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Spherical Functions for the Semisimple Symmetric Pair (Sp(2, ℝ), SL(2, ℂ))

Published online by Cambridge University Press:  20 November 2018

Tomonori Moriyama
Tomonori Moriyama*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-Ku, Tokyo 153-8914, Japan, email: moriy-to@ms.u-tokyo.ac.jp
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Abstract

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Let $\pi $ be an irreducible generalized principal series representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi $ to the representation induced from an irreducible admissible representation of $\text{SL}\left( 2,\,\mathbb{C} \right)$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

Keywords

22E4511F70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 828 - 865
Copyright
Copyright © Canadian Mathematical Society 2002

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