝÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÌ<strong>×</strong>Û<strong>×</strong>Ö<strong>×</strong>ØÓÒ<strong>×</strong>ØÖÙØÝÓÖÖ<strong>×</strong>ºÏÛÐÐ<strong>×</strong>ØØ<br />

r 1 =2,r 2 =1,r 3 =0(ÑÓ3)<br />

ØÒÓÑÒØÓÖÒØØÝÒØ<strong>×</strong>ÙÔÖ¹ÒÓÑÒØÓÖÒØØÝÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

ÛÖr 1 , r 2 , r 3 ∈ Z¸Òm= (r 3 2 − r 3 )n = 1(r 3 1 − r 2 )q= 1(r 3 3 − r<br />

ÒØÖÖ<strong>×</strong>ÖÖØÓØÐØÖØÙÖº ÒÜÔÐÒÓÛØ<strong>×</strong>ÓØÒ<strong>×</strong>ÒÖÐÞØÓÒÓØÒÓÑÒØÓÖÒØØÝÓÖÒÄ ÐÖ<strong>×</strong>º<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÒÜÑÔÐ<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>ÝÓÒØ<strong>×</strong>ÓÔÓØ<strong>×</strong>ÛÓÖ<br />

1<br />

ÝÔÖÔÐÒÔÖÔÒÙÐÖØÓαÛÒα<strong>×</strong>ÒÚÒ´Ö<strong>×</strong>ÔºÓµÖÓÓØÓÒÓÒ¹ÞÖÓÒÓÖѺÓÖÐÐ ÌÓÒØÏÝйùÓÖÖ<strong>×</strong>ÒÓÑÒØÓÖÓÖÑÙÐÛÖ<strong>×</strong>ØÒØÓÒØ´ÚÒµ αÐÓÒ ÏÝÐÖÓÙÔÓÃÅÄ<strong>×</strong>ÙÔÖÐÖº<strong>×</strong>ÓÖ¸ÛÒØÖØÓÒw ÀÖÓÖØÓÚÓÖÑÙÐØÓÓÐØÖÓÓØ<strong>×</strong>αÖÐ<strong>×</strong>ÓÖÕÙÖØÓ<strong>×</strong>Ø<strong>×</strong>ÝØØÓÒÐ ´¿º½½µ ÛØ<strong>×</strong>β∈ HÖC (α)<br />

(α,α)<br />

(α) 1 .<br />

(α,α)<br />

½¼<br />

−<br />

−<br />

+ ∑<br />

∑<br />

u 1 6 (r 1(r 1 −2)+r 2 (r 2 +2)+r3 2)<br />

0 u 1 6 (r2 1 +r 2(r 2 −2)+r 3 (r 3 +2))<br />

1 u 1 6 (r 1(r 1 +2)+r2 2+r 3(r 3 −2))<br />

2<br />

u 1 6 (r2 1 +r 2(r 2 +2)+r 3 (r 3 −2))<br />

0 u 1 6 (r 1(r 1 +2)+r 2 (r 2 −2)+r3 2)<br />

1 u 1 6 (r 1(r 1 −2)+r2 2+r 3(r 3 +2))<br />

2<br />

1<br />

⊗ Z Q<br />

u 1 6 (r2 1 +r 2(r 2 −2)+r 3 (r 3 +2))<br />

0 u 1 6 (r 1(r 1 +2)+r2 2+r 3(r 3 −2))<br />

1 u 1 6 (r 1(r 1 −2)+r 2 (r 2 +2)+r3 2)<br />

2 ,<br />

w α (β) =<br />

{<br />

β −<br />

2(β,α)<br />

β − 2(β,α)<br />

= 0 ,<br />

=

G¸ØÖ<strong>×</strong>ÒÓÒ¹ÒØÚÒØÖnÔÒÒÓÒxÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong> ÓÒØÓÒØØÓÖÒÝx ∈ G α<br />

0¿℄ºÌÏÝÐÖÓÙÔW<strong>×</strong>ÒÖØÝÐÐØÖØÓÒ<strong>×</strong><br />

, y ∈<br />

E<strong>×</strong>ÒØÓØÖÓÙÔÒÖØÝØ<br />

E<strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

y<strong>×</strong>ÙØØ(x) n y = 0ºÌÏÝÐÖÓÙÔW ÌÏÝÐÚØÓÖÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>Ò<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

+Ò<strong>×</strong>Ø<strong>×</strong><strong>×</strong>´¿º½½µÒ<strong>×</strong>ÓÒÓÒ¹ÞÖÓÒÓÖѺ ÒØÓÒ¿ºº¾ÌÚÒÏÝÐÖÓÙÔW ÖØÓÒ<strong>×</strong>w αi , i ∈ I<strong>×</strong>ÙØØa ><br />

EÒWÖØ<strong>×</strong>Ѻ<br />

w αiÛÖα ∈ L ∗<strong>×</strong>Ø<strong>×</strong>ÝÒ ÓÖÐÐÒÒØÑÒ<strong>×</strong>ÓÒÐÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ØÖÓÙÔ<strong>×</strong>W<br />

´¿º½½µ<br />

ÒØÓÒ¿ºº¿ÏÝÐÚØÓÖ<strong>×</strong>ÒØÓÚØÓÖρØÖÒØÙÐ<strong>×</strong>ÔC ⊗<br />

ÓÖÒHÓÖÒØ<strong>×</strong>ÙÐH ÖØÖÒ<strong>×</strong>ÙÔÖ¹ÖØÖÓÖØ<strong>×</strong>ÙÔÖ¹ÐÖºÓÖØ<strong>×</strong>¸ÛÒØÓÛÓÖÛØÓÖÑÐ ÐÖØÓÐØÓÖÒØØØÓÒÚÒÛØ<strong>×</strong>Ô<strong>×</strong>ºÏØÙ<strong>×</strong>¸ÖÕÙÖØÓÒØ ÏÖ<strong>×</strong>ØÒØÓÒÒÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÓÖØÑÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓØÛØ<strong>×</strong>Ô<strong>×</strong>ÓØ<strong>×</strong>ÙÔÖ<br />

(ρ, α i ) = 1(α 2 i, α i )ÓÖÐÐi ÓÐÐÓÛ<strong>×</strong> ÒØÓÒ¿ººÄØεØÓÑÑÙØØÚ<strong>×</strong><strong>×</strong>ÓØÚÐÖÓÓÖÑÐ<strong>×</strong>Ö<strong>×</strong><br />

λºÏÒØÖØÖÒ<strong>×</strong>ÙÔÖÖØÖÙ<strong>×</strong>ÒÓÖÑÐÜÔÓÒÒØÐ<strong>×</strong><strong>×</strong> ÜÔÓÒÒØÐ<strong>×</strong>e<br />

ØÐÖεÒÖ<strong>×</strong>ÔØÚÐÝØÓØÓÖÑÐ<strong>×</strong>ÙÑ<strong>×</strong><br />

OÖØÐÑÒØ<strong>×</strong>Ó ÓÖÛØÖÜ<strong>×</strong>ØÒØÐÝÑÒÝÐÑÒØ<strong>×</strong>λ i ∈ H¸i=1, . . .,m<strong>×</strong>ÙØØØÓÒØ<strong>×</strong>x ÖÒÓÒ¹ØÖÚÐÓÒÐÝλ ≤ λ iÓÖ<strong>×</strong>ÓÑ1 Ò<br />

i ≤ mºÅÙÐØÔÐØÓÒ<strong>×</strong>ÒÝe ÌÖØÖÒ<strong>×</strong>ÙÔÖ¹ÖØÖÓØH¹ÑÓÙÐV = V 0 ⊕ V 1<br />

´¿º½½µ<br />

∈<br />

ÆÓÛÛÛÖØØÒÓÑÒØÓÖÒØØ<strong>×</strong>ØØØÓÚÖØÖÒ<strong>×</strong>ÙÔÖ¹ÖØÖ 0º V =<br />

λ∈HÑV ∑ e λ ) <strong>×</strong>V =<br />

λ∈H(ÑV ∑ −ÑV )e λ .<br />

i<strong>×</strong>Ø(µ)º<br />

ÛÖV (Λ) ∈ O<strong>×</strong><strong>×</strong>ØÛØÑÓÙÐÓ<strong>×</strong>ØÛØΛ¸Ø<strong>×</strong><strong>×</strong><strong>×</strong>ÙÑØØ(Λ)<br />

ÓÖÑÙÐÐØÓºÓÖµ= ∑ i¸ÐØÙ<strong>×</strong>ÐÐ∑<br />

i∈I k iα<br />

i∈I\S k i<strong>×</strong>Ø0(µ)¸Ò∑<br />

i∈I k ½½¼<br />

i<br />

ii<br />

λ<br />

≤<br />

∑<br />

x λ e λ<br />

λ∈H<br />

∈ I<br />

Z Q<br />

λ<br />

λ<br />

0λ<br />

1λ<br />

e µ = e λ+µº<br />

=

ÏÒØÓÐÐÓÛÒ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

ÛÖ Ò ´¿º½¾¼µ<br />

T Λ = e(Λ + ρ) ∑ ǫ(µ)e −µ Λ ′ = e ∑ Λ+ρ ǫ ′ (µ)e −µ ,<br />

ÓÖÒÝÃÅÄ<strong>×</strong>ÙÔÖÐÖº ´¿º½¾½µ<br />

ÒØÓÒ¿ººÓÖÒÝÃÅÄ<strong>×</strong>ÙÔÖÐÖG¸ ÁÒØÖÑ<strong>×</strong>ÓØÓÚÒØÓÒ<strong>×</strong>¸ÛÒØÒÓÑÒØÓÖÒ<strong>×</strong>ÙÔÖ¹ÒÓÑÒØÓÖÓÖÑÙÐ<br />

ǫ(µ) ′ (µ) = (−1)Ø0(µ) .<br />

Ò ´¿º½¾¾µ<br />

´¿º½¾¿µ<br />

(1 − e −α )ÑÙÐØ0(α)<br />

(1 + e −α )ÑÙÐØ1(α)<br />

(1 − e −α )ÑÙÐØ0(α)<br />

Ø<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ÖÖ<strong>×</strong>ÔØÚÐÝØÒÓÑÒØÓÖÓÖÑÙÐÒØ<strong>×</strong>ÙÔÖ¹ÒÓÑÒØÓÖÓÖÑÙк ÌØÓÑÔÐØ<strong>×</strong>ÓÙÖÒØÓÒÓØÒÓÑÒØÓÖÒ<strong>×</strong>ÙÔÖ¹ÒÓÑÒØÓÖÓÖÑÙÐÓÖ<br />

(1 − e −α )ÑÙÐØ1(α)<br />

<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÜÑÔÐÓØÑÓÒ<strong>×</strong>ØÖÄÐÖ½¸℄Û<strong>×</strong>ÃÅÄÐÖ ¿ºº <strong>×</strong>ÒÜÑÔÐÓØÓÚÓÖÑÙÐÒØ<strong>×</strong>ØØÒÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ¸ÛÛÐÐÖÝ ÌÅÓÒ<strong>×</strong>ØÖÄÐÖ<br />

<strong>×</strong>ÖÒØÔÝ<strong>×</strong>Ð<strong>×</strong>ØØ<strong>×</strong>ÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒÓÒØÓÖÙ<strong>×</strong>ºÁØ<strong>×</strong>ÖÓÓØÐØØ<strong>×</strong>¾ÑÒ<strong>×</strong>ÓÒÐ<br />

1,1ÛÖΛ<strong>×</strong>ØÄÐØØ<br />

ÙÒÕÙÚÒÙÒÑÓÙÐÖÄÓÖÒØÞÒÐØØÓÖÒ2º<br />

2mn´Ø<strong>×</strong> 1,1<strong>×</strong>Ø ÚÒÙÒÑÓÙÐÖÄÓÖÒØÞÒÐØØÒÓØII<br />

(n)<strong>×</strong>ØÒÙÑÖÓÔÖØØÓÒ<strong>×</strong>ÓnÒØÓÔÖØ<strong>×</strong>Ó24ÓÐÓÖ<strong>×</strong>ºÌÙ<strong>×</strong>¸<br />

25,1 = Λ ⊕ II<br />

ÛØÐÑÒØ<strong>×</strong>α=(λ, m, (λ ∈ ΛÒ(m, ∈ II 1,1<br />

2ºÌÖÑÙÐØÔÐØÝ<strong>×</strong>ÚÒÝ<br />

)ÛØÒÓÖÑα = λ 2 −<br />

ØÙÒÕÙÔÓ<strong>×</strong>ØÚÒØÐØØÓÖÒ24ÛØÒÓÒÓÖÑ2ÚØÓÖ<strong>×</strong>℄µºÒII ÌÖÓÓØ<strong>×</strong>ÓII 25,1ÖØÒÓÒ¹ÞÖÓÚØÓÖ<strong>×</strong>αÛØα2 ≤<br />

p 24<br />

∗<strong>×</strong>ØÙÐÓLº 0´ÑÓ2µºÐ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÓÓºÌ m)º<br />

(1 − α 2 /2)¸ÛÖp ½½½<br />

ÒÒØÖÐÐØØL<strong>×</strong><strong>×</strong>ØÓÚÒÓÖÐÐv∈ L¸(v, v) ≡<br />

ÑÒ<strong>×</strong>ÓÒÒ<strong>×</strong>ÒØÙÖÓLÖØÑÒ<strong>×</strong>ÓÒÒ<strong>×</strong>ÒØÙÖ¸Ö<strong>×</strong>ÔØÚÐݸÓØÖÐÚØÓÖ<strong>×</strong>ÔL ⊗<br />

ÛØØÐÒÖÓÖÑÒÙÖÓÑLºÐØØ<strong>×</strong>ÐÐÄÓÖÒØÞÒØ<strong>×</strong><strong>×</strong>ÒØÙÖ(m, 1)ÓÖ(1,<br />

ÐØØ<strong>×</strong>ÙÒÑÓÙÐÖÓÒL = ∗¸ÛÖL<br />

24<br />

∏<br />

∏<br />

∏<br />

∏<br />

= (−1)Ø(µ)<br />

Ò<br />

α∈L + 0<br />

α∈L + 1<br />

α∈L + 0<br />

α∈L + 1<br />

L<br />

=<br />

=<br />

T<br />

ǫ<br />

e−ρ ∑ w∈W<br />

e−ρ<br />

∑<br />

w∈W<br />

det(w) w(T),<br />

det(w) w(T ′ )<br />

2<br />

Z R

ØÑÙÐØÔÐØ<strong>×</strong>ÓØÖÓÓØ<strong>×</strong>ÖÚÒÝ ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

.´¿º½¾µ<br />

ÒÛØÔÓÒØ<strong>×</strong><br />

p 24 (1 + n)q n = 1/∆(q) = q ∏ −1 − q<br />

n>0(1 n ) −24 = q −1 25,1¸ÛÖÒØÚÓÖÖ<strong>×</strong>ÔÓÒ¹<br />

+ 24 + 324q + 3200q + . . .<br />

−1)ºÌ ´¿º½¾µ ÌÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÖØÒÓÖÑ2ÚØÓÖ<strong>×</strong>αÒII<br />

(λ, − 1), λ ∈ Λ ´¿º½¾µ<br />

2<br />

ÒØÄÐØغÌÝÐÐ<strong>×</strong>Ø<strong>×</strong>Ý(ρ, α) = 1ÓÖØÏÝÐÚØÓÖρ=(0, ÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÖÓØÓÖÑ(0,<br />

ÛØØÏÝÐÚØÓÖº<br />

0ÓÖØÖÒÒÖÔÖÓÙØ<br />

n)ºÌÙ<strong>×</strong>¸ØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>Ö<br />

n)<strong>×</strong>ÙØØm>0ÓÖ<br />

0, n), n ∈ N<br />

ÒØÝÐÐÚÑÙÐØÔÐØÝp 24 (1) = 24ºÌ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>Ý(ρ, =<br />

Ì<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÖÚÒÝØ<strong>×</strong>ØÓÖÓÓØ<strong>×</strong>α=(λ,<br />

ÌÏÝÐÖÓÙÔÓØÐÖ<strong>×</strong>ÒÖØÝØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛØÒÓÖÑ2¸Ò ´¿º½¾µ<br />

m,<br />

α = (0, 0,<br />

ÓÚÒÓÖÑØÓÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÚÒØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>´¿º½¾µÒØØØØØÝ ÆÓÛÛÒÛÖØÓÛÒØÒÓÑÒØÓÖÒØØÝÓØÑÓÒ<strong>×</strong>ØÖÄÐÖÖÓÑØ<br />

25,1<strong>×</strong><strong>×</strong>ÓÑÓÖÔØÓØÖØÓÒÖÓÙÔÓØÄÐØغ<br />

ÒØØÝ<strong>×</strong><br />

/2)¸ÛÒÛÖØÓÛÒØÔÖÓÙØ<strong>×</strong>ÓØÒÓÑÒØÓÖ<br />

ØÙ<strong>×</strong>ØÏÝÐÖÓÙÔÓII<br />

ÌÏÝÐÖÓÙÔ<strong>×</strong>ÒÓÛÒÒÒÛÒÓÖÑØ<strong>×</strong>ÙÑ<strong>×</strong>ÓØÒÓÑÒØÓÖÒØØÝ ´¿º½¾µ ÐÐÚÑÙÐØÔÐØÝp<br />

ÙÒÒØØØØÐÐØÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÚÒÓÖÑ0ÒÖÑÙØÙÐÐÝÓÖØÓÓÒк<br />

24 (1 − α 2<br />

Ì<strong>×</strong>ÙÑ<strong>×</strong><strong>×</strong>ÚÒÝ<br />

∑<br />

∑<br />

n<br />

1, λ2<br />

0,<br />

α)<br />

α ∈ L + = {α ∈ II 25,1 |(α, ρ) > 0 or α = (0, 0, n)}<br />

∏<br />

α∈L + (1 − e −α ) p 24(1−α 2 /2) .<br />

n 1 ,n 2 ,...<br />

(−1) n 1+n 2 +... e n 1ρ<br />

= (1 − e ρ ) 24 (1 − e 2ρ ) 24 . . . . ´¿º½¾µ<br />

( ) ( )<br />

24<br />

e n 24<br />

2ρ<br />

. . .<br />

n 1 n 2<br />

½½¾

ÈÙØØÒØÓÚØÛÓÕÙØÓÒ<strong>×</strong>ØÓØÖÚ<strong>×</strong> ÔØÖ¿ºÃÅÄÐÖ<strong>×</strong><br />

´¿º½¿¼µ<br />

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e ∏ ρ .<br />

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(1 − e −α ) p 24(1−α 2 /2) = ∑ (<br />

e<br />

α∈L + w∈W<br />

n∈ZØ(w)w<br />

∏ )<br />

ρ (1 − e nρ ) 24<br />

n>0<br />

½½¿

4<br />

ÅÓÙÐÖÓÖÑ<strong>×</strong><br />

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0¸ØÐÑØ ´º½µ<br />

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ÆÓÛ¸ÐØUÒÓÔÒ<strong>×</strong>Ù<strong>×</strong>ØÓCºÙÒØÓÒf : U →<br />

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½½<br />

f ′ f(z) − f(z 0 )<br />

(z 0 ) = lim<br />

z→z0 z − z 0

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NÑØÖ<strong>×</strong>ÛØÒØÖ<strong>×</strong> ¿ºSL(N,<br />

Ø<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong> ÛØÒØÖ<strong>×</strong>ÖÓÑF¸ØÓØÖÛØØÓÔÖØÓÒÓÑØÖÜÑÙÐØÔÐØÓÒºÏÛÐÐ<br />

Z)¸Û<strong>×</strong>ÓÚÖØÐÓÒØÖ<strong>×</strong>¸Ò<strong>×</strong>ÓÑÓ<br />

F)´ËÔÐÄÒÖÖÓÙÔµÁØ<strong>×</strong>Ø<strong>×</strong>ØÓÐÐN<strong>×</strong><br />

ÑÓ<strong>×</strong>ØÐÝ<strong>×</strong>ØÙÝØÄÖÓÙÔSL(2,<br />

ºÍÔÔÖÀÐÈÐÒÌÙÔÔÖÐÔÐÒ¸H¸<strong>×</strong>Ø<strong>×</strong>ØÓÓÑÔÐÜÒÙÑÖ<strong>×</strong>ÛØ ´º¾µ<br />

ÔÓ<strong>×</strong>ØÚÑÒÖÝÔÖØ<strong>×</strong>ººº<br />

.<br />

R)ºÌ ´º¿µ<br />

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{ ( )<br />

a b<br />

}<br />

SL(2, Z) = : a, b, c, d ∈ FÒad − bc = 1<br />

c d<br />

H = {x + iy | y > 0; x, y ∈ R} .

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong> Z)ÒØ<strong>×</strong>ÓÒÖÙÒ<br />

º¾<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>º<br />

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ÁØ<strong>×</strong>Ø<strong>×</strong>ÕÙ<strong>×</strong>ØÓÒØØÛÛÐÐÝÓÒÖÒÛØÒØÓÐÐÓÛÒÔØÖ<strong>×</strong>¸ÒÛ ÛÐÐ<strong>×</strong>ÓÛØÒÓØÓÒ<strong>×</strong>ÛÒØÖÓÙÖØÒØÓØÒÒØÙÖÐÛݺ ÒÜÔÖ<strong>×</strong><strong>×</strong><strong>×</strong><strong>×</strong>ÙÑÓÐÑÒØ<strong>×</strong>ÓSÒ¸<strong>×</strong>Ó¸ÒÓÛÑÒÝÛÝ<strong>×</strong>ÒØ<strong>×</strong>ÓÒº<br />

ÓÖÐÖnºÏÛÐÐÐÖÒÑÓÖÓÙØØÓÚÔÖÓÐÑÒØÓÙÖ<strong>×</strong>ÓÓÙÖ<strong>×</strong>ØÙÝÓØ ÓSºÏ<strong>×</strong>ÓÖØÚÖÓÙ<strong>×</strong>ÔÖÓÔÖØ<strong>×</strong>Óp(n)¸<strong>×</strong>ÝÓÖÜÑÔиØ<strong>×</strong><strong>×</strong>ÝÑÔØÓØÚÓÙÖ ÄØp(n)ÒÓØØÒÙÑÖÓÛÝ<strong>×</strong>n¸ÒÒØÖ¸ÒÛÖØØÒ<strong>×</strong><strong>×</strong>ÙÑÓÐÑÒØ<strong>×</strong><br />

ØÓÖÝÖÒØÑØÐÝÖÐØØÓÐ<strong>×</strong><strong>×</strong>ÓÙÒØÓÒ<strong>×</strong>ÒÓÑÔÐÜÒÐÝ<strong>×</strong><strong>×</strong>ÐÐÐÐÔØÑÓÙÐÖ ÓÑÓÙÐÖÙÒØÓÒ<strong>×</strong>ºÌÔÖØØÓÒÙÒØÓÒp(n)ÒÓØÖÙÒØÓÒ<strong>×</strong>ÓØÚÒÙÑÖ Ò³<strong>×</strong>ØÙÒØÓÒÒÖÐØ<strong>×</strong>ºÓÖÒÓÛ¸ÛÐÓÓÓÖÛÝØÓÑÓØÚØØ<strong>×</strong>ØÙÝ<br />

ÙÒØÓÒ<strong>×</strong>ºËÓ¸Ø<strong>×</strong>ÖÓÙÒØ<strong>×</strong>ØØÛ<strong>×</strong>ØÖØÓÙÖ<strong>×</strong>ØÙÝÓÑÓÙÐÖÓÖÑ<strong>×</strong>ºÌ<strong>×</strong>ÔØÖ <strong>×</strong><strong>×</strong>ÑÓ<strong>×</strong>ØÐÝÓÒ½¸¾¸¿¸℄ º¾º½ ÙÒØÓÒf<strong>×</strong><strong>×</strong>ØÓÒÐÐÔØÙÒØÓÒ ÓÙÐÝÔÖÓÙÒØÓÒ<strong>×</strong><br />

¾ºf<strong>×</strong>ÑÖÓÑÓÖÔº ½ºf<strong>×</strong>ÓÙÐÝÔÖÓº<br />

ÓÒØÓÒºÏÛÐÐ<strong>×</strong>ØØÓÙÐÝÔÖÓÙÒØÓÒ<strong>×</strong>ÛÐÐÐÙ<strong>×</strong>ØÓØ<strong>×</strong>ØÓÐØØ<strong>×</strong>ÒC¸ ÒØ<strong>×</strong>ØÓÐØØ<strong>×</strong>ÒCÖÚÖÝÐÓ<strong>×</strong>ÐÝÖÐØØÓÑÓÙÐÖÓÖÑ<strong>×</strong>¸ÛÛÛÐÐÓÑØÓ ÏÐÖÝÒÓÛÛØÑÖÓÑÓÖÔÙÒØÓÒ<strong>×</strong>¸<strong>×</strong>ÓÛ<strong>×</strong>ØÖØÛØØÓÙÐÝÔÖÓ<br />

ÖÖÐØØÓÓØÖÚÖÝÐÓ<strong>×</strong>Ðݺ <strong>×</strong>ÓÖØÐݺÇÒØÛÓиÛÛÐÐÒØØÐÐÔØÙÒØÓÒ<strong>×</strong>¸ÐØØ<strong>×</strong>ÒC¸ÒÑÓÙÐÖÓÖÑ<strong>×</strong> ÙÒØÓÒfÓÚÖC<strong>×</strong>ÐÐÔÖÓ¸ÛØÔÖÓω¸ ´ºµ ½½<br />

f(z + ω) = f(z)

ωÖÒØÓÑÒÓfºÒÜÑÔÐÓ<strong>×</strong>ÙÙÒØÓÒÛÓÙÐØ ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

CÛØÔÖÓ2πiº<br />

)<strong>×</strong>ÐÐÙÒÑÒØÐÔÖºÌ<strong>×</strong>ØÓÐÐÐÒÖ<br />

2<strong>×</strong>ÙØØØÖØÓ ÛÒÚÖzÒz +<br />

ÜÔÓÒÒØÐÙÒØÓÒe z , z ∈<br />

ÑÓÙÐÖÓÖÑ<strong>×</strong>ÐØÖÒØ<strong>×</strong>ÔØÖº<br />

ÙÒØÓÒf<strong>×</strong>ÐÐÓÙÐÝÔÖÓØ<strong>×</strong>ØÛÓÔÖÓ<strong>×</strong>ω 1Òω ω 1 /ω 2<strong>×</strong>ÒÓØÖк½Áω 2ºÏÛÐÐ<strong>×</strong>ÜÑÔÐ<strong>×</strong>Ó<strong>×</strong>ÙÙÒØÓÒ<strong>×</strong>ÛÒÛÓÒ<strong>×</strong>Ö<strong>×</strong>ÓÑÓØÜÑÔÐ<strong>×</strong>Ó<br />

1Òω 2ÖÔÖÓ<strong>×</strong>Óf¸ØÒ<strong>×</strong>Ó<strong>×</strong>ÒÝÓÑÒØÓÒ(mω 1<br />

ÓÖÒÝm, n ∈ ZºÌÔÖ(ω , ω 2<br />

ÓÑÒØÓÒ<strong>×</strong>mω 1 + nω 2<strong>×</strong>ÒÓØΩ(ω , ω 2 ∗¸ÒLØ<strong>×</strong>ØÓÐÐÐØØ<strong>×</strong> C<strong>×</strong>ÙØØ<br />

)ºÌ<strong>×</strong><strong>×</strong>ÐÐØÐØØÒÖØÝω Òω ÐØØÒLÌÒ<strong>×</strong><strong>×</strong>ÖÝÒ<strong>×</strong>ÙÒØÓÒØÓÒÓÖØÛÓÐÑÒØ<strong>×</strong>ÓMØÓÓÖÖ<strong>×</strong>ÔÓÒØÓ<br />

ÄØMÒÓØØ<strong>×</strong>ØÓÔÖ<strong>×</strong>(ω 1 , ω 2 }ÓMÓÖÖ<strong>×</strong>ÔÓÒØÓØ<strong>×</strong>Ñ )¸Û<br />

)ÓÐÑÒØ<strong>×</strong>ÓC ÓCºÌÑÒÓÐC/L(ω 1 , ω 2 )<strong>×</strong>ÓØÒÝÒØÝÒØÔÓÒØ<strong>×</strong>z , z 2 ∈<br />

z 1 − z 2 = ω 1 m + ω 2 nÓÖ<strong>×</strong>ÓÑm, ∈ ZºÆÓÛ¸ÚÒM¸Ø<strong>×</strong>ØÓÐÐÔÖ<strong>×</strong>(ω , ω 2<br />

ÛÓÙÐÐØÓ<strong>×</strong>ÛÒÓØÛÓ<strong>×</strong>ÙÔÖ<strong>×</strong>{ω 1 , ω 2 }Ò{ω 1 , ω′ 2<br />

Z)º<br />

)<strong>×</strong> ´ºµ<br />

Ø<strong>×</strong>ÑÐØØÒLØÙÖÒ<strong>×</strong>ÓÙØØØØÝ<strong>×</strong>ÓÙÐÓÒÖÙÒØÑÓÐÙÐÓSL(2, ÌÔÖ(ω 1 ′, ω′ 2 )<strong>×</strong>ÕÙÚÐÒØØÓØÔÖ(ω , ω 2 )ÛÒÛÖØ(ω 1Òω 2<br />

τÚÖÐ<strong>×</strong><strong>×</strong> ÏÖØÒØÒ<strong>×</strong>ÐØÐÝÖÒØÓÖÑÐ<strong>×</strong>Ù<strong>×</strong>ØÓØÒÓØÓÒÓÙÒÑÓÙÐÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong><br />

ω 2 ′ = aω 2 + bω 1Òω 2ºÌÒ¸ØÓÚÕÙØÓÒÒØÖÑ<strong>×</strong>ÓØ ÛÖa, b, c, d ∈ Z<strong>×</strong>ÙØØad<br />

ÌØÖÒ<strong>×</strong>ÓÖÑØÓÒ ´ºµ ÒØÑÓÙÐÖÖÓÙÔºÄØτ = 2¸Òτ ´ºµ<br />

ω ω ′<br />

ÔÖÓÔÖØ<strong>×</strong>¸ÙÒÖÙÒÑÓÙÐÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>º ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>Ò<strong>×</strong>ØÙÝÙÒØÓÒ<strong>×</strong>ÛÖÒÚÖÒظÓÖÚ<strong>×</strong>ÔØÖÒ<strong>×</strong>ÓÖÑØÓÒ <strong>×</strong>ÐÐÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒºÁÒ<strong>×</strong>ØÙÝÒÑÓÙÐÖÓÖÑ<strong>×</strong>ÛÛÐÐÓÒÒØÖØÓÒ<strong>×</strong>Ù<br />

f(z)<br />

ÒÒ<strong>×</strong>ÓÒ<strong>×</strong>ØÒØÓÒÚÖÝÓÔÒÓÒÒØ<strong>×</strong>ØÓÒÛØ<strong>×</strong>ÒÐÝغ ÓØ<strong>×</strong>ÑÔÖÓ¸ÒØÖØÓ<strong>×</strong>ÖÐÒÖÖØÓÒиØÒ<strong>×</strong>ÓÛÒØØf<strong>×</strong>ÖØÖÖÐÝ<strong>×</strong>ÑÐÐÔÖÓ<strong>×</strong><br />

2ÖÒØÖÑÙÐØÔÐ<strong>×</strong> ½ÁØÖØÓÓØÔÖÓ<strong>×</strong><strong>×</strong>ÖÐÒÖØÓÒиØÒ<strong>×</strong>ÓÛÒØØÓØω 1Òω ½½<br />

1<br />

1<br />

n<br />

+ nω 2 )<br />

1<br />

1<br />

1 ′<br />

1 ′<br />

1º − bc =<br />

ω 1 ′ = ω′ 1<br />

τ ′ = aτ + b<br />

cτ + d .<br />

= az + b<br />

cz + d<br />

′ ′<br />

1 = cω 2 + dω 1 ,

º¾º¾ ÅÙ<strong>×</strong>ÌÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong> ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

Ì<strong>×</strong>ØÓÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>¸<strong>×</strong>ÒÓÚ¸ÛÐÐÑÔÓÖØÒØØÓÙ<strong>×</strong>ÛÒÛÒ ØØÓÒÓØÑÓÙÐÖÖÓÙÔÓÒØÙÔÔÖÐÔÐÒºËÓÖ¸ÛÚÙ<strong>×</strong>ØÒÛØ ÖÓÙÐÝÔÖÓÙÒØÓÒ<strong>×</strong>¸ÒÓÛØÔÖÓÓØÙÒØÓÒÒØÛÓÖÒØÖØÓÒ<strong>×</strong><br />

ÓÒ<strong>×</strong>ÖÙÒÑÓÙÐÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÛÖÐØÕÙÚÐÒØ<strong>×</strong>Ø<strong>×</strong>ÓÔÖ<strong>×</strong>ÓÖÔÖÓ<strong>×</strong>ºÌ<strong>×</strong> ÖØÖÞØ<strong>×</strong>ØÒØÔÖ<strong>×</strong>Ó<strong>×</strong>ÙÔÖÓ<strong>×</strong>ÛÒØ<strong>×</strong>ÑÐØØÒC¸ÛÑØÓ ÒÖØ<strong>×</strong>ÔÖÐÐÓÖÑÛÒ<strong>×</strong>ÐØØ<strong>×</strong>ÙÒØÓÒ<strong>×</strong>ÓØÔÖÓ<strong>×</strong>ºÁÒ<strong>×</strong>ÒØÓ<br />

ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÓÖÑÖÓÙÔ¸<strong>×</strong>ÛÛÐÐ<strong>×</strong>ÒÓÛ¸ÙØÓÖØØÛÒØÓÜØÒØ<br />

Ø<strong>×</strong>ÔÓÒØ<strong>×</strong><strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ´ººCØÓØÖÛØØÔÓÒØØ∞º˜C<strong>×</strong>Ð<strong>×</strong>ÓÐÐØÊÑÒÒ<strong>×</strong>ÔÖµºÌÓÓ<strong>×</strong>Ó¸Û<br />

Ò ∞ºÏÒØÚÐÙÓfØ ÓÑÒÓÒØÓÒÓØØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ØÓØÜØÒÓÑÔÐÜÔÐÒ˜C ≡<br />

´ºµ ÚØÓÜØÒØÒØÓÒØÓØÔÓÒØ<strong>×</strong>z= − d cÒz=<br />

f( −d)<br />

= ∞ = a c<br />

c , 2ÑØÖÜ 1ºÆÓÛ¸ ÛØØÙ<strong>×</strong>ÙÐÓÒÚÒØÓÒØØz/0 =<br />

´ºµ<br />

ÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒÖÑÒ<strong>×</strong>ÙÒÒÛÑÙÐØÔÐÝÐÐØÓÒØ<strong>×</strong>a, b,<br />

ÝØ<strong>×</strong>ÑÒÓÒÞÖÓÓÒ<strong>×</strong>ØÒغÌÙ<strong>×</strong>¸ÛÐÓ<strong>×</strong>ÒÓÒÖÐØÝÒ<strong>×</strong><strong>×</strong>ÙÑÒad − bc =<br />

ÐØÙ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒ´ºµ¸2 <strong>×</strong><br />

ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>¸Ø<strong>×</strong><strong>×</strong>ÝØÓÚÖݸ<strong>×</strong>ÚÒÝØÑØÖÜÔÖÓÙØÓØÑØÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØ<br />

)ÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓØÒØØÝØÖÒ<strong>×</strong>ÓÖÑØÓÒ<br />

1ºÌÒ¸ØÓÑÔÓ<strong>×</strong>ØÓÒÓØÛÓ<strong>×</strong>Ù<br />

.<br />

Ð<strong>×</strong>Ó<strong>×</strong>ÒÛÚ<strong>×</strong><strong>×</strong>ÙÑad − bc = 1¸ØA ´º½¼µ<br />

ØÓØØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>¸Ò<strong>×</strong>Ð<strong>×</strong>ÓÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒºÌÒØØÝÑØÖÜI =<br />

(<br />

1 0<br />

0 1<br />

f(∞)<br />

∞z≠º<br />

( )<br />

a b<br />

A =<br />

c d<br />

=<br />

f(z) = 1z + 0<br />

0z + 1 .<br />

C ∪ {∞}<br />

c, d<br />

ÁÒÚÖØÒ´ºµ¸Ò<strong>×</strong>ÓÐÚÒÓÖzÒØÖÑ<strong>×</strong>Óf(z)<br />

z = df(z) − b<br />

−cf(z) + a ,<br />

´º½½µ ½½

<strong>×</strong>ÓÛ<strong>×</strong>ØØfÑÔ<strong>×</strong>˜CØÓ˜CºÌÙ<strong>×</strong>¸ØÒÚÖ<strong>×</strong>Óf<strong>×</strong>Ð<strong>×</strong>ÓÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒÒØ ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÒÚÖ<strong>×</strong>ÑØÖÜ ´º½¾µ<br />

1ÓÖÑ<strong>×</strong>ÖÓÙÔºÌ<strong>×</strong><strong>×</strong>ÒÓ<strong>×</strong>ÙÖÔÖ<strong>×</strong><strong>×</strong>ØÑØÖ<strong>×</strong>A<strong>×</strong>ÒÓÚÖÙ<strong>×</strong>ØØ<strong>×</strong>ÙÖÓÙÔ<br />

)<br />

A<br />

ÑÓÙÐÖÖÓÙÔº 1ºÁÒ<strong>×</strong>ØÙÝÒÑÓÙÐÖÓÖÑ<strong>×</strong>Û<strong>×</strong>ØÙÝÒÑÔÓÖØÒØ<strong>×</strong>ÙÖÓÙÔ<br />

dÖØÒØÓÒØÖ<strong>×</strong>ºÁØ<strong>×</strong>ÐÐØ<br />

ÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓf −1 (z)ºÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØØ<strong>×</strong>ØÓÐÐÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÛØad−bc<br />

º¿<br />

ÓSL(2,<br />

ÌÅÓÙÐÖÖÓÙÔÒÙÒÑÒØÐÓÑÒ<br />

Z)ÛØØA ÓØ<strong>×</strong>ÖÓÙÔ¸ÛÖÐÐØÓÒØ<strong>×</strong>a, b, c,<br />

Ì<strong>×</strong>ØÓÐÐÅÙ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÓØÓÖÑ ´º½¿µ<br />

<strong>×</strong>ÐÐØÑÓÙÐÖÖÓÙÔ¸ÒÓØΓ(1)´ÌÖÙÑÒØ1ÓÑ<strong>×</strong>ÐÖÐØÖ¸ÛÒÛ <strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÖÙÒØ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>µºÖÓÑØ<strong>×</strong>ÒØÓÒ<strong>×</strong>ÛÒ<strong>×</strong>ØØØ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>ØØÖÓÙÔ<br />

1¸ÒÑØÖÜAÒØÛØØ<strong>×</strong>ÒØÚ¸−A¸<br />

z ′ = az + b<br />

cz + d ,<br />

ÛØa,<br />

ÜÑÔÐÓÑÓÙÐ<strong>×</strong>Ôº<br />

HÖÑÓÙÐÓÖØ<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÐÐÔØÙÖÚ<strong>×</strong>ÓÚÖCºÁØ<strong>×</strong>Ø<strong>×</strong>ÑÔÐ<strong>×</strong>Ø<br />

Z)¾ºÌÖÓÙÔØ<strong>×</strong>Ø<strong>×</strong>ÒÑÖÓÑØØØØØÔÓÒØ<strong>×</strong>ÓØÕÙÓØÒØ<strong>×</strong>Ô<br />

b, c, dÒØÖ<strong>×</strong>¸Òad bc =<br />

Z)ÓÒ˜C<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÄØ<br />

PSL(2, ˜CÒÝÔÓÒØÒ˜Cº<br />

Γ(1) \<br />

ÄØHØÙÔÔÖÐÔÐÒºÏÙÒÖ<strong>×</strong>ØÒØØÓÒÓSL(2,<br />

∈ SL(2, Z)ÒÝÐÑÒØÓSL(2, Z)ÒÐØz ∈<br />

ÏÖÖÔÖ<strong>×</strong>ÒØÒØØÖÒ<strong>×</strong>ÓÖÑØÓÒÝØÑØÖÜ<strong>×</strong><strong>×</strong>ÓØÛØغÏÒÓØØ ´º½µ ÌÒ¸ØØÓÒÓgÓÒz<strong>×</strong>ÚÒÝg<br />

)¸ÖÓÑØ<strong>×</strong>ÒØØ<strong>×</strong>ØÖÚÐÐÝÓÒHº Z)ØÑÓÙÐÖÖÓÙÔºÏÖÑÓÒÓÙØØÒØÖÓØÖÓÙÔ¸<br />

·<br />

¾ËÓÑÓÓ<strong>×</strong>ÐÐØÖÓÙÔSL(2,<br />

½½<br />

g ≡<br />

(<br />

(<br />

1 0<br />

±<br />

0 1<br />

(<br />

−1 d −b<br />

=<br />

−c a<br />

=<br />

=<br />

a b<br />

c d<br />

−<br />

)<br />

z ≡ az + b<br />

cz + d .

