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

Abstract

AbstractLet π be an irreducible generalized principal series representation of G = Sp(2, ℝ) induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from π to the representation induced from an irreducible admissible representation of SL(2, ℂ) 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.