Matrix Valued Spherical Functions and Orthogonal Polynomials Associated to the Three Dimensional Sphere

IGNACIO ZURRIÁN

Pucón

Congreso; First Joint Meeting American Mathematical Society - Sociedad de Matemática de Chile; 2010

In this paper, we determine all irreducible spherical functions  of any K-type associated tothe pair (G,K) = (SO(4), SO(3)). This is accomplished by associating to  a vector valuedfunction H = H(u) of a real variable u, which is analytic at u = 0 and its components are solutionsof two coupled systems of ordinary differential equations. By an appropriate conjugationinvolving Hahn polynomials we uncouple one of the systems. Then this is taken to a system ofhypergeometric equations, leading to express the entries of H in terms of Gegenbauer’s polynomials.Then, we identify those simultaneous solutions and use the representation theory ofSO(4) to characterize all irreducible spherical functions. Finally, we conjugate both operatorsby a polynomial matrix to get two hypergeometric operators, and using strongly [1] we finda sequence of matrix valued operators with respect to a matrix weight W.