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

2017/10/12 by Anton Mellit, Mellit, Anton · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1710.04513

openalex publication_date 2017/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an explicit formula for the Poincar 'e polynomials of parabolic\ncharacter varieties of Riemann surfaces with semisimple local monodromies,\nwhich was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an\napproach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of\na parabolic vector bundles and a nilpotent endomorphism of prescribed generic\ntype. The generating function counting these pairs is shown to be a product of\nMacdonald polynomials and the function counting pairs without parabolic\nstructure. The modified Macdonald polynomial H_\λ[X;q,t] is\ninterpreted as a weighted count of points of the affine Springer fiber over the\nconstant nilpotent matrix of type \λ.\n

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