Z)ÓÒ˜Cº<br />

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´º½µ ÓÐÐÓÛÒÓÙØØØÓÒÓSL(2,<br />

ØØÛÓÐÑÒØ<strong>×</strong>SÒTÚÒÝ z<strong>×</strong>ÖØÖØÒÞÖÓ¸ØÑÒÖÝÔÖØÓz<strong>×</strong>º<br />

Z)ºΓ(1)<strong>×</strong>ÒÖØÝ <strong>×</strong>ÓÛÒØØØÑÒÖÝÔÖØÓg ·<br />

ËÓ¸g.z ∈ Hz∈ HºÌÙ<strong>×</strong>¸H<strong>×</strong><strong>×</strong>ØÐÙÒÖØØÓÒÓSL(2,<br />

ÛØØÓÐÐÓÛÒÖÐØÓÒ<strong>×</strong>ØÛÒØÑ ´º½µ<br />

)<br />

Γ(1)<strong>×</strong>ÒÖØ<strong>×</strong>ØÖÔÖÓÙØÓØÝÐÖÓÙÔÓÓÖÖ2ÒÖØÝSÒØ ´º½µ<br />

ÝÐÖÓÙÔÓÓÖÖ3ÒÖØÝSTº ÌØÓÒÓØÒÖØÓÖ<strong>×</strong>ÓÒÒÝz∈H<strong>×</strong>ÚÒÝ<br />

ÓÖ<strong>×</strong>ÓÑA∈Γ(1)ºËÒΓ(1)<strong>×</strong>ÖÓÙÔ¸Ø<strong>×</strong><strong>×</strong>ÒÕÙÚÐÒÖÐØÓÒºÌ<strong>×</strong>ÕÙÚÐÒ<br />

ÐÔÐÒºÌ<strong>×</strong><strong>×</strong>ØØÙÒÓÒÓØÖÔÖ<strong>×</strong>ÒØØÚÔÓÒØ<strong>×</strong>ÓÓÖØ<strong>×</strong>ÐÐØ ÖÐØÓÒÚ<strong>×</strong>ØÙÔÔÖÐÔÐÒÒØÓ<strong>×</strong>ÓÒØÓÖØ<strong>×</strong>ÓØÖÓÙÔØÓÒÒØ<strong>×</strong>Ù<strong>×</strong><br />

ÙÒÑÒØÐ<strong>×</strong>ØÓΓ(1)ºÓÖ<strong>×</strong>Ø<strong>×</strong>ØØÚØÓÔÓÐÓÐ<strong>×</strong>ØÖÙØÙÖ¸ØÓÑ<strong>×</strong>ÚÒÒÖ ØÓÓÒ<strong>×</strong>ÖÓÒÔÓÒØÖÓÑÓÖØØÓÒÓÛØØÓÒÓØÛÓÐÖÓÙÔÓÒØÙÔÔÖ<br />

ÌÛÓÔÓÒØ<strong>×</strong>zÒz ′ÒØÙÔÔÖÐÔÐÒÖ<strong>×</strong>ØÓÕÙÚÐÒØÙÒÖΓ(1)z′ = Az<br />

ØÓÓÒ<strong>×</strong>ÖØØÓÔÓÐÓÐÔÖÓÔÖØ<strong>×</strong>ØÓ<strong>×</strong>ØÙÝØÖÓÙÔÒÖØÒغÌ<strong>×</strong><strong>×</strong>ØÒÓØÓÒ ÓÙÒÑÒØÐÓÑÒºÌ<strong>×</strong><strong>×</strong>ÒÔÒÛØÓÙÖÖÐÖÒØÓÒÓØÒÓØÓÒ<br />

<strong>×</strong>ØØÓØÖÛØ<strong>×</strong>ØÓÛØÓÒÔÓÒØ<strong>×</strong>´ÓÑ<strong>×</strong>ÙÖÞÖÓµº <strong>×</strong>ÔÝÓÑÓÑÓÖÔ<strong>×</strong>Ñ<strong>×</strong>ºÌÝÔÐÐݸØÙÒÑÒØÐÓÑÒÐÛÝ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÓÒÓÔÒ ÛÖÛ<strong>×</strong>ØÔØÙÖ<strong>×</strong>Ø<strong>×</strong>ÝÑÑØÖÝ<strong>×</strong>ØÖÙØÙÖÓØØÓÒÓÖÓÙÔÓÒØÓÔÓÐÓÐ<br />

ÔÖÓÔÖØ<strong>×</strong> ½ºÆÓØÛÓÔÓÒØ<strong>×</strong>ÓØÙÒÑÒØÐÓÑÒÖÕÙÚÐÒØÙÒÖØÖÓÙÔØÓÒº ÓÖÒÓÔÒ<strong>×</strong>ØØÓØÙÒÑÒØÐÖÓÒÓÖÓÙÔØ<strong>×</strong>ØÓÚØØÛÓÓÐÐÓÛÒ<br />

½¾¼<br />

S ≡<br />

(<br />

ÁÑ(g · z) =ÁÑ(z)<br />

|cz + d| 2,<br />

0 −1<br />

1 0<br />

)<br />

; T ≡<br />

(<br />

S 2 = (ST) 3 = ±1.<br />

1 1<br />

0 1<br />

S · z = − 1 z ; T · z = z + 1. ´º½µ

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÌÝÔÐÐÝÛÖÕÙÖØØÓÓÒÒØÛØ<strong>×</strong>ÓÑÖ<strong>×</strong>ØÖØÓÒÓÒØ<strong>×</strong>ÓÙÒÖݺÀÓÛÚÖ¸Ø ÒÒÓØÒ<strong>×</strong><strong>×</strong>ÖÐÝÓÒÒغ<br />

′<strong>×</strong>ÕÙÚÐÒØØÓzÙÒÖØÖÓÙÔØÓÒº<br />

′ÒØÐÓ<strong>×</strong>ÙÖÓØÙÒÑÒØÐÓÑÒ<strong>×</strong>ÙØØ ¾ºÓÖÒÝz∈HØÖ<strong>×</strong>ÔÓÒØz ÌÙÒÑÒØÐÓÑÒÓÖØØÓÒÓØÑÓÙÐÖÖÓÙÔÓÒH¸ÒÓØD¸<strong>×</strong><br />

z<br />

H<strong>×</strong>ÙØØ<br />

ÛÖØÖ<strong>×</strong>ØÔÖØ<strong>×</strong>ØÓÔÒ<strong>×</strong>ظÒØØÛÓÓØÖØÖÑ<strong>×</strong>ÖØÓÙÒÖ<strong>×</strong>ÓÒÓÒØ ´º½µ ÚÒÝÐÐz∈<br />

ÛÓ<strong>×</strong>ØÓÒÐÚ<strong>×</strong>ÚÒÐÑÒØÓDÒÚÖÒغÓÖÐÐØÔÓÒØ<strong>×</strong>ÒD¸ÜÔØØÓÚ<br />

{|Ê(z)| < 1, |z| > 1} ∪ {|z| ≥ 1,Ê(z) = − 1 2<br />

2 } ∪ {|z| = 1, −1

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÒÖÐÐݸÓÛÚÖ¸ØØÖÑÑÓÙÐÖÙÒØÓÒ<strong>×</strong>Ù<strong>×</strong>ÓÒÐÝÓÖÑÖÓÑÓÖÔÑÓÙÐÖ<br />

Z¸ÙÒØÓÒf<strong>×</strong><strong>×</strong>ØÓÛÐÝ<br />

∗º ºÙÒØÓÒÓÒÐØØ<strong>×</strong>ÒC<strong>×</strong>Ø<strong>×</strong>ÝÒF(λL) = F(L)ÓÖÐÐÐØØ<strong>×</strong>LÒÐÐλ ∈ C<br />

Γ(1)Ò<br />

ÙÒØÓÒ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÝÒÖØÒÖÓÛØÔÖÓÔÖØ<strong>×</strong>ºÓÖk∈<br />

ÞÖÓºÌÝÖÒÚÖÒØÙÒÖØØÓÒÓØÑÓÙÐÖÖÓÙÔºÌÔÖÓÙØÓØÛÓÛÐÝ ÖÓÑØÓÚÒØÓÒÛ<strong>×</strong>ØØØÓÒ<strong>×</strong>ØÒØÙÒØÓÒ<strong>×</strong>ÖÑÓÙÐÖÙÒØÓÒ<strong>×</strong>ÓÛØ ´º¾¼µ ÑÓÙÐÖÓÛØk¸f<strong>×</strong>ÑÖÓÑÓÖÔÙÒØÓÒÓÒH<strong>×</strong>ÙØØÓÖÐÐg∈<br />

ÖÒÓÑÓÙÐÖÙÒØÓÒ<strong>×</strong>ÓÓÛغÓÖÚÒk¸ØÓÚÕÙØÓÒ<strong>×</strong><strong>×</strong>Ñ<strong>×</strong> 2ºÌÖ ÑÓÙÐÖÙÒØÓÒ<strong>×</strong>ÓÛØ<strong>×</strong>k 1Òk 2<strong>×</strong>ÛÐÝÑÓÙÐÖÙÒØÓÒÓÛØk 1<br />

´º¾½µ<br />

+k<br />

ÑÓÙÐÖÙÒØÓÒ¸ÛÙ<strong>×</strong>ØÒØØÖÒ<strong>×</strong>ÓÖÑØÓÒÔÖÓÔÖØ<strong>×</strong>ÓØÑÖÓÑÓÖÔÙÒØÓÒ ËÒÛÒÓÛØØΓ(1)<strong>×</strong>ÒÖØÝSÒT¸ØÓÒÓÛØØÖÒ<strong>×</strong>ÓÖÑØÓÒÓÛÐÝ<br />

fÙÒÖSÒTºÑÖÓÑÓÖÔÙÒØÓÒfÓÒH<strong>×</strong><strong>×</strong>ØÓÛÐÝÑÓÙÐÖÓÛØ<br />

kØØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong>ÙÒÖSÒTÒØÓÐÐÓÛÒÛÝ<br />

k/2¸<strong>×</strong>ÒÚÖÒØÙÒÖØØÓÒÓΓ(1)º ÁÒÛÓÖ<strong>×</strong>¸ØÖÒØÐÓÖÑÓÛظf(z)(dz)<br />

´º¾¾µ<br />

ºº½ Õ¹ÜÔÒ<strong>×</strong>ÓÒ ´º¾¿µ<br />

f(− 1) = z zk f(z)<br />

′´ºº<br />

H/T¸ØØ<strong>×</strong>ØÕÙÓØÒØ<strong>×</strong>ÔÓHÑÓÙÐÓØÖÒ<strong>×</strong>ÐØÓÒÝÒØÖ<strong>×</strong>´ÝÐÒÖµºqÒÙ<strong>×</strong><br />

1ÛØØÓÖÒÖÑÓÚµºÌÙ<strong>×</strong>¸ÛÒÓÙÖÖÜÔÒf(z)<strong>×</strong><br />

Ò<strong>×</strong>ÓÑÓÖÔ<strong>×</strong>ÑØÛÒH/TÒØÔÙÒØÙÖ<strong>×</strong>ºÌÙ<strong>×</strong>¸ÑÖÓÑÓÖÔÙÒØÓÒÓÒ<br />

f(z)¸ÓÒ<strong>×</strong>ÖØ<strong>×</strong>Ô ÌÑÔz↦→ e2πizÒ<strong>×</strong>ÓÐÓÑÓÖÔÑÔÖÓÑHØÓØÔÙÒØÙÖÙÒØ<strong>×</strong>D HÛ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÓÒØÓÒ´º¾¿µÓÚ´ÒÚÖÒÙÒÖTµ¸ÒÙ<strong>×</strong>ÑÖÓÑÓÖÔ<br />

ÓÔÒÙÒØ<strong>×</strong>|q| <<br />

ÙÒØÓÒÓq(z) =<br />

∞ÜØÒ<strong>×</strong>ØÓ0¸Û<strong>×</strong>ÝØØf<strong>×</strong>ÑÖÓÑÓÖÔ´ÓÐÓÑÓÖÔµØÒÒØݺ<br />

−N a nq nºÌÒ¸<strong>×</strong>Òf(z+1) ∞<strong>×</strong>Ð<strong>×</strong>ÓÑÖÓÑÓÖÔØ0<strong>×</strong>ØØØÖÜ<strong>×</strong>Ø<strong>×</strong> f(z)ºÁØÑÖÓÑÓÖÔØÝ ÙÒØÓÒ¸f ∞¸ÓÒØÔÙÒØÙÖ<strong>×</strong><strong>×</strong>ÙØØf ∞ (q(z)) =<br />

´ÓÐÓÑÓÖÔØݵÓf Ò<strong>×</strong><strong>×</strong>ÖÝÒ<strong>×</strong>ÙÒØÓÒØÓÒØØf ½¾¾<br />

z ∈ H<br />

e 2πiz<strong>×</strong>f(z) = ∑ ∞<br />

f(z) = (cz + d) −k f(g · z) .<br />

f(g · z)(d(g · z)) k/2 = f(z)(dz) k/2 .<br />

f(z + 1) = f(z) .<br />

=

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÜÔÒ<strong>×</strong>ÓÒÒØÒÓÖÓÓÓØÓÖÒº<br />

∞ØÒÑØ<strong>×</strong>ÄÙÖÒØ <strong>×</strong>ÓÑÔÓ<strong>×</strong>ØÚÒØÖN<strong>×</strong>ÙØØf ∞<br />

´º¾µ<br />

(q)q N<strong>×</strong>ÓÙÒÒÖ0ºf nÖØÓÙÖÖ<br />

<strong>×</strong>ÑÖÓÑÓÖÔØ∞º ÓÒØ<strong>×</strong>Ófº ÒØÓÒºÛÐÝÑÓÙÐÖÙÒØÓÒÓÛØk<strong>×</strong>ÐÐÑÓÙÐÖÙÒØÓÒØ ÌÓÚ<strong>×</strong>ÐÐØÕ¹ÜÔÒ<strong>×</strong>ÓÒÓfÓÙØ∞ºÌÓÒØ<strong>×</strong>a ÐÐÑÓÙÐÖÓÖÑÓÛØk´ÒÐÚнµº ÒØÓÒºÑÓÙÐÖÙÒØÓÒÛ<strong>×</strong>ÓÐÓÑÓÖÔÚÖÝÛÖÓÒHÒØ∞<strong>×</strong><br />

(0)ºÌ<strong>×</strong><strong>×</strong>ØÚÐÙÓfØ∞º<br />

<strong>×</strong>Ö<strong>×</strong><br />

Áf<strong>×</strong>ÓÐÓÑÓÖÔØ∞¸Û<strong>×</strong>Øf(∞) = f ∞<br />

H¸f<strong>×</strong>ÚÒÝ Áf<strong>×</strong>ÑÓÙÐÖÓÖѸØÒØÖÖÒÙÑÖ<strong>×</strong>a n<strong>×</strong>ÙØØÓÖÐÐz∈ ´º¾µ<br />

ØØÓÒÓØÑÓÙÐÖÖÓÙÔÓÒfº HµºÑÓÙÐÖÓÖÑÓÛØk<strong>×</strong>ÐÐÙ<strong>×</strong>ÔÓÖÑ<br />

0ºÏÛÐÐÙ<strong>×</strong>ØÓÐÐÓÛÒÒÓØØÓÒÓÖ<br />

f(z)<br />

ÛÓÒÚÖ<strong>×</strong>ÓÖ|q| < 1´ººz ÓÛØk´ÒÐÚнµf(∞) = 0¸ºº¸a =<br />

º ´º¾µ<br />

<strong>×</strong>ØÒÑ<strong>×</strong>Ù<strong>×</strong>Ø<strong>×</strong>¸ÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>¸ÓÑØÖÜÖÓÙÔ¸Ö<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>ÒÝ ÓÒÖÙÒËÙÖÓÙÔ<strong>×</strong><br />

f[α] k = f(αz)(cz + d) −k (Øα)<br />

ÓÒÖÙÒÓÒØÓÒÓÒØÒØÖ<strong>×</strong>ÓØÑØÖܺÌÑØÖÜÖÓÙÔÛÖÒØÖ<strong>×</strong>ØÒ<strong>×</strong><br />

Z)¸ÛÒÖ<strong>×</strong>ØÖØØÒØÖ<strong>×</strong>ØÓÒZ/NZ¸ÓØÒÒØÓÑÓÑÓÖÔ<strong>×</strong>Ñ<br />

Z)Ö<strong>×</strong>ÒØÓÐÐÓÛÒÛݺÚÒØ<br />

PSL(2,<br />

ØÛÒØØÛÓÖÓÙÔ<strong>×</strong>ºÌÖÒдººØÒÚÖ<strong>×</strong>ÑÓØÒØØÝeµÓØ<strong>×</strong>ÑÔ<strong>×</strong> ´º¾µ<br />

Z)ºÌÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>ÓPSL(2,<br />

ÖÓÙÔPSL(2,<br />

ÒÜÑÔÐÓÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔÒ<strong>×</strong>ÐÐØÔÖÒÔÐÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<br />

0´ÑÓNµ´ÆÓÛÛ<strong>×</strong>ÛÖØ ÓÐÚÐÒ¸Γ(N)ºÁØ<strong>×</strong>ÚÒÝa≡d≡±1, b ≡ c ≡<br />

½¾¿<br />

∈<br />

f ∞ (q) =<br />

=<br />

0<br />

∞∑<br />

a n q n .<br />

−N<br />

∞∑<br />

a n q n<br />

n=0<br />

k/2<br />

PSL(2, Z) → PSL(2; Z/NZ)<br />

.

1ÒΓ(1)ÓÖØÙÐÐÑÓÙÐÖÖÓÙÔÓÑ<strong>×</strong>ÖÓѵºΓ(N)<strong>×</strong>¸ÒظÒÓÖÑÐ<strong>×</strong>ÙÖÓÙÔÓ ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

Ú<strong>×</strong>ÓØÖÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>ºÌÓÒ<strong>×</strong>ÛÛÐÐ<strong>×</strong>ØÙÝÒÒÖÐØÓÒØÓÓÙÒØÒÓ ÑÓÙÐÓNºÏÒØØÒÚÖ<strong>×</strong>ÑÓÒÝ<strong>×</strong>ÙÖÓÙÔ´ÒÓØÙ<strong>×</strong>ØØÒØØÝeµÒØØ<br />

Γ(N)ºÌÒÜÓΓ(N)ÒΓ(1)<strong>×</strong>ØÒÙÑÖÓÕÙÚÐÒÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÑØÖ<strong>×</strong><br />

Γ(1)¸<strong>×</strong>Ò<strong>×</strong>ÐÝÚÖÝ<strong>×</strong>ÒØØA−1 BA ∈ Γ(1)ÓÖÒÝØÛÓÑØÖ<strong>×</strong>A ÒB∈<br />

ÈË<strong>×</strong>ØØ<strong>×</strong>ÖØÓÐÐÓÛÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>ÓΓ(1)<br />

´º¾µ<br />

}<br />

ÛÖÑÒ<strong>×</strong>ÒÝÐÑÒغÌÒÙÑÖN<strong>×</strong>ÐÐØÐÚÐÓΓºÏÒÒÑÓÙÐÖ ´º¿¼µ<br />

(ÑÓN) ,<br />

ÙÒØÓÒ<strong>×</strong>ÓÖØÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<strong>×</strong>Ù<strong>×</strong>Ø<strong>×</strong>ÒØ<strong>×</strong>ÓØÙÐÐÑÓÙÐÖÖÓÙÔº<br />

(ÑÓN) ,<br />

ÓÐÐÐØØ<strong>×</strong>ÒCºÄØØ<strong>×</strong>ÒCÖÐÓ<strong>×</strong>ÐÝÖÐØØÓÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÓÚºÏÛÐÐ º ÏÖÐÖ<strong>×</strong>ÓÒÓØÛÝ<strong>×</strong>ÒÛØÑÓÙÐÖÖÓÙÔÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÝÓÒ<strong>×</strong>ÖÒØ<strong>×</strong>Ø ÄØØ<strong>×</strong><br />

ÓCºÅÓ<strong>×</strong>ØÓØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÛÐÐÒÓØØÓÓÓÖÑиÙØÛÚØÓÖÑÐÒØÓÒ<strong>×</strong>Ó Ý<strong>×</strong>ÓÖØ<strong>×</strong>ÓÓÑÔÐØÒ<strong>×</strong><strong>×</strong>ºÏÖ<strong>×</strong>ØÒÛØÛÑÒÝÐØØÒØÖÐ <strong>×</strong>ØظÙÔØÓÖØÒØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>¸ØÕÙÓØÒØH/Γ(1)ÒÒØÛØÐØØ<br />

ÚØÓÖ<strong>×</strong>ÔVºÌÖÖ<strong>×</strong>ÚÖÐÛÝ<strong>×</strong>ÓÒÒÐØØÒÚØÓÖ<strong>×</strong>Ô¸ÛÚÓÒ ØØ<strong>×</strong><strong>×</strong><strong>×</strong>ØØÓÙÒÖ<strong>×</strong>ØÒÐÓÛº <strong>×</strong>ÙØØV/L<strong>×</strong>ÓÑÔغËÑÐÖÐݸÓÒÒÒÐØØÒÓÑÔÐÜÚØÓÖ<strong>×</strong>Ôº<br />

2<strong>×</strong>ÙØØ ÐØØÒÖÐÚØÓÖ<strong>×</strong>ÔVÓÒØÑÒ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ÖØ<strong>×</strong>ÙÖÓÙÔ¸L¸ÓV ËÔÐÐݸÓÒ<strong>×</strong>ÖCºÚÒØÛÓÒÓÒ¹ÚÒ<strong>×</strong>ÒÓÑÔÐÜÒÙÑÖ<strong>×</strong>ω 1Òω )<strong>×</strong><br />

ω 1 /ω 2 /∈ R¸ÛÒ<strong>×</strong><strong>×</strong>ÓØÐØظL(ω , ω 2 )¸ØÓω Ï<strong>×</strong><strong>×</strong>ÙÑÁÑ(ω 2 /ω 1 ) > 0º{ω 1 , ω 2 }<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ÓLºÄØMÒÓØØ<strong>×</strong>ØÓÔÖ<strong>×</strong>(ω ÓÐÑÒØ<strong>×</strong>ÓC∗¸ÒÐØLØ<strong>×</strong>ØÓÐÐÐØØ<strong>×</strong>ÓCºÌÑÒÓÐC/L(ω 1 , ω 2<br />

½¾<br />

{ ( ) (<br />

a b<br />

Γ 1 (N) =<br />

∈ SL(2, Z) :<br />

c d<br />

{ ( ) ( ) ( )<br />

a b<br />

a b ∗ ∗<br />

Γ 0 (N) =<br />

∈ SL(2, Z) : ≡<br />

c d<br />

c d 0 ∗<br />

{ ( ) ( ) ( )<br />

Γ 0 a b<br />

a b ∗ 0<br />

(N) =<br />

∈ SL(2, Z) : ≡<br />

c d<br />

c d ∗ ∗<br />

∈ Γ(1)<br />

´º¾µ<br />

) ( )<br />

a b 1 ∗<br />

≡ (ÑÓN)}<br />

c d 0 1<br />

1 1Òω 2ÝL(ω<br />

1<br />

}<br />

}º<br />

1 , ω 2 ) ≡ {Zω 1 + Zω 2<br />

, ω 2 )

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong> nÓÖ<strong>×</strong>ÓÑ<br />

ÓÒØÓÒÓÖØÛÓÐÑÒØ<strong>×</strong>ÓMØÓÓÖÖ<strong>×</strong>ÔÓÒØÓØ<strong>×</strong>ÑÐØØÒLØÙÖÒ<strong>×</strong>ÓÙØØØØÝ<br />

2}ÓMÓÖÖ<strong>×</strong>ÔÓÒØÓØ<strong>×</strong>ÑÐØØÒLÌÒ<strong>×</strong><strong>×</strong>ÖÝÒ<strong>×</strong>ÙÒØ<br />

)¸ÛÛÓÙÐÐØÓ<strong>×</strong>ÛÒÓØÛÓ<strong>×</strong>ÙÔÖ<strong>×</strong> ÓØÒÝÒØÝÒØÔÓÒØ<strong>×</strong>z Z)ºÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØÛÒÒØÝØ<strong>×</strong>ØLÓÐØØ<strong>×</strong><br />

1 , z 2 ∈ C<strong>×</strong>ÙØØz 1 − z 2 = ω 1 m + ω 2<br />

ÆÓÛ¸ÚÒM¸Ø<strong>×</strong>ØÓÐÐÔÖ<strong>×</strong>(ω 1 , ω 2<br />

{ω 1 , ω 2 }Ò{ω<br />

)ÓM´Ö<strong>×</strong>ÔØÚÐÝLµ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

1, ω ∗ÓÒÒÝ<br />

′<br />

<strong>×</strong>ÓÙÐÓÒÖÙÒØÑÓÐÙÐÓSL(2,<br />

´º¿½µ<br />

ÓCÛØØÕÙÓØÒØÓMÝØØÓÒÓSL(2, Ð<strong>×</strong>Ó¸<strong>×</strong>ÒØ<strong>×</strong>ÓÒÐÝØÖØÓØØØÖÑÒ<strong>×</strong>ØÐØظÛÒØÝC ÐÑÒØ(ω 1 , ω<br />

ÓΓ(1)ÓÒHºÏÑØ<strong>×</strong>ÔÖÐÓÛ¸ÛÖÛÜÔÐÒÛØÛ<strong>×</strong>ÒØ<br />

2<br />

(ω 1 , ω 2 ) ↦→ (λω 1 , λω 2<br />

Z)ÓÒMÒØÓØØ<br />

), (Ö<strong>×</strong>ÔºL λL), λ ∈ C ∗ ,<br />

ÛØÓÙØÒÒØÖØÓºÌÙ<strong>×</strong>¸ÛÒÒØÝØÕÙÓØÒØM/C<br />

ÓΓ(1)\HÛØÐØØÓCÙÔØÓÓÑÓØØÝ´ÐØÓÒµº ÒÒÒÓÙØØÒØØÓÒÓÐØØÓCÛØØÕÙÓØÒØH/Γ(1)ºÌÑÔ<br />

∗ÛØHÝ(ω 1 , ω 2 ) ↦→<br />

z = ω 1 /ω 2¸ÒØÙ<strong>×</strong>¸Ø<strong>×</strong>ÒØØÓÒØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong>ØØÓÒÓSL(2,<br />

Z¸Û<strong>×</strong>ÝØØÓÑÔÐÜÚÐÙÙÒØÓÒ¸F¸ÓÒL<strong>×</strong>ÓÛØk<br />

∗ÓÒØÓΓ(1)\HºÌÙ<strong>×</strong>¸ÛÒÒØÝÒÐÑÒØ<br />

(ω 1 , ω 2 ) ↦→ ω 1 /ω 2Ú<strong>×</strong>ØÓÒÓL/C<br />

´º¿¾µ ÓÖk∈<br />

)ºÌÒØÓÚÓÖÑÙÐ<strong>×</strong>Ù<strong>×</strong>Ø<br />

)ØÚÐÙÓFÓÒØÐØØ<br />

F(λL) = λ −k F(L)<br />

ÓÖÐÐÐØØ<strong>×</strong>L∈LÒÐÐλ ∈ C ∗ºÄØÙ<strong>×</strong>ÒÓØÝF(ω , ω 2<br />

´º¿¿µ<br />

L(ω<br />

ÌÙ<strong>×</strong>¸ÛÒÐÛÝ<strong>×</strong>ÒÙÒØÓÒ¸f¸ÓÒH<strong>×</strong>ÙØØ<br />

1 , ω 2<br />

ÏÒÓØÒØÓÚÓÖÑÙÐØØØÔÖÓÙØω<br />

Z)ØÓÒÓÒM¸Û<strong>×</strong>ØØf<strong>×</strong>Ø<strong>×</strong><strong>×</strong>Ø<br />

2 −k F(ω 1 , ω 2<br />

´º¿µ<br />

)ÔÒ<strong>×</strong>ÓÒÐÝÓÒz 2 f(ω 1/ω 2 )<br />

Ð<strong>×</strong>Ó¸<strong>×</strong>ÒF(ω 1 , ω 2 )<strong>×</strong>ÒÚÖÒØÙÒÖÒSL(2, ½¾<br />

m, n ∈ Zº<br />

′<br />

↦→<br />

1<br />

F(λω 1 , λω 2 ) = λ −k F(ω 1 , ω 2 ) .<br />

F(ω 1 , ω 2 ) = ω −k<br />

2º = ω 1 /ω

ÓÐÐÓÛÒÒØØÝ ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´º¿µ ÓÖÐÐ(<br />

ÒÐØØÙÒØÓÒ<strong>×</strong>ÓÛغ Û<strong>×</strong>ÓÛØkºÏØÙ<strong>×</strong>ØÓÖÖ<strong>×</strong>ÔÓÒÒØÛÒÑÓÙÐÖÙÒØÓÒ<strong>×</strong>ÓÛØ<br />

º<br />

ÓÒÚÖ<strong>×</strong>ÐݸfÚÖ<strong>×</strong>ØÓÚÓÖÑÙиØÒÛÒ<strong>×</strong><strong>×</strong>ÓØ<strong>×</strong>ØØÓÙÒØÓÒFÓÒL<br />

ÏÖÒÓÛÖÝØÓ<strong>×</strong><strong>×</strong>ÓÑÜÑÔÐ<strong>×</strong>ÓÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÐÐØØÓÖÝÛÐÖÒØÒ ÔÙØØÓÙ<strong>×</strong>ºÏÛÐÐÐÖÒ<strong>×</strong>ÙÜÑÔÐ<strong>×</strong>ØØÛÛÐÐÚÓ<strong>×</strong>ÓÒØÓÙ<strong>×</strong>ÐØÖÒ<strong>×</strong>ØÙÝÒ ÜÑÔÐ<strong>×</strong>ÓÅÓÙÐÖÓÖÑ<strong>×</strong><br />

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K3ÑÒÓк<br />

¿ºÂÓÓÖÑ<strong>×</strong>¹ØØØÙÒØÓÒ<strong>×</strong>¸ÒØÓÙÖÖÓÒØ<strong>×</strong>ÓØËÐÑÓÙÐÖ <strong>×</strong>ØÖÒØÓÖÝØØÛÛÐÐÓÒ<strong>×</strong>Ö¸ÒÖØØÖØÓØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>º<br />

ÙÒØÓÒ¸Ø∆ÙÒØÓÒ´ÛÓÙÖÖ<strong>×</strong>ØÒÖØÒÙÒØÓÒÓØÑÙÐØÔÐØ<strong>×</strong>Ó ÓÖÑ<strong>×</strong>ÓÒ<strong>×</strong>ÖÓÚº<br />

