Arithmetic harmonic analysis on character and quiver varieties
Tamas Hausel
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 University Press
2011
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/1467
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>
info:eu-repo/semantics/altIdentifier/doi/10.1215/00127094-1444258
info:eu-repo/semantics/openAccess