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 nonclassical 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 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 "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.

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

“…This note is a sequel to [25] and [30]. The program of study of so-called enhanced reductive algebraic groups (and Lie algebras) is motivated by the study of modular representations of simple Lie algebras of non-classical type (see [25] and [30]). We always assume that k is an algebraically closed field.…”

“…To a large extent, the study of irreducible representations of L can be reduced to those of L 0 (see [28], [31], [32], etc.). Such a G of the form G ⋉U is called semi-reductive in [25]. Correspondingly, it becomes a vital topic for us to study representations of semi-reductive algebraic groups and their Lie algebras in prime characteristic.…”

“…Especially, when considering the enhanced product of a connected reductive algebraic group G and an arbitrarily given (finite-dimensional) rational representations (ρ, M), we have a natural semi-reductive algebraic group G ⋉ ρ M (see §1.1), which is called an enhanced reductive algebraic group. There are really lots of interesting new phenomena and questions related to different classical theories (see [25] and [30]). 0.2.…”

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This is a sequel to [25] and [30]. Associated with G := GL n and its rational representation (ρ, M ) over an algebraically closed filed k, we define an enhanced algebraic group G := G ⋉ ρ M which is a product variety GL n × M , endowed with an enhanced cross product.

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