The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

Abstract

AbstractWe exhibit a strong link between the Hall algebraHXof an elliptic curveXdefined over a finite field 𝔽l(or, more precisely, its spherical subalgebraU+X) and Cherednik’s double affine Hecke algebras$\ddot {\mathbf {H}}_n$of type GLn, for alln. This allows us to obtain a geometric construction of the Macdonald polynomialsPλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curveX.