The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

Abstract

We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field Fl (or, more precisely, its spherical subalgebra U+X) and Cheredniks double affine Hecke algebras Ḧn of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t -1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X. Copyright © 2010 Foundation Compositio Mathematica.