The Harish-Chandra isomorphism for quantum $GL_2$

Abstract

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra Dq(GL2) of quantum differential operators on GL2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all deformation parameters q ∈ C × and t = ±i, such that q is not a non-trivial root of unity. We also discuss its extension to this case.

The finiteness conjecture for skein modules

“…Example Let us consider the simplest interesting example, the case G = SL 2 , and = T 2 . Then the algebra SkAlg int G ( * ) coincides with the so-called "elliptic double" D q (G), a subalgebra of the Heisenberg double of U q (sl 2 ) (see [16] and for an expanded list of relations, [17]). This is the algebra generated by elements, a 1 1…”

“…The algebra D q (G) may be regarded simultaneously as a deformation quantization of the variety G ×G with its Heisenberg double Poisson structure [86], and as a q-analogue of the algebra D(G) of differential operators on the group G. The subalgebra of invariants in D q (G) surjects onto the usual skein algebra of the torus, via a very general procedure known as quantum Hamiltonian reduction [9,17,91].…”