The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$

TL;DR

This paper establishes an isomorphism between the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2 and the elliptic Hall algebra, connecting geometric representation theory with algebraic structures like DAHA.

Contribution

It demonstrates the isomorphism between the equivariant K-theory convolution algebra of the Hilbert scheme and the elliptic Hall algebra, linking geometric and algebraic frameworks.

Findings

Convolution algebra in K-theory is isomorphic to the elliptic Hall algebra.

Connection established between geometric Langlands correspondence and algebraic structures.

Results include actions on moduli spaces and relations to shuffle algebras.

Abstract

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA of GL_\infty. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P^2, virtual fundamental classes, shuffle algebras,...).