$q$-Analogue of the degree zero part of a rational Cherednik algebra

Abstract

Inside the double affine Hecke algebra $\mathbb{H}_{n, q, \tau}$ of type $GL_n$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-deformation of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra $\mathbb{H}^{\mathfrak{gl}_n}$ is a flat $\tau$-deformation of the semi-direct product of the group algebra $\mathbb{C} \mathfrak{S}_n$ of the symmetric group with the image of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{gl}_n)$ under the $q$-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra $\mathbb{H}^{\mathfrak{gl}_n}$. We describe its centre and establish a double centraliser property. Further, we develop the connection with integrable systems introduced by van Diejen, which we also generalise.Comment: 46 page