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

TLDR: In this article, the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2 was shown to be isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞.

Abstract: In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞. 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 P2, virtual fundamental classes, shuffle algebras, …).