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 $G=SL(2,\mathbb{C})$ of arbitrary type $\delta \in \hat{K}$, where $K=SU(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the three dimensional hyperbolic space $\mathbb{H}=G/K$. 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 ${}_2{F}_1$. Just as in the compact instance treated in [GPT] there is a useful role for a special class of generalized hypergeometric functions ${}_{p+1}{F}_p$.

55 pages