Composition Series and Intertwining Operators for the Spherical Principal Series. I

Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let be a not necessarily unitary principal series representation of G on the Hilbert space . If denotes the space of K-finite vectors of induces a representation of , the enveloping algebra of the Lie algebra of G, on . In this paper, we determine when is irreducible, and if is not irreducible we determine the composition series of and the structure of the induced representations on the subquotients. Explicit computation of the intertwining operators for the different principal series representations are obtained and their relationship to polynomials defined by B. Kostant are obtained.