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

2021/01/17 by Ou, Ke, Shu, Bin, Yao, Yu-Feng
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2101.06578

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 \mathbbF and \mathfrakg=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 \mathfrakg, which says that the G-invariant ring \mathbbF[\mathfrakg]^G is a polynomial ring if \mathfrakg 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.

Related

• Abelian Unipotent Subgroups of Reductive Groups
• Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations
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• Complete reducibility and subgroups of exceptional algebraic groups
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