Cramped subgroups and generalized Harish-Chandra modules

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$. We call a subgroup $H\subset G$ 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 $T^*(K/L)\to\mathfrak{k}^*$ 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.