# Quiver varieties and the character ring of general linear groups over finite fields

### Emmanuel Letellier

Université de Caen, Caen, France

## Abstract

Given a tuple $(X_{1},…,X_{k})$ of irreducible characters of $GL_{n}(F_{q})$ we define a star-shaped quiver $Γ$ together with a dimension vector $v$. Assume that $(X_{1},…,X_{k})$ is *generic*. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $X_{1}⊗⋯⊗X_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(Γ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity $⟨X_{1}⊗⋯⊗X_{k},1⟩$ is non-zero if and only if $v$ is a root of the Kac–Moody algebra associated with $Γ$. This is somehow similar to the connection between Horn's problem and the representation theory of $GL_{n}(C)$.

## Cite this article

Emmanuel Letellier, Quiver varieties and the character ring of general linear groups over finite fields. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1375–1455

DOI 10.4171/JEMS/395