Stable Graded Multiplicities for Harmonics on a Cyclic Quiver

Abstract

We consider Vinberg $\theta$-groups associated to a cyclic quiver on $k$ nodes. Let $K$ be the product of the general linear groups associated to each node. Then $K$ acts naturally on $\oplus \text{Hom}(V_i, V_{i+1})$ and by Vinberg's theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of $K$, and give a combinatorial formula for the stable graded multiplicity of each $K$-type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics.