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

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 Schlösser.

💡 Research Summary

The paper presents a completely algebraic construction of forward and backward shift operators for the non‑symmetric Heckman‑Opdam polynomials and for the non‑symmetric Macdonald‑Koornwinder polynomials. The authors start by recalling the basic objects: a reduced or non‑reduced root system R, its Weyl group W, the weight lattice P (or the lattice L in the q‑case), and the associated group algebras C