We construct a representation of the affine W -algebra of glr on the equivariant homology space of the moduli space of Ur-instantons, and we identify the corresponding module. As a corollary, we give a proof of a version of the AGT conjecture concerning pure N = 2 gauge theory for the group SU (r). Another proof has been announced by Maulik and Okounkov. Our approach uses a deformation of the universal enveloping algebra of W 1+∞ , which acts on the above homology space and which specializes to W (glr) for all r. This deformation is constructed from a limit, as n tends to ∞, of the spherical degenerate double affine Hecke algebra of GLn.

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∞. 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,...).

We prove a recent conjecture of Khovanov-Lauda concerning the categorification of one-half of the quantum group associated with a simply laced Cartan datum.

We prove the decomposition conjecture for the Schur algebra stated in [LT]. We also give a new approach to the Lusztig conjecture via canonical bases of the Hall algebra.