Drinfeld realization of the elliptic Hall algebra

Abstract

We give a new presentation of the Drinfeld double of the elliptic Hall algebra introduced in a previous work with I. Burban. This presentation is similar in spirit to Drinfeld's `new realization' of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relations satisfied by (principal, unramified) Eisenstein series for the groups GL(n) over a function field. It also provides proofs of some recent conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin.