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

Given a tuple $(\calX_1,\dots,\calX_k)$ of irreducible characters of $\GL_n(\F_q)$ we define a star-shaped quiver $\Gamma$ together with a dimension vector $\v$. Assume that $(\calX_1,\dots,\calX_k)$ is \emph{generic}. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $\calX_1\otimes\cdots\otimes\calX_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma,\v)$. 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 $\langle \calX_1\otimes\cdots\otimes\calX_k,1\rangle$ is non-zero if and only if $\v$ 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 $\GL_n(\C)$}.