Quantum mechanics

Character (mathematics)

Endomorphism

Affine transformation

Ecology

Matrix function

Difference polynomials

Pure mathematics

Biology

Eigenvalues and eigenvectors

Polynomial

Nilpotent matrix

Type (biology)

Physics

Mathematics

Symmetric matrix

Mathematical analysis

Geometry

Macdonald polynomials

Combinatorics

Koornwinder polynomials

Vector bundle

Generating function

Orthogonal polynomials

Annals of Mathematics

We prove an explicit formula for the Poincaré 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 $\tilde H_\lambda [X;q,t]$ is interpreted as a weighted count of points of the affine Springer fiber over the constant nilpotent matrix of type $\lambda$.

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