On the multiplicities of the irreducible highest weight modules over Kac-Moody algebras

TL;DR

This paper establishes a connection between weight multiplicities of integrable highest weight modules over Kac-Moody algebras and root multiplicities of related algebras, using quiver theory and geometric methods.

Contribution

It proves that these weight multiplicities equal root multiplicities of an enlarged quiver's Kac-Moody algebra, linking representation theory and quiver geometry.

Findings

Weight multiplicities match root multiplicities of an enlarged quiver's algebra.

Derived an explicit formula for Poincare polynomials of quiver varieties.

Connected Poincare polynomials with Kac polynomials via quiver representations.

Abstract

We prove that the weight multiplicities of the integrable irreducible highest weight module over the Kac-Moody algebra associated to a quiver are equal to the root multiplicities of the Kac-Moody algebra associated to some enlarged quiver. To do this, we use the Kac conjecture for indivisible roots and a relation between the Poincare polynomials of quiver varieties and the Kac polynomials, counting the number of absolutely irreducible representations of the quiver over finite fields. As a corollary of this relation, we get an explicit formula for the Poincare polynomials of quiver varieties, which is equivalent to the formula of Hausel.