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 $(G_0,\mathfrak{g}_1)$, where $G_0$ is the identity component of the subgroup $G^\theta$ of $\theta$-fixed points in $G$ and $\mathfrak{g}_1$ is the $\omega$-eigenspace of d$\theta$ in $\mathfrak{g}=\rm Lie(G)$, 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 $k_n[\mathcal{N}]$, where $\mathcal{N}$ is the variety of nilpotent elements in $\mathfrak{g}_1$ and $k_n[\mathcal{N}]$ is the space of polynomials on $\mathcal{N}$ of homogeneous degree $n$. We use this formula to compute the multiplicities of the simple highest weight modules in $k_n[\mathcal{N}]$. This multiplicity formula is also shown to hold for all $n$ up to a certain maximum when $k$ has positive characteristic.
@article{JOLT_2024_34_1_a8,
author = {J. A. Fox},
title = {Characters of the {Nullcone} {Related} to {Vinberg} {Groups
}},
journal = {Journal of Lie Theory},
pages = {193--206},
year = {2024},
volume = {34},
number = {1},
doi = {10.5802/jolt.1332},
zbl = {1536.20059},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1332/}
}
J. A. Fox. Characters of the Nullcone Related to Vinberg Groups. Journal of Lie Theory, Volume 34 (2024) no. 1, pp. 193-206. doi: 10.5802/jolt.1332
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