Kac's conjecture from Nakajima quiver varieties

TL;DR

This paper proves a generating function formula for Betti numbers of Nakajima quiver varieties, linking it to a q-deformation of the Weyl-Kac character formula and confirming Kac's conjecture from 1982.

Contribution

It establishes a new generating function formula for Betti numbers and confirms Kac's conjecture relating quiver representations to Kac-Moody algebra multiplicities.

Findings

Betti numbers of Nakajima quiver varieties are given by a specific generating function

The generating function is a q-deformation of the Weyl-Kac character formula

The constant term matches the multiplicity of a weight in a Kac-Moody algebra

Abstract

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.