Positivity for Kac polynomials and DT-invariants of quivers

• 成果类型: Article
• 署名作者: Hausel, Tamas; Letellier, Emmanuel; Rodriguez-Villegas, Fernando
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2013.177.3.8
• 发表日期: 2013
• 页码: 1147-1168
• 关键词: arithmetic harmonic-analysis; cohomological hall algebra; VARIETIES; REPRESENTATIONS; character

摘要

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 Poincare 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.