Non-symmetric Jacobi polynomials of type $BC_1$ as vector-valued polynomials Part 2: Shift operators

TL;DR

This paper introduces shift operators for non-symmetric Jacobi polynomials of type BC_1, using vector-valued and matrix-valued frameworks, and explores their algebraic and geometric properties.

Contribution

It develops a new set of differential-reflection shift operators and a Harish-Chandra-like homomorphism for these polynomials, linking algebraic and geometric perspectives.

Findings

Four fundamental shift operators generate all shift operators.

Symmetrizations of these operators correspond to symmetric Jacobi polynomials.

The Harish-Chandra homomorphism relates to the Lepowsky homomorphism in geometric cases.

Abstract

We study non-symmetric Jacobi polynomials of type $BC_1$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials allows us to introduce shift operators for the non-symmetric Jacobi polynomials. The shift operators are differential-reflection operators and we present four of these operators that are fundamental in the sense that they generate all shift operators. Moreover, the symmetrizations of these fundamental shift operators are the fundamental shift operators for the symmetric Jacobi polynomials of type $BC_1$. For the realization of non-symmetric Jacobi polynomials of type $BC_1$ as invariant $\mathbb{C}^2$-valued Laurent polynomials, we introduce a homomorphism that is analogous to the Harish-Chandra homomorphism for the symmetric Jacobi polynomials of type $BC_1$. For geometric root multiplicities, the non-symmetric Jacobi polynomials of type $BC_1$ can be interpreted as spherical functions and we show that our Harish-Chandra homomorphism in this context is related to the Lepowsky homomorphism via the radial part map.