On the classification of \(k\)-involutions
type: Item
instance of: scholarly article
title: On the classification of \(k\)-involutions
zbMATH Open document ID: 0974.20033
published in: Advances in Mathematics
publication date: 2001-01-30T00:00:00Z
review text: Let \(G\) be a connected reductive algebraic group defined over a field \(k\) of characteristic not \(2\), \(\theta\) an involution of \(G\) defined over \(k\), \(H\) a \(k\)-open subgroup of the fixed point group of \(\theta\), and \(G_k\) and \(H_k\) the set of \(k\)-rational points of \(G\) and \(H\), respectively. The variety \(G_k/H_k\) is called a symmetric \(k\)-variety. The author gives a characterization of the isomorphy classes of symmetric \(k\)-varieties together with their fine structure of restricted root systems. This restricted root system is the root system of a maximal \((\theta,k)\)-split torus of \(G\), i.e., a torus that is both \(\theta\)-split and \(k\)-split. The author gives a characterization of \((\Gamma,\theta)\)-indices, where \(\Gamma\) is the Galois group of a splitting extension of the maximal torus, and a classification of these for \(k\) the real numbers, the \(p\)-adic numbers, finite fields, and number fields.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1491732
zbMATH Keywords: symmetric spaces
zbMATH Keywords: symmetric varieties
zbMATH Keywords: \(k\)-involutions
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: restricted root systems
author: Aloysius G. Helminck
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: aima.1998.1884
OpenAlex ID: W2040919408
cites work: On a remarkable class of subvarieties of a symmetric variety
cites work: Lie algebras of type \(D_ 4\) over number fields
cites work: Q5735266
cites work: The principal series for a reductive symmetric space. I. $H$-fixed distribution vectors
cites work: Q3933105
cites work: \(H\)-fixed distribution vectors for generalized principal series of reductive symmetric spaces and meromorphic continuation of Eisenstein integrals
cites work: Les sous-groupes fermes de rang maximum des groupes de Lie clos
cites work: Valuations and real places in the theory of formally real fields
cites work: Q3999894
cites work: Q3941610
cites work: The most continuous part of the Plancherel decomposition for a reductive symmetric space
cites work: Groupes reductifs
cites work: Complements à l'article: ''Groupes reductifs''
cites work: Q5662096
cites work: Meromorphic basis of \(H\)-invariant distribution vectors for generalized principal series of reductive symmetric spaces: Functional equation
cites work: Q3268342
cites work: Plancherel formula for reductive symmetric spaces
cites work: Distributions spheriques sur les espaces hyperboliques
cites work: Algebras of type \(E_7\) over number fields
cites work: Lie algebras of type \(E_6\).II
cites work: Discrete series for semisimple symmetric spaces
cites work: Über die Galoiskohomologie halbeinfacher Matrizengruppen. I
cites work: Über die Galoiskohomologie halbeinfacher Matrizengruppen II
cites work: Q3897665
cites work: Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
cites work: Tori invariant under an involutorial automorphism. I
cites work: Q4308256
cites work: Tori invariant under an involutorial automorphism. II
cites work: A class of parabolic $k$-subgroups associated with symmetric $k$-varieties
cites work: Orbits and invariants associated with a pair of commuting involutions
cites work: Q4089001
cites work: On rationality properties of involutions of reductive groups
cites work: A trace formula for symmetric spaces
cites work: Galois-Kohomologie halb einfacher algebraischer Gruppen über \(p\)-adischen Körpern. I, II
cites work: Q3972765
cites work: Singularities of closures of K-orbits on flag manifolds
cites work: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups
cites work: Q3699041
cites work: Eigenspaces of invariant differential operators on an affine symmetric space
cites work: Q3704017
cites work: Lectures on formally real fields
cites work: Orbits, invariants, and representations associated to involutions of reductive groups
cites work: The Structure of Semisimple Symmetric Spaces
cites work: Spherical Characters on p-Adic Symmetric Spaces
cites work: Q5632777
cites work: On restricted roots of semi-simple algebraic groups
cites work: Q3708857
cites work: Q3205425
cites work: Linear algebraic groups
cites work: Q5442226
cites work: Endomorphisms of linear algebraic groups
cites work: Q5648549
cites work: Geometric realization of discrete series for semisimple symmetric spaces
cites work: Q5541427
cites work: Irreducible characters of semisimple Lie groups. III: Proof of Kazhdan-Lusztig conjecture in the integral case
cites work: Opération de groupes réductifs dans un type de cônes presque homogènes

On the classification of \(k\)-involutions (cites work)
cites work: Finiteness of orbit structure for real flag manifolds

On the classification of \(k\)-involutions (DOI)
DOI: 10.1006/AIMA.1998.1884