Mirror symmetry in the character table of SL_n(F_q) - GEOM - EPFL

Mirror symmetry in the character table of
SLn(Fq)
joint work with M. Mereb and F. R. Villegas
Tamás Hausel
Chair of Geometry, EPF Lausanne
http://geom.epfl.ch/Hausel/talks
De la géométrie algébrique aux formes automorphes :
une conférence en l’honneur de Gérard Laumon
Université Paris-Sud, Orsay
29 June 2012
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Quote from (Atiyah 2004)
”I would like to make a rash prediction that ideas from physics
will have a big impact on number theory as the ideas flow
across mathematics - on one extreme number theory, on the
other physics, and in the middle geometry: the wind is
blowing, and it will eventually reach to the farthest extremities
of number theory and give us a new point of view.”
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Cartoon I
∇ · E = 0 ∇ · B = 0
Maxwell’s equations:
∇ × E = − ∂B
∂E
∂t ∇ × B = ∂t
electro-magnetic duality: (E, B) ↔ (B, −E)
⇓
S-duality for N = 4 SUSY Yang–Mills gauge groups G and G L
(Montonen–Olive 1977)
⇓
MDol
T-duality between σ-models with target MHitchin MDR
MB
(Bershadsky–Johanssen–Sadov–Vafa 1995)
⇓
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Cartoon II
(Strominger–Yau–Zaslow 1996) mirror symmetry for
MDol(G)
MDol(G
χG
L )
χGL
A
(Hausel–Thaddeus 2003; Donagi–Pantev 2006 )
⇓
Homological mirror symmetry = Geometric Langlands
D b (Coh(MDR(G))) ∼ D b (Fuk(MDR(G L )))
(Kontsevich 1994; Laumon 1987)
(Beilinson–Drinfeld 1995; Kapustin–Witten 2007)
⇓
semi-classical limit fiberwise Fourier-Mukai transform
D b (Coh(MDol(G))) ∼ D b (Coh(MDol(G L )))
(Arinkin 2002, Donagi–Pantev 2009)
⇓
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Cartoon III
cohomological shadow of fiberwise Fourier–Mukai transform
H ∗ str (MDol(G)) H ∗ str (MDol(G L ))
(Hausel–Thaddeus 2003; Ngô 2010)
⇓
P = W conjecture of (de Cataldo–Hausel–Migliorini, 2012)
H ∗ str (MB(G)) H ∗ str (MB(G L ))
(Hausel–Villegas 2004)
⇓
Frobenius formula identities in Irr(G(Fq)) and Irr(G L (Fq))
#str(MB(G)(Fq)) = #str(MB(G L )(Fq))
(Hausel–Mereb–Villegas 2012)
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SLn & PGLn Higgs moduli spaces
C smooth projective curve; G = SLn G L = PGLn; Λ ∈ Pic d (C)
Λ-twisted SLn Higgs bundle (E, φ)
φ ∈ H 0 (C; End0(E) ⊗ K) and det(E) = Λ
MDol(SLn) moduli of semi-stable Λ-twisted SLn Higgs bundles
(n, d) = 1 ⇒ MDol(SLn) non-singular quasi-projective
Γ := Jac(C)[n] (Z/nZ) 2g acts by: (E, φ) ↦→ (E ⊗ L, φ)
PGLn Higgs moduli space is the orbifold:
MDol(PGLn) := MDol(SLn)/Γ
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Hitchin map and SYZ
Hitchin map: χSLn : MDol(SLn) → A := n
i=2 H0 (C; K i )
(E, φ) ↦→ charpol(φ)
Theorem (Hitchin 1987,1989; Nitsure 1991; Faltings 1993)
χSLn
is proper completely integrable system.
Theorem (Hausel–Thaddeus 2003)
MDol(SLn)
MDol(PGLn)
χSLn
χPGLn
A
satisfies (Strominger-Yau-Zaslow 1996) for mirror symmetry.
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Topological mirror symmetry
Conjecture (Hausel-Thaddeus 2003)
H∗ str (MDol(SLn); Q) H∗ str,B (MDol(PGLn); Q)
κ∈ˆΓ H∗ (MDol(SLn); Q)κ
γ∈Γ H∗−codimMγ
Results:
n = 2, 3 by (Hausel, Thaddeus 2003)
for all n in the middle degree ∗ = MDol(SLn)
by (Garcia-Prada–Heinloth–Schmidt, 2010)
using (Laumon, 1987)
for all n up to degree ∗ < minγ∈Γ∗ codimMγ
by (Hausel–Pauly 2012)
using symmetries of Hitchin fibers (Ngô, 2006)
(MDol(SLn) γ /Γ, L B γ )
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Perverse filtration
for any proper map f : X → Y (de Cataldo-Migliorini 2005)
introduce perverse filtration ⊂ Pi ⊂ Pi+1 ⊂ . . . Pk H k (X)
from the study of the Beilinson-Bernstein-Deligne-Gabber
decomposition theorem for f!(QX) into perverse sheaves
extending the geometric fundamental lemma of (Ngô, 2010)
from Aell over A
Conjecture (Refined topological mirror test)
Under the Weil pairing Γ ∗ Γ given by κ ↔ γ
Gr P
k H∗ (MDol(SLn))κ Gr P
k− F(γ) H
2
∗−F(γ) (MDol(SLn) γ /Γ, L B γ )
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Character varieties
the SLn-character variety:
MB(SLn) := {(Ai, Bi)i=1..g ∈ SL 2g
n | [A1, B1] . . . [Ag, Bg] = ζ d n In}//PGLn
non-singular, affine
for PGLn note that Γ (Zn) 2g acts on MB(SLn)
MB(PGLn) := MB(SLn)/Γ
Theorem (Non-Abelian Hodge Theorem; Simpson, Corlette)
MDol
diff
MB
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P = W and Betti Mirror Symmetry
Conjecture (”P=W”, de Cataldo-Hausel-Migliorini 2012)
Pk (MDol) W2k (MB) under the isomorphism
H ∗ (MDol) H ∗ (MB) from non-Abelian Hodge theory.
