Stationary subalgebras in general position for locally strongly effective actions
type: Item
instance of: scholarly article
title: Stationary subalgebras in general position for locally strongly effective actions
zbMATH Open document ID: 1230.22002
DOI: 10.1134/S0001434610110064
author: D. G. Il'inskii
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \(G\) be a reductive complex algebraic group, and let \(V\) be a finite-dimensional complex vector space equipped with a linear action of \(G\). For this action, there is a stabilizer in general position \(H\) (briefly, s.g.p.). We denote it by \(G_*(V)\). In invariant theory, a linear representation of an algebraic group \(G\) on a finite-dimensional space \(V\) can have several so-called ``good'' properties. First of all, these are the following properties: freedom of the algebra of invariants, constant dimension of the fibers of the factorization morphism, and finiteness of the family of orbits in each fiber of the factorization morphism. It turns out that the following condition correlates with these properties: the stabilizer in general position of the linear action \(G:V\) is nontrivial. In particular, for connected simple irreducible linear groups, the properties listed above are equivalent. In the general case, this assertion fails. Nevertheless, to clarify the general situation, the study of representations with nontrivial stabilizers in general position is of certain interest. Moreover, if the s.g.p. is nontrivial, then the stabilizers of typical closed orbits are nontrivial (and coincide with the s.g.p., if it is reductive and the group \(G\) is semisimple), and this gives a reduction to evaluate the algebra of invariants by the Luna--Richardson theorem. Denote by \(\mathfrak g\) the Lie algebra of the group \(G\). The Lie algebra \(\mathfrak h\) of the stabilizer in general position \(H = G_*(V)\) is the stationary subalgebra in general position (briefly, s.s.g.p.; we denote it by \({\mathfrak g}_*(V))\). This means that there is an open set \(U\subset V\) such that, for any vector \(v\in U\), the stationary subalgebra \({\mathfrak g}_v = \{\xi\in {\mathfrak g}\,|\,\xi_v= 0\}\) is conjugate to \(\mathfrak h\) with respect to the adjoint action of \(G\). The author somewhat simplifies the problem and study the linear actions \(G:V\) with nontrivial st…
Mathematics Subject Classification ID: 22-06
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 5980710
zbMATH Keywords: semisimple algebraic group
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: locally strongly effective action
zbMATH Keywords: stationary subalgebra
zbMATH Keywords: reductive complex algebraic group
zbMATH Keywords: irreducible representation
zbMATH Keywords: orthogonal and symplectic groups
MaRDI profile type: Publication
cites work: Q3363253
cites work: A generalization of the Chevalley restriction theorem
cites work: Q5566569
cites work: Canonical form and stationary subalgebras of points of general position for simple linear Lie groups
cites work: Canonical factorization of continuous operator functions relative to the circle
cites work: Q4040208
cites work: Q4002278
full work available at URL: s0001434610110064
OpenAlex ID: W2094604069
Recommended article: Local stationary algebras
Recommended article: The simplest stationary subalgebras, for compact linear Lie algebras
Recommended article: Localizing algebras and invariant subspaces
Recommended article: Stationary subsets and stabilizers of restrictive Menger \(P\)-algebras of multiplace functions
Recommended article: Local commutants and ultrainvariant subspaces
Recommended article: Q3320593
Recommended article: Stationarity of operator algebras
Recommended article: Effective discreteness radius of stabilizers for stationary actions
Recommended article: On stationary subgroups of compact homogeneous spaces
Recommended article: Stationary Subsets of Functional Menger ∩-Algebras of Multiplace Functions
description: scientific article; zbMATH DE number 5980710