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

Given a tuple (X1;:::; Xk) of irreducible characters of GLn(Fq) we define a star-shaped quiver together with a dimension vector v. Assume that (X1;:::; Xk) is generic. Our first result is a formula which expresses the multiplicity of the trivial character i n the tensor productX1 X k as the trace of the action of some Weyl group on the intersection cohomology of some (non-affi ne) quiver varieties associated to ( ; 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 m ultiplicityhX 1 X k; 1i is non-zero if and only if v is a root of the Kac-Moody algebra associated with . We conjecture that this remains true independently from the existence of the quiver variety. This is somehow similar to the connection between Horn’s problem and the representation theory of GL n(C) [20, Section 8].