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

Source 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 \langle \mathcal X_1\otimes\cdots\otimes\mathcal X_k,1\rangle 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) .