The elliptic Hall algebra and the K-theory of the Hilbert scheme of A2

Abstract

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of ${\mathbb{A}}^2$. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of ${\mathrm{GL}}_{\infty }$. 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 ${\mathbb{P}}^2$, virtual fundamental classes, shuffle algebras, …).