ØÖÓÓØ<strong>×</strong>ÓØÑÓÒ<strong>×</strong>ØÖÐÖ´¿º½¾µµ¸ÏÖ<strong>×</strong>ØÖ<strong>×</strong><strong>×</strong>℘ÙÒØÓÒ¸ÒÑÒÝÑÓÖ¸ ÙØÛÛÐÐÒÓØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ØÑÖºÌÓÚØÖÜÑÔÐ<strong>×</strong>ÖÒÓØÓÒÐÝÚÖÝÑÔÓÖØÒØ ÜÑÔÐ<strong>×</strong>ÓÑÓÙÐÖÓÖÑ<strong>×</strong>¸ÙØØÝÐ<strong>×</strong>ÓÔÐÝÚÖÝÑÔÓÖØÒØÖÓÐÒØÓÒ<strong>×</strong>ØÖÙØÓÒÓ ØÝÓÒÒÖÝÔÖØØÓÒÙÒØÓÒºÇØØÖ¸ÛÛÐÐ<strong>×</strong>ÔÒÓÒ<strong>×</strong>ÖÐØÑÓÒ<br />

ÌÖÖÑÒÝÑÓÖÑÔÓÖØÒØÒÐÐÙ<strong>×</strong>ØÖØÚÜÑÔÐ<strong>×</strong>ÓÑÓÙÐÖÓÖÑ<strong>×</strong>ÐØJ<br />

ËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÚÒØÖÑÔÓÖØÒÖÓÑØÔÓÒØÓÚÛÓØ<strong>×</strong>ÛÓÖºÏ<strong>×</strong>ØÖØ ÛØØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>º<br />

½¾<br />

f(z) = (cz + d) −k f( az + b<br />

cz + d )<br />

a b<br />

c d<br />

)<br />

∈ SL(2, Z)

ºº½ <strong>×</strong>Ò<strong>×</strong>ØÒËÖ<strong>×</strong> ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÛÖØ∑ÖÙÒ<strong>×</strong>ÓÚÖØÒÓÒÞÖÓÐÑÒØ<strong>×</strong>ÓΓºÌÙ<strong>×</strong>¸Ø<strong>×</strong>Ö<strong>×</strong><br />

2¸<br />

´º¿µ<br />

ÄØLÐØØÒCºÓÒ<strong>×</strong>ÖØ<strong>×</strong>Ö<strong>×</strong>∑γ∈L 1/|γ|σºÌ<strong>×</strong><strong>×</strong>Ö<strong>×</strong><strong>×</strong>ÓÒÚÖÒØÓÖσ><br />

G 2k<strong>×</strong>ÙÒØÓÒÓÒMÛØ<br />

1ºÁØ<strong>×</strong>ÐÐØ´ÒÓÒ¹ÒÓÖÑÐÞµ<strong>×</strong>Ò¹<br />

(L) = ∑ 1/γ γ∈Γ<br />

´º¿µ<br />

ÛÐÐ<strong>×</strong>ÓÐÙØÐÝÓÒÚÖÒØÓÖÒÝÒØÖk ><br />

<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÓÒÜ2kºÏÖØÒG ÔÖÑÒØ<strong>×</strong>ÙÔÖ<strong>×</strong>ÖÔغÖÓÑØÔÖÒ<strong>×</strong>ØÓÒ¸ØÙÒØÓÒÓÒHÓÖÖ<strong>×</strong>ÔÓÒÒØÓ<br />

)<strong>×</strong>ÚÒÝ 0)ÛÛÒØÝ<br />

´º¿µ<br />

ÛÖØ<strong>×</strong>ÙÑÑØÓÒ<strong>×</strong>ÓÚÖÐÐÔÖ<strong>×</strong>ÓÒØÖ<strong>×</strong>(m, n) ≠ (0,<br />

G 2k (ω 1 , ω 2<br />

1)<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> (z)ºÍÒÖTØÖÒ<strong>×</strong>ÓÖÑØÓÒ¸ 0)ºÄØÙ<strong>×</strong> ÛÖÒØ<strong>×</strong>ÙÑÑØÓÒ<strong>×</strong>ÓÚÖÔÖ<strong>×</strong>ÓÒØÖ<strong>×</strong>m, n<strong>×</strong>ÙØØ(m, n) ≠ (0,<br />

<strong>×</strong>ÓÛØTÒSØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ØÓÒØÓÖÑG k<br />

z ↦→ z + 1¸ØÖÓÖG (z) ↦→ G 2k<br />

´º¿µ<br />

(z + )<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ÍÒÖÒSØÖÒ<strong>×</strong>ÓÖÑØÓÒz↦→ − 1 z¸ØÙ<strong>×</strong>G (z) ↦→ G 2k<br />

´º¼µ<br />

(− 1 z<br />

ÇØÒ¸Ø<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong><strong>×</strong>ÖÒ¸<strong>×</strong>ÓØØØÓÒ<strong>×</strong>ØÒØØÖÑ<strong>×</strong>1¸ÝÚÒØÝ 2ζ(2k)¸ÛÖζ<strong>×</strong>ØÊÑÒÒÞØÙÒØÓÒº<br />

)µ<strong>×</strong>ÑÓÙÐÖÓÖÑÓÛØ ÌÙ<strong>×</strong>¸Û<strong>×</strong>ØØG 2k (z)´ÒÒ¸G (L)ÒG (ω 1 , ω 2<br />

2kÛØØÚÐÙØ∞ÚÒÝG 2k (∞) =<br />

½¾<br />

G 2k (z + 1) =<br />

G<br />

2k<br />

′∑<br />

m,n<br />

G 2k (ω 1 , ω 2 ) =<br />

′∑ 1<br />

(mω 1 + nω 2 ) 2k<br />

m,n<br />

2k (z) = ∑ m,n<br />

1<br />

′∑<br />

(m(z + 1) + n) = 2k<br />

m,n<br />

2k<br />

1<br />

(mz + n) 2k<br />

1<br />

′∑<br />

(mz + (n + m)) = 2k<br />

m,n<br />

1<br />

(mz + n ′ ) 2k = G 2k(z).<br />

G 2k (− 1) = ∑ 1<br />

z<br />

(−m/z + n) = ∑ z 2 k<br />

2k (−m + nz) = 2k z2 k ∑ 1<br />

(m ′ z + n ′ ) = 2k zk G 2k (z).<br />

m,n<br />

m,n<br />

m,n<br />

2k<br />

2k

2ζ(2k)ºÌ<strong>×</strong><strong>×</strong>ÐÐØÒÓÖÑÐÞ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong> ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´º½µ<br />

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E 2k (z) = G 2k(z)<br />

2ζ(2k) .<br />

<strong>×</strong>ÑÓÙÐÖÓÖÑØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÛÐÐÑØq¹ÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓÖÑÐÔÓÛÖ<strong>×</strong>Ö<strong>×</strong>Ò ºº¾ ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>Ó<strong>×</strong>Ò<strong>×</strong>ØÒËÖ<strong>×</strong><br />

<strong>×</strong>ÑÙÒØÓÒ<strong>×</strong>Ò<strong>×</strong>Ø<strong>×</strong>ÙÑÓØr¹ØÔÓÛÖ<strong>×</strong>ÓØÔÓ<strong>×</strong>ØÚÚ<strong>×</strong>ÓÖ<strong>×</strong>Ónººº ÒØÓÒºº½ËÑÙÒØÓÒÓÖÒÒØÖr≥0ÒÒÝÔÓ<strong>×</strong>ØÚÒØÖn¸Ø<br />

nÒØ<strong>×</strong>ÑÙÒØÓÒÛÛÒÐÓÛº ØÖÑ<strong>×</strong>Óq(z) = e 2πiτºÀÖÛÚØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>G ØÖÑ<strong>×</strong>ÓÖÒÓÙÐÐÒÙÑÖ<strong>×</strong>B ´º¾µ<br />

ØÓÐÐÓÛÒÕÙÐØÝÓÓÖÑÐÔÓÛÖ<strong>×</strong>Ö<strong>×</strong><br />

n¸ÖÒÝ ÏÐ<strong>×</strong>Ó<strong>×</strong>Øσ 0 (n) = d(n)ÓÖØÒÙÑÖÓÔÓ<strong>×</strong>ØÚÚ<strong>×</strong>ÓÖ<strong>×</strong>ÓnÒσ(n) ÒØÓÒºº¾ÖÒÓÙÐÐÆÙÑÖ<strong>×</strong>ÓÖn ≠ 0ØÖÒÓÙÐÐÒÙÑÖ<strong>×</strong>¸B Í<strong>×</strong>ÒØÓÚØÛÓÒØÓÒ<strong>×</strong>¸ÛÒÛÖØØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓØÒÓÖÑÐÞ<strong>×</strong>Ò¹ ´º¿µ<br />

2k<strong>×</strong>¿<br />

n! . ´ºµ <strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>E<br />

ÓÙØÖÒØÙ<strong>×</strong>Ô<strong>×</strong><strong>×</strong><strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÒÔÔÒܺÐÓÛ¸<strong>×</strong>ÒÜÑÔиÛÚØÓÙÖÖ ÓÖ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÓÖÐÚиÛÚØÓÓÑÔÙØØÖÜÔÐØÓÖÑÙ<strong>×</strong>ÒÚÖÓÙ<strong>×</strong><br />

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E<br />

ÖÐØÓÒ<strong>×</strong>ØÛÒØE 2<strong>×</strong>ÒÐÓÛ<strong>×</strong>ÒÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÛØØÛÓÙØÓÓÒÚÖÒºÐÓ<strong>×</strong>ÐÝÖÐØÒÓÒ¹<br />

)<strong>×</strong>ÛØØÛÓ´ËÔÔÒܵ ½¾<br />

¿E<br />

ÓÐÓÑÓÖÔÓÖÑE 2 ∗ = ( E 2 − 3<br />

ÁÑz<br />

k<br />

σ r (n) = ∑<br />

d r .<br />

1≤d|n<br />

= σ 1 (n)º<br />

∞<br />

x<br />

e x − 1 = ∑ x n<br />

B n<br />

2k = 1 − 4k<br />

B 2k<br />

n=0<br />

∞<br />

∑<br />

n=1<br />

σ 2k−1 (n)q n .<br />

(z)Ò

2º ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong> ÜÔÒ<strong>×</strong>ÓÒÓÖE (2)<br />

2 , E (3)<br />

2 , E 2ÒE (4) (5)<br />

·´ºµ ´ºµ<br />

ÒÙÑÖÓÔÐ<strong>×</strong>ºÀÖÛÚÐ<strong>×</strong>ØÓÒÐÝØÚÖÝ<strong>×</strong>Ø<strong>×</strong>ÓÙØØÑÒØÒØÖ<strong>×</strong>Ø Ì<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÖÚÖÝÑÔÓÖØÒØÒØØÓÖÝÓÙØÓÑÓÖÔÓÖÑ<strong>×</strong>ÒÓÙÖÒ ÖÖ<strong>×</strong>ÖÖØÓÒÝÓØÖÖÒ<strong>×</strong>ÓÖÑÓÖÓÑÔÐØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒº<br />

2 (τ) = 1 + 8q + 24q2 + 32q 3 + 24q 4 + 48q 5 + · · ·<br />

2 (τ) = 1 + 6q + 18q 2 + 24q 3 + 42q 4 + 6q 5 + 72q 6 + 48q 7 + 90q 8 + 78q 9 + · ·<br />

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ÚÐÓÔÑÒØÓËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÛÒÛ<strong>×</strong>ØÙÝËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÒØÒÜØ<strong>×</strong>ØÓÒº ÂÓÓÖÑ<strong>×</strong>ÖÖÓ<strong>×</strong><strong>×</strong>ØÛÒÐÐÔØÙÒØÓÒ<strong>×</strong>ÒÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÓÒÚÖÐÒØØ ÓÒÓØÚÖÐØØ<strong>×</strong><strong>×</strong>ÖÓÑC¸ÛÐØÓØÖ<strong>×</strong>Ö<strong>×</strong>ØÖØØÓHº<br />

Z)<strong>×</strong>ÓÐÓÑÓÖÔÙÒØÓÒ<br />

<strong>×</strong>Ø<strong>×</strong>ÝÒØØÛÓØÖÒ<strong>×</strong>ÓÖÑØÓÒÕÙØÓÒ<strong>×</strong> ´ºµ ÂÓÓÖÑÓÒSL(2,<br />

´ºµ<br />

φ : H <strong>×</strong> C → C<br />

ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓØÓÖÑ<br />

τ)<strong>×</strong> ´ºµ<br />

φ(z, τ + λz + µ) = e −2πim(λ2z+2λτ) φ(z, τ) λ, µ ∈ Z 2 ) .<br />

´º¼µ<br />

Ì<strong>×</strong>ØÛÓ<strong>×</strong>Ø<strong>×</strong>ÓØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÒØÂÓÖÓÙÔ´ËÔÔÒܵºφ(z,<br />

NÖÐÐØÛØÒÒÜÓφ¸Ö<strong>×</strong>ÔØÚÐݸÒØÓÙÖÖÓÒØ<strong>×</strong>¸<br />

0)<strong>×</strong>ÒÓÖÒÖÝ ÛÖk, m ∈<br />

½¾<br />

c(n, r) = 0¸ÙÒÐ<strong>×</strong><strong>×</strong>n≥0Ò4mn r 2 ≥ 0ºÆÓØØØØÙÒØÓÒφ(z,<br />

E (2)<br />

2 (τ) = 1 + 24q + 24q 2 + 96q 3 + 24q 4 + 144q 5 + 96q 6 + 192q 7 + 24q 8 + 312q 9 + · · ·<br />

E (3)<br />

2 (τ) = 1 + 12q + 36q 2 + 12q 3 + 84q 4 + 72q 5 + 36q 6 + 96q 7 + 180q 8 + 12q 9 + · · ·<br />

E (4)<br />

E (5)<br />

( az + b<br />

φ<br />

cz + d , τ<br />

)<br />

= (cz + d) 2k e 2πimcτ<br />

cz + d<br />

φ(z, τ) =<br />

−<br />

∞∑<br />

∑<br />

n=0 r∈Z,r 2 ≤4mn<br />

cz+d φ(z,τ)<br />

c(n, r)e 2πi(nz+rτ)

ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ØÓØÙ<strong>×</strong>ÙÐÒÓØÓÒÓÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÓÒÚÖк<br />

0¸ØÒφ<strong>×</strong>ÒÔÒÒØÓτÒØÒØÓÒÖÙ<strong>×</strong><br />

ÒÖ¿℄ÒÜØÒØÓйÒØÖÐÒ<strong>×</strong>ÝÖØ<strong>×</strong>ÒÓ℄º<br />

r)ÖÒÓÒ¹ÚÒ<strong>×</strong>ÒÓÒÐÝÛÒn≥0ÖÐÜÒ ÑÓÙÐÖÓÖÑÓÛØ2kºÁm ÌÐÐÔØÒÙ<strong>×</strong>ÓйÙÑÒÓÐ<strong>×</strong>ÖÛÂÓÓÖÑ<strong>×</strong>ºÜÑÔÐ<strong>×</strong>ÒÐÙ<br />

)ºÂÓÓÖÑ<strong>×</strong>ÓÒØÖÒÜÛÖÓÒ<strong>×</strong>ÖÝÐÖ<br />

=<br />

ÓÖÛÂÓÓÖÑ<strong>×</strong>¸ØÓÒØ<strong>×</strong>c(n, ØÓÒØÓÒÒÚÓÐÚÒ(4nt − l 2 ´º½µ<br />

) 2 JÒÛÖØÒ ÏÛÐÐ<strong>×</strong>ØÔÔÖÒÓÛØÞÖÓÂÓÓÖÑ<strong>×</strong>ÓØÖÓÙÔΓ 0<br />

´º¾µ<br />

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(N)ºÌÓÙÖÖ ÛØα (N) (τ) =<br />

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´º¿µ<br />

ÜÔÒ<strong>×</strong>ÓÒÓÖφ 0,1ØØÙ<strong>×</strong>ÔØi∞<br />

(N)<br />

(<br />

φ −2,1 (z 1 , z 2 ) = E<strong>×</strong>Ø<strong>×</strong>T 2(z iϑ1 (z 1 , z 2 )<br />

1, z 2 ) =<br />

η 3 (z 1 )<br />

4∑<br />

(<br />

ϑi (z 1 , z 2 )<br />

φ 0,1 (z 1 , z 2 ) = E K3 (z 1 , z 2 ) = 8<br />

ϑ<br />

i=2 i (z 1 , 0)<br />

(Z)º<br />

φ (N) 2N<br />

0,1 (τ, z) =<br />

N + 1 α(N) (τ) φ −2,1 (τ, z) + 1<br />

N + 1 φ 0,1(τ, z) ,<br />

φ (2)<br />

0,1(τ, z) =<br />

φ (3)<br />

0,1 (τ, z) = (<br />

φ (5)<br />

0,1(τ, z) =<br />

) 2<br />

0<br />

(<br />

2r + 4 + 2 )<br />

+<br />

(4r 2 − 8 + 4 )<br />

q + O ( q 2)<br />

r<br />

r 2<br />

2r + 2 + 2 )<br />

+<br />

(2r 2 − 2r − 2 r<br />

r + 2 )<br />

q + O ( q 2)<br />

r<br />

(<br />

2 2r + 2 ) (<br />

+ 2r − 4 + 2 )<br />

q + O ( q 2) .<br />

r<br />

r<br />

½¿¼

ÒÓÙØØÙ<strong>×</strong>ÔØ0<strong>×</strong><br />

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(<br />

φ (2)<br />

0,1 =8 + − 16 ) ( 8<br />

r + 32 − 16r q 1/2 +<br />

r − 64<br />

)<br />

2 r + 112 − 64r + 8r2 q + O ( ´ºµ<br />

q 3/2)<br />

φ (3)<br />

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(− 6 ) (<br />

r + 12 − 6r q 1/3 + − 18 )<br />

r + 36 − 18r q 2/3<br />

( 6<br />

+<br />

r − 42<br />

)<br />

2 r + 72 − 42r + 6r2 q + O ( q 4/3)<br />

(<br />

φ (5)<br />

0,1 =4 + − 2 )q<br />

r + 4 − 2r 1/3 +<br />

(− 6 )<br />

r + 12 − 6r q 2/5 +<br />

(− 8 )<br />

ÌØÙÒØÓÒ<strong>×</strong><br />

r + 16 − 8r q 3/5<br />

(<br />

+ − 14 ) ( 4<br />

r + 28 − 14r q 4/5 +<br />

r − 26<br />

)<br />

inCÒÝ 2 r + 44 − 26r + 4r2 q + O ( q 6/5) .<br />

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∈ H¸ÁÑτ > 0<br />

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n∈Z<br />

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n<br />

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ϑ(z, τ) = ∑ n∈Z<br />

a n (τ)e 2πinz) , ÛÖa<br />

(τ) = e πin2 τ<br />

´ºµ

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A B<br />

C D<br />

)<br />

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= Ω T = −Ω.º<br />

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A B<br />

C D<br />

)<br />

∈ Sp(2, Z)ÓÒτ ∈<br />

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ÐÐËÐÑÓÙÐÖÓÖÑÓÛØρ ÆÓÛ¸ÛÖÖÝØÓÒËÐÑÓÙÐÖÓÖѺ<br />

V<strong>×</strong><br />

ρ<br />

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→<br />

Ø∞º<br />

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ρÓÖÓÒÐÝØ ρÖÒØ<br />

ÑÓÙÐÖÓÖÑ<strong>×</strong>¸ÐÓÛº<br />

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ØÓØÖØ<strong>×</strong>ÙÑM ρ1 ⊕ M<br />

C)º<br />

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kÒÖg<strong>×</strong>ÓÐÓÑÓÖÔÙÒØÓÒf : H g →<br />

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∞µºÏÒÓØÝM kÓMÓÐ<strong>×</strong><strong>×</strong>ÐËÐÑÓÙÐÖÓÖÑ<strong>×</strong>º<br />

k = M k (Γ g<br />

kºÌ<strong>×</strong><strong>×</strong>Ô<strong>×</strong>ÓÖÑÖÖÒMcl := ⊕M<br />

Z)º<br />

½¿ ÏÒg= 1¸Ø<strong>×</strong><strong>×</strong>ÑÔÐÝÖÙ<strong>×</strong>ØÓØÙ<strong>×</strong>ÙÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÒSL(2,<br />

(<br />

A B<br />

C D<br />

)<br />

: GL(g, C) → GL(V )<br />

f(γ(τ)) = ρ(Cτ + D)f(τ)<br />

∈<br />

=<br />

f(γ(τ)) = det(cτ + d) k f(τ)

ºº ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong> ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

Z)¸ÛÒÜÔÒØÚØÓÖÚÐÙ<br />

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gÑØÖ<strong>×</strong>sÑØ<strong>×</strong>ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<br />

CØØ<strong>×</strong>ÔÖÓÒØ<strong>×</strong>Ò<strong>×</strong>ØØ<br />

ÌÖ(nτ)<br />

йÒØÖÐÑØÖܺÆÓÛ¸ÙÒØÓÒf : H g →<br />

CÚÒÝØÓÙÖÖØÖÒ<strong>×</strong>ÓÖÑÓf(τ)<strong>×</strong> ´º¼µ<br />

f(τ + s) = f(τ)ÓÖÐÐ<strong>×</strong>ÝÑÑØÖÜg nÐÒØÖÐa(n)e 2πiÌÖ(nτ)<br />

ÛÖdx<strong>×</strong>ØÙÐÒÚÓÐÙÑÓØ<strong>×</strong>ÔÓx¹ÓÓÖÒØ<strong>×</strong>ÒØÒØÖÐÖÙÒ<strong>×</strong>ÓÚÖ ´º½µ<br />

ÛØa(n) ∈<br />

ÛÖØÓÒØ<strong>×</strong>a(n)ÛÐÐØÚÐÙ<strong>×</strong>ÒØÚØÓÖ<strong>×</strong>ÔÒ<strong>×</strong>ØÓC<strong>×</strong>ÒØ<strong>×</strong>Ó<br />

2ºÌ<strong>×</strong><strong>×</strong>Ö<strong>×</strong><strong>×</strong>ÙÒÓÖÑÐÝÓÒÚÖÒØÓÒÓÑÔØ<strong>×</strong>Ù<strong>×</strong>Ø<strong>×</strong>º<br />

xÑÓ1<br />

ÔÖÓÙÒØÓÒ<strong>×</strong>ÒÓÚÒ<strong>×</strong>Ø<strong>×</strong>Ý<br />

ρÛÚ<strong>×</strong>ÑÐÖÓÙÖÖ<strong>×</strong>Ö<strong>×</strong><br />

− 1 ≤ x 2 ij ≤ 1 ´º¾µ<br />

ÓÖØ<strong>×</strong>ÓÚØÓÖ¹ÚÐÙÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÒM a(uÌnu) = ρ(uÌ)a(n)ÓÖÐÐu∈GL(g, Z) .<br />

½¿<br />

=<br />

<strong>×</strong><br />

f(τ) =<br />

∫<br />

a(n) =<br />

g∑<br />

n ii τ ii + 2 ∑<br />

i=1<br />

∑<br />

1≤i≤j≤g<br />

n ij τ ij .<br />

f(τ)e −2πiÌÖ(nτ) dx,

2πiÌÖ(nτ)ÒÛÖØ´º¼µ<strong>×</strong> ÔØÖºÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´º¿µ ÄÓÖÛÛÖØqn = e<br />

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0µº nйÒØÖÐa(n)q n<br />

ÑÓÙÐÖÓÖÑf = ∑ n a(n)e2πiÌÖ(nr) ∈ M k (Γ g )<strong>×</strong>ÐÐ<strong>×</strong>ÒÙÐÖa(n) 0ÑÔÐ<strong>×</strong>ØØn <strong>×</strong><strong>×</strong>ÒÙÐÖÑØÖÜ´ººØ(n) =<br />

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ØÐÒ<strong>×</strong>ÓØg=<br />

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1<strong>×</strong>¸ÛÑÓÚÓÒØÓ<strong>×</strong>ØÙÝÓØÖÚÐÓÔÑÒØ<strong>×</strong>ØØÜ<strong>×</strong>ØÓÖg> 1<br />

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1¸ØÖÖÓØÖÚÐÓÔÑÒØ<strong>×</strong>ÓËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÛÖÑÓÖÒÖÐØÒ<br />

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ÓÖg =<br />

g ><br />

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C¸Û<strong>×</strong>ÔÖÓÒØÔÖÑØÖ Cº ÓÒ<strong>×</strong>Ö´º¼µ¸ÛÖØÙÒØÓÒf(τ) : H g →<br />

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)ºÌÒÐÓÓØÓÒØ<strong>×</strong>a(n)ÛÓÙÐÒÓÛÓÖÖ<strong>×</strong>ÔÓÒØÓÙÒØÓÒ<strong>×</strong>Û<br />

∈<br />

ÀÖτ ∈ H<br />

g−1ºËÔÐÞÒØÓÓÙÖ<strong>×</strong>Óg=2¸ÛÒÓÙÖÖÜÔÒØËÐ<br />

f(τ)¸ÓÛØkÓÒΓg¸ÙÒÖτ ′ ∈ H 1¸<strong>×</strong>Ò<strong>×</strong>Øτ H<br />

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′<br />

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f(τ) =<br />

f(Z) =<br />

∑<br />

∈<br />

.<br />

≠<br />

∞∑<br />

φ m (z 1 , z 2 )e 2πimz 3<br />

.<br />

m=0<br />

( )<br />

z 1 z 2<br />

∈ H 2<br />

z 2 z 3

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m<strong>×</strong>ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓØÓÖÑ<br />

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´ºµ<br />

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<br />

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<br />

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∑<br />

n=0 r∈Z,r 2 ≤4mn<br />

c(n, r)e 2π(nz 1+rz 2 ) .<br />

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q 1 2 (l 1+ a 1<br />

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a 2<br />

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b 1<br />

( )<br />

z 1 z 2<br />

z 2 z 3<br />

=<br />

1<br />

¼<br />

( m ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )<br />

a 0000 0100 1000 1100 0001 1001 0010 0110 0011 1111<br />

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≡ 1 2 q2<br />

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−<br />

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n<br />

´º½µ<br />

<strong>×</strong>ÙØØ1<br />

2 q2 e =<br />

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) (k+2)/2 (ÚÓÐ⊥ ) 1/2 + · · ·<br />

½<br />

1<br />

lim<br />

µ→0<br />

1 2 q2 e + N L = 1 ,<br />

= 1 2 q2 e = N L − 1 .<br />

16<br />

η(z 1 ) = ∑ ∞<br />

d(n) q n ,<br />

24<br />

n=−1<br />

16<br />

g ρ (z 1 /N) ≡<br />

1<br />

g ρ (iµ/2πN) = 16 e4π2 /µ<br />

∞<br />

∑<br />

n=−1<br />

( µ<br />

2π<br />

d(n) q n/N ,

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20,4 ⊂ Γ<br />

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r =<br />

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1Òa ´º¾¿µ<br />

a 1 = a<br />

º¿º¾<br />

a<br />

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24<br />

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24<br />

N+1º<br />

g ρ (z 1 ) =<br />

N∏<br />

η(rz 1 ) ar = η(z 1 ) a 1<br />

η(2z 1 ) a2 · · ·η(Nz 1 ) a N<br />

,<br />

r=1<br />

( )<br />

Na1 + N a 2<br />

2<br />

+ · · · + a N = 24 ,<br />

a 1 + a 2 + · · · + a N = 2(k + 2) ,<br />

(<br />

1<br />

a 1<br />

2 a2 · · ·N a N<br />

.<br />

1 + 2a 2 + · · · + Na N = 24 .<br />

1<br />

= a N =

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23½℄º<br />

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ÓÖÖ¸n´Ø<strong>×</strong>ÒÓÛÒØØn ≤<br />

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24ÒÖÔÖ<strong>×</strong>ÒØ<strong>×</strong>ÔÖÑÙØØÓÒÖÓÙÔØÒÓÒ<strong>×</strong>Ø<br />

ε(n)<br />

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ρ ≡ 1 a 1<br />

24¸ÓÒÐ<strong>×</strong>Ó<strong>×</strong>ØÓÒØÓÒ ´º¾µ<br />

2 a2 · · ·n an .<br />

) an<strong>×</strong>Ø<strong>×</strong>Ñ<strong>×</strong>ρºËÒÑ(H ´º¾µ<br />

(K3)) =<br />

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4Ò<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÓÚ<br />

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ρ (z 1<br />

´º¾µ ÆÓØØØÛÒØÓÒØÓÒ´º¾µ<strong>×</strong><strong>×</strong>Ø<strong>×</strong>¸g ρ (z 1<br />

½<br />

( M<br />

1<br />

) a1<br />

( M<br />

2<br />

) a2<br />

· · ·(<br />

M<br />

n<br />

(K3)º<br />

24<br />

=<br />

n ∏ p|n (1 + 1) ,<br />

p<br />

∗<br />

23º<br />

∑<br />

i a i = 24 .<br />

i<br />

=<br />

ρ ↦−→ g ρ (z 1 ) ≡ η(z 1 ) a 1<br />

η(2z 1 ) a2 · · ·η(nz 1 ) an .<br />

(K3)¸H 2,0 (K3)¸

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1<br />

ØÚØݺÙÒØÓÒg(z 1 ) = ∑ n a nq n<strong>×</strong>ÑÙÐØÔÐØÚa = a n a mÛÒ(n,<br />

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16µ¸ØÓÖÖ<strong>×</strong>ÔÓÒÒ<br />

=<br />

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MÒØØÖÙ 2ºÁÒ<br />

º¿º¿<br />

ØÓÒÓÆÙÐÒÒÚÓÐÙØÓÒ<strong>×</strong>ÓÒH ∗ (K3)Ø<strong>×</strong>ÒÐÙ<strong>×</strong>ÔÖÓÙØÖÓÙÔ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>Z <strong>×</strong> Z<br />

ÜÑÔÐ<strong>×</strong>ÒÚÓÐÚÒÔÖÓÙØÖÓÙÔ<strong>×</strong>¸Øη¹ÔÖÓÙØ<strong>×</strong>ÖØÙÐÐÝÓÐÚÐN <<br />

Ù<strong>×</strong>ÒÕº´ºµºÓÖÔÖÑNØ<strong>×</strong>ÖÔÖÓÙ<strong>×</strong>ØÖ<strong>×</strong>ÙÐØÓÂØÖÒËÒ¾℄º<br />

)ÒÖØÖχ(d)ÖÓÑÌк¿ØÓØÖÑÒØ ÓÖÑÙÐÓÖΦ k (Z)Ò˜Φk ÏÒÙ<strong>×</strong>ØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÓÖg ρ (z 1<br />

(Z)¸ÓÙØÒÓØÖÒÕÙÚÐÒØÙ<strong>×</strong>ÔºÄØ<br />

ØÚ<strong>×</strong>ÒÒÓÒ<strong>×</strong>ØÖÙØØÑÓÙÐÖÓÖÑΦ k<br />

(Z)¸<br />

(Z)ÓÖN ÛØ ´º¾µ<br />

<strong>×</strong><strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÝÂØÖÒËÒ¾℄¸ØÒÖØÒÙÒØÓÒÓÝÓÒÒÖ<strong>×</strong>¸˜Φk <strong>×</strong>ÚÒÝÜÔÒ<strong>×</strong>ÓÒÓØÑÓÙÐÖÓÖÑ¸Φ k<br />

˜Φ k<br />

(Z)ØØÖ<strong>×</strong>ÖÓÑØÓÒÙ<strong>×</strong>Ò¾℄ÙØÖ<strong>×</strong>ÛØ<br />

(Z) ≡ (ÚÓÐ⊥ 1 Φ k (˜Z) ,<br />

m)ÓÝÓÒ<strong>×</strong>ÛØØ<strong>×</strong>Ö<strong>×</strong><strong>×</strong><br />

˜z<br />

ÏÚÓ<strong>×</strong>ÒÒÓÖÑÐÞØÓÒÓÖ˜Φk ØÓÒÙ<strong>×</strong>Ò¿¼℄ºÓÒ<strong>×</strong>Ö1 4¹ÈËÝÓÒ<strong>×</strong>ÛØÖ<strong>×</strong>q eÒq m<strong>×</strong>ÙØØ2n =<br />

2m = q 2 mÒl=q e · q mºÌÒ¸ØÒÖÝd(n, ½<br />

ÛÖq= exp(2πiz<br />

g ρ (z 1 ) =<br />

∞∑<br />

m=1<br />

1<br />

nm<br />

= 1 ,<br />

m)<br />

2<br />

(Z)<br />

)<br />

1/2 z −k<br />

=<br />

1 = −1/z 1 , ˜z 2 = z 2 /z 1 , ˜z 3 = z 3 − z 2 2 /z 1 .<br />

l,<br />

1, 2, 3, 4, 5, 6, 7, 8, 11<br />

Nq 2 e¸

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

(k ÝÐ<strong>×</strong>Ôρ 1 24 12 1 1<br />

+ 2) χ( a b<br />

c d ) M N G<br />

1 8 2 8 8 2 2 Z 2<br />

1 6 3 6 6 3 3 Z 3<br />

2 12 6 4 2 Z 2 <strong>×</strong> Z 2<br />

1 4 2 2 4 4 5<br />

( −1<br />

d<br />

)<br />

4 4 Z 4<br />

1 4 5 4 4 5 5 Z 5<br />

1 2 2 2 3 2 6 2 4 6 6 Z 6<br />

2 4 4 4 4 8 4 Z 2 <strong>×</strong> Z 4<br />

3 8 4 9 3 Z 3 <strong>×</strong> Z 3<br />

(<br />

1 3 7 3 3 −7<br />

)<br />

7 7 Z<br />

d 7<br />

(<br />

1 2 2 1 4 1 8 2 3 −2<br />

)<br />

8 8 Z<br />

d<br />

´ØÖÙÐÚÐNÒÖØÖχµºÇÒÐÝÒÓÒ¹ØÖÚÐÖØÖ<strong>×</strong>ÖÒØÒÓÐÙÑÒ¿ºÌ<br />

8<br />

(<br />

2 3 6 3 3 −3<br />

)<br />

12 6 Z<br />

d 2 <strong>×</strong> Z<br />

11ÜÑÔÐ<strong>×</strong>ÒÓØ<strong>×</strong>ÝÑÔÐØÒÚÓÐÙØÓÒÓÿº<br />

6<br />

(<br />

4 6 3 −1<br />

)<br />

16 4 Z<br />

d 4 <strong>×</strong> Z 4<br />

1 2 3 2 11 11 Z 11<br />

Ìк¿ÌÙÒØÓÒg ρ (z 1 )<strong>×</strong>ÑÓÙÐÖÓÖÑÓÛØ(k + 2)¸ÒÖÐÞÐÚÐM<br />

N =<br />

(Z)<strong>×</strong>ÚÒÝØÓÐÐÓÛÒ<strong>×</strong> ´º¾µ<br />

˜Φ k = ∑ d(n, l, m) q n/N r l s m .<br />

n,l,m<br />

<strong>×</strong>ÑÐÖØÚÐØÓÖ˜Φk NÓÖÓÐ<strong>×</strong>º<br />

˜φ k,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2<br />

(Z)ÓÖØÀÄ<br />

) 2<br />

ÏÒÓÛÔÖÓÚØÐÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÓÖØÒÙ<strong>×</strong>¹ØÛÓÑÓÙÐÖÓÖÑ<strong>×</strong>Φ k<br />

Z<br />

ÒÖØÝ<br />

64<br />

η(z 1 ) 6 g ρ (z 1 /N) . ´º¿¼µ<br />

ÓÖÔÖÑNÒN+ 1|24¸ØØÚ<strong>×</strong><strong>×</strong>ÚÒÝ´k + 2 = 24/(N + 1)<br />

φ k,1 (z 1 , z 2 ) = ϑ 1 (z 1 , z 2 ) 2 η(z 1 ) k−4 η(Nz 1 ) k+2 = ∑ a(n, l) q n r l .<br />

n,l<br />

´º¿½µ ½¼<br />

N = 1, 2, 3, 5

l)<strong>×</strong><strong>×</strong>ÒÝØÓÚÕÙØÓÒµ ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

