On Chevalley Restriction Theorem for Semi-reductive Algebraic Groups and Its Applications

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 be a connected semi-reductive algebraic group over an algebraically closed field \(\mathbb{F}\) and \(\mathfrak{g}=Lie{\rm{(}}G{\rm{)}}\). 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 “positivity” 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.