Positivity for Kac polynomials and DT-invariants of quivers

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 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.

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Authors

Tamás Hausel

École Polytechnique Fédérale de Lausanne
1015 Lausanne
Switzerland

Emmanuel Letellier

Laboratoire LMNO
Université de Caen
14032 Caen
France

Fernando Rodriguez-Villegas

Department of Mathematics
University of Texas at Austin
78712 Austin, TX
and
International Centre for Theoretical Physics
Strada Costiera, 11
34151 Trieste
Italy