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

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.