Coadjoint orbits and the beginnings of a geometric representation theory

Abstract

This is a review of Banach Poisson manifolds with special emphasis on Banach–Lie–Poisson spaces. Unlike the finite-dimensional case, the existence of the (weak) symplectic leaves is not guaranteed even for coadjoint orbits in the predual of the associated Lie algebra. A significant part of the paper is hence devoted to examples of such coadjoint orbits. A special class of K¨ahler orbits is isolated for which the classical Borel–Weil theorem extends to the Banach case by the use of reproducing kernels.