The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials

TL;DR

This paper establishes an isomorphism between the Hall algebra of coherent sheaves on an elliptic curve and the stable limit of spherical double affine Hecke algebras, linking Macdonald polynomials to geometric Eisenstein series.

Contribution

It provides a new geometric construction of Macdonald polynomials via the Hall algebra and double affine Hecke algebras, connecting algebraic and geometric frameworks.

Findings

Hall algebra equals the stable limit of DAHA for GL(k) as k approaches infinity

Hecke operators correspond to Macdonald operators under the isomorphism

Macdonald polynomials are realized through specialized Eisenstein series

Abstract

We show that the Hall algebra of the category of coherent sheaves on an elliptic curve (or, equivalently, the algebra of unramified automorphic forms for GL(n) for all n) is equal to the stable limit of spherical double affine Hecke algebras for GL(k) as k goes to infinity. The two parameters correspond to the size of the finite field and the modulus of the elliptc curve. Under this isomorphism the Hecke operators are mapped to the Macdonald operators. This allows us to give a geometric construction of Macdonald polynomials (eigenvectors for the Macdonald operator) in terms of a suitable specialization of Eisenstein series (eigenvectors for the Hecke operators).