Positivity for Kac polynomials and DT-invariants of quivers

• Autores: Tamás Hausel, Emmanuel Letellier, Fernando Rodríguez-Villegas
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 177, Nº 3, 2013, págs. 1147-1168
• Idioma: inglés
• DOI: 10.4007/annals.2013.177.3.8
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• Resumen

  • 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