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

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 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.