Matrix valued spherical functions associated to the three dimensional hyperbolic space

Abstract

The main purpose of this paper is to compute all irreducible spherical functions on G = SL(2, ℂ) of arbitrary type δ ∈ K̂, where K = SU(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the three dimensional hyperbolic space ℍ = G/K. The entries of H are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of H in terms of Gauss' functions 2F1. Just as in the compact instance treated in [7], there is a useful role for a special class of generalized hypergeometric functions p+1Fp.