Concepedia

On orbits of algebraic groups and Lie groups

Bulletin of the Australian Mathematical Society · 1982 · 94 citations · 8 references

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Concepts

Natural SciencesMathematicsAlgebraGroup DynamicsRepresentation TheoryAbstract AlgebraAlgebraic GeometryFixed PointsGroup TheoryLie AlgebrasLie GroupsAlgebraic Lie GroupsLie Group ActionsLie Group AutomorphismsOrbit ClosuresOrbit Spaces

TL;DR

The study examines a closed subgroup D of a group G with the homogeneous space X = G/D, and a subgroup S whose identity component of the centralizer, Z = (GS)⁰, is the group of interest. The paper investigates the orbits of the subgroup Z on the homogeneous space X. The authors analyze the action of Z = (GS)⁰ on the fixed‑point set X_S of S in X, including the general case where S acts by automorphisms on G and X.

Abstract

In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G . More precisely, let D be a closed subgroup of G and let X denote the coset space G/D . Let S be a subgroup of G and let Z denote ( G S ) 0 the identity component of G S , the centralizer of S in G . We consider the orbits of Z on X S , the set of fixed points of S on X . We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G ; again we consider the orbits of ( G S ) 0 on X S .