Degeneration of horospheres in spherical homogeneous spaces
type: Item
instance of: scholarly article
title: Degeneration of horospheres in spherical homogeneous spaces
zbMATH Open document ID: 1402.22016
publication date: 2018-10-11T00:00:00Z
review text: Horospheres in a certain class (containing all simply connected symmetric spaces) of spherical homogeneous spaces \(G/H\) for semisimple algebraic groups \(G\) are studied. Here, the stationary subgroups \(H\) are supposed to be excellent (this is a notion introduced in the article). Generic horospheres are ``generic'' orbits of a maximal unipotent subgroup \(U \subset G\). But there are some other horospheres, which are not generally orbits of \(U\). They are called here degenerate horospheres and they are formed by \(U\)-orbits of lower dimensions. The main result of the article is a description of such degenerate horospheres. Two examples for \(G=\mathrm{SL}_n({\mathbf C}), H=\mathrm{SO}_n(\mathbf C)\) and for \(G=\mathrm{SL}_2({\mathbf C}) \times\mathrm{SL}_2({\mathbf C}) \times\mathrm{SL}_2({\mathbf C})\), \(H=\mathrm{SL}_2(\mathbf C)\) (diagonally embedded into \(G\); here the subgroup \(H\) is not excellent) are considered.
reviewed by: V. V. Gorbatsevich
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6951616
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: homogeneous space
zbMATH Keywords: horosphere
zbMATH Keywords: spherical variety
zbMATH Keywords: symmetric space
author: È. B. Vinberg
author: S. G. Gindikin
MaRDI profile type: Publication
full work available at URL: s10688-018-0214-7
OpenAlex ID: W2823576044
cites work: Quelques propriétés des espaces homogènes sphériques. (Some properties of spherical homogeneous spaces)
cites work: Complexity of action of reductive groups
cites work: Commutative homogeneous spaces and co-isotropic symplectic actions
cites work: Equivariant symplectic geometry of cotangent bundles
cites work: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups
cites work: Q5662023
cites work: Invariant Theory
cites work: On valuations invariant under a reductive group
cites work: Q4186561
cites work: Q5758393
cites work: Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory
cites work: Q3754107
cites work: Q3260091
Wikidata QID: Q129546570
DOI: 10.1007/S10688-018-0214-7
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 6951616