0 Representations of quantum tori and double - affine Hecke algebras

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. 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 •• H , introduced by Cherednik, at v = 1. We propose a DeligneLanglands-Lusztig type conjecture relating irreducible •• H -modules 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.