Ì<strong>×</strong>Ö<strong>×</strong>ÙÐØ<strong>×</strong>ÓÖÒÐÐÝÙØÓÂØÖÒËÒ¾℄ ´º¿¾µ ÌØÚÐØ<strong>×</strong>´a(n,<br />

, l d d)<br />

q n r l s m . 2<br />

4ºÌ<strong>×</strong>ÓÖØØÚÐØ<strong>×</strong><br />

l)<strong>×</strong><strong>×</strong>ÒÝØÓÚÕÙØÓÒµ ´º¿¿µ ÖÓÑÌк¿¸Û<strong>×</strong>ØØk= 3ÓÖN =<br />

´º¿µ ÌØÚÐØ<strong>×</strong>´a(n,<br />

, l d d)<br />

q<br />

(γ)ÛÖ<br />

n r l s m , 2<br />

ÛÖØÂÓ<strong>×</strong>ÝÑÓÐ( )<strong>×</strong>+1ÛÒd=1ÑÓ4−1ÛÒd=3ÑÓ4Ò0<br />

−1<br />

d<br />

ÓØÖÛ<strong>×</strong>ºÌ<strong>×</strong>ËÐÑÓÙÐÖÓÖÑÛØÐÚÐÓÙÖÒÖØÖψ Z)℄º ´º¿µ<br />

4<br />

)ÓÖγ=<br />

(<br />

ÛÖG 0<br />

6ºÌ<strong>×</strong>ÓÖØØÚÐØ<strong>×</strong><br />

(4)<strong>×</strong>ØÐÚÐÓÙÖ<strong>×</strong>ÙÖÓÙÔÓSp(2,<br />

l)<strong>×</strong><strong>×</strong>ÒÝØÓÚÕÙØÓÒµ ´º¿µ ÖÓÑÌк¿¸Û<strong>×</strong>ØØk= 2ÓÖN =<br />

ÌØÚÐØ<strong>×</strong>ØÒ´a(n,<br />

N = 4<br />

N = 6<br />

Φ k (Z) ≡<br />

∑<br />

(n,l,m)>0<br />

∑<br />

d|(n,l,m)<br />

( −1<br />

d<br />

)<br />

d k−1 a ( nm<br />

φ 3,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2 ) 2<br />

η(2z<br />

η(z 1 ) 2 1 ) 2 η(4z 1 ) 4 = ∑ a(n, l) q n r l .<br />

n,l<br />

Φ 3 (Z) ≡<br />

∑<br />

(n,l,m)>0<br />

( −1<br />

ψ 4 (γ) =<br />

detD<br />

∑<br />

d|(n,l,m)<br />

( −1<br />

d<br />

)<br />

d k−1 a ( nm<br />

)<br />

A B<br />

∈ G 0 (4) ,<br />

C D<br />

φ 2,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2 ) 2<br />

η(2z<br />

η(z 1 ) 4 1 ) 2 η(3z 1 ) 2 η(6z 1 ) 2 = ∑ a(n, l) q n r l .<br />

n,l<br />

Φ 2 (Z) ≡<br />

∑<br />

(n,l,m)>0<br />

∑<br />

d|(n,l,m)<br />

d=1,5ÑÓ6<br />

d k−1 a ( nm<br />

d 2 , l d<br />

)<br />

q n r l s m . ´º¿µ<br />

½½

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

8ºÌ<strong>×</strong>ÓÖØØÚÐØ<strong>×</strong><br />

l)<strong>×</strong><strong>×</strong>ÒÝØÓÚÕÙØÓÒµ ´º¿µ ÖÓÑÌк¿¸Û<strong>×</strong>ØØk= 1ÓÖN =<br />

´º¿µ ÌØÚÐØ<strong>×</strong>ØÒ´a(n,<br />

, l d d)<br />

q n r l s m , 2<br />

º¿º<br />

ÛÖØÂÓ<strong>×</strong>ÝÑÓÐ( −2<br />

5ºÓÖÓØÖÚÐÙ<strong>×</strong>ÓN¸ÛÒÒÓÒ¹ÒØÖÐ<br />

ÓØÖÛ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>Ð<strong>×</strong>ÓËÐÑÓÙÐÖÓÖÑØÐÚÐØÒÖØÖ( −2<br />

ÓÒ<strong>×</strong>ØÖÙØÒØÑÓÙÐÖÓÖÑ∆ k/2<br />

(Z)ºÁÒØ<strong>×</strong><strong>×</strong><strong>×</strong>¸ØØÚ Ì<strong>×</strong>ÕÙÖ¹ÖÓÓØÛÓÖ<strong>×</strong>ÓÒÐÝÓÖN = 1, 2, 3, 4,<br />

ÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>Ö<strong>×</strong>ÒÖÓÑØÒØ<strong>×</strong>ÕÙÖ¹ÖÓÓسÓΦ k<br />

´º¼µ <strong>×</strong>ÓÖØÑÓÙÐÖÓÖÑ∆ ÝØÐØÓØØÚ<strong>×</strong>¾<br />

)ÖØη¹ÔÖÓÙØ<strong>×</strong>ÓØÒÖÓÑÌк¿ºÌ<strong>×</strong>ÔÔÒ<strong>×</strong>ØÓØ<strong>×</strong>ÕÙÖ<br />

k/2<br />

(Z)<strong>×</strong>ÚÒ ÛÖg ρ (z 1<br />

ÖÓÓØÓØÂÓÓÖÑØØÒÖØ<strong>×</strong>Φ k<br />

´º½µ<br />

(Z)ºËÑÐÖÐݸØÑÓÙÐÖÓÖј∆k/2<br />

ÓÒ´<strong>×</strong>Õº´ºµµÒØÓÖÖ<strong>×</strong>ÔÓÒÒÑÓÙÐÖÓÖÑ<strong>×</strong>ÓØÝØÒØÔÔÖÓÔÖØ<br />

5¸ØÖØÖχ(d)<strong>×</strong>ØØÖÚÐ ÏÚÐÖÝ<strong>×</strong>ÒØ<strong>×</strong>ÓN = 1ºÓÖN 2,<br />

½¾<br />

N = 8<br />

φ 1,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2 ) 2<br />

η(2z<br />

η(z 1 ) 4 1 )η(4z 1 )η(8z 1 ) 2 = ∑ a(n, l) q n r l .<br />

n,l<br />

Φ 1 (Z) ≡<br />

∑<br />

(n,l,m)>0<br />

∑<br />

d|(n,l,m)<br />

( −2<br />

d<br />

)<br />

d k−1 a ( nm<br />

detD)º<br />

d<br />

)<strong>×</strong>+1ÛÒd=1, 3ÑÓ8−1ÛÒd=5, 7ÑÓ8Ò0<br />

(Z)<strong>×</strong><br />

(Z)<br />

ψ k/2,1/2 (z 1 , z 2 ) = θ √<br />

1(z 1 , z 2 )<br />

g<br />

η(z 1 ) 3 ρ (z 1 ) ,<br />

˜ψ k/2,1/2 (z 1 , z 2 ) = θ √<br />

1(z 1 , z 2 )<br />

g<br />

η(z 1 ) 3 ρ (z 1 /N) .<br />

=<br />

¾ÌÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓØÂÓÓÖÑÖ<strong>×</strong>ÔÓÛÖ<strong>×</strong>Óq 1/NÌÙ<strong>×</strong>ÓÒ<strong>×</strong>nN ∈<br />

ZÒÕº´º½¼µº

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong> )ºº¸<br />

´º¾µ<br />

ÚÐÙ<strong>×</strong>ÓkÒmºÓÖN = 3¸ÛÒÒÓÒ¹ØÖÚÐÖØÖχψ (d) = ( −3<br />

d<br />

3¸ØÛØÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ÚÒ<strong>×</strong>k=2ºÀÓÛÚÖ¸Ø<strong>×</strong><br />

ψ¸ÓÒÓØÒ<strong>×</strong> ÌÙ<strong>×</strong>¸ÛÒN =<br />

ÂÓÓÖѸψ 2,1/2 (z 1<br />

´º¿µ<br />

, z)¸ØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong>ÛØÖØÖw w ψØÙ<strong>×</strong>ÚÒØÖ<strong>×</strong>ØÖØÓÒÓÒk ÒÓºÌÒÒØÓÓÙÒØØØÓÒÐÖØÖ¸w , l α α)<br />

ØØÚÐØ<strong>×</strong>ÒÓØÒ<strong>×</strong><strong>×</strong>Öݺ<br />

(Z)ÒÒ<strong>×</strong>ØÔÖÓÙØÓkÚÒÒÙ<strong>×</strong>¹ØÛÓØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong>Ò<br />

q n/2 r l/2 s m/2 , 2<br />

ÛØχψ (α)<strong>×</strong>ÒÒÕº´º¾µÖÔÐÒχ(α)ºÓÖN 1, 2, 4¸ÛÛÐÐ<strong>×</strong>ØØ∆ ÁÒÔØÖ4Û<strong>×</strong>ÛØØËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÜÔÖ<strong>×</strong><strong>×</strong>ÒØÖÑ<strong>×</strong>ÓÔÖÓÙØ<strong>×</strong>ÓÚÒ º¿º ÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓÒÙ<strong>×</strong>¹ØÛÓØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong><br />

<strong>×</strong>ÛÐÐ<strong>×</strong>˜∆k/2<br />

ÑØ<strong>×</strong>ÙÒÜÔÖ<strong>×</strong><strong>×</strong>ÓÒÒÛÚØÖºÏÑÒØÓÒÖÐÖØØØËÐÑÓÙÐÖ ÒÙ<strong>×</strong>¹ØÛÓØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong>ºËÓÑÓØËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÙÖÒÒÓÙÖ<strong>×</strong>ØÙÝÐ<strong>×</strong>Ó<br />

ØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong> (Z)ÒÛÖØØÒ<strong>×</strong>Ø<strong>×</strong>ÕÙÖÔÖÓÙØÓÐÐØÚÒÒÙ<strong>×</strong>¹ØÛÓ ´ºµ ÓÖÑÓÖN = 1¸Φ ØØÓÒ<strong>×</strong>ØÒ<strong>×</strong><strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

10<br />

2<br />

(Z)<strong>×</strong>ÔÖÓÙØ<strong>×</strong>ÓÚÒÒÙ<strong>×</strong>¹ØÛÓ<br />

Φ θ m (Z))<br />

≡ [∆5 (Z)] 2 .<br />

ËÑÐÖÐݸÓÖN = 2ÓÒÒÛÖØØÑÓÙÐÖÓÖÑΦ (Z)ØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÚÒÝ ´ºµ<br />

2<br />

θ m (Z))<br />

≡ [∆3 (Z)] 2 ´ºµ<br />

.<br />

m=1ÑÓ2 ÛÐÓÖ˜Φ6<br />

) 2 ] 2<br />

≡<br />

[˜∆3 (Z) ,<br />

½¿<br />

∆ 2 (Z) =<br />

⎧<br />

⎪⎨ 0 d = 0 mod 3 ,<br />

χ ψ (d) = 1 d = 1 mod 3 ,<br />

⎪⎩<br />

−1 d = 2 mod 3 .<br />

γ<br />

∑<br />

(n,l,m)>0<br />

∑<br />

α|(n, l, m)<br />

α > 0<br />

χ ψ (α) α k−1 g ( nm<br />

=<br />

( 1<br />

10 (Z) =<br />

64<br />

9∏<br />

m=0<br />

6<br />

Φ 6 (Z) =<br />

( 1<br />

64 θ 2(Z)<br />

∏<br />

˜Φ 6 (Z) =<br />

( 1<br />

16 θ 1(Z) θ 3 (Z) θ 6 (Z) θ 7 (Z) θ 8 (Z) θ 9 (Z)<br />

k/2 (Z)

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÜÔÖ<strong>×</strong><strong>×</strong>Ð<strong>×</strong>ÔÖÓÙØ<strong>×</strong>ÓÚÒÒÙ<strong>×</strong>¹ØÛÓØØÙÒØÓÒ<strong>×</strong> 4ÜÑÔÐÛÙ<strong>×</strong>ØÓÒ<strong>×</strong>ÖÖÐ<strong>×</strong>Ó<br />

´ºµ<br />

ËÑÐÖÐݸØËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÖØN =<br />

( ) 2 1<br />

Φ 3 (Z) =<br />

8 θ 5 (2Z) θ 7 (2Z) θ 9 (2Z) ≡ [ ∆ 3/2<br />

´ºµ<br />

(Z) ]2 .<br />

]<br />

ÓÚÒÒÙ<strong>×</strong>¹ØÛÓØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong>Ø<strong>×</strong>ÔÖÓÚ<strong>×</strong>ÒÒÔÒÒØÛÝØÓ<strong>×</strong>ØØØÖ<br />

(Z)Ö<strong>×</strong>ÕÙÖ<strong>×</strong>ÓÔÖÓÙØ<strong>×</strong><br />

2<br />

.<br />

ÛÖZ ′<br />

ÓÒ<strong>×</strong>ØÒØ<strong>×</strong>ØÓÖÐÝÔÓÛÖºÌÖÔÖ<strong>×</strong>ÒØØÓÒÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØ<br />

4ºÏÚÚÖØ<strong>×</strong>ÓÖÑÙÐÝÓÑÔÖÒ<br />

ÚÒÒÙ<strong>×</strong>¹ØÛÓÑÓÙÐÖÓÖÑ<strong>×</strong>Ú<strong>×</strong>Ù<strong>×</strong>ÝØÒÓØÖÛÝØÓÓØÒØÑÓÙÐÖÓÖÑ<strong>×</strong>º ØÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>ÖÓÑØØÚÐØØÓØÓÒÚÒÒØÖÑ<strong>×</strong>ÓÚÒÒÙ<strong>×</strong>¹ØÛÓØØ<br />

ÁØ<strong>×</strong>ÔÐ<strong>×</strong>ÒØÓÒÓØØØØÓÖÑÙÐÓÖΦ k (Z)Ò˜Φk <strong>×</strong>ÕÙÖ¹ÖÓÓØ<strong>×</strong>ÖÛÐйÒÓÖN = 1, 2,<br />

º (Z)ºÌÔÖÓÙØ<br />

K3<strong>×</strong>Ò<strong>×</strong>¿ (Z)ÒÚÒÒØÖÑ<strong>×</strong>ÓØÓÒØ<strong>×</strong>ÓØÓÙÖÖ<br />

ÈÖÓÙØÓÖÑÙÐÓÖΦ k (Z)Ò˜Φk N¹ÓÖÓÐÓ<br />

(Z)ÒΦ ÆÜØÛÓÑØÓØÔÖÓÙØÓÖÑÓØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk k<br />

ÓÖÑÙÐÓÖΦ k (Z)<strong>×</strong>ÛÐÐ<strong>×</strong>˜Φk ÜÔÒ<strong>×</strong>ÓÒÓØØÛ<strong>×</strong>ØÐÐÔØÒÖ¿¾℄ºÌØÛ<strong>×</strong>ØÐÐÔØÒÙ<strong>×</strong>ÓÖZ ÓÛØÞÖÓ¸ÒÜÓÒÒÐÚÐN¿¾℄º )ºÌØÛ<strong>×</strong>ØÐÐÔØÒÖÖÛÂÓÓÖÑ<strong>×</strong> ÛÖgÒÖØ<strong>×</strong>Z NÒq = exp(2πız 1<br />

Z)ºÌÒ¸ÓÒ<strong>×</strong><br />

)ÝÙ<strong>×</strong>ÓØÖØÖÒ<strong>×</strong>ÓÖÑØÓÒÔÖÓÔÖØ<strong>×</strong>ÙÒÖ ÏÛÐÐÒØÓÓÑÔÙØØF m,n (z 1 , z<br />

½¼¾℄ ¿ÌÓÖÒÓØ<strong>×</strong>ØÛ<strong>×</strong>ØÐÐÔØÒÖÖÒØÖ<strong>×</strong>ÓÐÓÖÖØÓÒ<strong>×</strong>Ò<strong>×</strong>ØÖÒØÓÖݸ¸½¼¼¸½¼½¸<br />

2<br />

´º¼µ ØÑÓÙÐÖÖÓÙÔºÄØγ= ( ) ∈ SL(2,<br />

½<br />

Ò<br />

˜Φ3<br />

)º<br />

( ) 2 1<br />

(Z) =<br />

4 θ 8 (Z ′ ) θ 3 (Z ′ ) θ 9 (Z ′ ) ≡<br />

(<br />

1<br />

2<br />

=<br />

z 1 z 2<br />

z 2 2z 3<br />

[˜∆3/2 (Z)<br />

(Z)<br />

´ºµ<br />

F m,n (z 1 , z 2 ) = 1 ((−)<br />

NÌÖRR,g F L+F R m g n q L0¯q ) ¯L 0<br />

e 2πızF L<br />

, 0 ≤ m, n ≤ (N − 1) ,<br />

a<br />

c d b<br />

F m,n (z 1 , z 2 ) ∣ = F am+cn,bm+dn (z 1 , z 2 ) .<br />

γ

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

1¸ÓÒ<strong>×</strong> ´º½µ ÁÒÔÖØÙÐÖ¸ÙÒÖT NºÍ<strong>×</strong>ÒØÖØÖÒ<strong>×</strong>ÓÖÑØÓÒÔÖÓÔÖØ<strong>×</strong>ÙÒÖØÑÓÙÐÖÖÓÙÔÛÒ<strong>×</strong>ØÙÝØÖÓÖØ<strong>×</strong><br />

: z 1 → z 1 + 1ÒS: z 1 → −1/z<br />

ÙÒÖØØÓÒÓØÒÖØÓÖ<strong>×</strong>TÒSÓØÑÓÙÐÖÖÓÙÔÒØ<strong>×</strong>ÚÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong><br />

)ÖÛÂÓÓÖÑ<strong>×</strong>ÓÛØÞÖÓÒÒÜÓÒØÐÚÐ<br />

ÓÒ<strong>×</strong>ÖØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒ<br />

ÅÓÖÒÖÐÐݸØF r,s (z 1 , z 2<br />

)º ÓÒØÓÖÑÓØF m,n (z 1 , z 2<br />

´º¾µ<br />

)ºÚ¸ÂØÖÒËÒÔÖÓÚØÓÐÐÓÛÒÔÖÓÙØÓÖÑÙÐÙ<strong>×</strong>Ò<br />

)ØÛ<strong>×</strong>ØÐÐÔØÒÖ¿¾℄ºÇÒ<strong>×</strong>l)ÓÖØÓÙÖÖ<br />

ÛÖq = exp(2πiz 2 )ºÏÛÐÐÐ<strong>×</strong>ÓÛÖØc (n,<br />

ÓÒØca,b<br />

m (4n −l2 (4n − l 2<br />

F 0,n (z 1 , z 2 ) ∣ = F 0,n (z 1 , z 2 ) , F 0,n (z 1 , z 2 ) ∣ = F n,0 (z 1 , z 2 ) .<br />

T S<br />

F a,b (z 1 , z 2 ) =<br />

exp(2πiz 1 )Òr =<br />

1∑<br />

∑<br />

c a,b<br />

m (4n − l 2 ) q n r l ,<br />

m=0 l∈2Z+m,n∈Z/N<br />

a,b<br />

ØÓÙÖÖÓÒØ<strong>×</strong>¸ca,b ´º¿µ<br />

Ò<br />

,<br />

´ºµ<br />

Φ<br />

ÔØÖ2º ÌÔÖÓÙØÓÖÑÙÐÓÖ˜Φk<strong>×</strong>ÐÖÝÒÓØÒÖÓÑØÑÖÓ<strong>×</strong>ÓÔÓÙÒØÒÓÒ<strong>×</strong>ÖÒ<br />

.<br />

½<br />

˜Φ k (q, r, s) =(q 1/N rs) <strong>×</strong><br />

<strong>×</strong><br />

N−1<br />

∏<br />

m=0<br />

N−1<br />

∏<br />

m=0<br />

∏<br />

l,b∈Z,k∈Z− r N<br />

k,l,b>0<br />

k (q, r, s) =(qrs) <strong>×</strong><br />

<strong>×</strong><br />

N−1<br />

∏<br />

∏<br />

l,b∈Z,k∈Z+ r N<br />

k,l,b>0<br />

N−1<br />

∏<br />

(<br />

1 − q k r b s l)1 2<br />

P N−1<br />

n=0 e−2πiln/N c(m,n) (kl,b)<br />

(<br />

1 − q k r b s l)1 2<br />

P N−1<br />

n=0 e2πiln/N c(m,n) (kl,b)<br />

∏<br />

m,n=0 (k,l,b)∈Z<br />

(k,l,b)>0<br />

∏<br />

m,n=0 (k,l,b)∈Z<br />

(k,l,b)>0<br />

{<br />

1 − e 2πim/N q k r b s l}1 2 c(m,n) (kl,b)<br />

{<br />

1 − e −2πim/N q k r b s l}1 2 c(m,n) (kl,b)

ºº½ ØÖÑÒÒØØÛ<strong>×</strong>ØÐÐÔØÒÖ ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÊØÖØÒÖÖÝÓÙØÒÜÔÐØÓÑÔÙØØÓÒ¸ÛØÖÑÒØØÛ<strong>×</strong>ØÐÐÔØÒÖÙ<strong>×</strong>Ò ÓÒ<strong>×</strong><strong>×</strong>ØÒÝÓÒØÓÒ<strong>×</strong><strong>×</strong>ÓÒØÖÑÓÙÐÖÔÖÓÔÖØ<strong>×</strong>ºÏÒN<strong>×</strong>ÔÖѸØ<strong>×</strong>ÓÒ¹<br />

<strong>×</strong>ÓÑÔØÐÛØØÔÖÓÙØÓÖÑÓØ<strong>×</strong>ÓÖØØÚÐØÚÒÒÕº´ºµºÏ ÔÖÑØÖ<strong>×</strong>ºÌ<strong>×</strong>ÔÖÑØÖ<strong>×</strong>ÖÜÝÑÔÓ<strong>×</strong>ÒØÓÒØÓÒØØØÔÖÓÙØÓÖÑÙÐ ØÓÒ<strong>×</strong>ÙÒÕÙÐÝÜØØÛ<strong>×</strong>ØÐÐÔØÒÖºÓÖÓÑÔÓ<strong>×</strong>ØN¸ØÖÖÑÒÙÒØÖÑÒ<br />

<strong>×</strong>ØÒØÓØÒÖÐ<strong>×</strong>ÖÓÑغ<br />

(Z)¸Ò<br />

ÓÖÑÒ̹ÓÖØ<strong>×</strong><br />

ÛÐÐÐÐÙ<strong>×</strong>ØÖØØÔÖÓÙÖÓÖÓÑÔÓ<strong>×</strong>ØNØÒØÜÑÔÐÓΦ )ÖØÑÙÔÒØÓÓÖØ<strong>×</strong>º<br />

2ÖT¹ÒÚÖÒغº¸ØÝÓÖÑÓÖØ<strong>×</strong>ÓÐÒØ<br />

3 (Z)Ò˜Φ3<br />

ÌØÓÒÓTÓÒØF r,s (z 1 , z<br />

ØÓÒÓTµº ÓÒº<br />

2<br />

ÏÚÐÖÝ<strong>×</strong>ÒØØF 0,s (z 1<br />

)ÓÖÑ<strong>×</strong>ÒÐÓÖØÓÐÒØN´ÙÒÖÖÔØ<br />

, z<br />

ÏÒ(r, N) = 1¸ÐÐØF (z 1 , z 2<br />

)ÖÙÔÒØÓm<strong>×</strong>ØÒØÓÖØ<strong>×</strong>ÓÐÒØ ÏÒ(r, N) = m¸ØÒØF (z 1 , z 2<br />

)º<br />

ÓÔÖØÓÒ<strong>×</strong>º<br />

)ÓØÒÖÓÑØØÓÒÓØTÒÖØÓÖ¸Ø<strong>×</strong>Ù<strong>×</strong>ØÓÛÓÖ ÏÛÐÐÙ<strong>×</strong>Ø<strong>×</strong>Ö<strong>×</strong>ÙÐØ<strong>×</strong>ØÓÑÔÓ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>ÓÒØÓÖÑÓØF<br />

ÒÛÖØØÒ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>℄<br />

r,s (z 1 , z 2<br />

ÐÓÒÛØØF r,s (z 1 , z 2<br />

ÓÙØF 0,s (z 1 , z 2 )ÒØÓØÖF (z 1 , z 2 )ÒÓØÒÝØØÓÒÓ<strong>×</strong>ÙØÐSL(2, ÄØÙ<strong>×</strong>ÛÖØØÑÓ<strong>×</strong>ØÒÖÐF 0,s (z 1 , z 2<br />

´ºµ<br />

)ºÓÖÛÂÓÓÖÑ<strong>×</strong>ÓΓ F 0,0 (z 1 , z 2 ) = 8 A(z N 1, z 2<br />

(N)Ò ´ºµ<br />

) ,<br />

´ºµ ÛÖα N (z 1 )<strong>×</strong>ÛعØÛÓÑÓÙÐÖÓÖÑÓΓ ) 2<br />

.<br />

½<br />

N/mº<br />

r,s<br />

r,s r,s<br />

Z)<br />

J<br />

F 0,s (z 1 , z 2 ) = a A(z 1 , z 2 ) + α N (z 1 ) B(z 1 , z 2 ) , s ≠ 0 ,<br />

0<br />

A(z 1 , z 2 ) =<br />

4∑<br />

i=2<br />

(<br />

ϑi (z 1 , z 2 )<br />

ϑ i (z 1 , 0)<br />

) 2 (<br />

ϑ1 (z 1 , z 2 )<br />

, B(z 1 , z 2 ) =<br />

η 3 (z 1 )<br />

0(N)¸F 0,s (z 1 , z 2 )

ÏÒN<strong>×</strong>ÓÑÔÓ<strong>×</strong>ظØÑÒ<strong>×</strong>ÓÒÓÑÓÙÐÖÓÖÑ<strong>×</strong>ØÛØØÛÓ<strong>×</strong>ÖØÖØÒÓÒº ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´ºµ ´ºµ ´º¼µ<br />

ÏÐ<strong>×</strong>ØØÔÓ<strong>×</strong><strong>×</strong>ÐØ<strong>×</strong>ÓÖN =<br />

)<strong>×</strong>Ø<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÓÛعØÛÓÒÐÚÐN<br />

ÒÓÖÑÐÞ<strong>×</strong>ÓØØØ<strong>×</strong>ÓÒ<strong>×</strong>ØÒØÓÒØ<strong>×</strong>ÓÒ´ËÔÔÒܵº<br />

ÛÖE N (z 1<br />

1¸ØÖÖÒÓÓÚÓÙ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>º )ÓÑÔØÐÛØØ<strong>×</strong>ÓÖØ<strong>×</strong>Þº<br />

)ÒÓÐÐÓÛØ<strong>×</strong>ØÖÒ<strong>×</strong>ÓÖ¹<br />

1¸ØÒØÖÛÐÐÓÒ<strong>×</strong>ØÖÒØ<strong>×</strong>º<br />

ÆÜØÛÙ<strong>×</strong>ØSØÖÒ<strong>×</strong>ÓÖÑØÓÒÓÒØÒ<strong>×</strong>ØÞÓÖF 0,s (z 1 , z 2<br />

ÑØÓÒÙÒÖÔÓÛÖ<strong>×</strong>ÓTÒÑØÒ<strong>×</strong>ØÞÓÖα N (z 1<br />

ÏÒ(s, N) =<br />

<strong>×</strong>ÞÓØÖº<br />

0<strong>×</strong>ÛÒØÓÚÒÓÖØÓ<strong>×</strong>ÞØÛÓº ÏÒ(s, N) = m ><br />

0<strong>×</strong>ÓØØØ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØÒÓÖØ<br />

0<strong>×</strong>ÓØØØ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØÒÓÖØ ÏÒN = 4Òs 2¸ØÒb =<br />

ÏÒN = 6Òs 2, 4¸ØÒb = b 3 =<br />

ÓÓÙÖº <strong>×</strong>ÞÓØÛÓº<br />

0<strong>×</strong>ÓØØØ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØÒÓÖØ<strong>×</strong>Þ ÏÒN = 6Òs=3¸ØÒb = b 3 =<br />

<strong>×</strong>ÞÓØÛÓº<br />

0<strong>×</strong>ÓØØØ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØÒÓÖØ ÏÒN = =<br />

ÏÒN = 8Òs=4¸ØÒb = b 3<br />

)º<br />

=<br />

ÙÖØÖ<strong>×</strong>ÑÔÐØÓÒÓÙÖÖÓÑ<strong>×</strong>ÝÑÑØÖÝÓÒ<strong>×</strong>ÖØÓÒ<strong>×</strong>ºF r,s (z 1 , z 2<br />

)º )ºÁØ<br />

) = F −r,−s (z 1 , z 2<br />

ÑÔÐ<strong>×</strong>ØØÛÚØÕÙÚÐÒF 0,s (z 1 , z 2 ) = F 0,N−s (z 1 , z 2<br />

)º ÓÖN = 4¸ÛÒØÓÓÒÐÝÛÓÖÓÙØF (z 1 , z 2 )¸F 0,1 (z 1 , z 2 )ÒF (z 1 , z 2<br />

ÓÖN = 6¸ÛÒØÓÓÒÐÝÛÓÖÓÙØF (z 1 , z 2 )¸F 0,1 (z 1 , z 2 )¸F 0,2 (z 1 , z 2 )ÒF 0,3 (z 1 , z 2<br />

½<br />

8º 4, 6,<br />

α 4 (z 1 ) = b 1 E 2 (z 1 ) + b 2 E 4 (z 1 ) ,<br />

α 6 (z 1 ) = b 1 E 2 (z 1 ) + b 2 E 3 (z 1 ) + b 3 E 6 (z 1 ) ,<br />

α 8 (z 1 ) = b 1 E 2 (z 1 ) + b 2 E 4 (z 1 ) + b 3 E 8 (z 1 ) ,<br />

E N (z 1 ) =<br />

=<br />

=<br />

2<br />

1<br />

2<br />

3<br />

2<br />

8Òs=2, 6¸ØÒb<br />

0,0 0,0<br />

12i<br />

π(N−1) ∂ z 1<br />

[<br />

ln η(z1 ) − ln η(Nz 1 ) ] ,<br />

0,2

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

)º<br />

ÖÐØØÓØÓÔÓÐÓÐÓØ<strong>×</strong>ÓÒK3ÒÒÒØÖÑÒÝ<strong>×</strong>ØÙÝÒØØÓÒÓ ÏÒÓÛÒØÓÜØÙÒØÖÑÒÓÒ<strong>×</strong>ØÒØ<strong>×</strong>Û<strong>×</strong>ÓÒÝÐÓÓÒØØÓÒØÓÒ<strong>×</strong> ÓÖN = 8¸ÛÒØÓÓÒÐÝÛÓÖÓÙØF 0,0 (z 1 , z 2 )¸F 0,1 (z 1 , z 2 )¸F 0,2 (z 1 , z 2 )¸F 0,3 (z 1 , z 2 )<br />

ÒF 0,4 (z 1 , z 2<br />

)ºÌ<strong>×</strong>ØÛÓ<strong>×</strong>Ø<strong>×</strong>ÓÒÙÑÖ<strong>×</strong>Ö ÓÒØÓÙÖÖÓÒØ<strong>×</strong>¸c 0,s<br />

b<br />

(0)ÓF 0,s (z 1 , z 2<br />

´º½µ<br />

ØÖÓÙÔÓÒH∗ (K3, Z)¿½℄ºÄØQ 0,sØÒÙÑÖÓg s¹ÒÚÖÒØÐÑÒØ<strong>×</strong>ÓH ´ÛÖgÒÖØ<strong>×</strong>Z NµºÐ<strong>×</strong>ÓQ 0,s = Nc 0,s<br />

0 (0) + 2Nc 0,s<br />

1 (−1) .<br />

(−1)ÓÙÒØ<strong>×</strong>ØÒÙÑÖÓgsÒÚÖÒØ(0, 0,s<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> 2)ÓÖÑ<strong>×</strong>ÓÒK3ºÓÖ<strong>×</strong>ÝÑÔÐØ<br />

2ºÏØÙ<strong>×</strong>ÓØÒØÖÐØÓÒ ´º¾µ<br />

1 0)Ò(0,<br />

ÒÚÓÐÙØÓÒ<strong>×</strong>¸Ø<strong>×</strong>ÓÖÑ<strong>×</strong>ÖÒÚÖÒØÒÒNc (−1) =<br />

0¸ÐÐÓÖÑ<strong>×</strong>ÓÒØÖÙØÒ<br />

0 (0) = Q0,s − 4 .<br />

ÚÒØÝÐ<strong>×</strong>ÔÓÒÒÓÑÔÙØØQ 8ºÏ<br />

ÓÖÔÖÑNÌÝÐ<strong>×</strong>Ô<strong>×</strong>1k+2 N k+2ºÏÒ¸s ÒQ0,0 = 24ºÓÖÒÝs 0¸ÓÒ<strong>×</strong>Q = k + 2ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØNc ÒNc 0,s<br />

0 (0) = k − 2ÓÖs N = 4ÌÝÐ<strong>×</strong>Ô<strong>×</strong>1 2 2 4 4ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØQ = Q 0,3 = 4ÒQ =<br />

ØÙ<strong>×</strong>ÓØÒ4c 0,s<br />

0 (0) = 0ÓÖs 1, 3ÛÐ4c<br />

N= 6ÌÝÐ<strong>×</strong>Ô<strong>×</strong>1 2 2 3 2 6 2ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØQ = Q 0,5 = 2ÒQ Q 0,4 = 6ÒQ = 8ºÌÙ<strong>×</strong>ÓÒ<strong>×</strong>6c<br />

0 (0) = 2ÓÖs ÙÖØÖ¸ÓÒ<strong>×</strong> 4º<br />

N = 8ÌÝÐ<strong>×</strong>Ô<strong>×</strong>1 2 1 4 1 8 2ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØQ ÒQ0,2 = Q 0,6 = 4ÒQ = 8ºÌÙ<strong>×</strong>ÓÒ<strong>×</strong>8c<br />

(0)ºÓÖÔÖÑNÐÐØ<br />

0 (0) = 0ÓÖs 2, 6Ò8c0 (0) =<br />

)ÓÖÔÖÑNÖ ´º¿µ<br />

c 1 (−1) = 2 N<br />

Ð<strong>×</strong>Ó¸<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒÝÓÒØc 0,s (0)ÓÒ<strong>×</strong>k= 0 0<br />

ÓÒØ<strong>×</strong>ÖÜÝØÓÚÓÒØÓÒ<strong>×</strong>ÒÓÒÒ<strong>×</strong>ØF r,s (z 1 , z<br />

½<br />

2<br />

Nc 0,s<br />

b<br />

(−1)Òc 0,s<br />

∗<br />

Nc 0,s<br />

1<br />

4º<br />

=<br />

0 (0) =<br />

≠ 0,s 0,0<br />

0<br />

≠ 0<br />

4º<br />

4 0,1 0,2 = 0,2<br />

2 0,1<br />

0,3 0,s<br />

0 (0) = −2ÓÖs=1, 5¸6c 0,3<br />

0<br />

6c 0,s = 2, 2 0,1 0,4 0,s<br />

8c 0,s = 0,4<br />

0,0<br />

0 (0) = 20<br />

N<br />

, c 0,s<br />

1<br />

2<br />

(K3, Z)<br />

(0) = 20<br />

4Ò<br />

0,2 =<br />

7¸<br />

(0) =<br />

= Q 0,3 = Q 0,5 = Q 0,7 = 2<br />

0 (0) = −2ÓÖs=1, 3, 5,<br />

∑ N−1<br />

s=0 c0,s

ÚÒÝ ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´ºµ<br />

F 0,0 (z 1 , z 2 ) = 8 N A(z 1, z 2 )<br />

F 0,s 8<br />

(z 1 , z 2 ) =<br />

N(N + 1) A(z 1, z 2 ) − 2<br />

N + 1 B(z 1, z 2 )E N (z 1 )<br />

F r,rk 8<br />

(z 1 , z 2<br />

8¸ØÖÖÚÙÒØÖÑÒÔÖÑØÖ<strong>×</strong>ºÌ<strong>×</strong><br />

) =<br />

N(N + 1) A(z 2<br />

1, z 2 ) +<br />

N(N + 1) B(z 1, z 2 )E N<br />

6¸ØÖÖØÛÓ )ºÓÖ<br />

( z 1 + k<br />

N )<br />

ÓÖÓÑÔÓ<strong>×</strong>ØN¸ÓÛÚÖ¸ÓÒÒ<strong>×</strong>ÑÓÖÓÒØÓÒ<strong>×</strong>ØÓÓÑÔÙØØF r,s (z 1 , z 2<br />

