A formula for the $q$-character of functions on the nilpotent cone of some Lie algebra representations

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra and $V$ a finite-dimensional $\mathfrak{g}$-representation. When $V$ is the representation of a cyclic quiver with equal dimensions, the representation of a cyclic quiver with two vertices, or a representation of a product of copies of $\mathfrak{sl}_2$ we call an acyclic extended quiver representation of trivial type, we prove a $q$-character formula for the nilpotent cone of $V$ analogous to Hesselink's $q$-character formula of the usual nilpotent cone of $\mathfrak{g}$. We also define a new class of representations we call Hesselink-type representations, for which we make a conjecture in relation to our formula and describe a geometric interpretation.