Bulletin of the Australian Mathematical Society

Research Article

On orbits of algebraic groups and Lie groups

R.W. Richardsona1

a1 Department of Mathematics, Institute of Advanced Studies, Australian National University, PO Box 4, Canberra, ACT 2600, Australia.

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 (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, 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 (GS)0 on XS.

(Received July 20 1981)