4Ò<br />

N = 4¸ØÖ<strong>×</strong>ÓÒÙÒØÖÑÒÔÖÑØÖÒF 0,1 (z 1 , z 2 )ºÓÖN =<br />

ÙÒØÖÑÒÔÖÑØÖ<strong>×</strong>ÒÓÖN =<br />

ÛÐÐÚØÓÐØÛØÓÒ<strong>×</strong>Ý<strong>×</strong><strong>×</strong><strong>×</strong>ºÄØÙ<strong>×</strong>ÓÓ<strong>×</strong>ØÜÑÔÐÓN =<br />

)ÒÖÓÑØÑØÔÖÓÙØÓÖÑÓ<br />

ºº¾<br />

ÐÐÙ<strong>×</strong>ØÖØØÔÖÓÙÖÓÖÓÑÔÙØÒØF Ï<strong>×</strong>ØÖØÝÒÒ r,s (z 1 , z 2<br />

(Z)ÒΦ ØÓÖÖ<strong>×</strong>ÔÓÒÒÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φ3 3<br />

ÈÖÓÙØÓÖÑÓΦ 3<br />

´ºµ<br />

(Z)<br />

ÒØÓÒ<strong>×</strong>¿¾℄ l)Ø<strong>×</strong>ÓÙÖÖÓÒØ<strong>×</strong>ºÌÔÖÓÙØÓÖÑÖÛÖØØÒÙ<strong>×</strong>ÒØÓÚ<br />

̂F<br />

´ºµ<br />

ÒÐØĉa (n,<br />

ØÐÒØÓØÕÙØÓÒº<br />

l)ÒÐÐÓÙÖÖÒ<strong>×</strong>ÓÚØÓÖÙ<br />

4¸ÛÓØÒ<br />

1 − ( q n r l s m) ) 4 ĉ 1<br />

ÛÖÛÚÓÑØØØÖÙÑÒØÓĉaØ<strong>×</strong>(nm, ´ºµ<br />

ËÔÐÞÒØÒÖÐÓÖÑÙÐÓÚØÓØ<strong>×</strong>ÓN =<br />

)Ö<strong>×</strong>ÒÒÕº´ºµºÌ<strong>×</strong>Ð<strong>×</strong>ØÓÓÖÑÙÐÓÖØ − b)E 12 4(z 1 )B(z 1 , z 2 )<br />

ÛÖA(z 1 , z 2 )ÒB(z 1 , z 2<br />

½<br />

Φ 3 (Z) = qrs ∏<br />

(n,l,m)<br />

a (z 1 , z 2 ) =<br />

(Z)º<br />

3∑<br />

F a,b (z 1 , z 2 ) ,<br />

b=0<br />

( )ĉ0 1 − q n r l s m −ĉ 2 (<br />

<strong>×</strong> 1 − ( q n r l s m) ) 2 ĉ 2 −ĉ 1 (<br />

<strong>×</strong><br />

̂F 0 (z 1 , z 2 ) = 10 3 A(z 1, z 2 ) + (2b + 1 3 )E 2(z 1 )B(z 1 , z 2 ) + ( 5 6 − 2b)E 4(z 1 )B(z 1 , z 2 )<br />

̂F 1 (z 1 , z 2 ) = 4 3 A(z 1, z 2 ) − 2bE 2 (z 1 )B(z 1 , z 2 ) − ( 5<br />

̂F 2 (z 1 , z 2 ) = 2A(z 1 , z 2 ) + 1 2 E 2(z 1 )B(z 1 , z 2 ) − ( 5 6 − 2b)E 4(z 1 )B(z 1 , z 2 ) ,

Ö<strong>×</strong>ØØÛÓÓÙÖÖÓÒØ<strong>×</strong> ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

´ºµ<br />

ĉ 0 (−1) = 5 + 1 + 5 = 2 , 6 3 6 ĉ0 (0) = 25 − 7 = 6 3 3<br />

ÛÖØÓÙØÐÐØØÖÑ<strong>×</strong>ÛØm=0ÒØÔÖÓÙØÓÖÑÙÐ<strong>×</strong>ÛÒÓÛÚØÖÑÒØØ )Ò<br />

+ 2b 6<br />

6ºÌ<strong>×</strong>ÚÖ<strong>×</strong>ØÓØÖÑ<strong>×</strong>ÓØÓÖÑ<br />

(Z)ºÌ<strong>×</strong>Ü<strong>×</strong>ØÙÒÜÓÒ<strong>×</strong>ØÒØb=−1/12ºÏÒÒÓÛ ÏÒĉ1 (−1) = ĉ 2 (−1) = 0Ð<strong>×</strong>ÛÛÐÐÚØÖÑ<strong>×</strong>ÓØØÝÔ(1 r 2 )Ò(1 r 4<br />

ØÔÖÓÙØÜÔÒ<strong>×</strong>ÓÒÓÖΦ 3<br />

ĉ<br />

Ì<strong>×</strong>Ö<strong>×</strong>ÛØØ´ÒÒØ<strong>×</strong>ØÓµØÖÑ<strong>×</strong>ØØÔÔÖÖÓÑØÔÖÓÙØÜÔÒ<strong>×</strong>ÓÒÓØ<br />

1 (0) = 4Òĉ (0) =<br />

ØÚ<strong>×</strong> )º<br />

φ<br />

ËÒÛÚÜØÓÒ<strong>×</strong>ØÒØb¸ÛÒÒÓÛÛÖØÜØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÓÖØF a,b (z 1<br />

´ºµ<br />

, z 2<br />

Ò<br />

´º¼µ<br />

̂F<br />

2<br />

ĉ 1 (−1) = 1 − 5 − b = −b − 1 , ĉ 1 (0) = 25<br />

3 12 12<br />

ĉ 2 (−1) = 1 + 1 − 5 + 2b = 2b + 1 , ĉ 2 (0) = 17 − 4b<br />

2 2 6 6 3<br />

−<br />

∞∏<br />

(1 − q n ) 0 (1 − q 2n ) 2 (1 − q 4n ) 4 .<br />

n=1<br />

3,1 (z 1 , z 2 ) = ϑ2 1 (z 1, z 2 )<br />

η(z 1 ) 6 η(z 1 ) 4 η(2z 1 ) 2 η(4z 1 ) 4 .<br />

F 0,0 (z 1 , z 2 ) = 2A(z 1 , z 2 )<br />

[<br />

F 0,1 (z 1 , z 2 ) = F 0,3 (z 1 , z 2 ) = 1A(z 3 1, z 2 ) +<br />

F 0,2 (z 1 , z 2 ) = 2A(z 3 1, z 2 ) + 1E 3 2(z 1 )B(z 1 , z 2 )<br />

[<br />

F 1,k (z 1 , z 2 ) = F 3,3k (z 1 , z 2 ) = 1A(z 3 1, z 2 ) + − 1 E (<br />

z1 +k<br />

24 2 2<br />

F 2,2k (z 1 , z 2 ) = 2A(z 3 1, z 2 ) − 1E (<br />

z1<br />

)<br />

+k<br />

6 2 B(z1 , z<br />

2 2 )<br />

[<br />

]<br />

F 2,2k+1 (z 1 , z 2 ) = 1A(z 5<br />

3 1, z 2 ) + E 12 2(z 1 ) − 1E 2 4(z 1 ) B(z 1 , z 2 )<br />

−<br />

]<br />

− 1 E 12 2(z 1 ) + 1E 2 4(z 1 ) B(z 1 , z 2 )<br />

)<br />

+<br />

1<br />

8 E 4<br />

(<br />

z1 +k<br />

4<br />

0 (z 1 , z 2 ) = 10 3 A(z 1, z 2 ) + 1 6 E 2(z 1 )B(z 1 , z 2 ) + E 4 (z 1 )B(z 1 , z 2 )<br />

̂F 1 (z 1 , z 2 ) = 4 3 A(z 1, z 2 ) + 1 6 E 2(z 1 )B(z 1 , z 2 ) − 1 2 E 4(z 1 )B(z 1 , z 2 )<br />

) ] B(z 1 , z 2 )<br />

̂F 2 (z 1 , z 2 ) = 2A(z 1 , z 2 ) + 1 2 E 2(z 1 )B(z 1 , z 2 ) − E 4 (z 1 )B(z 1 , z 2 )<br />

½¼

ºº¿ ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong> ÈÖÓÙØÓÖÑÙÐÓÖ˜Φ3 (Z)<br />

(Z)<strong>×</strong><br />

´º½µ ÌÔÖÓÙØÓÖÑÙÐÓÖ˜Φ3<br />

)ÖØÓÙÖÖÓÒØ<strong>×</strong>ÓØ<br />

b=0 ω−bm c (a,b) (4nm−l 2 )<br />

ÓÐÐÓÛÒÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong> )ºÇÒÒÐ<strong>×</strong>ÓÔÖÓÚØØÐÐØÜÔÓÒÒØ<strong>×</strong>ØØÔÔÖ<br />

(Z)ÖÐÐÚÒÒØÖ<strong>×</strong>ºÇÒÒ<strong>×</strong>ÓÛØØØ<br />

ÛÖω=exp( 2πı)<strong>×</strong>ÙÖÓÓØÓÙÒØݸc<br />

(4nm −l 2 3<br />

ØÛ<strong>×</strong>ØÐÐÔØÒÖ¸F (a,b) (z 1 , z 2<br />

ÒØÔÖÓÙØÓÖÑÙÐÓÖΦ ÐÐÚÒØÖÐÓÙÖÖÓÒØ<strong>×</strong>¸<strong>×</strong>ÔÔÒÜ℄º<strong>×</strong>ØÖØÓÖÛÖÙØØÓÙ<strong>×</strong>Óѹ<br />

3 (Z)Ò˜Φ3<br />

ÔÙØØÓÒØÒ<strong>×</strong>ÓÛ<strong>×</strong>ØØÐÐÜÔÓÒÒØ<strong>×</strong>ÖÚÒÒØÖ<strong>×</strong>ºÌ<strong>×</strong>ÛÐÐÑÔÓÖØÒØØÓÙ<strong>×</strong>ÛÒ<br />

ÓÔÖØÓÒÓØÒ<strong>×</strong>ÕÙÖÖÓÓØ<strong>×</strong>ØÓÚк (Z)ÒÒØÜÔÓÒÒØ<strong>×</strong>ØÓÚÒÒØÖ<strong>×</strong>ÓÖØ<br />

(Z)<strong>×</strong><strong>×</strong>ÕÙÖÖÓÓØ<strong>×</strong>³<br />

ÒØÖÐØÝÔÖÓÔÖØ<strong>×</strong>ÓØÒÙ<strong>×</strong>¹ØÛÓØØÓÒ<strong>×</strong>ØÒØ<strong>×</strong>º ÇÒØ<strong>×</strong>ÙÑ<strong>×</strong>¸ØÒØÖÐØÝÓÓÒØ<strong>×</strong>ÒØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓÐÐÓÛ<strong>×</strong>ÖÓÑØ<br />

(Z)Ò∆ ÛÓÒ<strong>×</strong>ØÖÙØØÔÖÓÙØÓÖÑ<strong>×</strong>ÓØÑÓÙÐÖÓÖÑ<strong>×</strong>˜∆k/2 k/2<br />

(Z)ÒΦ ÓØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk<br />

º ÌØÚ<strong>×</strong>ÓÖØÝÔÁÁÑÓÐ<strong>×</strong><br />

k<br />

(Z)¸ÓÖØØÝÔÁÁÑÓÐ<strong>×</strong>ÚØ<br />

ØØÚ<strong>×</strong><strong>×</strong><strong>×</strong>ÒØÀÄÑÓÐ<strong>×</strong>´Õº´ºµµºÏÛÐÐ<strong>×</strong>ØØØÑÙÐØÔÐØÚ ØÚÐغÌ<strong>×</strong><strong>×</strong><strong>×</strong>ÑÐÖØÓÛØÛ<strong>×</strong>ÓÒÓÖØÀÄÑÓÐ<strong>×</strong>ºÏÖ<strong>×</strong>ØÓØÒ<br />

)<strong>×</strong>ÓÖºÌÒ ÏÛÐÐÒÓÛÓÒ<strong>×</strong>ØÖÙØØÑÓÙÐÖÓÖÑ<strong>×</strong>¸Ψ k (Z)ÒΨ ØØÝÔÁÁ<strong>×</strong>ØÖÒ´ËÐ<strong>×</strong>ÓØ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÒ¾ºº½µº ØØØÐØÖÐÐÝÖ<strong>×</strong>ØØ<strong>×</strong>ÒØØÝÔÁÁÑÓÐÖ<strong>×</strong>ÖÓÑÓ<strong>×</strong>ÓÒÐعÑÓÚÖ<strong>×</strong>Ó<br />

η¹ÔÖÓÙØ<strong>×</strong>ØØÔÔÖÒØÀÄÑÓÐØÖÔÐÝη¹ÕÙÓØÒØ<strong>×</strong>ºÌ<strong>×</strong>ÖØ<strong>×</strong>Ø ØÒÖØÒÙÒØÓÒÓÖÐØÖÐÐÝÖ1 2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÐÐØg ρ (z 1<br />

½½<br />

˜Φ 3 (Z) = q 1/4 rs<br />

3∏<br />

∏<br />

a l,m∈Z,<br />

n∈Z+ a 4<br />

(a,b)<br />

(1 − q n r l s m ) P 3<br />

[4A(z 1 , z 2 ) − B(z 1 , z 2 )] /12 , [E 2 (z 1 ) − 1]/24Ò[E 4 (z 1 ) − 1]/8<br />

k

ºº½ ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÐØÖÐÐÝÖ<strong>×</strong>ØØ<strong>×</strong>ÔÔÖ<strong>×</strong>ÜØØÓÒ<strong>×</strong>ÓØØÝÔÁÁ<strong>×</strong>ØÖÒºÁÒÔÖØÙÐÖ¸Ø <strong>×</strong>ÑÒØÓÒÖÐÖ¸ÛÛÐÐÒÓÙÖÖÒØ<strong>×</strong>ÓÒ<strong>×</strong>ÖÔØÓÒºÁÒØ<strong>×</strong><strong>×</strong>¸<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong><br />

ÒÖÝ<strong>×</strong>ÓÑÒØÝØÓÒØÖÙØÓÒÖÓÑØØÛ<strong>×</strong>Ø<strong>×</strong>ØÓÖ<strong>×</strong>ØØ<strong>×</strong>ºÏÛÐÐÓÑÔÙØ<br />

ÓÙÒØÒÐØÖÐÐÝÖ1<br />

ÖÒØÖÓÙÒ<strong>×</strong>ØØÒÛÐÐÓÛÐÐÜØØÓÒ<strong>×</strong>ØØÖÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØÐÚÐÑØÒº<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>Ö<strong>×</strong>ÛÒØÖعÑÓÚÖ<strong>×</strong> ØÐØÖÐÐÝÖ<strong>×</strong>ØØ<strong>×</strong>ÒØÛ<strong>×</strong>Ø<strong>×</strong>ØÓÖº1<br />

Ó<strong>×</strong>ÐÐØÓÖ<strong>×</strong>¸Ó<strong>×</strong>ÓÒÒÖÑÓÒ¸ÚÒØÖÑÓÒÒØÏØØÒÒÜ<strong>×</strong>ÚÒÝØ <strong>×</strong>ØÓÖÒÒØÐعÓÒÙ¸ÓÒ<strong>×</strong>ØÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÒÔÖÓÖÑÓÒ<strong>×</strong>ºÐÐ<br />

6ºÁÒØÊÑÓÒ <strong>×</strong>ÛÖѹÙÔ¸ÓÒ<strong>×</strong>ÖØÐعÑÓÚÖ<strong>×</strong>ÓØØÝÔÁÁ<strong>×</strong>ØÖÒÓÒT Fµ ÔÖÓÙØÓØÓ<strong>×</strong>ÓÒ´ÒØÝW ÆÓØØØÛÚÒÓØÓÒ<strong>×</strong>ÖØÞÖÓ¹ÑÓ<strong>×</strong>ºÌ<strong>×</strong><strong>×</strong>ÜÔØ<strong>×</strong>ØÖ<strong>×</strong>ÔÖØ<br />

BµÒÖÑÓÒÓÒØÖÙØÓÒ<strong>×</strong>´ÒØÝW ´º¾µ<br />

8<br />

(1 − q ))<br />

Ó<strong>×</strong>ÐÐØÓÖÔÖØØÓÒÙÒØÓÒ<strong>×</strong>ÒÓØÙÒØÝÒÕÙÐ<strong>×</strong> ÒÐÐØÓÒÓÓ<strong>×</strong>ÓÒÒÖÑÓÒÓÒØÖÙØÓÒ<strong>×</strong>ÒØÏØØÒÒܺÇÓÙÖ<strong>×</strong>¸Ø<br />

n = 1 .<br />

ÁÒØÖ<strong>×</strong>ØÒÐݸØ<strong>×</strong><strong>×</strong>ÕÙÓØÒØÓη¹ÙÒØÓÒ<strong>×</strong>ØÐÚÐ2´Ì<strong>×</strong>ÔÔÖ<strong>×</strong>ÒØÓÒ<strong>×</strong>ØÖÙØÓÒÓ ´º¿µ<br />

ØÅÓÒ<strong>×</strong>ØÖÄ<strong>×</strong>ÙÔÖÐÖ℄µ<br />

ØÏØØÒÒÜÛÐØØÒعÔÖÓÖÑÓÒ<strong>×</strong>ÚйÒØÖÑÓÒÒ −1ØÓ ÌØÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÚÒØÖÑÓÒÒÓÒØÖÙØØÓÖÓη(τ)<br />

½¾<br />

N = 1<br />

N = 2<br />

( ) ( 8<br />

1 ∏<br />

W B <strong>×</strong> W F = ∏<br />

n (1 − <strong>×</strong><br />

qn )<br />

n<br />

( ) ( ) 8 8<br />

1 ∏<br />

Z B <strong>×</strong> Z F = ∏<br />

n (1 − <strong>×</strong> (1 + q n ) = η(2τ)8<br />

qn )<br />

η(τ) . 16 n

ÓÒØÖÙØη(τ/2)/η(τ)ºÇÒ<strong>×</strong><br />

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

( ) ( ) 8 8<br />

1 ∏<br />

´ºµ<br />

W B <strong>×</strong> W F = ∏<br />

n (1 − <strong>×</strong> (1 − q n+1/2 )<br />

qn )<br />

n<br />

ÒÖÐÞØ<strong>×</strong>ÒÓØÓÒØÓÓÒÙÝÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÐÑÒØ<strong>×</strong>ÓÖØÖÖÝ<strong>×</strong>ÖØÖÓÙÔ<strong>×</strong>ºÁÒÓÙÖ ÊÐÐØØÝÐ<strong>×</strong>Ô<strong>×</strong>ÖÔÖ<strong>×</strong>ÒØÓÒÙÝÐ<strong>×</strong><strong>×</strong><strong>×</strong>ÓÔÖÑÙØØÓÒºÖÑ<strong>×</strong>Ô<strong>×</strong><br />

16º<br />

= η(τ/2)8 1<br />

=<br />

η(τ) 16 g˜ρ (τ/2) ,<br />

ÛÖØÖÑ<strong>×</strong>Ô˜ρ = 1 −8 2<br />

1½¼¿℄<strong>×</strong>Û<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÐØÖº ÜÑÔиØ<strong>×</strong>ÖØÖÓÙÔØÙÖÒ<strong>×</strong>ÓÙØØÓØÓÒÛÝÖÓÙÔCo<br />

ÖÑÓÒ<strong>×</strong>ÚÖØÓÒÐÑÓÒÓ±1/3ÒÓÒØÖÙØη(τ/2)/η(τ)ºÇÒ<strong>×</strong> ÏØØÒÒܺÏÐØØÛÓÓØÖÓ<strong>×</strong>ÓÒ<strong>×</strong>ÚÑÓÒÖØÓÒÐÑÓÒÓ±1/3ºÌ<br />

−1ØÓØ<br />

N = 3<br />

Ì<strong>×</strong>ÜÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÚÒØÖÑÓÒÒÓÒØÖÙØØÓÖÓη(τ)<br />

W B <strong>×</strong> W F 9º ´ºµ<br />

1<br />

= ∏<br />

n (1 − qn ) 6 (1 − q n+1/3 )(1 − q n−1/3 ) <strong>×</strong> ∏ (1 − q n+1/3 ) 4 (1 − q n−1/3 ) 4<br />

n<br />

= η(τ/3)3 1<br />

=<br />

η(τ) 9 g˜ρ (τ/3) ,<br />

ÛÖØÖÑ<strong>×</strong>Ô˜ρ = 1 −3 3<br />

N = 4<br />

η(τ/2)/η(τ)ºÇÒ<strong>×</strong><br />

ÏØØÒÒܺÏÐØØÛÓÓØÖÓ<strong>×</strong>ÓÒ<strong>×</strong>ÖÒØÔÖÓÒÚÑÓÒÖØÓÒРйÒØÖÐÑÓÒºÌÖÑÓÒ<strong>×</strong>ÚÖØÓÒÐÑÓÒÓ±1/4ÒÓÒØÖÙØ<br />

−1ØÓØ Ì<strong>×</strong>ÜÔÖÓÓ<strong>×</strong>ÓÒ<strong>×</strong>ÚÒØÖÑÓÒÒÓÒØÖÙØØÓÖÓη(τ)<br />

W B <strong>×</strong> W F 4º ´ºµ<br />

1<br />

= ∏<br />

n (1 − qn ) 6 (1 − q n+1/2 ) <strong>×</strong> ∏ (1 − q n+1/4 ) 4 (1 − q n−1/4 ) 4<br />

2<br />

n<br />

= η(τ/4)4<br />

η(τ) 4 η(τ/2) = 1<br />

½¿<br />

6 g˜ρ (τ/4) ,<br />

ÛÖØÖÑ<strong>×</strong>Ô˜ρ = 1 −4 2 6 4

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

ÓÙÖÓ<strong>×</strong>ÓÒ<strong>×</strong>ÚÒØÖÑÓÒÛÐØÓØÖÓÙÖÚÖØÓÒÐÑÓÒÓr/5ÛØ ÒÔÖ<strong>×</strong>ºÇÒ<strong>×</strong><br />

4ÓÙÖÖÒ<br />

r = 1, 2, 3, 4ºÌÖÑÓÒ<strong>×</strong>ÔÔÖÛØÖØÓÒÐÑÓÒÓr/5ÛØr 1, 2, 3,<br />

´ºµ<br />

ÅÙÐØÔÐØÚη¹ÕÙÓØÒØ<strong>×</strong> 5º ÛÖØÖÑ<strong>×</strong>Ô˜ρ = 1 −1 5<br />

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2¹ÈË<strong>×</strong>ØØ<strong>×</strong><strong>×</strong>ÚÒÝη¹ÕÙÓØÒØ<strong>×</strong>ØØÖ<strong>×</strong><strong>×</strong>ÓØÛØØÖÑ<br />

ÌÔÔÖÒÓØη¹ÕÙÓØÒØ<strong>×</strong>ÒÖÑ<strong>×</strong>Ô<strong>×</strong>ÑÝÙÒÖ<strong>×</strong>ØÓÓ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºÁØ<strong>×</strong><br />

ÌÓÙÒØÒÓ1<br />

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ÖÐ<strong>×</strong>ÙÔÖ<strong>×</strong>ØÖÒÑÙ<strong>×</strong>ØØÙ<strong>×</strong>ÒÐÑÒØÓCo<br />

ÓCo<br />

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ÌÓÒ³<strong>×</strong>ÓÖN ÏØÒÅÖØÒÓÖÙ<strong>×</strong>ÙÐÓÖÖ<strong>×</strong>ÔÓÒÒÛÐÖØ<strong>×</strong>ÔÓÒغ<br />

3ÚÒÖÚÒ¿½℄ÒÓÙÖÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÖÛØØ<br />

= 2ÖÒÓØÒÓÖÑ<strong>×</strong>ÓÖT Ìη¹ÕÙÓØÒØ<strong>×</strong>ÓÖN = 2,<br />

½<br />

N = 5<br />

=<br />

W B <strong>×</strong> W F =<br />

1<br />

∏n (1 − qn ) 4 ∏ 4<br />

r=1 (1 − qn+r/5 ) <strong>×</strong> (∏ n<br />

= η(τ/5)<br />

η(τ) 5 =<br />

1<br />

g˜ρ (τ/5) ,<br />

4∏<br />

(1 − q n+r/5 )) 2<br />

r=1<br />

=

ÔØÖºÓÒ<strong>×</strong>ØÖÙØÒØÅÓÙÐÖÓÖÑ<strong>×</strong><br />

k ˜ρ ρ χ ( a c d b ) N G<br />

2 1 −8 2 16 1 16 2 −8 2 Z 2<br />

(<br />

1 1 −3 3 9 1 9 3 −3 −3<br />

)<br />

3 Z<br />

d<br />

3<br />

(<br />

1 1 −4 2 6 4 4 1 4 2 6 4 −4 −1<br />

)<br />

ºº¾ ÈÖÓÙØÓÖÑÙÐÓÖØØÝÔÁÁÑÓÐ<strong>×</strong><br />

5ρ<strong>×</strong>ØÖÑ<strong>×</strong>Ô¸k+2<strong>×</strong>ØÛØÓØη¹ÕÙÓØÒغ<br />

4 Z<br />

d<br />

4<br />

0 1 −1 5 5 1 5 5 −1 5 Z 5<br />

3ØÝÔÁÁÑÓÐ<strong>×</strong>¿½℄º<br />

Ìкη¹ÕÙÓØÒØ<strong>×</strong>ÛØN ≤<br />

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(Z)ÖÒØÐØÓØÓ<strong>×</strong>ÔÔÖÒÒØÀÄÑÓÐ<strong>×</strong><br />

z)¸Ú¸ÂØÖÒËÒÒ<br />

4ºÌ Ú¸ÂØÖÒËÒÚÔÖÓÚÔÖÓÙØÓÖÑÙÐÓÖØN = 2,<br />

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F (0,0) (τ, z) = 0<br />

F (0,s) (τ, z) = 16 ( πs<br />

)<br />

( ) ( )<br />

ϑ1 τ, z +<br />

s ϑ1<br />

N sin4 N τ, −z +<br />

s<br />

N<br />

(<br />

N ϑ s<br />

) 2<br />

1 N<br />

ÓÖ1 ≤ s ≤ N − 1,<br />

ÌØÛ<strong>×</strong>ØÐÐÔØÒÖÓÖØÝÔÁÁÑÓÐ<strong>×</strong>ÒÖÛÖØØÒÒØÖÑ<strong>×</strong>ÓØÐÐÔØÒÖ<br />

(<br />

F (r,s) 4 N ϑ 1 τ, z +<br />

s<br />

(τ, z) =<br />

+ r τ) (<br />

ϑ<br />

N N 1 τ, −z +<br />

´ºµ<br />

s<br />

+ r τ) N N<br />

(N − 1) 2 (<br />

ϑ s<br />

1 + r τ) 2<br />

,<br />

ÛÖØØÒ<strong>×</strong> z)Ò<br />

N N<br />

ÓÖ1 ≤ r ≤ N − 1¸0 ≤ s ≤ N − 1 .<br />

ØØÔÔÖÒØÀÄÑÓÐ<strong>×</strong>ºÓÖØZ 3ÓÖÓÐÓØØÝÔÁÁÑÓиØÝÒÛÖØØÒ<strong>×</strong><br />

2ÓÖÓÐÓØØÝÔÁÁÑÓиF (r,s) (τ,<br />

F (r,s)<br />

II<br />

(τ, z) = 2F (r,s)<br />

(r,s)<br />

N=2 CHL<br />

(τ, z) − F<br />

ËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÖØÀÄÑÓÐ<strong>×</strong>ºÁÒÓÖÖØÓ<strong>×</strong>Ø<strong>×</strong>¸ÛÖÛÖØØη¹ÕÙÓØÒØ<strong>×</strong> 3ÒÛÖØØÒÒØÖÑ<strong>×</strong>ÓØ<br />

ÒÓÖØZ F (r,s)<br />

II<br />

(τ, z) = 3 2 F (r,s)<br />

N=3 CHL (τ, z) − 1 2 F (r,s)<br />

Ì<strong>×</strong>ÑÔÐ<strong>×</strong>ØØØØÝÔÁÁÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÖN = 2,<br />

½<br />

´ºµ<br />

´º¼µ<br />

N=1 Het. (τ, z),<br />

N=1 Het. (τ, z).

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.<br />

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Ψ 2 (Z) = Φ 6(Z) 2<br />

Φ 10 (Z) .<br />

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3<strong>×</strong>¸ÛÒØØg 4ØÝÔÁÁÑÓдºµ<br />

´ºµ<br />

º<br />

ÏÓÒÐÙØ<strong>×</strong><strong>×</strong>ØÓÒÛØÓÒØÙÖÐÓÖÑÙÐÓÖØN =<br />

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ÒΨ .<br />

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∆ 5 (Z)<br />

½ ÝÓÒ<strong>×</strong>ØØ<strong>×</strong>ÒN =<br />

g 4 (τ) = η16 (2τ)<br />

= η16 (2τ)η 16 (τ)<br />

,<br />

η 8 (τ) η 24 (τ)<br />

g 3 (τ) = η9 (3τ)<br />

η 3 (τ) = η9 (3τ)η 9 (τ)<br />

η 12 (τ)<br />

˜Ψ 1 (Z) = ∆ 3(Z)∆ 3/2 (Z) 2<br />

Ψ 1 (Z) = ∆ 2(Z) 3<br />

∆ 5 (Z) ,<br />

˜Ψ 2 (Z) = ˜Φ 6 (Z) 2<br />

Φ 10 (Z)<br />

˜Ψ 1 (Z) = ˜∆ 2 (Z) 3<br />

∆ 5 (Z)<br />

1<br />

(Z) = ˜∆ 3 (Z)˜∆ 3/2 (Z) 2<br />

∆ 5 (Z)

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g r ho(z<br />

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(Z)Ò∆ ØÔÖÓÙØÓÖÑÓ˜∆k<br />

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(z 1 ,z 2 ) 2<br />

η(z 1 ) 6<br />

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6<br />

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ÒÓÑÒØÓÖÓÖÑÙÐ<strong>×</strong><strong>×</strong>ÔÐ<strong>×</strong>ÓØÑÓÖÒÖÐÏÃÖØÖÓÖÑÙÐÓÖÄ<br />

(Z)¸ØØÒÖØ<strong>×</strong> ÒÓÑÒØÓÖÓÖÑÙÐÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖG <strong>×</strong>ÙÔÖÐÖ<strong>×</strong>℄ºÌÏÃÖØÖÓÖÑÙÐÔÔÐØÓØØÖÚÐÖÔÖ<strong>×</strong>ÒØØÓÒÚ<strong>×</strong><br />

1<strong>×</strong>ÖÐØØÓØÑÓÙÐÖÓÖÑ∆ 5<br />

ÓËÔ(2,<br />

ØÏÃÒÓÑÒØÓÖÓÖÑÙкÄØGÃÅÄ<strong>×</strong>ÙÔÖÐÖÒWØ<strong>×</strong>ÏÝÐÖÓÙÔº ÐÖ<strong>×</strong>ÛÚ<strong>×</strong>ØÖØÖ<strong>×</strong>ÓÒØÖÐ<strong>×</strong>ØÛØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓÃÅÄ<br />

2ºÌÏÝйùÓÖÖ<strong>×</strong>´Ïõ¹<br />

Z)¸ØØØÖÒ<strong>×</strong>ÓÖÑ<strong>×</strong>ÛØÖØÖ℄ºÌÑÓÙÐÖÓÖÑ¸Φ 10<br />

ØÒÖÝÓ1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong><strong>×</strong>ÕÙÐØÓ∆ 5 (Z)<br />

ÌÒ¸ØÏÃÒÓÑÒØÓÖÒØØÝÓÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖ½℄G<strong>×</strong><br />

+ÒÓØØ<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖÒρØÏÝÐÚØÓÖº<br />

´º½µ<br />

ÄØL Ø´wµ<strong>×</strong>ÒØÓ±1ÔÒÒÓÒÛØÖw<strong>×</strong>ØÔÖÓÙØÓÒÚÒÓÖÓÒÙÑÖ<br />

(1 − e −α )ÑÙÐØ(α) (Øw) +¸½¸½½¼¸½½½℄ºÁÒØÓÚÕÙØÓÒ¸<br />

w(e<br />

α∈L +<br />

Z)½℄º nα<strong>×</strong>Ø<strong>×</strong>ÙÑÓnÔÖÛ<strong>×</strong>ÒÔÒÒظ ÛÖÑÙÐØ(α)<strong>×</strong>ØÑÙÐØÔÐØÝÓÖÓÓØα∈L (N)º<br />

ÓÖØÓÒ<strong>×</strong>Òǫ(α)<strong>×</strong>ÒØÓ(−1) ¾ÌÜØÒ˹ÙÐØÝÖÓÙÔ<strong>×</strong>ÒÝÒÐÙÒZ 2ÔÖØÝÓÔÖØÓÒØÓØ˹ÙÐØÝÖÓÙÔΓ 1<br />

½¿<br />

ÓÖN = 1¸Ø<strong>×</strong><strong>×</strong>ØÖÓÙÔPGL(2, ¿Ì<strong>×</strong><strong>×</strong>ÕÙÚÐÒØØÓØÓÜØÖÖÓÙÔÒÖØÝØÖØÒÑØÖÜA<br />

∏<br />

=<br />

W(A (N) ) ⋊ D N ,<br />

e −ρ<br />

∑ w∈W<br />

ρ ∑<br />

α∈L +<br />

ǫ(α)e α ) ,<br />

(Z)¸<br />

(N)º

ÓÖØÓÓÒÐÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸Ò0ÓØÖÛ<strong>×</strong>ºÁÒØ<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong> ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ØÖÓÓØ<strong>×</strong>ÔÔÖÛØÖÑÙÐØÔÐØÝÖÑÓÒÖÓÓØ<strong>×</strong>ÔÔÖÛØÒØÚÑÙÐØÔÐØÝ ÛÐÓ<strong>×</strong>ÓÒÖÓÓØ<strong>×</strong>ÔÔÖÛØÔÓ<strong>×</strong>ØÚÑÙÐØÔÐØݺÓÐÐÓÛÒØ<strong>×</strong>ÓÓÖÖ<strong>×</strong>¸½℄¸<br />

1ÝÒÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><br />

ØÓØÖÛÓ<strong>×</strong>ҳظ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> <strong>×</strong>ÓÑÓ<strong>×</strong>ÓÒÒÓØÖ<strong>×</strong>ÖÑÓÒºÄØÙ<strong>×</strong>ÛÖØØÏÝйùÓÖÖ<strong>×</strong>ÒÓÑÒØÓÖ ÓÖÑÙÐ<strong>×</strong>ÔÖØÒØÒØÓØÛÓÔÖØ<strong>×</strong>ÓÒÓÛÒÚÓÐÚ<strong>×</strong>ØÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ò ÖØ<strong>×</strong>ÒÓÒÆÙÐÒÓÒ<strong>×</strong>ØÖÙØ<strong>×</strong>ÙÔÖÐÖ¸G<br />

e ∏ ( −πı(ρ,z) 1 − e<br />

⎠´º½µ<br />

−πı(α,z) )ÑÙÐØ(α)<br />

α∈L +<br />

2ÓØÒ<br />

Ö<strong>×</strong>ØØÖÑØÙ<strong>×</strong>Ö<strong>×</strong><strong>×</strong><strong>×</strong>Ø<strong>×</strong>ÙÑ<strong>×</strong>ÓØÄÐÖÛØÒÓÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌ <strong>×</strong>ÓÒØÖÑ<strong>×</strong><strong>×</strong>ÔØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÙØÓØÔÖ<strong>×</strong>ÒÓÑÒÖÝ<strong>×</strong>ÑÔÐ<br />

+ºÇØØÛÓØÖÑ<strong>×</strong>ÒØ<strong>×</strong>ÙÑ<strong>×</strong>¸ÓÒÖ<strong>×</strong><strong>×</strong>ÖÓÑØ 0µºÌ ÛÖØÐÑÒØZ=z 3 f 2 +z 2 f 3 +z<br />

ZºÌ<strong>×</strong>ÑÙÐØÔÐØ<strong>×</strong>ÖØÖÑÒÝØÓÒÒØÓÒ<br />

1 f −2ÐÓÒ<strong>×</strong>ØÓØ<strong>×</strong>Ù<strong>×</strong>ÔR +ıV + ∼ H<br />

ÙÔÓÒÓÑÔÐÜØÓÒÓØÓÒV ÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>´η = 0µÒØÓØÖÖ<strong>×</strong>ÒÖÓÑØÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>´η ÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><strong>×</strong>ÙØØØÙØÓÑÓÖÔÔÖÓÔÖØ<strong>×</strong>ÖØØÒº ÓÖÑÙÐØÓØ<strong>×</strong>ÙÑÓÖÑÓØÑÓÙÐÖÓÖÑÓØÒÖÓÑØØÚÐØÒ<strong>×</strong>ÒÓÙ<br />

(Z)ÚÞº´ºµºÇÒÓÑÔÖ<strong>×</strong>Ø<strong>×</strong>ÙÑ<strong>×</strong>ÓØÒÓÑÒØÓÖ ÖÓÓØ<strong>×</strong>ÛØÑÙÐØÔÐØ<strong>×</strong>³m(η) ∈<br />

ÛØØÙØÓÑÓÖÔÓÖÑ∆ +ÐÓÒÛØØÖÑÙÐØÔÐØ<strong>×</strong>ÒÖÑÓÒÖÓÓØ<strong>×</strong>ÔÔÖÛØ<br />

5<br />

(Z)¸ÒØ<strong>×</strong>ØÖÑÒ<strong>×</strong><br />

ØÖÒØÛÒØÑÙÐØÔÐØ<strong>×</strong>ÓØÓ<strong>×</strong>ÓÒÒÖÑÓÒÒÖØÓÖ<strong>×</strong>ÒÒ ÑÙÐØÔÐØ<strong>×</strong>ÖÓÑØÜÔÓÒÒØÒØÔÖÓÙØÓÖÑÙÐØÔÖÓÙØÓÖÑÙÐÚ<strong>×</strong>ÓÒÐÝ ÒØÚÑÙÐØÔÐØÝÒØÜÔÓÒÒغÀÓÛÚÖ¸ØÖ<strong>×</strong><strong>×</strong>ÙØÐ<strong>×</strong><strong>×</strong>ÙÒÜØÖØÒØ<br />

<strong>×</strong>ÑÓÖÐÏØØÒÒܺÓÑÔÖÒÛØØÒÓÑÒØÓÖÒØØÝ´º½µ¸ØÓÑÑÓÒ<br />

ÌÄÀËÓ´º½µ<strong>×</strong>ÒØÛØØÔÖÓÙØÓÖÑÙÐÓÖ∆ 5<br />

ØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>L (Z)¸ÓÒÒ<strong>×</strong>Ý<strong>×</strong>ØÑØÐÐÝÓÒ<strong>×</strong>ØÖÙØØÃÅÄ<strong>×</strong>ÙÔÖ¹<br />

z))ÚÒÙ<strong>×</strong>ØÏÝÐÚØÓÖρº ØÓÖq 1/2 r 1/2 s 1/2ÒÒØÛØÜÔ(−πı(ρ, 1ÒØ<br />

−ρÒÕº´º½µ´ËÐ<strong>×</strong>Ó<strong>×</strong>ØÓÒº¿º½µº <strong>×</strong>Ñ<strong>×</strong>Ù<strong>×</strong>ØÓÓÒ<strong>×</strong>ØÖÙØØÓØÖÐÖ<strong>×</strong>ØØÓÙÖÒØ<strong>×</strong>ÔØÖºÓÖÛ<strong>×</strong>ÙÑÑÖÞ ÏÖØØÒÖÒØÒÓØØÓÒÓÖØ<strong>×</strong>ÒÓÒÆÙÐÒ¸ÛÖÒÔÖØÙÐÖ¸ÓÒÒ<strong>×</strong>ØÓÖÔÐρÝ<br />

ÚÒØÑÓÙÐÖÓÖÑ∆ 5<br />

ÐÖG 1ÖÓÑغÏÛÐÐÐÐÙ<strong>×</strong>ØÖØØ<strong>×</strong>ÔÖÓÙÖÓÖØ<strong>×</strong>ÓØÐÖG ½<br />

⎛ ⎧<br />

= ⎝<br />

w∈WØ(w)<br />

∑ ⎨<br />

⎩ e−πı(w(ρ),z) −<br />

∑<br />

η∈M II ∩R + M II<br />

m(η) e −πı(w(ρ+η),z) ⎫<br />

⎬<br />

⎭<br />

2,1<br />

≠<br />

⎞

ØÔÖÓÙÖØÓÖÓÒÞØÐÖ¸ÚÒØÒÓÑÒØÓÖÒØØݸØÛÐÐÙ<strong>×</strong>ÙÐØÓ ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

Ð<strong>×</strong>Ø<strong>×</strong>ÓÑÓØÓ<strong>×</strong>ÖÚØÓÒ<strong>×</strong>ØØÒÑÓÙØØÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓØÑÓÙÐÖÓÖÑ<br />

º¿º ÒÐÝÞÒØÅÓÙÐÖÓÖÑ<strong>×</strong> )ÒØÖÒÒÖÔÖÓÙØÛØ ½ºÍ<strong>×</strong>ÒØÜÔÖ<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÓÖØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸(δ 1 , δ 2 , δ 3<br />

Z¸ÓÒ<strong>×</strong><strong>×</strong>ØØe 1/2ºÙÖØÖ¸ÓÒ<strong>×</strong>ØÒØØÓÒÖÐØÒØÖÓÓØ<br />

−πi(δ1,Z) = qr , e −πi(δ2,Z) = r −1Òe ´ÊÐÐØØq =<br />

exp(−πi(ρ,Z)) = q 1/2 r 1/2 s<br />

ÒØ<strong>×</strong>ÒÓØØÓÒº<br />

m −8nm)ºÌÖÐ<br />

r l s<br />

ÛÖØÖÓÓØα[n, l, m] = nδ 1 +(−l+m+n)δ 2 +mδ<br />

ÝØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÝÓÒÓØÒÚÓÐÚØÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÓØ Ö<strong>×</strong>ÛØÓÒر1Ö<strong>×</strong>ÝØØÓÒÓÐÐÐÑÒØ<strong>×</strong>ÓØÏÝÐÖÓÙÔÒÖØ<br />

(Z)¸ÐÐØÖÑ<strong>×</strong>´ÒØÜÔÒ<strong>×</strong>ÓÒÚÒÒØÔÔÒܵØØ<br />

3<strong>×</strong>ÒÓÖÑ(2l <strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ö(α[1, 1, 0], α[0, −1, 0], α[0, 1, 1])ÒØÏÝÐÚØÓÖ<strong>×</strong>ρ=α[<br />

ÃÅÄ<strong>×</strong>ÙÔÖÐÖºÓÖÒ<strong>×</strong>ØÒ¸ØØÖÑ<strong>×</strong>Ö<strong>×</strong>ÒÖÓÑÏÝÐÖØÓÒ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØ<br />

¾ºÁÒØÜÔÒ<strong>×</strong>ÓÒÓÖ∆ 5<br />

1Ö<br />

ÒÓÑÒØÓÖÓÖÑÙÐØÓÜØÖØØÖÓÓØ<strong>×</strong>º<br />

ÛØØ<strong>×</strong>ÑÔÐÖÐÖÓÓØ<strong>×</strong>ÓG (q 3/2 r 3/2 s 1/2 , q 1/2 r −1/2 s 1/2 , q 1/2 r 3/2 s 3/2 ) = q 1/2 r 1/2 s (qr, r −1 , sr) .<br />

ÆÓØØØÛÒØÓÔÙÐÐÓÙØÒÓÚÖÐÐØÓÖÓq (Z)ºÓÖÑÐÔÖÓÓÒÚÒÝÓÐÐÓÛÒÖØ<strong>×</strong>ÒÓÒ<br />

1/2ÒØ<strong>×</strong>ÙÑ<strong>×</strong>ÓØ<br />

3<strong>×</strong>ÝÑÑØÖÝÛÔÖÑÙØ<strong>×</strong>ØØÖ<br />

1/2 r 1/2 s<br />

¿ºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖG 1¸<strong>×</strong>ÈÖÓÔº¾º½℄ºÌÖÔÖÓÓÑ<strong>×</strong>Ù<strong>×</strong>ÓØÒÓÒ¹ØÖÚÐ sµ<br />

1<strong>×</strong>ÒÓÙØÖS ÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÁØ<strong>×</strong><strong>×</strong>ÝØÓ<strong>×</strong>ÓÒÐÝØδ 1<br />

(Z)´<strong>×</strong>Õº´º½µº<br />

↔ δ 3´ÓÖÕÙÚÐÒØÐÝØq <strong>×</strong>ÝÑÑØÖÝÒØ∆ 5<br />

ÆÙÐÒ³<strong>×</strong>ÖÙÑÒØÓÖG ÖØÖvΓÔÔÖÒÒØÑÓÙÐÖØÖÒ<strong>×</strong>ÓÖÑ∆ 5<br />

½<br />

∆ 5 (Z)º<br />

α[n, l, m]ØÓq<br />

n<br />

q<br />

)ºµÌÙ<strong>×</strong>¸ÓÒ<strong>×</strong><br />

−πi(δ3,Z) = sr .<br />

exp(2πiz 1 )¸r=exp(2πiz 2 )Òs=exp(2πiz 3<br />

n r l s m = e −πi(α[n,l,m],Z) ,<br />

2 1<br />

↔<br />

2 , 1 2 , 1 2 ]

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ÇÒÒ<strong>×</strong>ÓÛØØÙÒÖØ<strong>×</strong>ÜÒ ºÔÖØÐÓØÓÙØÖS 3Ò<strong>×</strong>Ù<strong>×</strong>ØÓÚÖÝØδ 1 ↔ δ 2ÒÚÖÒÓ∆ ÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓØÑÓÙÐÖÓÖѸÓÒÒÒØÝØÃÅÄ<strong>×</strong>ÙÔÖÐÖØØÓÖÖ¹ <strong>×</strong>ÔÓÒÒØÓØÔÖØÙÐÖÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ºËØÖØÒÛØØÔÖÓÙØÖÔÖ<strong>×</strong>ÒØØÓÒ ÀÚÒÒØØØÛÓ<strong>×</strong><strong>×</strong>ÓØÒÓÑÒØÓÖÒØØÝÛØØ<strong>×</strong>ÙÑÒÔÖÓÙØ ÓÖÒ<strong>×</strong>ØÒ¸ØÐعÐÖÓÓØα[0, 0, 1]<strong>×</strong>ÑÔÔØÓÒÓØÖÐعÐÖÓÓØα[1,<br />

ÓØÑÓÙÐÖÓÖѸÒÓÑÔÖÒÛØØÓÚÕÙØÓÒ¸Ú<strong>×</strong>Ù<strong>×</strong>Ø<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚ<br />

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ØÔÖÓÙØ<strong>×</strong>ÖÒØÖÐ<strong>×</strong>ØÑÙÐØÔÐØ<strong>×</strong>ÒØÔÖÓÙØÓÖÑÙÐÖÚÒÒØÖ<strong>×</strong> <strong>×</strong><strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÖÐÖºÐ<strong>×</strong>Ó¸ÜÔÒÒØÑÓÙÐÖÓÖѸÛÕÙØØÜÔÒ<strong>×</strong>ÓÒØÓØ<strong>×</strong>ÙÑ ÖÓÓØ<strong>×</strong>αÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ¸ØÓØÖÛØØÖÑÙÐØÔÐØ<strong>×</strong>ºÐÐÑÙÐØÔÐØ<strong>×</strong>Ò<br />

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ÜØÖØØÔÓ<strong>×</strong>ØÚÖÓÓØ<strong>×</strong>ÒÓÖÖ<strong>×</strong>ÔÓÒÒÑÙÐØÔÐØ<strong>×</strong>¸Ø<strong>×</strong>ØÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸ØÏÝÐ ÖÓÙÔ¸ØÏÝÐÚØÓÖÒÖÓÑØÓÚÒÓÖÑØÓÒ¸ØÖØÒÑØÖÜÓØÃÅÄ <strong>×</strong>ÙÔÖÐÖº<br />

1¸ÛÙ<strong>×</strong>ØÑÔ<strong>×</strong>ÞØÛÓÔÓÒØ<strong>×</strong>½µÌÓÙØ<strong>×</strong>Ø<br />

Ò¾µÇÒÒ<strong>×</strong>ÓØØ<strong>×</strong>ÙÑÒÔÖÓÙØÖÔÖ<strong>×</strong>ÒØØÓÒ<strong>×</strong>ÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ØÓÓÑÔÖ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>¸Ø<strong>×</strong>ØÑÓÙÐÖ<br />

ØÛØØÒÓÑÒØÓÖÒØØÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖÒÖÓÒ<strong>×</strong>ØÖÙØØÐÖÖÓÑ<br />

ÓÖÓÒÐÙÒÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓG ÑÓÙÐÖÓÖÑΦ (Z)ØØÒÖØ<strong>×</strong>ØÒÖ<strong>×</strong>ÓØ1<br />

10<br />

ÓÖÑ∆ 5 (Z) = (Φ 10 (Z)) 1/2ØØÓÙÖ<strong>×</strong><strong>×</strong>ØÒÓÑÒØÓÖÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖG N¹ÓÖÓÐÓØØÓÖÝÚÒØÚÖÓÙ<strong>×</strong>ÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>º<br />

(Z)ºÆÜØÛÐÓÓØØ<strong>×</strong>Ó ØÒÓÑÒØÓÖÒØØݺÌ<strong>×</strong>ÓÒÐÙ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖG 1<br />

ÓÑÒ<strong>×</strong>ØÒÓÑÒØÓÖÒØØÝÓØÑÓÙÐÖÓÖÑ∆ <strong>×</strong>ÑÒØÓÒÒØÔÖÚÓÙ<strong>×</strong>ÔØÖ¸ÓÒÐÝÓÖØ<strong>×</strong>ÓØÙÒÓÖÓÐØÓÖÝØ º<br />

5<br />

ØÑÐ<strong>×</strong>ÓØÒÝØÒZ ÓÖÖØÓÒ<strong>×</strong>ØÓØØÚØÓÒÖØ<strong>×</strong>ѺÓÖÐÐØÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>ÒÖØÝØÒ½<br />

ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>G NÒ˜GN ÑÓÙÐÖÓÖÑÒÖØÒØÒÖÝÓØ1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØØÒÖØÒØR2<br />

5 (Z)º<br />

α[n, l, m] ↔ α[−l + m + n, −l + 2m, m] .<br />

2,<br />

(Z)º<br />

1]º<br />

5

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ÏÛÐÐÖ<strong>×</strong>ØÐÓÓØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>˜GNÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÑÓÙÐÖÓÖÑ<strong>×</strong><br />

(Z)¸ÖÖÐØ<strong>×</strong>ÒÕº´ºµ¸ÙØÖÒغ<br />

ÓÒ<strong>×</strong>ØÖÙØÒØÃÅÄ<strong>×</strong>ÙÔÖÐÖÖÓÑØÑÓÙÐÖÓÖÑ<strong>×</strong><strong>×</strong>ÐÓÒØ<strong>×</strong>ÑÐÒ<strong>×</strong><strong>×</strong><br />

Z NÓÖÓиØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk (Z)ÒΦ k<br />

(Z)ºÌÑØÓÓÖ<br />

(Z)¸ÓÖÓÒ ÒÖØÒØÒÖÝÓ1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>¸ººØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk ØÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>G NÐÓÛº<br />

NÖÐØØÓØÑÓÙÐÖÓÖÑ<strong>×</strong>Φ k<br />

<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÖØ<strong>×</strong>ÓΦ 10 (Z)ºÏÛÐÐÒÓÛ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜GN<br />

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ÒG ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G2<br />

ÖØÖºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G2<strong>×</strong>ÚÒÝØÖØÒÑØÖÜ<br />

(Z)<strong>×</strong>Ð<strong>×</strong>ÓÐÚÐ2ÑÓÙÐÖÓÖÑÛØ<br />

(Z)ÒÖØÒØÒÖ<strong>×</strong><br />

˜∆ 3 (Z)℄Û<strong>×</strong>Ø<strong>×</strong>ÕÙÖÖÓÓØÓØÑÓÙÐÖÓÖјΦ6 ÓØ1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØZ 2ÓÖÓÐØÓÖݺ˜∆3<br />

´º½µ<br />

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´º¾¼µ<br />

<strong>×</strong>ÓÙÖÖÓÓØ<strong>×</strong>Û¸ÒØÓÒÚÒØÓÒÒØÖÓÙÓÚÓÖG ÓØ̹ÙÐØÝÒÚÖÒØ<strong>×</strong>¸ÙØÓØÔÖ<strong>×</strong>ÒÓØØÛ<strong>×</strong>Ø<strong>×</strong>ØØ<strong>×</strong>¸ÒÓØÐÐØ<strong>×</strong>ÔÐØ<strong>×</strong>ÓØ Ì<strong>×</strong>ÒÙÒÖ<strong>×</strong>ØÓÓ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong>ÏÒÓÒØ<strong>×</strong>ØÓÖÓÐÙ<strong>×</strong>ÓØÕÙÒØÞØÓÒ<br />

δ .<br />

1 ≡<br />

⎛<br />

⎞<br />

2 −2 −6 −2<br />

A 2,II ≡<br />

−2 2 −2 −6<br />

⎜<br />

⎝−6 −2 2 −2<br />

⎟<br />

⎠ .<br />

−2 −6 −2 2<br />

Ö<strong>×</strong>Ò´¾ºµÖÐÐÓÛºÁÒ<strong>×</strong>ØÓÒ<strong>×</strong>ØÓÖ<strong>×</strong>ØÖØÓÒ<strong>×</strong>ÐØÓØÓÒÖÙÒ<strong>×</strong>ÙÖÓÙÔ<br />

( ) ( ) ( ) ( )<br />

0 −1 2 1 2 3 0 1<br />

, δ 2 ≡ , δ 3 ≡ , δ 4 ≡<br />

−1 0 1 0 3 4 1 4<br />

{ ( )<br />

a b<br />

Γ 0 (N) =<br />

c d<br />

| ad − bc = ±1, c = 0 mod N<br />

}<br />

/{±1} . ´º¾½µ<br />

½<br />

ÓPGL(2, Z)

Í<strong>×</strong>ÒØÖÐØÓÒØÛÒØ<strong>×</strong>ÔÐØÓÖ<strong>×</strong>¸Ø<strong>×</strong>ØÓÔÓ<strong>×</strong>ØÚÖÐÖÓÓØ<strong>×</strong>ÖÐÚÒØÓÖØ ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

NÓÖÓÐØÓÖÝÖÓØÓÖÑ ÛÐÐÖÓ<strong>×</strong><strong>×</strong>ÒÓÖØZ<br />

ÚÒÝ rºÌÜØÒ˹ÙÐØÝÖÓÙÔ<strong>×</strong><br />

2ÓÙÖÓÖÐÐNºÁÒØÖÑ<strong>×</strong> ÖÓÑØ<strong>×</strong>¸ÓÒ<strong>×</strong><strong>×</strong>ØØØØÛÓÖÓÓØ<strong>×</strong>α (N)<br />

1 = δ 1Òα ´º¾¿µ<br />

2 = δ<br />

ÓØÚÖÐ<strong>×</strong>q, r, sØ<strong>×</strong>ÖÓÓØ<strong>×</strong>Ör , qr, qr 3 s 2Òs<br />

ØÓØÏÝÐÑÖº 2<strong>×</strong>ØÖÐÖÓÙÔØØ<strong>×</strong>Ø<strong>×</strong>ÝÑÑØÖÝÖÓÙÔÓØÔÓÐÝÓÒÓÖÖ<strong>×</strong>ÔÓÒÒ<br />

)<strong>×</strong>ØÏÝÐÖÓÙÔÒÖØÝÏÝÐÖØÓÒ<strong>×</strong>ÓÐÐØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><br />

Γ 1 (2) = W(A 2,II<br />

)Ò<strong>×</strong><strong>×</strong>Թк<br />

) ⋊ D 2 ,<br />

ÛÖW(A (2)<br />

´º¾¼µÒD ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G3<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÖÓÑØ<strong>×</strong>ÕÙÖÖÓÓØÓØÑÓÙÐÖÓÖÑ ºº¾<br />

ÌÏÝÐÚØÓÖ<strong>×</strong>ÚÒÝρ =<br />

(Z)℄ºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G3<strong>×</strong>ÚÒÝØÖØÒÑØÖÜ ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G3<br />

˜Φ 4<br />

´º¾µ<br />

(Z)¸ÒÓؘ∆2<br />

.<br />

ÚÒÝ 2¸Ø<strong>×</strong>4ÓØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛÖ ÁÒØÓÒØÓØØÛÓÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>δ 1Òδ ´º¾µ δ .<br />

½<br />

(<br />

α (N) 2n<br />

=<br />

l<br />

l<br />

2m<br />

)<br />

−1<br />

, (α, α) = 2, (n, m, l) > 0, m mod N . ´º¾¾µ<br />

(<br />

1/2 1/2<br />

1/2 1<br />

(N)<br />

2<br />

⎛<br />

⎞<br />

2 −2 −10 −14 −10 −2<br />

−2 2 −2 −10 −14 −10<br />

−10 −2 2 −2 −10 −14<br />

A 3,II ≡<br />

−14 −10 −2 2 −2 −10<br />

⎜<br />

⎝−10 −14 −10 −2 2 −2<br />

⎟<br />

⎠<br />

−2 −10 −14 −10 −2 2<br />

( )<br />

4 5<br />

3 ≡ , δ 4 ≡<br />

5 6<br />

(<br />

4 7<br />

7 12<br />

)<br />

, δ 5 ≡<br />

(<br />

2 5<br />

5 12<br />

)<br />

( )<br />

0 1<br />

, δ 6 ≡<br />

1 6

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ÜØÒ˹ÙÐØÝÖÓÙÔÒØ<strong>×</strong><strong>×</strong><strong>×</strong>ÚÒÝ<br />

rºÌ<br />

)Ò<strong>×</strong><strong>×</strong>Թк ´º¾µ<br />

ÁÒØÖÑ<strong>×</strong>ÓØÚÖÐ<strong>×</strong>q, r, sØ<strong>×</strong>ÜÖÓÓØ<strong>×</strong>Ör−1 , qr, q 2 r 5 s 3 , q 2 r 7 s 6 , qr 5 s 6Òs<br />

ºº¿<br />

ÌÏÝÐÚØÓÖ<strong>×</strong>ÚÒÝρ =<br />

ØNÓØÓÖÓÐÒÖÓÙÔ<strong>×</strong>ÒÓØÔÖѺ<strong>×</strong>Û<strong>×</strong>ÛÒØÔÖÚÓÙ<strong>×</strong>ÔØÖ¸ÛÒÛ<br />

4ÓÖÓÐÒ<strong>×</strong>ÚÖÝÒØÖ<strong>×</strong>ØÒÓÒºÌ<strong>×</strong><strong>×</strong>ØÖ<strong>×</strong>ØÜÑÔÐÛÖ<br />

ÓÖÑ<strong>×</strong>ÓÖÔÖÑN<strong>×</strong>ÖÐØÚÐÝ<strong>×</strong>ÑÔÐÖÙ<strong>×</strong>ØÐÒÝÐ<strong>×</strong>ÔÓÒØÓÒ<strong>×</strong>Ú<br />

(Z)ÜÔÐØÐݸØÓÒ<strong>×</strong>ØÖÙØÓÒÓØÑÓÙÐÖ<br />

ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G4 Ì<strong>×</strong>ÓØZ ÓÒØÓÒ<strong>×</strong>ÐÓÒÒÓÒÒ<strong>×</strong>ØÓÙ<strong>×</strong>ÓØÖÓÒ<strong>×</strong><strong>×</strong>ØÒÝÓÒØÓÒ<strong>×</strong>ØÓÜØѺÌÃÅ<br />

0¸ÐÚÒÒÓÙÒØÖÑÒÓÒØ<strong>×</strong>ºÓÖØ<strong>×</strong>Ó<br />

(Z)ÒΦ ÓÒ<strong>×</strong>ØÖÙØØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk rÛÖÒÓØÓÑÔÐØÐÝÜÝØÝÐ<strong>×</strong>Ô<br />

k<br />

N+1ÒÐÐÓØÖa r<br />

4ÑÓÐ<strong>×</strong>ÚÖÝ (Z)Û<br />

=<br />

ÒÓÒ¹ÔÖÑN¸ÓÛÚÖ¸ØÖÖÑÒa 4Ò6ÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ö<strong>×</strong>ÔØÚÐݺÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G4<strong>×</strong>ÓÔÖÓÐ<br />

3ÛÖÐÐÓÐÐÔØØÝÔÛØÒØÚÓÐÙÑÓØÏÝÐÑÖ<strong>×</strong>Ò<br />

2Ò3ºÏ<strong>×</strong>ÛØØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong> Ä<strong>×</strong>ÙÔÖÐÖ˜G4ÓÖN = 4<strong>×</strong>ÒÖØÝØÑÓÙÐÖÓÖÑ(˜∆ (Z)) 2 = ˜Φ 3<br />

Û<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÒÔØÖ5ºÚÒØÃÅÄ<strong>×</strong>ÙÔÖÐÖÓÖØN =<br />

ÒÒعÑÒ<strong>×</strong>ÓÒÐÚØÓÖ<strong>×</strong> Ò3<strong>×</strong><strong>×</strong>ºÌÓÛÖØØÖØÒÑØÖÜÓ˜G4¸ÐØÙ<strong>×</strong>ÓÖÖØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÒØÓÒ<br />

ÖÒØÒÒØÙÖØÓØÓÒ<strong>×</strong>ÓÖN = 1,<br />

ÓÖN = 1, 2,<br />

3,<br />

ØÝÔÛØÒÒØÒÙÑÖÓÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Û<strong>×</strong>ÑÖÐÝ<strong>×</strong>ØÒØÖÓÑØN =<br />

ÕÙÚÐÒØÐݸÐØ<br />

X = (. . .,x −2 , x −1 , x 0 , x 1 , x 2 , x 3 , . . .) = (. . .,α 1 , β −1 , α 0 , β 0 , α −1 , β 1<br />

´º¾µ<br />

, . . .) .<br />

ØÒÒعÑÒ<strong>×</strong>ÓÒÐÑØÖÜ<br />

〉Ò<strong>×</strong>ÚÒÝ<br />

x<br />

ÌÖØÒÑØÖÜ<strong>×</strong>ÚÒÝØÑØÖÜÓÒÒÖÔÖÓÙØ<strong>×</strong>a mn ≡ 〈x n , x m<br />

´º¾µ A (4) = (a nm )ÛÖa ½<br />

a 1 = a N = 24<br />

Γ 1 (3) = W(A 3,II ) ⋊ D 3 .<br />

(<br />

1/3 1/2<br />

1/2 1<br />

3/2<br />

m =<br />

{<br />

α −m/2 ,<br />

m ∈ 2Z<br />

β (m−1)/2 , m ∈ 2Z + 1 .<br />

nm = 2 − 4(n − m) 2 ,<br />

3<br />

1, 2

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ÖØÒÑØÖÜÛØÞÖÓÒÚÐÙº ZºÁØ<strong>×</strong><strong>×</strong>ÝØÓ<strong>×</strong>ÓÛØØØÓÐÐÓÛÒÑÐÝÓÚØÓÖ<strong>×</strong>ÖÒÚØÓÖ<strong>×</strong>ÓØ ÛØm, n ∈<br />

´º¾µ<br />

⎛<br />

ÛغÒØÒ<strong>×</strong>ѹÒÒØ<strong>×</strong>ÕÙÒÓÞÖÓ<strong>×</strong>ºÇÒÒ<strong>×</strong>ÓÛØØA<strong>×</strong>ÖÒØÖº<strong>×</strong><br />

)Ò<strong>×</strong>ÐعкÊÐÐØØØÏÝÐÚØÓÖ<strong>×</strong> ´º¿¼µ Ù<strong>×</strong>ÙиØÏÝÐÚØÓÖρ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>〈ρ,<br />

ÄØÙ<strong>×</strong>ÜÔÐØÐÝÛÖØØÖ<strong>×</strong>ØØÖÓÓØ<strong>×</strong>ÓØÒÒØÒÙÑÖÓÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ó<br />

2Ò3ØÓÖ<strong>×</strong>ÛÖ<strong>×</strong>Թк<br />

x m 〉 = −1 , ∀m .<br />

Z)ÑØÖ<strong>×</strong><br />

ÌÏÝÐÚØÓÖ<strong>×</strong>ÚÒÝρ =<br />

ÓÖØN = 1,<br />

˜G 4ÒØÖÑ<strong>×</strong>ÓPGL(2,<br />

´º¿½µ<br />

sØ<strong>×</strong>ÖÓÓØ<strong>×</strong>ÖÚÒÝ<br />

,<br />

Ì<strong>×</strong>Ö<strong>×</strong>ÙÐØ<strong>×</strong>ÖÓÑÔØÐÛØÜÔØØÓÒ<strong>×</strong><strong>×</strong>ÓÒØÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓÖ ´º¿¾µ ÁÒØÖÑ<strong>×</strong>ÓØÚÖÐ<strong>×</strong>q, 4¹ÓÖÓÐ<strong>×</strong>ÓÒËÒ³<strong>×</strong>ÖÙÑÒØ<strong>×</strong>¸<strong>×</strong>ÛÛÐÐ<strong>×</strong>ÐÓÛºÓÖØظÓÛÚÖ¸ÐØÙ<strong>×</strong><br />

r,<br />

(Z)<strong>×</strong>Ø<strong>×</strong>ÒÓÑÒØÓÖÓÖÑÙк<br />

½¼<br />

ØZ ÚÖÝØØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>˜∆3/2<br />

(<br />

1/4 1/2<br />

1/2 1<br />

º⎞<br />

º1<br />

−3<br />

3<br />

⎜<br />

⎝<br />

−1⎟<br />

⎠<br />

( ) ( ) ( )<br />

0 −1 2 1 0 1<br />

α 0 ≡ , β 0 ≡ , β −1 ≡ ,<br />

−1 0 1 0 1 8<br />

( ) ( ) ( )<br />

2 7 6 17 6 7<br />

α 1 ≡ , β −2 ≡ , α −1 ≡<br />

7 24 17 48 7 8<br />

( ) ( )<br />

12 17 20 31<br />

β 1 ≡ , α −2 ≡ .<br />

17 24 31 48<br />

r −1 , qr , rs 4 , qr 7 s 12 , q 3 r 17 s 24 , q 3 r 7 s 4 , q 6 r 17 s 12 , q 10 r 31 s 24 .

