  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.
