  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,...).