Conjecture (Hausel–Villegas 2004)
Under the Weil pairing Γ ∗ Γ given by κ ↔ γ
Gr W
k H∗ (MB(SLn))κ Gr W
2k−F(γ) H∗−F(γ) (MB(SLn) γ /Γ, L B γ )
in particular
Eκ(MB(SLn)) :=
E(MB(SLn) γ /Γ, L B γ )q F(γ) =
k,i
i,k
|Gr W
k Hi (MB(SLn))κ|q k (−1) i
|Gr W
2k Hi (MB(SLn) γ /Γ, L B γ )|q k+F(γ) (−1) i
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Arithmetic harmonic analysis
Theorem (Katz, 2008)
When X/Z has polynomial-count
#(X(Fq)) = E(X/C, q) = |GrW k Hi (X/C)|qk (−1) i
Theorem (Frobenius, 1896)
For any finite group G:
# a1, b1, . . . , ag, bg ∈ G| [ai, bi] = z =
i,k
χ∈Irr(G)
|G| 2g−1
χ(1) 2g−1 χ(z)
for G = GLn(Fq) (Hausel–Letellier–Villegas, 2006–2012)
have evaluated this and more general character formulas
complete conjectures about the mixed Hodge polynomials
i,k |GrW k Hi (MB(GLn))|qk ti using Macdonald polynomials
for G = SLn(Fq) and z = exp( 2πid
n ) the character formula was
evaluated in (Mereb, 2010)
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Character formulas for refined Betti mirror symmetry
κ ∈ ˆΓ and ɛ ∈ ˆF × q such that ord(κ) = ord(ɛ) = k
Eκ(MB(SLn)) =
|θ| −2g
2g−2 |SLn(Fq)| θ(ζIn)
θ(1) θ(1)
when k = ord(γ)
=
θ∈Irr(SLn(Fq))
k||θ|
1
(q−1) 2g−1
E(MB(SLn) γ /Γ, L B γ )q F(γ) =
s|k
n2
k−1
µ(s)q k (g−1)
k(q−1) 2g−1
χ∈Irr(GLn/k (F q s ))
χ=χ Frobq
χ∈Irr(GLn(Fq))
χ=ɛχ
|GLn/k (F q s )|
χ(1)
2g−2 |GLn(Fq)| χ(ζIn)
χ(1) χ(1)
(2g−2)k/s k/s χ(ζn1)
χ(1)
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Main result
Theorem (Hausel–Mereb–Villegas 2012)
When κ ∈ ˆΓ and γ ∈ Γ such that ord(κ) = ord(γ) = k
E(MB(SLn) γ /Γ, L B γ )q F(γ) =
= qn2 (g−1)
k(q−1) 2g−2
u|k ∞
1
u
s|k
= Eκ(MB(SLn)).
(g−1)n2
q− uk
µ(s)
(qus−1) 2
E
M
k ( s )
B (GL n ); qus
ku
one can compute from this χ(MB(SLn)) = µ(n)n 4g−3
matching (Mereb 2010)
this has a natural t-deformation giving a conjecture for the
mixed Hodge polynomial of MB(SLn) and in turn the mixed
Hodge polynomial of MDol(SLn)
=
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Betti mirror symmetry for generic reductive group G?
let G be complex reductive
(Hausel–Letellier–Villegas 2012) definition for
H∗ str (T ∗V////G) for the stringy cohomology of symplectic
quotient attached to any representation G → GL(V)
G
µ :
2g → G
(Ai, Bi) g
group valued moment map
↦→ Πi[Ai, Bi] 1
equivariant
usual quotient MB(G) := µ −1 (1)//G
C ⊂ G generic regular semisimple conjugacy class
˜MB(G) := µ −1 (C)//G orbifold and the Weyl group W of G acts
on H∗ str ( ˜ MB(G))
∗+dim C
H∗ str (MB(G)) := Hstr ( ˜ MB(G))ɛ where ɛ is the sign
representation of W on Hmid (C)
Conjecture
H ∗ str (MB(G)) H ∗ str (MB(G L ))
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