Hausel, TamasIST Austria; Letellier, Emmanuel; Rodríguez Villegas, Fernando
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 Mathematical Journal
Hausel’s work was supported by National Science Foundation grants DMS-0305505 and DMS-0604775, by an Alfred Sloan Fellowship, and by a Royal Society University Research Fellowship. Letellier’s work supported by Agence Nationale de la Recherche grant ANR-09-JCJC-0102-01. Rodriguez-Villegas’s work supported by National Science Foundation grant DMS-0200605, by an FRA from the University of Texas at Austin, by EPSRC grant EP/G027110/1, by visiting fellowships at All Souls and Wadham Colleges in Oxford, and by a Research Scholarship from the Clay Mathematical Institute.
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Hausel T, Letellier E, Rodríguez Villegas F. Arithmetic harmonic analysis on character and quiver varieties. Duke Mathematical Journal. 2011;160(2):323-400. doi:10.1215/00127094-1444258
Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2011). Arithmetic harmonic analysis on character and quiver varieties. Duke Mathematical Journal. Duke University Press. https://doi.org/10.1215/00127094-1444258
Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Arithmetic Harmonic Analysis on Character and Quiver Varieties.” Duke Mathematical Journal. Duke University Press, 2011. https://doi.org/10.1215/00127094-1444258.
T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Arithmetic harmonic analysis on character and quiver varieties,” Duke Mathematical Journal, vol. 160, no. 2. Duke University Press, pp. 323–400, 2011.
Hausel T, Letellier E, Rodríguez Villegas F. 2011. Arithmetic harmonic analysis on character and quiver varieties. Duke Mathematical Journal. 160(2), 323–400.
Hausel, Tamás, et al. “Arithmetic Harmonic Analysis on Character and Quiver Varieties.” Duke Mathematical Journal, vol. 160, no. 2, Duke University Press, 2011, pp. 323–400, doi:10.1215/00127094-1444258.