Funciones Esféricas Matricales en S^3

IGNACIO ZURRIÁN

Tandil

Congreso; Reunión anual de la Unión Matemática Argentina 2010; 2010

Union Matemática Argentina

In this paper, we determine all irreducible spherical functions$\Phi$ of any $K$-type associated to the pair$(G,K)=(\mathrm{SO}(4),\mathrm{SO}(3))$. This is accomplished by associating to$\Phi$ a vector valued function $H=H(u)$ of a real variable $u$,which is analytic at $u=0$ and its components are solutions of twocoupled systems of ordinary differential equations. By anappropriate conjugation involving Hahn polynomials we uncouple oneof the systems. Then this is taken to a system of hypergeometric 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 of $\mathrm{SO}(4)$ to characterize all irreducible spherical functions. Finally, we conjugate both operators by a polynomial matrix to get two hypergeometric operators, and using strongly \cite{T4} we find a sequence of matrix valued operators with respect to a matrix weight $W$.