ABSTRACT. We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C 2 . Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m n shuffle conjecture.

In this paper we introduce certain new features of the shuffle algebra of [3] that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra of [1], [7]. These results are necessary for our work in [4] and [6], where they will be applied to the study of the Hilbert scheme and to computing knot invariants.

We define an integral form of the deformed W -algebra of type gl r , and construct its action on the K-theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S, under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W -algebras by generators and relations is a new feature. Physically, this action encodes the AGT correspondence for 5d supersymmetric gauge theory on S×circle.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of G L n . We construct the action of the Yangian of sl n in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sl n [s ±1 , t]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space M n,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is 574 B. Feigin et al.naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of gl n naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on M n,d is the image of a noncommutative power sum in Z .

We construct the action of the q-deformed W -algebra on its level r representation geometrically, using the moduli space of U (r) instantons on the plane and the double shuffle algebra. We give an explicit LDU decomposition for the action of W -algebra currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson-Okounkov Ext operator and intertwiners for the deformed W -algebra. We interpret this result as a q-deformed version of the AGT-W relations.