Intertwining Operators for Representations Induced from a Maximal Parabolic Subgroup

We present a new method of calculating intertwining operators between principal series representations of semisimple Lie groups G. Working in the compact realization we nd the eigenvalues of the operators on the K-types, and give several examples. Among the advantages of our method is its applicability to bundle-valued cases. 0. Introduction Intertwining operators of various forms play an important role in the theory of representations of semisimple Lie groups. This is in particular the case for principal series representations, where intertwinors have been applied to classiication and unitarity questions. In the Knapp-Stein theory, one starts with a non-compact realization of the principal series in which the intertwinors appear as singular integral operators depending on a parameter. Analytic continuation in the parameter is then performed in a compact picture. On the other hand, all information is also encoded in the behavior of intertwinors on the K-isotypic submodules. In this work we present a new way of constructing intertwining operators between principal series representations induced from maximal parabolic subgroups P, in the case where K-types occur with multiplicity at most 1. The method amounts to setting up an apparatus which will calculate explicitly the spectra of the operators; i.e., the eigenvalues on each K-type. The search for these eigenvalues is by no means a new project (at least in the spherical, or line bundle case, in which one induces from a 1-dimensional representation). Notable progress along these lines, in the spherical case, was made by Johnson and Wallach for rank one groups, and more recently by Kostant and Sahi, in connection with the Capelli identity. In our approach, one is excused from some of the hard work; for example, that of computing radial parts of diierential operators. This would be a formidable undertaking in the bundle-valued case, and the fact that we can bypass some of this analysis is related to the relative utility of our apparatus in the bundle case. An important application of the results obtained from the spectrum generating technique is to the determination of composition series of principal series representations. This, of course, should be a consequence of any analysis of the intertwinors, together with some analysis of irreducibility questions for the subquotients obtained from the (rational) spectral function. We include a brief discussion of irreducibil-ity questions here, and present a method of deriving strong irreducibility results