Cramped subgroups and generalized Harish-Chandra modules

Abstract

Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup H of G {\bf cramped} if there is an integer b(G,H) such that each finite dimensional representation of G has a non-trivial invariant subspace of dimension less than b(G,H). We show that a subgroup is cramped if and only if the moment map from T^*(K/L) to k^* is surjective, where K and L are compact forms of G and H. We will use this in conjunction with sufficient conditions for crampedness given by Willenbring and Zuckerman (2004) to prove a geometric lemma on the intersections between adjoint orbits and Killing orthogonals to subgroups. We will also discuss applications of the techniques of symplectic geometry to the generalized Harish-Chandra modules introduced by Penkov and Zuckerman (2004), of which our results on crampedness are special cases.