Spherical functions, orthogonal polynomials and the complex projective space

I. PACHARONI

Córdoba

Congreso; IV CLAM; 2012

In the scalar case, it is well known that the classical orthogonal polynomials can be obtained from the zonal spherical functions of compact Riemannian symmetric spaces of rank one. In this talk we will exhibit the interplay among the matrix orthogonal polynomials, the matrix hypergeometric function and the matrix spherical functions. We will concentrate on the determination of all irreducible spherical functions associated to the complex projective space G=K = SU(n + 1)=U(n). We will also describe the construction of a sequence of matrix valued orthogonal polynomials using all the irreducible spherical functions of a given K-type . A three term recursion relation satised by these matrix polynomials are obtained explicitly from the representation theory of the group SU(n + 1). The semi-innite matrix associated with this three term recursion relation is an stochastic matrix and it is the transition matrix of a random walk in the -spherical dual of the group SU(n + 1), for a xed nite dimensional irreducible representation  of U(n). Two stochastic models of this random walk are given in [1].