Cramped Subgroups and Generalized Harish-chandra Modules

Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup H ⊂ G 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 T ∗(K/L) → 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 proposition 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.