$q$-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 $GL_n$, 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.