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

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(Z)Ú<strong>×</strong>Ö<strong>×</strong>ØÓØÒÓÑÒØÓÖÒØØÝÓÖØ<strong>×</strong>ÃÅÄ<strong>×</strong>ÙÔÖÐÖº<br />

(Z)º<br />

∞¹ÁÒÚÖÒÓ˜∆3/2 ÄØÙ<strong>×</strong><strong>×</strong>˜∆3/2<br />

ÓÒÒ<strong>×</strong>ÐÝÔÖÓÚØÓÐÐÓÛÒØÛÓÒØØ<strong>×</strong>ÓÙؘ∆3/2 )ºÌÒ¸˜∆3/2 ½ºÄØZ ′ −z 2 z 3 ′ ) = −˜∆ 4º ´º¿¿µ<br />

3/2<br />

3ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØØÑÓÙÐÖÓÖÑ<br />

(Z) .<br />

Ì<strong>×</strong>ÑÔÐ<strong>×</strong>ØØØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÓÙÒØÓÒÙÒÖr →<br />

¾º˜∆3/2 (Z)<strong>×</strong>ÒÚÖÒØÙÒÖØÜÒz ↔ 4z<br />

mÛÖØØÒ<strong>×</strong>2<strong>×</strong>2ÑØÖÜ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong> ´º¿µ<br />

<strong>×</strong>ÒÓÙÒØÓÒÙÒÖØÜÒq↔ s<br />

ÆÜظØD ∞¹ÒÖØÓÖ<strong>×</strong>γÒδØÓÒØÖÓÓØ<strong>×</strong>x (2)<br />

(Z)<strong>×</strong>ÒÚÖÒØÙÒÖØ<strong>×</strong>ÝÑÑØÖÝÒÖØ ´º¿µ ÌÑØÖÜγ<strong>×</strong>ÒÓØÝγ ÇÒÒ<strong>×</strong>ÓÛØØØ<strong>×</strong>ÒÑÙ<strong>×</strong>Ø+1ÝÓ<strong>×</strong>ÖÚÒØØÒÝÔÖÓØÖÑ<strong>×</strong>ÒØÓÙÖÖ<br />

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ÝØØÓÒÓδÔÔÖÛØÓÒØ+1ºËÑÐÖÐݸØØÖÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØØÖÐ<br />

(Z)ÖÐØÝØØÓÒÓγ´δÖ<strong>×</strong>ÔºµÔÔÖÛØØ<strong>×</strong>ÑÓÙÖÖ 0ÖÐØ ÜÔÒ<strong>×</strong>ÓÒÓ˜∆3/2<br />

ÔÖÓÓØØØÒÒØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÚÒÝØÚØÓÖXÐÐÔÔÖÒØÓÙÖÖ<br />

−1ÖÐØÝγ¹ØÖÒ<strong>×</strong>ÐØÓÒÐ<strong>×</strong>ÓÔÔÖÛØÓÒØ+1ºÌÙ<strong>×</strong>¸Û ÓÒغÓÖÒ<strong>×</strong>ØÒ¸ØØÖÑ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØØØÛÓ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>α 0Òβ <strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>β 0Òβ (Z)º<br />

∞ºÌ<strong>×</strong>ÔÖÓÚ<strong>×</strong>ÒÐйÓÖÖ<strong>×</strong><br />

½½<br />

(Z)<strong>×</strong>ÒÚÖÒØÙÒÖØÙÐÐÖÐÖÓÙÔD <strong>×</strong>Øؘ∆3/2 (2)<br />

ÜÔÒ<strong>×</strong>ÓÒÓ˜∆3/2<br />

D (2)<br />

Òδ¸<br />

˜∆3/2<br />

=<br />

(<br />

z 1 −z 2<br />

(Z<br />

1<br />

( ) (<br />

1 −1 1 −1<br />

γ : x m −→ · x m ·<br />

)Ì,<br />

4 −3 4 −3<br />

( ) (<br />

−1 1 −1 1<br />

δ : x m −→ · x m ·<br />

)Ì.<br />

0 1 0 1<br />

(Z) → ± ˜∆ 3/2 (Z) .<br />

r −1º

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

ØÖÐÒÖØÓÖ¸y¸<strong>×</strong>ÒÒÕº´¾º¿µº<br />

4<strong>×</strong>ÝÑÑØÖÝÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ÕÙÚÐÒØØÓØ<strong>×</strong>ÝÑÑØÖÝÒÖØÝ Ìq→s<br />

´<strong>×</strong>Õº´º¿¿µµ¸ØÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÙÒÖØÏÝÐÖØÓÒºÇÒ<strong>×</strong> 0Ò<strong>×</strong><strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÖÐÖ<br />

(Z) ÏÝÐÌÖÒ<strong>×</strong>ÓÖÑØÓÒÓ˜∆3/2 ÌØÖÒ<strong>×</strong>ÓÖÑØÓÒr → r −1<strong>×</strong>ØÏÝÐÖØÓÒÓÙØØÖÓÓØα<br />

ÌÖØÓÒÙØÓÒÝÓØÖÐÑÒØÖÝÏÝÐÖØÓÒÛÐÐÐ<strong>×</strong>ÓÚØ<strong>×</strong>Ñ<strong>×</strong>ÒºÏ ´º¿µ<br />

ÖÔØÒÖÙÑÒØÖÓÑØÔÔÒÜÓ℄ØÓ<strong>×</strong>ÓÛØ<strong>×</strong>ºÖ<strong>×</strong>ظØÖØÓÒÙØÓ<br />

.<br />

ØÓÒÝØÑØÖÜ℄ )ºÌØÓÒÓÒZ<strong>×</strong>ÕÙÚÐÒØØÓSp(2, α<br />

ÌÑÒÙ<strong>×</strong><strong>×</strong>ÒÙØÓØÏÝÐÖØÓÒÑÔÐ<strong>×</strong>ØØØÖØÖ¸v(M)¸<strong>×</strong><strong>×</strong>ÓØÛØ<br />

0<strong>×</strong>ÖÔÖ<strong>×</strong>ÒØÝØÑØÖÜw Z)<br />

−1ºÆÜظÒÝÓØÖÐÑÒØÖÝÏÝÐ ∞º<br />

M<br />

ÙÒÖÐÐÐÑÒØÖÝÖØÓÒ<strong>×</strong>ºÀÒÓÒ<strong>×</strong><br />

−1ÓÖ<strong>×</strong>ÓÑÒÚÖØÐÑØÖÜsºÁØÓÐÐÓÛ<strong>×</strong>ØØØÖØÖ (Z)<strong>×</strong>Ó<br />

(Z)<strong>×</strong><strong>×</strong>ÙØØv(M) ØÑÓÙÐÖÓÖј∆3/2 =<br />

ÖØÓÒ¸w¸ÑÙ<strong>×</strong>ØÓÒÙØØÓw 0Ø<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong>ÕÙÒÓÖÐ<strong>×</strong>ÝÑÑØÖݸD (2)<br />

ÀÒ¸ÓÒ<strong>×</strong>w=s · w 0 · s<br />

<strong>×</strong><strong>×</strong>ÓØÛØØÏÝÐÖØÓÒw<strong>×</strong>Ø<strong>×</strong>Ñ<strong>×</strong>ØØÓÖw 4<strong>×</strong>ÚÒÝ 0ºÁÒÓØÖ<strong>×</strong>¸˜∆3/2<br />

´º¿µ<br />

´º¿µ ÏØÙ<strong>×</strong><strong>×</strong>ØØØÜØÒ˹ÙÐØÝÖÓÙÔÓÖN =<br />

ÐØÓÙØ<strong>×</strong>ØÖÙØÙÖÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G4<strong>×</strong>ÑÓÖÓÑÔÐØ<strong>×</strong>ÓÑÔÖ<br />

∞<strong>×</strong>ØÒÒعÑÒ<strong>×</strong>ÓÒÐÖÐÖÓÙÔÒÖØÝγÒδº<br />

∞ , Z)ÒØÙ<strong>×</strong><strong>×</strong>ÒÓØÒ<br />

3℄º<br />

ÛÖW(A (4) )<strong>×</strong>ØÓÜØÖÖÓÙÔÒÖØÝØÖØÓÒ<strong>×</strong>ÝÐÐÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>x ÒD (2)<br />

½¾<br />

ÌÒÖØÓÖy<strong>×</strong>ÒÓØÖÐÞ<strong>×</strong>ÒÐÑÒØÓÐÚÐ4<strong>×</strong>ÙÖÓÙÔÓPGL(2, ÐÑÒØÓØÜØÒ˹ÙÐØÝÖÓÙÔºÌ<strong>×</strong><strong>×</strong><strong>×</strong>ÑÐÖØÓÛØÔÔÒ<strong>×</strong>ÓÖN = 2,<br />

0<br />

( ( )<br />

1 0 1 0<br />

w α0 · Z =<br />

)Ì· Z ·<br />

0 −1 0 −1<br />

(<br />

1 0<br />

≡<br />

0 −1<br />

=<br />

(<br />

(w −1<br />

)<br />

0 )Ì0<br />

,<br />

0 w 0<br />

˜∆ 3/2 (w · Z) = det(w) ˜∆ 3/2 (Z) .<br />

W(A (4) ) ⋊ D (2)<br />

m

ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

4Ò<strong>×</strong>ÒÓÖÒÛØËÒ³<strong>×</strong>ÜÔØØÓÒ<strong>×</strong>º<br />

3¸ØÓÖÖ<strong>×</strong>ÔÓÒÒØÛÒØÛÐÐ<strong>×</strong>ÓØÏÝÐÑÖ<strong>×</strong>ÒØÛÐÐ<strong>×</strong>Ó ØÓ˜G1 , ˜G 2<br />

(Z)ÒØÓÖÖ<strong>×</strong>ÔÓÒÒÃÅÄ<br />

3ØÓÖ<strong>×</strong>ÓÒØÒÙ<strong>×</strong>ØÓÓÐÚÒÓÖ<br />

, ˜G<br />

(Z)ÒØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

ÑÖÒÐ<strong>×</strong>ØÐØÝÛÛ<strong>×</strong>ÔÖ<strong>×</strong>ÒØÒØN = 1, 2,<br />

N =<br />

ºº NÓÖÖ<strong>×</strong>ÔÓÒÒØÓØѺ Ì<strong>×</strong>ÓÒÐÙ<strong>×</strong>ÓÙÖ<strong>×</strong>ØÙÝÓØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk <strong>×</strong>ÙÔÖÐÖ<strong>×</strong>˜GNºÏÒÓÛ<strong>×</strong>ØÙÝØÑÓÙÐÖÓÖÑ<strong>×</strong>Φ k<br />

G<br />

(Z)ÖÒÚÒÝØÒÓÑÒØÓÖÓÖÑÙÐ<br />

(Z)ºÁØÛ<strong>×</strong>Ö<strong>×</strong>Ø ÌÑÐÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>G <strong>×</strong>ÓÛÒØØ<br />

N<br />

ÌÐ<strong>×</strong><strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>G ½ºÐÐØÐÖ<strong>×</strong>Ö<strong>×</strong><strong>×</strong>´ÖÒصÙØÓÑÓÖÔÓÖÖØÓÒ<strong>×</strong>ØÓØÄÐÖ<strong>×</strong><strong>×</strong>Ó¹<br />

NÖ<strong>×</strong>ÖÓÑØÑÓÙÐÖÓÖÑ<strong>×</strong>Φ (Z)ºÁÒÔÖØÙÐÖ¸ØÛ<strong>×</strong><br />

k<br />

<strong>×</strong>ÓÛÒÒ℄ØØØÑÓÙÐÖÓÖÑ<strong>×</strong>∆ k/2<br />

ÓÖÃÅÄ<strong>×</strong>ÙÔÖÐÖG N¸ØØÖÐÓ<strong>×</strong>ÐÝÖÐØØÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖG 1<br />

ÓÒ<strong>×</strong>ØÖÙØÝÖØ<strong>×</strong>ÒÓÒÆÙÐÒÖÓÑØÑÓÙÐÖÓÖÑ∆ 5<br />

1<strong>×</strong>Ð<strong>×</strong>ÓÓÒ<strong>×</strong>ØÖÙغ<br />

<strong>×</strong>ÒÓÒØÖ<strong>×</strong>ØØÓØ<strong>×</strong>ÓÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>˜GNÛÖØÖÓÓØ<strong>×</strong>Ý<strong>×</strong>ØѸ<br />

NÖÒØÐØÓ ØÛØØÖÒØÖÖØÒÑØÖÜA<br />

ÖØÒÑØÖܸÒÒÐ<strong>×</strong>ÓØÏÝÐÖÓÙÔÛ<strong>×</strong>ÖÒØÓÖÖÒØNºÀÓÛÚÖ¸<br />

)ºÌ<strong>×</strong>ÑÔÐ<strong>×</strong>ØØØÏÝÐÖÓÙÔ<strong>×</strong>ÒØÐ<strong>×</strong>ÛÐкÌ<strong>×</strong><br />

1,II¸ÖÓÑÛG ¾ºÌÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>´ÒÒØÖØÒÑØÖÜA 1,IIµÓÖØG ØÖÐÖÓÓØ<strong>×</strong>Óg(A 1,II<br />

1¸Ø<strong>×</strong>ÏÝÐÖÓÙÔ<strong>×</strong>ÒÓÐÓÒÖØ<strong>×</strong>ÝÑÑØÖÝÖÓÙÔ<br />

¿ºÌÑÙÐØÔÐØ<strong>×</strong>ÓØÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>Ö¸ÓÛÚÖ¸ÖÒغÓÖÒ<strong>×</strong>ØÒ¸<br />

1ºÌÖ<strong>×</strong>ÓÒ<strong>×</strong>ØØØÐØØÓ Ø<strong>×</strong><strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÙ<strong>×</strong>ÓÖN ><br />

ÓØÐØØÓÝÓÒÖ<strong>×</strong><strong>×</strong>ØÛ<strong>×</strong>ÓÖN )ÚÒÝØÓÖÑÙÐ<br />

=<br />

ÝÓÒÖ<strong>×</strong><strong>×</strong>ÒÓØÒÖØÝ1/Φ k (Z)¸ÙØÒ<strong>×</strong>ØÝ1/˜Φ 0<strong>×</strong>ÔÖÑØÚÐعÐ<strong>×</strong>ÑÔÐÖÓÓØ¸Ú ÑÒÖÝÖÓÓØ<strong>×</strong>ÓØÓÖÑtη 0¸ÛÖη ÑÙÐØÔÐØÝm(tη<br />

ºÌÖÖÐ<strong>×</strong>ÓÓØÖÑÒÖÝ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÛÖÒÓØÐعÐÛÓ<strong>×</strong>ÑÙÐØÔÐØ<strong>×</strong> ÆÓØØØØ<strong>×</strong>ÓÖÑÙÐÓÖÖØÐÝÖÔÖÓÙ<strong>×</strong>ØÑÙÐØÔÐØ<strong>×</strong>ÓØÑÒÖÝÖÓÓØ<strong>×</strong><br />

1<strong>×</strong>ÓÙÒÝÖØ<strong>×</strong>ÒÓÒÆÙÐÒ℄º<br />

0<br />

½¿<br />

ÓÖG ÖØÖÑÒÑÔÐØÐÝÝØÑÓÙÐÖÓÖÑ∆ k/2<br />

1 − ∑ t∈N<br />

m(tη 0 ) q n = ∏ n∈N<br />

k<br />

(Z)º<br />

(1 − q n ) k−4<br />

2 (1 − q Nn ) k+2<br />

2<br />

(Z)º

Ì<strong>×</strong>ÓÑÔÐØ<strong>×</strong>ÓÙÖ<strong>×</strong>Ù<strong>×</strong><strong>×</strong>ÓÒÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong><strong>×</strong>ÓØØÓØÑÓÙÐÖ ÔØÖºÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÖÓÑÝÓÒËÔØÖ<br />

º (Z)ºÏÓÒÐÙÛØÛÓÑÑÒØ<strong>×</strong>º<br />

ÏÚ<strong>×</strong>ÒØØØ<strong>×</strong>ÕÙÖÖÓÓØÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ØØÒÖØØÝÓÒÒÖ<strong>×</strong> ÓÒÐÙ<strong>×</strong>ÓÒ ÓÖÑ<strong>×</strong>˜Φk (Z)ÒΦ k<br />

<strong>×</strong>ÓÙÐÚÒÝÑÓÙÐÖÓÖÑ<strong>×</strong>¸<strong>×</strong>Ø<strong>×</strong>ÐÚÖÝÖÑÖÐÖ<strong>×</strong>ÙÐظÓÖØÖ<strong>×</strong>ÒÓÓÚÓÙ<strong>×</strong> <strong>×</strong>ØÖÙØÙÖØÓØØÓÖÝ<strong>×</strong>ÒÓØÑÑØÐÝÔÔÖÒغÌØØÒÖÝÓÈË<strong>×</strong>ØØ<strong>×</strong> Ì<strong>×</strong><strong>×</strong>ÚÖÝÒØÖ<strong>×</strong>ØÒÖ<strong>×</strong>ÙÐظÓÖØÓÖÒÓØÙÒÖÐÝÒÃÅÄ<strong>×</strong>ÙÔÖÐÖ<br />

2ÓÖÖØÓÒ<strong>×</strong>ØÓØ<strong>×</strong>ØÖÒØÚØÓÒÖÖÐØØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>º ÒØR <strong>×</strong>ÓÙÐ<strong>×</strong>ÙØØØÝÙÔÜØÐÝØÓÚÒÝËÐÑÓÙÐÖÓÖѺ <strong>×</strong>ØØ<strong>×</strong>ÖÚÒÝËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÒØ<strong>×</strong>ÚÖÝÖÑÖÐØØØÒÖ<strong>×</strong> 4¹ÈË<br />

ÒÓØÖÑÔÓÖØÒØ<strong>×</strong>ÔØØÓÒÓØ<strong>×</strong>ØÔÒÒÓØÑÓÙÐÖÔÖÓÔÖØ<strong>×</strong>ÓÒØ<br />

Ö<strong>×</strong>ÓÒØØØ<strong>×</strong>ÓÙÐÚØÙÖÒÓÙØØÓ<strong>×</strong>ÓºÁÒÔÖØÙÐÖ¸ØÒÖÝÓØ1<br />

ØÖÒ<strong>×</strong>ÓÖÑØÓÒÔÖÓÔÖØ<strong>×</strong>ÖÑÓÖÒÚÓÐÚØØÒØηÙÒØÓÒ<strong>×</strong>ºÁÒÖ<strong>×</strong>ÑÓÙÒØÓ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÖÚÒÝÔÖÓÙØ<strong>×</strong>Óη¹ÙÒØÓÒ<strong>×</strong>¸ÛÐ<br />

<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝ<strong>×</strong>Ñ<strong>×</strong>ØÓÔÐÝÖÙÐÖÓÐÒØÒÓÑÓÙÐÖÓÖÑ<strong>×</strong>ØØÒÖØØ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÖÚÒÝÑÓÖÒÓÒ¹ØÖÚÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÛÓ<strong>×</strong><br />

ÒÖÝÓ<strong>×</strong>ØØ<strong>×</strong>ÔÖ<strong>×</strong>ÖÚÒØ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺ<br />

<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖݺÌÒÖÝÓ1<br />

Ð<strong>×</strong>Ó¸ØØØÑÓÙÐÖÓÖÑ<strong>×</strong><strong>×</strong>ÓÙÐÖÐØØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><strong>×</strong>ÒÕÙÐÐÝ<br />

ØÒÖ<strong>×</strong>ÓØ1<br />

ÒÓÒ¹ØÖÚÐÒÖÑÖкҸ<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝ<strong>×</strong>Ñ<strong>×</strong>ØÓÔÐÝÒÑÔÓÖØÒØÖÓÐÒØ ÒÓÐÖ<strong>×</strong>ØØÖÖÐØØÓØ<strong>×</strong>ØÖÙØÙÖºÓÖÜÑÔÐØÒÒعÑÒ<strong>×</strong>ÓÒÐÄ<br />

ÖÑÓÖÒÚÓÐÚ<strong>×</strong>ØÖÙØÙÖØÒØÒùÅÓÓÝÄÐÖ<strong>×</strong>ºÌÔÔÖÒÓ ÐÖ<strong>×</strong>ÖÐØØÓØÒÙ<strong>×</strong>¹ÓÒÑÓÙÐÖÓÖÑ<strong>×</strong>ÖØÒùÅÓÓÝÄÐÖ<strong>×</strong>º<br />

ØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÔÔÖ<strong>×</strong>ÒÓØÑÖÐÝØÓÒÒØи<strong>×</strong>Ò<strong>×</strong>ÒÖÓÑØ<br />

4<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÝÖÙØ<strong>×</strong>Ø<strong>×</strong>ØÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÛÚ<br />

ØÀÄØÓÖÝØÝÓÑÖÓѺÁØÛÐÐÒØÖ<strong>×</strong>ØÒØÓÜÔÐÓÖØ<strong>×</strong>ÖØÓÒÙÖØÖØÓ ÓÖÖ<strong>×</strong>ÔÓÒÒØÛÒØÛÐÐ<strong>×</strong>ÓØÏÝÐÑÖ<strong>×</strong>ÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Ò<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>¸Ò<strong>×</strong>ÑØÓÓÒØÒÒÓÖÑØÓÒÓÙØ<br />

ÊÕÙÖÒN =<br />

ÖØÓÒ<strong>×</strong>ÒÛÓÒÒÐÓÓغÀÖÚÝÒÅÓÓÖÚÓÒ<strong>×</strong>ÖØÐÖÓÈË ÙÒÖØÑÓÖÓÒÒØÓÒ<strong>×</strong>ØÛÒØÑÐÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒØÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>º Ð<strong>×</strong>Ó¸ÓÒÒ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>ØÖÙØÙÖ<strong>×</strong>Ü<strong>×</strong>ØÓÖÓØÖÑÓÐ<strong>×</strong>ºÌ<strong>×</strong>ÖÐÐÒÛÒÒØÖ<strong>×</strong>ØÒ<br />

ØÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓØ1<br />

ÓÙÒÚÒÝÖÐØÓÒØÓØÐÖÓÈË<strong>×</strong>ØØ<strong>×</strong>º <strong>×</strong>ØØ<strong>×</strong>½½¾¸½½¿℄ºÁØ<strong>×</strong>ÓÒØÖ<strong>×</strong>ØØÓ<strong>×</strong>ÛØÖØÃÅÄ<strong>×</strong>ÙÔÖÐÖØØÛÚ<br />

½

7<br />

Ê<strong>×</strong>ÙÐØ<strong>×</strong>ÓØÌ<strong>×</strong><strong>×</strong><br />

ÁÒØ<strong>×</strong>ÔØÖÛÐ<strong>×</strong>ØØ<strong>×</strong>ØÓÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÒØ<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØØÖÙØÓØÙØÓÖÓØ Ø<strong>×</strong><strong>×</strong>¸ÓØÒ<strong>×</strong>ÔÖØÓÛÓÖÓÒÛØÓÐÐÓÖØÓÖ<strong>×</strong>ºÌ<strong>×</strong>Ö<strong>×</strong>ÙÐØ<strong>×</strong>ÚÒÔÖ<strong>×</strong>ÒØ Ò¸¸½¼℄ºÏÐ<strong>×</strong>ØØÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÐÓÒÛØØÓÒØÜØÒÛØÝÛÖÛÓÖº<br />

ÛÓ<strong>×</strong>ÏÝйùÓÖÖ<strong>×</strong>ÒÓÑÒØÓÖÓÖÑÙÐÚ<strong>×</strong>Ö<strong>×</strong>ØÓÒÙ<strong>×</strong>¹ØÛÓ<br />

N¸ÛÖ<strong>×</strong>ÓÛÒ<br />

•ÁÒ℄ØÜ<strong>×</strong>ØÒÓÑÐÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>¸G ÒÛØØÛÓÖÓÖ¸ÎÖÐÒ¸ÒÎÖÐÒ½℄ØÛ<strong>×</strong>ÓÙÒÝÂØÖÒ ÄØÙ<strong>×</strong>ÖÝÖÐиÖÓÑØÔÖÚÓÙ<strong>×</strong>ÔØÖ<strong>×</strong>¸ØÓÒØÜØÓØÓÚÖ<strong>×</strong>ÙÐغËØÖع ÑÓÙÐÖÓÖÑØÐÚÐN¸∆ k/2 (Z)¸ÓÖ(N,<br />

<strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖ<strong>×</strong>ØÖÒØÓÖ<strong>×</strong>ÒÓÙÖ<strong>×</strong>Ô¹ØÑÑÒ<strong>×</strong>ÓÒ<strong>×</strong>℄¸Û<strong>×</strong>ÚÒÝÒÙ<strong>×</strong>¹<br />

ÃÅÄ<strong>×</strong>ÙÔÖÐÖºÁÒ℄Ø<strong>×</strong>Û<strong>×</strong>ÜØÒØÓØÑÐÝÓÑÓÙÐÖÓÖÑ<strong>×</strong> <strong>×</strong>ÕÙÖÖÓÓسÓØÑÓÙÐÖÓÖÑÒÕÙ<strong>×</strong>ØÓÒÔÔÖ<strong>×</strong><strong>×</strong>ØÒÓÑÒØÓÖÒØØÝÓ<br />

(Z)¾℄ºÁØÛ<strong>×</strong>Ð<strong>×</strong>ÓÓ<strong>×</strong>ÖÚÝÎÎØØØ ËÒØØØÒÖØÒÙÒØÓÒÓØÒÖ<strong>×</strong>Ó1<br />

ÓÒ<strong>×</strong>ØÖÙØÝÂØÖÒËÒºÑÐÝÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÛÖÓÒ<strong>×</strong>ØÖÙظ<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒÐ<strong>×</strong><strong>×</strong>ÓN =<br />

ÐÓÒØÐÒ<strong>×</strong>ÓØÛÓÖÝÖØ<strong>×</strong>ÒÓÒÆÙÐÒ℄¸ÛÓ<strong>×</strong>ÒÓÑÒØÓÖÒØØ<strong>×</strong><br />

ØÛÓËÐÑÓÙÐÖÓÖÑ<strong>×</strong>¸ÒÓؘΦk<br />

ÚÒÝØÖØÒÑØÖÜ<br />

2¹ÓÖÖØÓÒ<strong>×</strong>ØÓØ<strong>×</strong>ØÖÒØÚØÓÒºÐÐØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>Ö (Z)ÒÖØÒ ÛÖÚÒÝ<strong>×</strong>ÕÙÖÖÓÓØ<strong>×</strong>¸∆ k/2 (Z)¸ÓØÒÙ<strong>×</strong>¹ØÛÓÑÓÙÐÖÓÖÑ<strong>×</strong>Φ ØR<br />

(5, 2)º<br />

)Ò<br />

k) = (1, 10)¸(2, 6), (3, 4)¸(4, 3 2<br />

k<br />

4<br />

⎛ ⎞<br />

2 −2 −2<br />

⎜ ⎟<br />

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[<br />

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16<br />

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2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒÐÐÀÄÑÓÐ<strong>×</strong>ÛÖØÝÐ<strong>×</strong>Ô<strong>×</strong>Ö<strong>×</strong>ÖÓÑ<br />

ρ<br />

(K3)´ÒØÙÐ<strong>×</strong>ÖÔØÓÒÓØÀÄÓÖÓÐ<br />

2µ¸ÒÐÙÒÔÖÓÙØ ÒÖÝÓÖØ1 ØØÓÒÓÆÙÐÒÒÚÓÐÙØÓÒ<strong>×</strong>ÓÒH∗ <strong>×</strong><strong>×</strong>ÙÔÖ<strong>×</strong>ÝÑÑØÖÓÖÓÐÓØÝÔÁÁ<strong>×</strong>ØÖÒØÓÖÝÓÒK3 <strong>×</strong> T<br />

ÖÓÙÔ<strong>×</strong><strong>×</strong>Ù<strong>×</strong>Z M<br />

r<strong>×</strong>Ø<strong>×</strong>Ý 2)¸<strong>×</strong>Ø<strong>×</strong><strong>×</strong>ØÓÐÐÓÛÒÓÒØÓÒ<strong>×</strong> ÌÑÓÙÐÖÓÖÑg ρ (τ)¸ÓÛØ(k<br />

´º½µ<br />

½ºÌÓÒØ<strong>×</strong>a ) −1<br />

½¼<br />

=ÚÓÐ⊥<br />

z 2 2z 3<br />

)º<br />

g ρ (τ/N) ≡<br />

∞<br />

∑<br />

n=−1<br />

d(n) q n/N .<br />

g ρ (τ)<br />

Nº<br />

N∏<br />

= η(rτ) ar = η(τ) a 1<br />

η(2τ) a2 · · ·η(Nτ) a N<br />

.<br />

r=1<br />

<strong>×</strong> Z<br />

+<br />

(<br />

Na1 + N a 2<br />

2<br />

+ · · · + a N<br />

)<br />

= 24 ,<br />

a 1 + a 2 + · · · + a N = 2(k + 2) ,<br />

(<br />

1<br />

a 1<br />

2 a2 · · ·N a N<br />

,<br />

2Z¸<br />

Z N<br />

e ∈

ÔØÖºÊ<strong>×</strong>ÙÐØ<strong>×</strong>ÓØÌ<strong>×</strong><strong>×</strong><br />

⊥º ÛÖÚÓÐ⊥ØÚÓÐÙÑÓØÙÒØÐÐÒΓ NÖÒÓÒ¹ÞÖÓÛÖ<strong>×</strong>ÛØ 0ÙÒÐ<strong>×</strong><strong>×</strong><br />

ØÒ<strong>×</strong>ºÁØÐ<strong>×</strong>ÓÑÔÐ<strong>×</strong>ØØØÖ<strong>×</strong>ØÕÙØÓÒÒÕº´º¾¾µÒÖÛÖØØÒ<strong>×</strong> ÒÓÛÒÖ<strong>×</strong>ÙÐØ<strong>×</strong>º ¾ºÌÓÒÐÝÔÖÑØØÝÐ<strong>×</strong>ÖÓÐÒØr<strong>×</strong>ÙØØr|N¸ÒÒa r =<br />

r|NºÌÙ<strong>×</strong>¸ÛÒN<strong>×</strong>ÔÖѸÓÒÐÝa 1Òa NÑÓÒÓØÖ<br />

Ì<strong>×</strong>ÓÒÒØ<strong>×</strong>ØÓØÖ<strong>×</strong>ÙÐØ<strong>×</strong>ÓÙÑÑظÃ<strong>×</strong>ÐÚ<strong>×</strong>ÝÒÅÃÝ¿℄ÓÒØÑÙÐØÔÐØÚ<br />

¿ºÌÖÕÙÖÑÒØØØØÝÐÐÒÑÔÐ<strong>×</strong>ØØa 1<br />

´º½µ<br />

= a<br />

a 1 + 2a 2<br />

24¾℄º<br />

+ · · · + Na N = 24<br />

(Z)<strong>×</strong>ÒÒÒØ<strong>×</strong>ÙÑ<strong>×</strong>ÚÒ<strong>×</strong> )ºÌØÚÐØÚÒØÑÓÙÐÖ<br />

ÐÒÝÐ<strong>×</strong>Ô<strong>×</strong>ÓÐÑÒØ<strong>×</strong>ÓM Ð<strong>×</strong>Ó¸ØÛ<strong>×</strong>ÓÒØÙÖÒ℄ØØØÂÓÓÖÑÓÛØk¸ÒÜ1ÒÐÚÐN ØØ<strong>×</strong>Ø<strong>×</strong>ÓÖØØڴŵÐØÐÒØÓØËÐÑÓÙÐÖÓÖÑΦ k<br />

ÛÒG=Z NÓÖÐÐN<strong>×</strong>ÚÒÝϑ 1<br />

g ρ (z 1<br />

´º¾¼µ ÓÖÑ<strong>×</strong>Φ k<br />

(Z)<strong>×</strong>ÚÒÝØÓÐÐÓÛÒØÚ<strong>×</strong><br />

(Z)<strong>×</strong><strong>×</strong>ÓÛÒØÓ<strong>×</strong>ÑÐÖØÓØÓÒ<strong>×</strong> ´º¾½µ <strong>×</strong>ÑÐÖØÚÐØÓÖ˜Φk<br />

g<br />

η(z 1 ) 6 ρ (z 1 /N) .<br />

<strong>×</strong>ÑÔÐÖÓÓØÓÔÒÓÒØÓÖÓÐÒºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖÓÖ ÙÒÒÝØÓÖÓÐÒºÀÓÛÚÖ¸ØÑÙÐØÔÐØ<strong>×</strong>ÓØÑÒÖÝ ÔÔÖÒÒ℄ºÌÖØÒÑØÖܸÏÝÐÚØÓÖÒÏÝÐÖÓÙÔÖÑÒ<br />

•ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖÓÖ∆ ÒØÖµÛØÖØÒÑØÖÜ<br />

(Z)<strong>×</strong>ÓÔÖÓÐØÝÔÛØÒÒØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>´ÐÐÐÝÒ<br />

3/2<br />

˜∆ 3/2<br />

´º¾µ<br />

ÑÓÐØÑÔÔØÓØÛÐÐ<strong>×</strong>ÓØÙÒÑÒØÐÏÝÐÑÖÓØ<br />

A (4) = (a nm )ÛÖa = 2 − 4(n − m) 2 ,<br />

ÒÐعÐÏÝÐÚØÓÖºÌÛÐÐ<strong>×</strong>ÓÑÖÒÐ<strong>×</strong>ØÐØÝÓÖØN= 4<br />

½½<br />

(z 1 ,z 2 ) 2<br />

η(z 1 ) 6<br />

φ k,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2 ) 2<br />

g<br />

η(z 1 ) 6 ρ (z 1 ) = ∑ a(n, l) q n r l .<br />

n,l<br />

˜φ k,1 (z 1 , z 2 ) = ϑ 1(z 1 , z 2 ) 2<br />

nm<br />

(Z)

ÃÅÄ<strong>×</strong>ÙÔÖÐÖº ÔØÖºÊ<strong>×</strong>ÙÐØ<strong>×</strong>ÓØÌ<strong>×</strong><strong>×</strong><br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong><br />

ØÓÓØÖÚÐÙ<strong>×</strong>ÓNÒ℄Ò℄º<strong>×</strong>ÑÒØÓÒÓÚØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong><br />

1<strong>×</strong>Û<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÝÖØ<strong>×</strong>ÒÓÒÆÙÐÒ℄ÒÛ<strong>×</strong>ÜØÒ 1¸<br />

3Ò5ÛÖÓÒ<strong>×</strong>ØÖÙØÒ<br />

ÏÐÓÓØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÙÒÖÐÝÒØÒÖÝÓØ1 ÒØÑÐÝÓÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>ÒØÔÖÚÓÙ<strong>×</strong>ÔØÖ<strong>×</strong>ºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ¸G ÓÖØN =<br />

ÝÒÒÓÐÖ℄º<br />

3ÛÖÓÒ<strong>×</strong>ØÖÙØÝÖØ<strong>×</strong>ÒÓÒÆÙÐÒ½¼℄½¼℄ÒØÖÖÐØÓÒ<br />

G NÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÑÓÙÐÖÓÖÑ<strong>×</strong>Φ 3Û<strong>×</strong>ÔÓÒØÓÙØ<br />

(Z)ÓÖN 2,<br />

(Z) ℄¸ÛÐØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>˜GNÓÖÖ<strong>×</strong>ÔÓÒÒØÓØÑÓÙÐÖÓÖÑ<strong>×</strong>˜Φk ÓÖN = 2,<br />

ÑÓÐÒÓØÒÜÔÐØÐÝÓÒ<strong>×</strong>ØÖÙØÓÖ℄ØÓÙÐÒÓØÚÖºÁÒ℄Ø<br />

3ÀÄÑÓÐ<strong>×</strong>ÑÝ ØÓØÒÖ<strong>×</strong>Ó1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÓØÀÄÑÓÐ<strong>×</strong>ÓÖN = 2,<br />

ÓÖÑ<strong>×</strong>º ÀÄÑÓÐÛ<strong>×</strong><strong>×</strong>ÓÛÒØÓÜ<strong>×</strong>ظÒÛ<strong>×</strong>ÓÒ<strong>×</strong>ØÖÙØÖÓÑØÓÖÖ<strong>×</strong>ÔÓÒÒÑÓÙÐÖ<br />

ÁØÛ<strong>×</strong>ÔÖØÒ℄ØØØÃÅÄ<strong>×</strong>ÙÔÖÐÖÓÖN ><br />

ÒÓØÜ<strong>×</strong>ØÙØ<strong>×</strong>ÒØÑÓÙÐÖÓÖÑ<strong>×</strong>Φ 3 (Z)ÓÖÖ<strong>×</strong>ÔÓÒÒØÓØN = (Z)Ò˜Φ3 ÃÅÄ<strong>×</strong>ÙÔÖÐÖÙÒÖÐÝÒØÒÖÝÓØ1 4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØN =<br />

ÖÓÓØ<strong>×</strong>ÒØÓÒÒÒعÑÒ<strong>×</strong>ÓÒÐÚØÓÖ<strong>×</strong> ÓÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÌÓÛÖØØÖØÒÑØÖÜÓ˜G4¸ÐØÙ<strong>×</strong>ÓÖÖØÖÐ<strong>×</strong>ÑÔÐ<br />

(Z)ºÌÃÅÄ<strong>×</strong>ÙÔÖÐÖÐÖ˜G4<strong>×</strong>ÓÔÖÓÐØÝÔÛØÒÒØÒÙÑÖ ÌÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G4ÓÖN = 4<strong>×</strong>ÒÖØÝØÑÓÙÐÖÓÖÑ(˜∆<br />

˜Φ<br />

ÕÙÚÐÒØÐݸÐØ<br />

3<br />

´º¾¾µ<br />

ÚÒÝØÒÒعÑÒ<strong>×</strong>ÓÒÐÑØÖÜ<br />

〉Ò<strong>×</strong><br />

x<br />

ÌÖØÒÑØÖÜ<strong>×</strong>ÚÒÝØÑØÖÜÓÒÒÖÔÖÓÙØ<strong>×</strong>a<br />

ZºÁØ<strong>×</strong><strong>×</strong>ÝØÓ<strong>×</strong>ÓÛØØØÓÐÐÓÛÒÑÐÝÓÚØÓÖ<strong>×</strong>ÖÒÚØÓÖ<strong>×</strong><br />

mn ≡ 〈x n , x m<br />

´º¾¿µ<br />

A (4) = (a nm )ÛÖa ÛØm, n ∈<br />

½¾<br />

k<br />

=<br />

4<br />

4<br />

3/2 (Z)) 2 =<br />

X = (. . .,x −2 , x −1 , x 0 , x 1 , x 2 , x 3 , . . .) = (. . .,α 1 , β −1 , α 0 , β 0 , α −1 , β 1 , . . .) .<br />

m =<br />

{<br />

α −m/2 ,<br />

m ∈ 2Z<br />

β (m−1)/2 , m ∈ 2Z + 1 .<br />

nm<br />

= 2 − 4(n − m) 2 ,

ÓØÖØÒÑØÖÜÛØÞÖÓÒÚÐÙº ÔØÖºÊ<strong>×</strong>ÙÐØ<strong>×</strong>ÓØÌ<strong>×</strong><strong>×</strong><br />

´º¾µ<br />

⎛º⎞<br />

º1<br />

−3<br />

ÛغÒØÒ<strong>×</strong>ѹÒÒØ<strong>×</strong>ÕÙÒÓÞÖÓ<strong>×</strong>ºÇÒÒ<strong>×</strong>ÓÛØØA<strong>×</strong>ÖÒØÖº<br />

3<br />

<strong>×</strong>Ù<strong>×</strong>ÙиØÏÝÐÚØÓÖρ<strong>×</strong>Ø<strong>×</strong><strong>×</strong><br />

