
Journal of Lie Theory 34 (2024), No. 1, 193206 Copyright Heldermann Verlag 2024 Characters of the Nullcone Related to Vinberg Groups Joseph A. Fox Mathematics Department, Aquinas College, Grand Rapids, U.S.A. jaf005@aquinas.edu [Abstractpdf] \newcommand{\g}{\mathfrak{g}} \newcommand{\N}{\mathcal{N}} \newcommand{\Lie}{{\rm Lie}} Let $G$ be a reductive linear algebraic group defined over an algebraically closed field $k$ of characteristic 0, and let $\theta$ be an automorphism of $G$ of order $m$. We consider the Vinberg pair $(G_0,\g_1)$, where $G_0$ is the identity component of the subgroup $G^\theta$ of $\theta$fixed points in $G$ and $\g_1$ is the $\omega$eigenspace of d$\theta$ in $\g=\Lie(G)$, where $\omega$ is a primitive $m$th root of 1 in $k$. In particular, we derive a formula for the formal characters of the $G_0$modules $k_n[\N]$, where $\N$ is the variety of nilpotent elements in $\g_1$ and $k_n[\N]$ is the space of polynomials on $\N$ of homogeneous degree $n$. We use this formula to compute the multiplicities of the simple highest weight modules in $k_n[\N]$. This multiplicity formula is also shown to hold for all $n$ up to a certain maximum when $k$ has positive characteristic. Keywords: Reductive groups, Vinberg pairs, nullcone, formal characters, good characteristic. MSC: 20G05. [ Fulltextpdf (150 KB)] for subscribers only. 