On Chevalley restriction theorem for semi-reductive algebraic groups and its applications

TL;DR

This paper extends the Chevalley restriction theorem to semi-reductive algebraic groups, showing their invariant rings are polynomial under certain conditions, and explores applications to nilpotent cones and singularity resolutions.

Contribution

It establishes an analogue of the Chevalley restriction theorem for semi-reductive Lie algebras, broadening understanding of their invariant theory and geometric properties.

Findings

The invariant ring [rg]^G is polynomial under a positivity condition.

Semi-reductive groups share properties with reductive groups, such as Bruhat decomposition.

Applications include analysis of nilpotent cones and singularity resolutions.

Abstract

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let $G$ ba a connected semi-reductive algebraic group over an algebraically closed field $\mathbb{F}$ and $\mathfrak{g}=Lie(G)$. It turns out that $G$ has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for $\mathfrak{g}$, which says that the $G$-invariant ring $\mathbb{F}[\mathfrak{g}]^G$ is a polynomial ring if $\mathfrak{g}$ satisfies a certain "posivity" condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.