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

Abstract

Given a tuple $({\mathcal{X}}_1,\dots ,{\mathcal{X}}_k)$ of irreducible characters of ${\mathrm{GL}}_n({\mathbb{F}}_q)$ we define a star-shaped quiver $\Gamma$ together with a dimension vector $\mathbf{v}$. Assume that $({\mathcal{X}}_1,\dots ,{\mathcal{X}}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product ${\mathcal{X}}_1\otimes \cdots \otimes {\mathcal{X}}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,\mathbf{v})$. 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 $⟨{\mathcal{X}}_1\otimes \cdots \otimes {\mathcal{X}}_k,1⟩$ is non-zero if and only if $\mathbf{v}$ is a root of the Kac–Moody algebra associated with $\Gamma$. This is somehow similar to the connection between Horn's problem and the representation theory of ${\mathrm{GL}}_n(\mathbb{C})$.