The Harish-Chandra isomorphism for quantum GL_2

TL;DR

This paper constructs an explicit Harish-Chandra isomorphism linking quantum differential operators on GL_2 to the spherical double affine Hecke algebra, valid for generic parameters and extended to roots of unity.

Contribution

It provides a new explicit isomorphism in quantum algebra connecting differential operators and Hecke algebras for GL_2.

Findings

Explicit isomorphism for generic parameters

Extension to root of unity cases

Conditions on parameters q and t

Abstract

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all deformation parameters non-zero q, t, such that t does not equal to +/-i, and q is not a non-trivial root of unity. We also discuss its extension to the root of unity case.