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.