Letters in Mathematical Physics

, Volume 56, Issue 1, pp 17–30

Poisson Geometry of Discrete Series Orbits, and Momentum Convexity for Noncompact Group Actions

  • Alan Weinstein

DOI: 10.1023/A:1010913023218

Cite this article as:
Weinstein, A. Letters in Mathematical Physics (2001) 56: 17. doi:10.1023/A:1010913023218


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.

momentum mappingsconvexitynoncompact semisimple Lie algebrasstability of linear Hamiltonian systems

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Alan Weinstein
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyU.S.A.