Mirror symmetry in the character table of SL_n(F_q) - GEOM - EPFL
Mirror symmetry in the character table of SL_n(F_q) - GEOM - EPFL
Mirror symmetry in the character table of SL_n(F_q) - GEOM - EPFL
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Arithmetic harmonic analysis<br />
Theorem (Katz, 2008)<br />
When X/Z has polynomial-count<br />
<br />
#(X(Fq)) = E(X/C, q) = |GrW k Hi (X/C)|qk (−1) i<br />
Theorem (Frobenius, 1896)<br />
For any f<strong>in</strong>ite group G:<br />
# a1, b1, . . . , ag, bg ∈ G| [ai, bi] = z =<br />
i,k<br />
<br />
χ∈Irr(G)<br />
|G| 2g−1<br />
χ(1) 2g−1 χ(z)<br />
for G = GLn(Fq) (Hausel–Letellier–Villegas, 2006–2012)<br />
have evaluated this and more general <strong>character</strong> formulas <br />
complete conjectures about <strong>the</strong> mixed Hodge polynomials<br />
<br />
i,k |GrW k Hi (MB(GLn))|qk ti us<strong>in</strong>g Macdonald polynomials<br />
for G = <strong>SL</strong>n(Fq) and z = exp( 2πid<br />
n ) <strong>the</strong> <strong>character</strong> formula was<br />
evaluated <strong>in</strong> (Mereb, 2010)<br />
12 / 15
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