Isomorphisms of type $A$ affine Hecke algebras and multivariable orthogonal polynomials

Abstract

We examine two isomorphisms between affine Hecke algebras of type $A$ associated with parameters $q^{-1}$, $t^{-1}$ and $q$, $t$. One of them maps the non-symmetric Macdonald polynomials $E_{\eta}(x;q^{-1},t^{-1})$ onto $E_{\eta}(x;q,t)$, while the other maps them onto non-symmetric analogues of the multivariable Al-Salam & Carlitz polynomials. Using the properties of $E_{\eta}(x;q^{-1},t^{-1})$, the corresponding properties of these latter polynomials can then be elucidated.