⎜<br />

⎝<br />

−1⎟<br />

⎠<br />

)Ò<strong>×</strong>Ðعк<br />

4<strong>×</strong>ÚÒÝ<br />

1/4 1/2<br />

ÌÏÝÐÚØÓÖ<strong>×</strong>ÚÒÝρ =<br />

)<strong>×</strong>ØÓÜØÖÖÓÙÔÒÖØÝØÖØÓÒ<strong>×</strong>ÝÐÐÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong><br />

1/2 1<br />

ÌÜØÒ˹ÙÐØÝÖÓÙÔÓÖN =<br />

2ÑØÖÜ<strong>×</strong>ÓÐÐÓÛ<strong>×</strong><br />

W(A (4) ) ⋊ D (2)<br />

ÛÖW(A (4)<br />

x mÒD (2)<br />

ÓÒØÖÓÓØ<strong>×</strong>x mÛÖØØÒ<strong>×</strong>2 ´º¾µ<br />

<strong>×</strong><br />

( ) (<br />

1 −1 1 −1<br />

γ : x m −→ · x m ·<br />

)Ì,<br />

Ð<strong>×</strong>Ó¸ØÛÐÐ<strong>×</strong>ÓØÏÝÐÑÖÓØÃÅÄ<strong>×</strong>ÙÔÖÐÖ˜G4Û<strong>×</strong><strong>×</strong>ØÙ ´º¾µ<br />

4 −3 4 −3<br />

ÒÓÙÒØÓÓÑÔØÐÛØËÒ³<strong>×</strong>ÜÔØØÓÒ<strong>×</strong>ºÌÙÒÑÒØÐÓÑÒ»ÏÝÐ<br />

( ) (<br />

−1 1 −1 1<br />

δ : x m −→ · x m ·<br />

)Ì.<br />

4<strong>×</strong>ÓÙÒÝÒÒÒØÒÙÑÖÓ<strong>×</strong>ѹÖÐ<strong>×</strong><strong>×</strong>ØÃÅÄ<br />

0 1 0 1<br />

<strong>×</strong>ѹÖÐ<strong>×</strong>º <strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ÒÒØÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ºÓØ<strong>×</strong>ѹÖÐ<strong>×</strong>ÖÔÖ<strong>×</strong>ÒØ<strong>×</strong>ÖÐ<br />

2<strong>×</strong>ÔÔÖÓ<strong>×</strong>ÐÑØÔÓÒØÓØÒÒØ<strong>×</strong>ÕÙÒÓ ÑÖÓÖN =<br />

½¿ <strong>×</strong>ÑÔÐÖÓÓغÌÔÓÒØ1<br />

´º¾µ<br />

〈ρ, x m 〉 = −1 , ∀m .<br />

(<br />

´º¾µ<br />

∞<strong>×</strong>ØÒÒعÑÒ<strong>×</strong>ÓÒÐÖÐÖÓÙÔÒÖØÝγÒδÛØ<br />

∞ ,

ÔØÖºÊ<strong>×</strong>ÙÐØ<strong>×</strong>ÓØÌ<strong>×</strong><strong>×</strong><br />

Ö<strong>×</strong>ÔÓÒÒØÓØÛ<strong>×</strong>Ð<strong>×</strong>ÓÓÒ<strong>×</strong>ØÖÙØÒ℄ºÌÏÝÐÚØÓÖρ<strong>×</strong>Ø<strong>×</strong>Ñ<strong>×</strong>ÓÖØ 4ÓÖ¹<br />

ÙÒÒ<strong>×</strong>ÓÖ℄ºÌÑÒÖÝÖÓÓØ<strong>×</strong>ÖÑÒÙÒÒ<strong>×</strong>ÛÐиÙØØÖÑÙй<br />

5ÓÖÔÖÑNºÐ<strong>×</strong>ÓØØÖÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>ÖÑÒ ËØÖØÒÖÓÑØÔÖÓÙØÜÔÒ<strong>×</strong>ÓÒÓÖ∆ 3/2 (Z)ØÃÅÄ<strong>×</strong>ÙÔÖÐÖ¸G ÖÓÓØ<strong>×</strong>ºÌÓÖÓÐÒÓÒÐÝÒØÑÙÐØÔÐØ<strong>×</strong>ÓØÑÒÖÝÖÓÓØ<strong>×</strong>ÓÖ¹ ÒØ<strong>×</strong>Ñ<strong>×</strong>ØÓÖÐ<strong>×</strong>ÑÔÐÖÓÓØ<strong>×</strong>¸ÏÝÐÖÓÙÔ¸ÏÝÐÚØÓÖ¸ÒÑÒÖÝ<br />

ÐÖ<strong>×</strong>G NÓÖN = 1, 2, 3,<br />

NÛÖÐÐÚÒÝØ<strong>×</strong>ÑÖØÒÑØÖܸ<br />

ÖÒØÚÐÙ<strong>×</strong>ÓNºÁØÛ<strong>×</strong><strong>×</strong>ÒØØØ<strong>×</strong>ÑÔØØÖÒÓÒØÒÙ<strong>×</strong>ØÓÓÐÓÖØÃÅ<br />

1/2ÓÖÔÖÑN¸Û ØÔÐØ<strong>×</strong>ÖÒÝØÓÖÓÐÒºÓÖ∆ k/2 (Z) = (Φ k (Z))<br />

ÖÐÐØØØÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>G ÑÙÐØÔÐØÚη¹ÕÙÓØÒØ<strong>×</strong>ØØÖ<strong>×</strong><strong>×</strong>ÓØÛØØÖÑ<strong>×</strong>Ô<strong>×</strong>˜ρÚÒ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong><strong>×</strong>ÚÒÝ Ä<strong>×</strong>ÙÔÖÐÖÚÒÛÒN<strong>×</strong>ÒÓÒ¹ÔÖÑÓÖ∆ η¹ÔÖÓÙØ<strong>×</strong>ÓÖÖ<strong>×</strong>ÔÓÒÒØÓÝÐ<strong>×</strong>Ô<strong>×</strong>º ÒÌк¸ÒÖÐÞÒØÓÖÖ<strong>×</strong>ÔÓÒÒÖ<strong>×</strong>ÙÐØÓÖÀÄ<strong>×</strong>ØÖÒ<strong>×</strong>ÛÖ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÛÖÚÒÝÑÙÐØÔÐØÚ<br />

3/2<br />

•ÁÒ½¼℄Ø<strong>×</strong>Ò<strong>×</strong>ÓÛÒØØØÓÙÒØÒÓ1<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>Ö ØÒÖØÒÙÒØÓÒ<strong>×</strong>ÓÖØ1<br />

ØÔÐØÚη¹ÕÙÓØÒØ<strong>×</strong>ØÖÑÒÝØÖÑ<strong>×</strong>Ô<strong>×</strong><strong>×</strong><strong>×</strong>ÓØÛØØÓÒÙÝ ÚÒÝÑÙÐØÔÐØÚη¹ÔÖÓÙØ<strong>×</strong>ºÌÛ<strong>×</strong>ÜØÒØÓØØÝÔÁÁÑÓÐ<strong>×</strong>ÛÖ<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÛÖ<strong>×</strong>ÓÛÒØÓÚÒÝÑÙй<br />

2¹ÈË<strong>×</strong>ØØ<strong>×</strong>¸<br />

ÁØÛ<strong>×</strong><strong>×</strong>ÓÛÒÒ℄ØØØÒÖÝÓØÐØÖÐÐÝÖ1<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong><br />

ØÒÖÝÓØÐØÖÐÐÝÖ1<br />

ØÝÔÁÁÑÓÐ<strong>×</strong>ÚÒÓÙÒÒØÖÑ<strong>×</strong>ÓØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒ<br />

Ð<strong>×</strong><strong>×</strong><strong>×</strong>ÓCo 1ºÍ<strong>×</strong>ÒØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒØÒÖÝÓØ1 4º ØØÚÐØÓÖØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒØÒÖÝÓØ1<br />

4¹ÈË<strong>×</strong>ØØ<strong>×</strong>ÒØÀÄÑÓÐ<strong>×</strong>º<strong>×</strong>ÑÐÖÖÐØÓÒ<strong>×</strong><br />

ÖÓÒ<strong>×</strong>ØÖÙØÓÖN = 2, 3ÒÓÒØÙÖ<strong>×</strong>ÔÖÓÚÓÖN •ÁÒ½¼℄ØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒØÒÖÝÓ1<br />

ÚÒÜÔÖ<strong>×</strong><strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØÚÖÓÙ<strong>×</strong>ËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÓÙÖÒÒØÀÄ<br />

2ÓÖÖØÓÒ<strong>×</strong> 3ØÝÔÁÁ ØÒÖÝÓ1 Ð<strong>×</strong>ÓÒÓÙÒÓÖØÑÓÙÐÖÓÖÑ<strong>×</strong>ÒÖØÒØ<strong>×</strong>ØÖÒR ÒØÖÔÖØ<strong>×</strong>ØÒÓÑÒØÓÖÒØØ<strong>×</strong>ÓÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ºÜÔÖ<strong>×</strong><strong>×</strong>ÒØ ÑÓÐ<strong>×</strong>ºÌÑÓÙÐÖÓÖÑ<strong>×</strong>ÒØÀÄÑÓÐ<strong>×</strong>ÚÒÛÐÐ<strong>×</strong>ØÙÒÚÒ<br />

4ºÁÒ½¼℄Ø<strong>×</strong>ÑÓÙÐÖÓÖÑ<strong>×</strong> Ú¸ÂØÖÒËÒÚÔÖÓÚÔÖÓÙØÓÖÑÙÐÓÖØN<br />

ÑÓÙÐÖÓÖÑ<strong>×</strong>ÓØØÝÔÁÁÑÓÐ<strong>×</strong>ÒØÖÑ<strong>×</strong>ÓØÓÒ<strong>×</strong>ÓÙÖÒÒØÀÄÑÓÐ<strong>×</strong><br />

= 2,<br />

ÑÓÐ<strong>×</strong>¿½℄ÒØÖÑ<strong>×</strong>ÓØØÛ<strong>×</strong>ØÐÐÔØÒÙ<strong>×</strong>ÓÖT <strong>×</strong>ÓÙÐÐÔÒ<strong>×</strong>ØÙÝÒØÙÒÖÐÝÒÃÅÄ<strong>×</strong>ÙÔÖÐÖ<strong>×</strong>ØÖÙØÙÖ¸Òݺ ½<br />

(Z)º<br />

=

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2<br />

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A<br />

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2ºÇÒ<strong>×</strong>ϑ (z1 , z 2<br />

]<br />

(z1 , z 2 1]<br />

)Òϑ (z1 , z 2<br />

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(1 − q n ) .<br />

ϑ 3 (z 1 , z 2 ) ≡ θ [ 0<br />

0<br />

[ a<br />

θ (z 1 , z 2 ) =<br />

b] ∑ q 1 2 (l+ a 2 )2 r (l+ a 2 ) e iπlb ,<br />

l∈Z<br />

1<br />

4<br />

(z 1 , z 2 ) ≡ θ [ 0<br />

ÛØq= exp(2πiτ)Òr=exp(2πiz)º<br />

(z 1 , z 2 ) ≡ θ [ ]<br />

1 (z1 , z 2 )¸ϑ 2 (z 1 , z 2 ) ≡ θ [ 1<br />

T : ϑ 1 (τ + 1, z) = e iπ/4 ϑ 1 (τ, z) ,<br />

S : ϑ 1 (−1/τ, −z/τ) = − 1<br />

q 1/2 r eπiz2 /τ ϑ 1 (τ, z) ,<br />

η(τ) = e 2πiτ/24 ∞<br />

∏<br />

n=1<br />

1<br />

T : η(τ + 1) = e πi/12 η(τ) ,<br />

S : η(−1/τ) = e −πi/4 (τ) 1/2 η(τ) . ´ºµ<br />

½

ÌØÖÒ<strong>×</strong>ÓÖÑØÓÒÓη(Nτ)<strong>×</strong>ÚÒÝ ÔÔÒܺÌØÙÒØÓÒ<strong>×</strong><br />

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T : + N) = e Nπi/12 η(τ) ,<br />

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(τ)ºÇÒ<strong>×</strong> ´ºµ<br />

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´ºµ<br />

+ j) , j = 0, 1, . . ., N − 1 mod N .<br />

N<br />

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T : ψ j (τ + 1) = e πi/12 ψ j+1 (τ)<br />

S : ψ j (−1/τ) = e (j+j′)πi/12 (τ) 1/2 ψ −j<br />

(τ)Ò<br />

′(τ) ,<br />

ØÓÒ<strong>×</strong>ÒØÖØÛÂÓÓÖÑ<strong>×</strong>ØØÖÙ<strong>×</strong>ØÓÓÒ<strong>×</strong>ØÖÙØØËÐÑÓÙÐÖÓÖÑ<strong>×</strong> Ø<strong>×</strong><strong>×</strong>ÕÙÖÖÓÓØÒØÖÒ<strong>×</strong>ÓÖÑÛØÒÓÒ¹ØÖÚÐÖØÖºÁÒÔÖØÙÐÖ¸ÓÒÒ<strong>×</strong>ÓÛØØ<br />

ÛÖjj ′ = 1 mod<br />

(Z)Ö<strong>×</strong>ÔØÚÐݸØ<strong>×</strong>ØÛÓËÐÑÓÙÐÖÓÖÑ<strong>×</strong>ÛÐÐØÖÒ<strong>×</strong>ÓÖÑÛØÒÓÒ¹ØÖÚÐ<br />

(τ)ØÖÒ<strong>×</strong>ÓÖÑÛØÖØÖº<strong>×</strong>Ø<strong>×</strong>ØÛÓÙÒ¹ ÌØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ÓØØÖÐØÙÒØÓÒ<strong>×</strong><strong>×</strong>ÓÛÙ<strong>×</strong>ØØØÙÒØÓÒ<strong>×</strong>fk ÓÖN = 7¸f 3¸√ (1) (τ)ÒÓÖN f<br />

(Z)ÑÙ<strong>×</strong>ØØÖÒ<strong>×</strong>ÓÖÑÛØÒÓÒ¹ØÖÚÐ<br />

(4)<br />

Φ 1 (Z)Ò∆ ÖØÖ¿℄ºÌ<strong>×</strong><strong>×</strong>Ø<strong>×</strong><strong>×</strong>ÓÖÓÙÖÐÑØØ∆ 2<br />

ÖØÖÒ<strong>×</strong>ÓÒ<strong>×</strong><strong>×</strong>ØÒØÛØØÓ<strong>×</strong>ÖÚØÓÒÓÂØÖ¹ËÒÖÖÒΦ 1<br />

2<br />

S :<br />

ψ j (τ) ≡ η ( τ+j<br />

=<br />

η(−1/τ) = e−πi/4<br />

√ (τ) 1/2 η(τ/N) .<br />

N<br />

(Z)º<br />

¾¼¼

B<br />

<strong>×</strong>Ò<strong>×</strong>ØÒËÖ<strong>×</strong>ØÐÚÐN<br />

º½ Z)ºÁØ<strong>×</strong>ÚÒÝ ÈÖÑN ÄØE 2 ∗(τ)ÒÓØØÛØØÛÓÒÓÒ¹ÓÐÓÑÓÖÔÑÓÙÐÖÓÖÑÓSL(2, dlºÌÓÑÒØÓÒ ´º½µ<br />

πÁÑτ ,<br />

´º¾µ ÛÖσ l (n) = ∑ 1≤d|n<br />

Ø<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÔÖÓÙ<strong>×</strong>ÛØØÛÓÑÓÙÐÖÓÖѺÓÖÜÑÔн¸<br />

(N)ÛØÓÒ<strong>×</strong>ØÒØÓÒØÕÙÐØÓ1½¾¾¸<br />

∂<br />

]<br />

π(N−1) τ[<br />

ln η(τ) − lnη(Nτ)<br />

1¸ <strong>×</strong>ÛØØÛÓÓÐÓÑÓÖÔÑÓÙÐÖÓÖÑÓΓ 0<br />

ÌÓÖѺ℄ºÆÓØØÒÐÐØÓÒÓØÒÓÒ¹ÓÐÓÑÓÖÔÔ<strong>×</strong>ºÌÙ<strong>×</strong>¸ØÐÚÐN <strong>×</strong>ØÛعØÛÓ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ØÐÚÐ2ºØÐÚÐ<strong>×</strong>3Ò5¸ÓÒ<strong>×</strong> ´º¿µ<br />

><br />

˽¾¿℄ºÏÖÖØÙÐØÓØÙØÓÖ<strong>×</strong>ÓËÓÖÑÒØÖ<strong>×</strong>ÓØÛÖÖÐÝÚÐкÁØÛ<strong>×</strong><strong>×</strong>Ý ÓÖÙ<strong>×</strong>ØÓÚÖÝÕº´º½¼µÙ<strong>×</strong>ÒËØÓØ<strong>×</strong>ÖÓÖÖº ½ÐÐÜÔÒ<strong>×</strong>ÓÒ<strong>×</strong>ÓÖØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÚÒÖÚÒÓØÒÙ<strong>×</strong>ÒØÑØÑØ<strong>×</strong><strong>×</strong>ÓØÛÖ ´ºµ<br />

¾¼½<br />

∞<br />

E2 ∗ (τ) = 1 − 24 ∑<br />

σ 1 (n) q n + 3<br />

n=1<br />

E N (τ) = 1<br />

)<br />

(NE 2 ∗ N − 1<br />

(Nτ) − E∗ 2 (τ)<br />

= 12i<br />

E 2 (τ) = 1 + 24q + 24q 2 + 96q 3 + 24q 4 + 144q 5 + 96q 6 + · · ·<br />

E 3 (τ) = 1 + 12q + 36q 2 + 12q 3 + 84q 4 + 72q 5 + 36q 6 + · · ·<br />

E 5 (τ) = 1 + 6q + 18q 2 + 24q 3 + 42q 4 + 6q 5 + 72q 6 + · · ·

º¾<br />

ÔÔÒܺ<strong>×</strong>Ò<strong>×</strong>ØÒËÖ<strong>×</strong>ØÐÚÐN<br />

ÐÚÐM<strong>×</strong>Ð<strong>×</strong>ÓÑÓÙÐÖÓÖÑØÐÚÐNºÓÖÒ<strong>×</strong>ØÒØÐÚÐÓÙÖ¸ÓÒ<strong>×</strong>ØÛÓ<strong>×</strong>Ò<strong>×</strong>ØÒ<br />

(τ)Ò (M)ºÌÙ<strong>×</strong>¸ÓÖÓÑÔÓ<strong>×</strong>ØN¸Ø<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>Ø ÓÑÔÓ<strong>×</strong>ØN ËÙÔÔÓ<strong>×</strong>M|N¸ØÒÓÒ<strong>×</strong>Γ 0 (N) ⊂ Γ 0<br />

´ºµ <strong>×</strong>Ö<strong>×</strong>E 2<br />

(τ)Ò ´ºµ ØÐÚÐ<strong>×</strong>ܸÓÒ<strong>×</strong>ØÖ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>E 2 (τ)¸E 3<br />

(τ)Ò ´ºµ ØÐÚÐظÓÒ<strong>×</strong>ØÖ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>E 2 (τ)¸E <strong>×</strong>ØÒØØÖѽ¾¾℄º ØØÐÐ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÒØ<strong>×</strong>ÒÓÖÑÐÞØÓÒÚÒØÖÐÓÒØ<strong>×</strong>ÜÔØÓÖØÓÒ¹<br />

(τ)ÖÖØÓ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÒÓÖÑÐÞ<strong>×</strong>ÙØØØÓÒØÓq<strong>×</strong>+1ºÁØ<strong>×</strong>ÒÓÛÒ<br />

4<br />

º¿<br />

Ê N<br />

ÓÙÖÖØÖÒ<strong>×</strong>ÓÖÑÓÙØØÙ<strong>×</strong>Ôؼ<br />

ÜÔÒ<strong>×</strong>ÓÒÓÙØØÙ<strong>×</strong>ÔØ0Ø<strong>×</strong><strong>×</strong>ÓÒÝÑÔÔÒ0ØÓi∞Ù<strong>×</strong>ÒØSØÖÒ<strong>×</strong>ÓÖѺÌÓ<br />

−1/τÑÔ<strong>×</strong>ØÙ<strong>×</strong>ÔØ0ØÓØÙ<strong>×</strong>ÔØ<br />

ÓØÒØØÖÒ<strong>×</strong>ÓÖÑÓØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>¸Ö<strong>×</strong>ØÓÒ<strong>×</strong>Ö<br />

(N)<strong>×</strong>ÓÒÐÝØ<strong>×</strong>ØÛÓÙ<strong>×</strong>Ô<strong>×</strong>ºÇÒÑÝÛ<strong>×</strong>ØÓÓØÒØÓÙÖÖ ÌÑÓÙÐÖØÖÒ<strong>×</strong>ÓÖÑØÓÒ¸S¸ÙÒÖÛτ →<br />

i∞ºÏÒN<strong>×</strong>ÔÖÑ¸Γ Í<strong>×</strong>ÒØ<strong>×</strong>Ö<strong>×</strong>ÙÐظØ<strong>×</strong><strong>×</strong>ÝØÓ<strong>×</strong>Øؾ<br />

0<br />

.´ºµ<br />

)<br />

ÐÐ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>ÓÒ<strong>×</strong>ÖÒØ<strong>×</strong>ÔÔÒÜÖÓÛØØÛÓº ¾ÏÙØÓÒØÖÖØØØ<strong>×</strong>Ù<strong>×</strong>ÖÔØNÒÓØ<strong>×</strong>ØÐÚÐÒÒÓØØÛØÓØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong>º ´ºµ<br />

)<br />

.<br />

¾¼¾<br />

E<br />

E2 ∗ (Nτ)∣ ∣<br />

S<br />

= (τ) −2 E2 ∗ (NS · τ)<br />

4 (τ) = 1 + 8q + 24q 2 + 32q 3 + 24q 4 + 48q 5 + · · ·<br />

Ê 6 (τ) = 5/24 + q + 3q 2 + 4q 3 + 7q 4 + 6q 5 + · · ·<br />

Ê 8 (τ) = 7/24 + q + 3q 2 + 4q 3 + 7q 4 + 6q 5 + · · ·<br />

= (τ) −2 E ∗ 2(−N/τ) = (τ) −2 (τ/N) 2 E ∗ 2(τ/N) = 1<br />

E N (τ) ∣ ∣<br />

S<br />

= − 1 N E N<br />

( τ<br />

N<br />

( τ<br />

N 2 E∗ 2 N

ÚÒØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒØi∞¸ÛÒÓØÒØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓÙØ0ºÆÓØØ<br />

i∞ÒØÊÀËÓØÓÚÕÙØÓÒºÌÙ<strong>×</strong>¸ ÔÔÒܺ<strong>×</strong>Ò<strong>×</strong>ØÒËÖ<strong>×</strong>ØÐÚÐN<br />

ØÛØÓØÙ<strong>×</strong>ÔØ0<strong>×</strong>NºÐ<strong>×</strong>Ó¸ÒÓØØØØÓÚÓÖÑÙÐ<strong>×</strong>ÚÐÓÖÐÐN¸ÒÓØ Ò<strong>×</strong><strong>×</strong>ÖÐÝÔÖѺ<br />

ÆÓØØØτ ÒÓØÖÙ<strong>×</strong>ÙÐØÓÒÓÖÑÙÐÓÖØ<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong><strong>×</strong>ØÓÐÐÓÛÒ<br />

1/NØÓÔÖ<strong>×</strong>¸ØØ<strong>×</strong>Ù<strong>×</strong>ÔºÌ<strong>×</strong><strong>×</strong>ÜÔØ<strong>×</strong><br />

= 0ÒØÄÀËÓÖÖ<strong>×</strong>ÔÓÒ<strong>×</strong>ØÓτ =<br />

ÔÔÖÒÓÖØÓÒÐÔÓÛÖ<strong>×</strong>Óq¸q Ì<strong>×</strong>ÓÖÑÙÐÛ<strong>×</strong>ÜÔÖÑÒØÐÐÝÓØÒÝÙ<strong>×</strong>ÒØ<strong>×</strong>ÚÖØÝ<strong>×</strong>ÒØÓÖÓÙÒ ´º½¼µ<br />

ØÛÒØÝÓÖÖ<strong>×</strong>ÒØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒº º ØÓÑÔØÙ<strong>×</strong>ÔØÓi∞ÒØÒØÖØØÖÒ<strong>×</strong>ÓÖÑØÓÒÓØÒÓÒ¹ÓÐÓÑÓÖÔ<strong>×</strong>Ò<strong>×</strong>ØÒ Ì<strong>×</strong>ÑÑØÓÒÙ<strong>×</strong>ØÓÓØÒØÜÔÒ<strong>×</strong>ÓÒÓÙØÓØÖÙ<strong>×</strong>Ô<strong>×</strong>ºÒÛÛÐÐÒ ÓÙÖÖØÖÒ<strong>×</strong>ÓÖÑÓÙØÓØÖÙ<strong>×</strong>Ô<strong>×</strong><br />

)ÑÔ<strong>×</strong>1/2ØÓi∞º<br />

4ÒÓÒ<strong>×</strong>Ö <strong>×</strong>Ö<strong>×</strong>ºÄØÙ<strong>×</strong>Ó<strong>×</strong>ÔÜÑÔÐØØ<strong>×</strong>ÓÒØÖ<strong>×</strong>ØÒØ<strong>×</strong>ÔÔÖºÄØN =<br />

(τ))´º½½µ<br />

ØÙ<strong>×</strong>ÔØ1/2ºγ = 2 −1<br />

ÓÓØ<strong>×</strong>ÛØÒÓÛÒS¹ØÖÒ<strong>×</strong>ÓÖÑØÓÒ<strong>×</strong>ºÌÒÐÒ<strong>×</strong>ÛÖ<strong>×</strong>ÒØÖÑ<strong>×</strong>Ó<strong>×</strong>Ò<strong>×</strong>ØÒ<strong>×</strong>Ö<strong>×</strong><br />

(<br />

2E2 ( τ )∣ ∣<br />

2 S<br />

− E 4 ( τ )∣ )<br />

∣<br />

4 S = (E2 (τ) − E<br />

8¸ØÔÔÖ<strong>×</strong>ØØØÖÖÒÓ<strong>×</strong>ØÒÖ<br />

(τ)ÓÙØØÙ<strong>×</strong>Ô<br />

4<br />

ÁÒØÔÒÙÐØÑØ<strong>×</strong>ØÔ¸ÛÑÙ<strong>×</strong>ÓÕº´º½¼µÒÓÖÖØÓÛÖØE<br />

(τ)ÓÙØÐÐØÙ<strong>×</strong>Ô<strong>×</strong>Ø<strong>×</strong><br />

)ÒØÖÑ<strong>×</strong><br />

4 ( τ + 1 4 2<br />

ÛÓ<strong>×</strong>ÓÙÖÖÓÒØ<strong>×</strong>ÖÒÓÛÒØÙ<strong>×</strong>ÚÒÙ<strong>×</strong>ØÜÔÒ<strong>×</strong>ÓÒÓE ÓØÒÒØØ<strong>×</strong><strong>×</strong>ÑÐÖØÓØÓÒÚÒÒÕº´º½¼µº ØØÛ<strong>×</strong>ØÐÐÔØÒÙ<strong>×</strong>ÒØÓÖÖ<strong>×</strong>ÔÓÒÒÀÄÑÓÐ<strong>×</strong>ºÁØÛÓÙÐÐÔÙÐÓÒÒ <strong>×</strong>ÑÒÓÖØÒÐÙÖÐØØÒ<strong>×</strong>ØÓ<strong>×</strong>ÙÖÑÓÙÒØØÓÓÑÔÐØØÓÑÔÙØØÓÒÓ<br />

4<br />

ÓÖØÀÄÑÓÐ<strong>×</strong>ÛØN = 6ÒN ÑØÓ<strong>×</strong>ØÓØÖÑÒØÓÙÖÖÜÔÒ<strong>×</strong>ÓÒÓE 6 (τ)ÒE ¾¼¿<br />

( 1 −1<br />

E 4 (τ) + E 4 (τ + 1 2 ) = 2 E 2(2τ) .<br />

E 4 (τ) ∣ ∣<br />

ST 2 S = −1 4 E 4( τ 4 )∣ ∣<br />

ST 2<br />

= − 1 4 E 4( τ 4 + 1 2 )∣ ∣<br />

S<br />

Ø1/2º<br />

= − 1 4<br />

=<br />

8

2ÓÖÐÐÚÐÙ<strong>×</strong>Ókº ÏÒÓØØØ∆ k/2 (Z)<strong>×</strong><strong>×</strong>ÝÑÑØÖÙÒÖØÜÒz z 2 → −z<br />

∆ 5 =<br />

+<br />

+<br />

+<br />

+<br />

+<br />

+<br />

r 3 2<br />

C<br />

ÜÔÐØÓÖÑÙÐÓÖ∆ 3Ò<strong>×</strong>Òع<strong>×</strong>ÝÑÑØÖÙÒÖ<br />

k/2 (Z)<br />

↔ z<br />

1<br />

(<br />

−√ 1 + √ ) ( √q √ 9<br />

r s + − 93 + √ 90 − 90 √ )<br />

r + 93 r 3 5<br />

r r 5 2 r 3 2 − 9 r 2<br />

2 r<br />

(<br />

9<br />

r −3 2 + √r − 9 √ ( √<br />

r − r 2) 3 q 3 √<br />

)<br />

3<br />

2 s + q s 2<br />

( −9<br />

− √ 27 + 27 √ ( √<br />

r + 9 r 2) 3 q 5 √<br />

)<br />

5<br />

2 s + q s 2<br />

r<br />

(<br />

27<br />

−r −5 2 +<br />

r 3 2<br />

( 9<br />

− 12<br />

r 5 2 r 3 2<br />

( −27<br />

− 90<br />

r 5 2 r 3 2<br />

+ 12 √ r<br />

− 12 √ r − 27 r 3 2 + r<br />

5<br />

2) (<br />

q 7 2<br />

+ 90 √ r<br />

− 90 √ r + 12 r 3 2 − 9 r<br />

5<br />

2) (<br />

q 9 2<br />

− 135 √ r<br />

+ 135 √ r + 90 r 3 2 + 27 r<br />

5<br />

2) (<br />

q 11 2<br />

√ √<br />

)<br />

7<br />

s + q s 2<br />

√ √<br />

)<br />

9<br />

s + q s 2<br />

√ √<br />

)<br />

11<br />

s + q s 2<br />

(<br />

12<br />

r −7 2 + + 135 − √ 54 + 54 √ r − 135 r 3 5 7<br />

(<br />

r 5 2 r 3 2 − 12 r 2 − r 2)<br />

q 13 2<br />

2 r<br />

q 3 2 s<br />

3<br />

2<br />

√ √<br />

)<br />

13<br />

s + q s 2 + . . .<br />

¾¼

ÔÔÒܺÜÔÐØÓÖÑÙÐÓÖ∆ k/2 (Z)<br />

∆ 3 =<br />

+<br />

+<br />

+<br />

+<br />

+<br />

+<br />

(<br />

−√ 1 + √ ) ( √q √ 5<br />

r s + r −5 2 − r r 3 2<br />

(<br />

1<br />

r −3 2 + √r − √ (<br />

r − r 2) 3 q 3 2<br />

(<br />

5<br />

−r −3 2 + √r − 5 √ (<br />

r + r 2) 3 q 5 2<br />

(<br />

5<br />

−r −5 2 −<br />

r 3 2<br />

(<br />

4<br />

r −5 2 +<br />

r 3 2<br />

( 5<br />

+ 6<br />

r 5 2 r 3 2<br />

(<br />

r −7 2 −<br />

4<br />

r 5 2<br />

√ s +<br />

√ q s<br />

3<br />

2<br />

− 6 √ r<br />

+ 6 √ r + 5 r 3 2 − r<br />

5<br />

2<br />

)<br />

√ √<br />

)<br />

5<br />

s + q s 2<br />

− √ 4 + 4 √ r + 5 r 3 5<br />

(<br />

2 + r 2)<br />

q 7 2<br />

r<br />

− √ 6 + 6 √ r − 4 r 3 5<br />

(<br />

2 − r 2)<br />

q 9 2<br />

r<br />

+ √ 1 − √ r − 6 r 3 5<br />

(<br />

2 − 5 r 2)<br />

q 11 2<br />

r<br />

√ √<br />

)<br />

7<br />

s + q s 2<br />

√ √<br />

)<br />

9<br />

s + q s 2<br />

√ √<br />

)<br />

11<br />

s + q s 2<br />

− r −(3 2) 6 − √r + 6 √ r + r 3 5 7<br />

(<br />

2 + 4 r 2 − r 2)<br />

q 13 2<br />

)<br />

q 3 2 s<br />

3<br />

2<br />

√ √<br />

)<br />

13<br />

s + q s 2 + . . .<br />

∆ 2 =<br />

+<br />

+<br />

+<br />

( ) (<br />

√r 1 √q √ − √r s + 3r 3/2 − 3 ) ( ) 1 (√sq<br />

q 3/2 s 3/2 +<br />

r 3/2 r − 3/2 r3/2 3/2 + s 3/2√ q )<br />

(<br />

r 5/2 − 3 √ r + √ 3 − 1 ) (√sq 7/2 + s 7/2√ q ) (<br />

+ 3r 3/2 − 3 ) (√sq 9/2 + s 9/2√ q )<br />

r r 5/2 r<br />

(<br />

3/2 −r 7/2 − 3r 5/2 + 3<br />

r + 1 ) (√sq 13/2 + s 13/2√ q )<br />

5/2 r<br />

(<br />

7/2 3r 7/2 + 5 √ r − √ 5 − 3 ) (√sq 19/2 + s 19/2√ q ) + . . .<br />

r r 7/2<br />

¾¼

ÔÔÒܺÜÔÐØÓÖÑÙÐÓÖ∆ k/2 (Z)<br />

∆ 1 =<br />

+<br />

+<br />

+<br />

+<br />

+<br />

+<br />

( √r −<br />

1 √r<br />

) √q √ s +<br />

(<br />

−r 3/2 + √ r − √ 1 + 1 ) (√sq 3/2 + s 3/2√ q )<br />

r r<br />

(<br />

3/2 r 5/2 − 2r 3/2 + 5 √ r − √ 5 + 2<br />

r r − 1 )<br />

q 3/2 s 3/2<br />

3/2 r<br />

(<br />

5/2 −r 3/2 + 2 √ r − √ 2 + 1 ) (√sq 5/2 + s 5/2√ q )<br />

r r<br />

(<br />

3/2 r 5/2 − 2r 3/2 + 3 √ r − √ 3 + 2<br />

r r − 1 ) (√sq 7/2 + s 7/2√ q )<br />

3/2 r<br />

(<br />

5/2 r 5/2 − 3r 3/2 + 5 √ r − √ 5 + 3<br />

r r − 1 ) (√sq 9/2 + s 9/2√ q )<br />

3/2 r<br />

(<br />

5/2 2r 5/2 − 5r 3/2 + 5 √ r − √ 5 + 5<br />

r r − 2 ) (√sq 11/2 + s 11/2√ q )<br />

3/2 r<br />

(<br />

5/2 −r 7/2 + 3r 5/2 − 5r 3/2 + 9 √ r − √ 9 + 5<br />

r r − 3<br />

3/2 r + 1 ) (√sq 13/2 + s 13/2√ q ) + . . .<br />

5/2 r 7/2<br />

∆ 3/2 =<br />

( ) (<br />

√r 1 √s √ − √r q + r 5/2 − r 3/2 + 2 √ r − √ 2 + 1<br />

r r − 1 )<br />

s 3/2 q 3/2<br />

3/2 r<br />

(<br />

5/2<br />

+ −r 3/2 + √ r − √ 1 + 1 ) ( √ qs 3/2 + √ sq 3/2)<br />

r r<br />

(<br />

3/2<br />

+ −r 3/2 + √ r − √ 1 + 1 ) ( √ qs 5/2 + √ sq 5/2)<br />

r r<br />

(<br />

3/2<br />

+ r 5/2 − r 3/2 + 2 √ r − √ 2 + 1<br />

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