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

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

2006-09-13

arxiv.org/abs/math/0609349v2

Mathematics

Representation Theory

8 pages

Scientific paper

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.

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