Matrix valued orthogonal polynomials and spherical functions.

I. PACHARONI

Universidad Carlos III. Madrid

Congreso; Recent Trends in Constructive Approximation Theory.; 2006

Satellite Conference of ICM06

 The representation theory of $SU(n+1)$ and the  geometry of the complex projective space $P_n(C)=SU(n+1)/U(n)$ can be used to produced examples of matrix valued orthogonal  polynomials which are eigenfunctions of a matrix valued hypergeometric differential operator. Starting from the theory of spherical functions of any type associated to the pair $(SU(n+1), U(n))$ we obtain in a natural way, families of examples of matrix valued second order ordinary differential operators $D$ which  are symmetric with respect to certain matrix valued weight functions $W$. This implies that the elements of any sequence of orthogonal polynomials with respect to these $W$´s are eigenfunctions of the corresponding $D$´s. The aim  of this talk is to describe this procedure for a class of representations of $U(n)$ and for each dimension L to obtain explicitly a sequence of orthogonal polynomials in terms of matrix valued hypergeometric functions.