An elementary approach to non-symmetric shift operators and their q-analogs

TL;DR

This paper presents an algebraic method to construct shift operators for non-symmetric Heckman-Opdam and Macdonald-Koornwinder polynomials, unifying and extending previous results across different cases.

Contribution

It introduces a unified algebraic framework for shift operators associated with all linear characters of the Weyl group, encompassing known cases and providing new constructions.

Findings

Constructed shift operators for non-symmetric polynomials.

Unified approach recovers known operators in special cases.

Extends the theory to new algebraic settings.

Abstract

We give an algebraic construction of shift operators for the non-symmetric Heckman-Opdam polynomials and the non-symmetric Macdonald-Koornwinder polynomials. To each linear character of the finite Weyl group, we associate forward and backward shift operators, which are differential-reflection and difference-reflection operators that satisfy certain transmutation relations with the (Dunkl-)Cherednik operators. In the Heckman-Opdam case, the construction recovers the non-symmetric shift operators of Opdam and Toledano Laredo for the sign character. Furthermore, in rank one, we recover the rank-one non-symmetric shift operators previously obtained by the authors and Schl\"osser.