Vector-valued Heckman-Opdam polynomials: a Steinberg variation

TL;DR

This paper introduces a new class of vector-valued Jacobi polynomials associated with finite reflection groups, extending classical theories and connecting to spherical functions on symmetric pairs, with potential for generating matrix-valued orthogonal polynomials.

Contribution

It develops a unified framework for vector-valued Jacobi polynomials for parabolic subgroups, combining Steinberg, Heckman, and Opdam results to produce new examples and properties.

Findings

Polynomials are eigenfunctions of a commutative algebra of differential operators.

Identification with spherical functions for higher K-types on symmetric pairs.

Method to generate new matrix-valued orthogonal polynomials in multiple variables.

Abstract

We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate cases we combine results of Steinberg and Heckman and Opdam to obtain new examples of families of vector-valued orthogonal polynomials with properties similar to those of the usual Jacobi polynomials. Most notably we show that these polynomials, when suitably interpreted as vector-valued polynomials, are determined up to scaling as simultaneous eigenfunctions of a commutative algebra of differential operators. We establish an example in which the vector-valued Jacobi polynomials can be identified with spherical functions for a higher $K$-type on a compact symmetric pair with restricted root system of Dynkin type $A_{2}$. We also describe how to obtain new examples of matrix-valued orthogonal polynomials in several variables.