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

n 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 𝔽 and 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 𝔤 , which says that the G -invariant ring 𝔽[𝔤]^G is a polynomial ring if 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.