Positivity of Kac polynomials and DT-invariants for quivers

Understand

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 isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group.
• The generating function of the corresponding Poincar\'e polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions.