Positivity of Kac polynomials and DT-invariants for quivers

TL;DR

This paper provides a cohomological framework for understanding Kac polynomials and DT-invariants of quivers, proving a longstanding conjecture and connecting to recent advances in quiver variety geometry.

Contribution

It offers a new cohomological interpretation that proves Kac's conjecture and links to Kontsevich-Soibelman's work, expanding understanding of quiver invariants.

Findings

Proof of Kac's conjecture from 1982.

Derived explicit formulas for Kac polynomials and DT-invariants.

Connected generating functions to Hua's formula and Hall-Littlewood functions.

Abstract

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. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.