Matrix Valued Spherical Functions Associated to the Three Dimensional Hyperbolic Space

Grunbaum, F. A.; Pacharoni, I.; Tirao, J.

Matrix Valued Spherical Functions Associated to the Three Dimensional Hyperbolic Space

Authors

Grunbaum, F. A.

Pacharoni, I.

Tirao, J.

Description

The main purpose of this paper is to compute all irreducible spherical functions on $G={SL}(2,{\mathbb C})$ of arbitrary type $δ\in \hat K$, 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 ${\mathbb H}=G/K$. 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 ${}_2F_1$. 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