Elliptic curves, Cherednik Hecke algebras and Macdonald polynomials I

We provide a realization of Cherednik's double affine Hecke

algebras (for GL_n) as a convolution algebra of functions on moduli spaces

of coherent sheaves on an elliptic curve. As an application we give a

geometric construction of Macdonald polynomials as (traces of) certain

natural perverse sheaves on these moduli spaces. We will discuss the

possible extensions to higher (or lower !) genus curves and the relation

to the Hitchin nilpotent variety. This is (partly) based on joint work

with I. Burban and E. Vasserot.