$q$-Analogue of the degree zero part of a rational Cherednik algebra and generalised Van Diejen's Hamiltonians

Abstract

Inside the double affine Hecke algebra of type $G{L}_n$, which depends on two parameters $q$ and $\tau$, we define a subalgebra ${\mathbb{H}}^{{\mathfrak{gl}}_n}$ that may be thought of as a $q$-analogue 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 crossed product of the group algebra 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. As an application, we also obtain new integrable generalisations of a Macdonald-Ruijsenaars system with an external Morse (exponential) potential introduced by Van Diejen. In particular, we extend Van Diejen's Hamiltonian to the case of a system with two different types of interacting particles, and thus generalise the deformed Macdonald-Ruijsenaars operators of Chalykh-Sergeev-Veselov to the setting with an external field.

55 pages, many small improvements: introduction expanded, other details throughout and references are added, the title is modified. Accepted by Comm. Math. Physics