  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.
