Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras

TL;DR

This paper develops a basic representation for certain confluent Cherednik algebras and introduces non-symmetric versions of various continuous $q$-orthogonal polynomials to establish faithfulness.

Contribution

It introduces a new basic representation for confluent Cherednik algebras and defines non-symmetric polynomials to prove the representation's faithfulness.

Findings

Established faithfulness of the basic representation.

Introduced non-symmetric versions of key $q$-orthogonal polynomials.

Connected representation theory with non-symmetric polynomial structures.

Abstract

In this paper we introduce a basic representation for the confluent Cherednik algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials.