Poincare polynomials of character varieties, Macdonald polynomials and affine Springer fibers

• 成果类型: Article
• 署名作者: Mellit, Anton
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.1.3
• 发表日期: 2020
• 页码: 165-228
• 关键词: moduli space; higgs; COHOMOLOGY; bundles

摘要

We prove an explicit formula for the Poincare polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial (H) over tilde (lambda)[X; q, t] is interpreted as a weighted count of points of the affine Springer fiber over the constant nilpotent matrix of type lambda.