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

Max van Horssen, Maarten van Pruijssen

TL;DR

The paper develops an elementary algebraic framework to construct non-symmetric shift operators for the non-symmetric Heckman--Opdam and Macdonald--Koornwinder polynomials, parameterized by Weyl-group characters. It defines forward and backward shift operators that satisfy transmutation relations with Dunkl--Cherednik (or Cherednik) operators, shifting multiplicities by the fundamental shifts and mapping $E_\mu(k)$ to $E_{\mu_{\varepsilon,\pm}}(k\pm l)$ (or their $q$-analogs with $l^\wedge$) with explicit shift factors. The main contributions include explicit operator constructions, adjoint relationships, and $L^2$-norm recursions for HO polynomials, as well as the first higher-rank non-symmetric shift operators for Macdonald--Koornwinder polynomials, complete with transmutation properties. The approach unifies symmetric and non-symmetric shift operators via the shift principle and KZ-type correspondences, enriching the spectral theory of DAHA-related families and offering practical tools for norms, evaluations, and eigenfunction analysis. Overall, the work advances shift-operator techniques in integrable systems and algebraic combinatorics, with potential applications in special functions and representation theory.

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 Schlösser.

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

TL;DR

The paper develops an elementary algebraic framework to construct non-symmetric shift operators for the non-symmetric Heckman--Opdam and Macdonald--Koornwinder polynomials, parameterized by Weyl-group characters. It defines forward and backward shift operators that satisfy transmutation relations with Dunkl--Cherednik (or Cherednik) operators, shifting multiplicities by the fundamental shifts and mapping to (or their -analogs with ) with explicit shift factors. The main contributions include explicit operator constructions, adjoint relationships, and -norm recursions for HO polynomials, as well as the first higher-rank non-symmetric shift operators for Macdonald--Koornwinder polynomials, complete with transmutation properties. The approach unifies symmetric and non-symmetric shift operators via the shift principle and KZ-type correspondences, enriching the spectral theory of DAHA-related families and offering practical tools for norms, evaluations, and eigenfunction analysis. Overall, the work advances shift-operator techniques in integrable systems and algebraic combinatorics, with potential applications in special functions and representation theory.

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 Schlösser.
Paper Structure (22 sections, 43 theorems, 168 equations)

This paper contains 22 sections, 43 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Structure
  5. Background on Heckman--Opdam theory
  6. Preliminaries on root systems
  7. Orthogonal polynomials
  8. Dunkl--Cherednik operators
  9. Graded Hecke algebra
  10. Symmetric shift operators
  11. KZ-equations, Cherednik--Matsuo correspondence, and the KZ-shift operator
  12. Shift operators for non-symmetric Heckman--Opdam polynomials
  13. Non-symmetric shift operators for fixed weights
  14. Genuine non-symmetric shift operators
  15. Properties of non-symmetric shift operators
  16. ...and 7 more sections

Key Result

Theorem A

For each linear character $\varepsilon$ of $W$, there exist non-symmetric shift operators $\mathcal{G}^{(\varepsilon)}_\pm(k)$ with shifts $\pm l$, i.e., which are called the $\varepsilon$-forward/backward non-symmetric shift operators. These operators shift the multiplicity of the non-symmetric Heckman--Opdam polynomials by $\pm l$: for certain explicit shift factors $\mathcal{H}^{(\varepsilon)

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem A: See Theorem \ref{['thm:transmutation_prop']}
  • Definition 1.2
  • Theorem B: See Theorem \ref{['thm:transmutation_prop_q']}
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.4
  • Proposition 2.5: Shift principle
  • Theorem 2.6
  • Corollary 2.7
  • ...and 47 more