Representations of quantum tori and double-affine Hecke algebras

Abstract

We study a BGG-type category of infinite dimensional representations of H[W], a semi-direct product of the quantum torus with parameter `q' built on the root lattice of a semisimple group G, and the Weyl group of G. Irreducible objects of our category turn out to be parameterized by semistable G-bundles on the elliptic curve C^*/q^Z. In the second part of the paper we construct a family of algebras depending on a parameter `v' that specializes to H[W] at v=0, and specializes to the double-affine Hecke algebra introduced by Cherednik, at v=1. We propose a Deligne-Langlands-Lusztig type conjecture relating irreducible modules over the double-affine Hecke algebra to Higgs G-bundles on the elliptic curve. The conjecture may be seen as a natural `v-deformation' of the classification of simple H[W]-modules obtained in the first part of the paper. Also, an `operator realization' of the double-affine Hecke algebra, as well as of its Spherical subalgebra, in terms of certain `zero-residue' conditions is given.