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

Abstract

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 $(\Gamma ,\check{)}$. 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 \( \GL_n(\C) \)}.