Vinberg's \(\theta \)-groups in positive characteristic and Kostant-Weierstrass slices
type: Item
instance of: scholarly article
title: Vinberg's \(\theta \)-groups in positive characteristic and Kostant-Weierstrass slices
zbMATH Open document ID: 1209.17016
author: Paul Levy
published in: Transformation Groups
publication date: 2009-10-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0712.1725
review text: Let \(G\) be a reductive algebraic group over the algebraically closed field \(k\) and let \(\mathfrak g:={\text{Lie}}(G)\). Let \(\theta: G\to G\) be an automorphism of order \(m<\infty\) such that \({\text{char}}(k)\nmid m\) if \({\text{char}}(k)>0\). Fix a primitive \(m\)th root of unity \(\zeta\) in \(k\). Then \(\mathfrak g=\mathfrak g(0)\oplus\mathfrak g(1)\oplus\ldots\oplus\mathfrak g(m-1)\), where \(\mathfrak g(i)\) is the eigenspace of \(d\theta\) associated to the eigenvalue \(\zeta^i\). Let \(G(0)\) be the identity component of the fixed point set \(G^\theta\) of \(\theta\). Then \(G(0)\) is a reductive group and the adjoint action of \(G(0)\) stabilizes each \(\mathfrak g(i)\). For \({\text{char}}(k)=0\), in [\textit{È. B. Vinberg}, Math. USSR, Izv. 10(1976), 463--495 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 488--526 (1976; Zbl 0363.20035)], some properties of the action of \(G(0)\) on \(\mathfrak g(1)\) have been established, namely: (1) any two maximal commutative subspaces of \(\mathfrak g(1)\) (called Cartan subspaces) are \(G(0)\)-conjugate and any semisimple element of \(\mathfrak g(1)\) is contained in such a subspace; (2) closed \(G(0)\)-orbits in \(\mathfrak g(1)\) are precisely \(G(0)\)-orbits of semisimple elements; (3) for any Cartan subspace \(\mathfrak c\), the restriction map is an isomorphism between polynomial \(G(0)\)-invariants on \(\mathfrak g(1)\) and polynomial \(W_{\mathfrak c}\)-invariants on \(\mathfrak c\), where \(W_{\mathfrak c}\), the little Weyl group, is the quotient of the normalizer of \(\mathfrak c\) in \(G(0)\) by its centralizer; (4) \(W_{\mathfrak c}\) is a finite linear group generated by pseudoreflections and the algebra of polynomial \(W_{\mathfrak c}\)-invariants on \(\mathfrak c\) is free. In the paper under review these results are generalized to the case of \({\text{char}}(k)>0\). In particular, for any \(G\) satisfying the so called ``standard hypotheses'' (namely, \({\text{char}}(k)\) is g…
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 5617607
zbMATH Keywords: Lie algebra
zbMATH Keywords: invariant theory
zbMATH Keywords: adjoint representation
zbMATH Keywords: Kostant--Weierstrass section
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
OpenAlex ID: W1972800709
arXiv ID: 0712.1725
cites work: Q4273960
cites work: Q3999894
cites work: Q5668957
cites work: Q3160253
cites work: Block representation type of reduced enveloping algebras.
cites work: Reductive groups are geometrically reductive
cites work: Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
cites work: Q4089001
cites work: Q3795864
cites work: Instability in invariant theory
cites work: Orbits and Representations Associated with Symmetric Spaces
cites work: Commuting varieties of Lie algebras over fields of prime characteristic
cites work: Involutions of reductive Lie algebras in positive characteristic
cites work: The map \(V\to V//G\) need not be separable.
cites work: Component groups of unipotent centralizers in good characteristic
cites work: Q3506242
cites work: On invariant theory of \(\theta\)-groups
cites work: Period, index and potential \(\text Ш\)
cites work: Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms
cites work: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory
cites work: Zassenhaus varieties of general linear Lie algebras
cites work: Orbits, invariants, and representations associated to involutions of reductive groups
cites work: Q4154996
cites work: Q3977493
cites work: Q4214815
cites work: Q3798874
cites work: Endomorphisms of linear algebraic groups
cites work: The center of the universal enveloping algebra of a Lie algebra in characteristic $p$
cites work: Opération de groupes réductifs dans un type de cônes presque homogènes
cites work: Fixed Points and Stable Subgroups of Algebraic Group Automorphisms
DOI: 10.1007/S00031-009-9056-Y
description: scientific article