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

作　　者：Ke OU Bin SHU Yu Feng YAO 

机构地区：[1]School of Statistics and Mathematics,Yunnan University of Finance and Economics,Kunming 650221,P.R.China [2]School of Mathematical Sciences,East China Normal University,Shanghai 200241,P.R.China [3]Department of Mathematics,Shanghai Maritime University,Shanghai 201306,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2022年第8期1421-1435,共15页数学学报（英文版）

基　　金：Supported by NSFC(Grant Nos.12071136,11671138,11771279,12101544);Shanghai Key Laboratory of PMMP(Grant No.13dz2260400);the Fundamental Research Funds of Yunnan Province(Grant No.2020J0375)。

摘　　要：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.

关 键 词：Semi-reductive algebraic groups semi-reductive Lie algebras Chevalley restriction theorem nilpotent cone Steinberg map Springer resolution 

分 类 号：O152[理学—数学]