Kac's conjectures on quiver representations via arithmetic harmonic analysis -- Tamas Hausel (University of Oxford and University of Texas at Austin)

In 1982 Kac introduced the A-polynomial, as the number of absolutely indecomposable representations of a quiver with a dimension vector over a finite field. He conjectured that (1) the constant term of this polynomial is a certain weight multiplicity in the corresponding Kac-Moody algebra and that (2) the coefficients of the A-polynomial are non-negative integers. In this talk I will mention how doing Fourier transform on quiver varieties completes a proof of (1). I also explain a conjecture of Hausel-Letellier-Villegas on the mixed Hodge polynomials of all character varieties of Riemann surfaces using Macdonald polynomials. A further conjecture will relate it to the A-polynomial, and will imply (2) in the case of star-shaped quivers.