On the maximal primitive ideal corresponding to the model nilpotent orbit

Abstract

Let be a Cartan decomposition of a simple complex Lie algebra corresponding to the Chevalley involution. It is well known that among the set of primitive ideals with the infinitesimal character, there is a unique maximal primitive ideal J. Let. Let K be a connected compact subgroup with Lie algebra so that the notion of-modules is well defined. In this paper, we show that is isomorphic to. In particular, is commutative. A consequence of this result is that if W is an irreducible-module annihilated by J, then W is K-multiplicity free and two such irreducible-modules with a common nonzero K-type are isomorphic. © 2012 The Author(s).