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

Given a tuple (X-1, .... , X-k) of irreducible characters of GL(n)(F-q) we define a star-shaped quiver Gamma 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 circle times ... circle times 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, 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 circle times ... circle times X-k, 1 > is non-zero if and only if v is a root of the Kac-Moody algebra associated with Gamma. This is somewhat similar to the connection between Horn's problem and the representation theory of GL(n)(C) [28, Section 8].