Resumen de Positivity for Kac polynomials and DT-invariants of quivers

Tamás Hausel, Emmanuel Letellier, Fernando Rodríguez-Villegas

• We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas-invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich�Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isotypical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua�s formula for Kac polynomials of quivers involving Hall�Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties