Composition series and intertwining operators for the spherical principal series. I

Johnson, Kenneth D.; Wallach, Nolan R.

doi:10.1090/S0002-9947-1977-0447483-0

Composition series and intertwining operators for the spherical principal series. I
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by Kenneth D. Johnson and Nolan R. Wallach PDF

Trans. Amer. Math. Soc. 229 (1977), 137-173 Request permission

Abstract:

Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let $\pi$ be a not necessarily unitary principal series representation of G on the Hilbert space ${H^\pi }$. If ${X^\pi }$ denotes the space of K-finite vectors of ${H^\pi },\pi$ induces a representation ${\pi _0}$ of $U(g)$, the enveloping algebra of the Lie algebra of G, on ${X^\pi }$. In this paper, we determine when ${\pi _0}$ is irreducible, and if ${\pi _0}$ is not irreducible we determine the composition series of ${X^\pi }$ 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.

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