The Harish-Chandra isomorphism for quantum $G{L}_2$

Abstract

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra ${\mathcal{D}}_q(G{L}_2)$ of quantum differential operators on $G{L}_2$, to the spherical double affine Hecke algebra associated to $G{L}_2$. The isomorphism holds for all deformation parameters $q\in {\mathbb{C}}^{\times }$ and $t\ne \pm i$, such that $q$ is not a non-trivial root of unity.