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Journal of Lie Theory 34 (2024), No. 1, 193--206
Copyright Heldermann Verlag 2024

Characters of the Nullcone Related to Vinberg Groups

Joseph A. Fox
Mathematics Department, Aquinas College, Grand Rapids, U.S.A.


\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.

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