Drinfeld realization of the elliptic Hall algebra

TL;DR

This paper introduces a new Drinfeld-style presentation of the elliptic Hall algebra, addressing a key question in the theory of Eisenstein series for function fields and confirming recent conjectures.

Contribution

It provides a novel Drinfeld double presentation of the elliptic Hall algebra, linking it to quantum affine algebra realizations and solving an open problem posed by Kapranov.

Findings

New presentation similar to Drinfeld's realization

Answers Kapranov's question on Eisenstein series relations

Proves conjectures by Feigin et al.

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.