article
Arithmetic harmonic analysis on character and quiver varieties
published
Tamas
Hausel
author 4A0666D8-F248-11E8-B48F-1D18A9856A87
Emmanuel
Letellier
author
Fernando
Rodríguez Villegas
author
We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn(C) with fixed generic semisimple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of GLn(Fq). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of gln(Fq). In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of GLn(Fq). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of GLn(Fq).
Duke University Press2011
Duke Mathematical Journal10.1215/00127094-1444258
1602323 - 400
yes
T. Hausel, E. Letellier, F. Rodríguez Villegas, Duke Mathematical Journal 160 (2011) 323–400.
Hausel, Tamás, et al. “Arithmetic Harmonic Analysis on Character and Quiver Varieties.” <i>Duke Mathematical Journal</i>, vol. 160, no. 2, Duke University Press, 2011, pp. 323–400, doi:<a href="https://doi.org/10.1215/00127094-1444258">10.1215/00127094-1444258</a>.
T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Arithmetic harmonic analysis on character and quiver varieties,” <i>Duke Mathematical Journal</i>, vol. 160, no. 2. Duke University Press, pp. 323–400, 2011.
Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Arithmetic Harmonic Analysis on Character and Quiver Varieties.” <i>Duke Mathematical Journal</i>. Duke University Press, 2011. <a href="https://doi.org/10.1215/00127094-1444258">https://doi.org/10.1215/00127094-1444258</a>.
Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2011). Arithmetic harmonic analysis on character and quiver varieties. <i>Duke Mathematical Journal</i>. Duke University Press. <a href="https://doi.org/10.1215/00127094-1444258">https://doi.org/10.1215/00127094-1444258</a>
Hausel T, Letellier E, Rodríguez Villegas F. Arithmetic harmonic analysis on character and quiver varieties. <i>Duke Mathematical Journal</i>. 2011;160(2):323-400. doi:<a href="https://doi.org/10.1215/00127094-1444258">10.1215/00127094-1444258</a>
Hausel T, Letellier E, Rodríguez Villegas F. 2011. Arithmetic harmonic analysis on character and quiver varieties. Duke Mathematical Journal. 160(2), 323–400.
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