On the Existence of Slices for Actions of Non-Compact Lie Groups

Abstract

If G is a topological group then by a G-space we mean a completely regular space X together with a fixed action of G on X. If one restricts consideration to compact Lie groups then a substantial general theory of G-spaces can be developed. However if G is allowed to be anything more general than a compact Lie group, theorems about G-spaces become extremely scarce, and it is clear that if one hopes to recover any sort of theory, some restriction must be made on the way G is allowed to act. A clue as to the sort of restriction that should be made is to be found in one of the most fundamental facts in the theory of G-spaces when G is a compact Lie group; namely the result, proved in special cases by Gleason 12], Koszul [5], Montgomery and Yang [6] and finally, in full generality, by Mostow [8] that there is a through each point of a G-space (see 2.1.1 for definition). In fact it is clear from even a passing acquaintance with the methodology of proof in transformation group theory that if G is a Lie group and X a G-space with compact isotropy groups for which there exists a slice at each point, then many of the statements that apply when G is compact are valid in this case also. In ? 1 of this paper we define a G-space X (G any locally compact group) -to be a Cartan G-space if for each point of X there is a neighborhood U such that the set of g in G for which g U n U is not empty has compact closure. In case G acts freely on X (i.e., the isotropy group at each point is the identity) this turns out to be equivalent to H. Cartan's basic axiom PF for principal bundles in the Seminaire H. Cartan of 1948-49, which explains the choice of name. In ? 2 we show that if G is a Lie group then the Cartan G-spaces are precisely those G-spaces with compact isotropy groups for which there is a slice through every point. As remarked above this allows one to extend a substantial portion of the theory of G-space that holds when G is a compact Lie group to Cartan G-spaces (or the slightly more restrictive class of proper G-spaces, also introduced in ? 1) when G is an arbitrary Lie group. Part of this extension is carried out in ? 4, more or less by way of showing what can be done. In particular we prove a generalization of Mostow's equivariant embed-