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

Understand

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,...).