  The main result of this paper is a convexity theorem for momentum mappings of
certain hamiltonian actions of noncompact semisimple Lie groups. The image is
required to fall within a certain open subset D of the (dual of the) Lie
algebra, and the momentum map itself is required to be proper as a map to D.
The set D corresponds roughly, via the orbit method, to the discrete series of
representations of the group,
  Much of the paper is devoted to the study of D itself, which consists of the
Lie algebra elements which have compact centralizer. When the group is Sp(2n),
these elements are the ones which are called "strongly stable" in the theory of
linear hamiltonian dynamical systems, and our results may be seen as a
generalization of some of that theory to arbitrary semisimple Lie groups.
  As an application, we prove a new convexity theorem for the frequency sets of
sums of positive definite hamiltonians with prescribed frequencies.
