One-step spherical functions of the pair (SU(n+1),U(n))

TL;DR

This paper classifies all irreducible spherical functions for the pair (SU(n+1),U(n)), introduces a matrix-valued polynomial framework, and connects these functions to orthogonal Jacobi polynomials, enriching the theory of spherical functions and matrix orthogonal polynomials.

Contribution

It provides a complete classification of spherical functions of a specific type for (SU(n+1),U(n)), introduces a matrix polynomial approach, and links to orthogonal Jacobi polynomials.

Findings

Spherical functions are parameterized by pairs (w,r) in Z x Z with constraints.

Existence of a matrix-valued polynomial al of degree l that diagonalizes the problem.

Construction of matrix-valued orthogonal Jacobi polynomials from the spherical functions.

Abstract

The aim of this paper is to determine all irreducible spherical functions of the pair (G,K)=(SU(n+1),U(n)), where the highest weight of their K-types are of the form (m+l,...,m+l,m,...,m). Instead of looking at a spherical function \Phi of type \pi we look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of P_n(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type \pi turn out to be parameterized by S={(w,r)\in Z x Z : 0\leq w, 0 \leq r \leq l, 0\leq m+w+r}. A key result to characterize the associated function H_{w,r} is the existence of a matrix-valued polynomial function \Psi of degree l such that F_{w,r}(t)=\Psi(t)^{-1}H_{w,r}(t) becomes an eigenfunction of a matrix hypergeometric operator with eigenvalue \lambda(w,r), explicitly given. In the last section we assume that m\ge 0 and define the matrix polynomial P_w as the (l+1) x (l+1) matrix whose r-row is the polynomial F_{w,r}. This leads to interesting families of matrix-valued orthogonal Jacobi polynomials P_w^{\alpha,\beta} for \alpha,\beta>-1.