The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

Abstract

AbstractWe 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 Cherednik’s double affine Hecke algebras $\ddot {\mathbf {H}}_n$ 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.