On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.Comment: AMS-TeX, 9 page

Affine quantum groups and equivariant K-theory

We give complete proofs of the K-theoretic construction of the quantized enveloping algebra of affine gl(n) sketched with V. Ginzburg in Internat. Math. Res. Notices, 3 (1993).Comment: 30 pages, plain te

From double affine Hecke algebras to quantized affine Schur algebras

We prove that the double affine Hecke algebra of type A is Morita equivalent to the quantized affine Schur algebra.Comment: 27 page

On cohomological Hall algebras of quivers : generators

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.Comment: 80 page

On cohomological Hall algebras of quivers : Yangians

We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov, and we construct an embedding of Y in the Yangian, intertwining the respective actions of both algebras on the cohomology of quiver varieties.Comment: 41 page

On the K-theory of the cyclic quiver variety

We compute the convolution product on the equivariant K-groups of the cyclic quiver variety. We get a q-analogue of double-loop algebras, closely related to the toroidal quantum groups previously studied by the authors. We also give a geometric interpretation of the cyclic quiver variety in terms of equivariant torsion-free sheaves on the projective plane.Comment: 22 pages, AMS-te

The cohomological Hall algebra of a surface and factorization cohomology

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we identify R(S), the Hecke algebra of 0-dimensional sheaves. For the flat case S=A^2, we identify R(S) explicitly. For a general S we find the graded dimension of R(S), using the techniques of factorization cohomology.Comment: 48 page

Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A^2

We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and 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 suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r). This deformation is in turn constructed from a limit, as n tends to infinity, of the spherical degenerate double affine Hecke algebra of GL_n.Comment: 95 pages, Latex2

Perverse sheaves and quantum Grothendieck rings

We define a quantum analogue of the Grothendieck ring of finite dimensional modules of a quantum affine algebra of simply laced type. The construction is based on perverse sheaves on a variety related to quivers. We get also a new geometric construction of the tensor category of finite dimensional modules of a finite dimensional simple Lie algebra of type A-D-E.Comment: 20 page

Formal loops III: Factorizing functions and the Radon transform

To any algebraic variety X and and closed 2-form \omega on X, we associate the "symplectic action functional" T(\omega) which is a function on the formal loop space LX introduced by the authors in math.AG/0107143. The correspondence \omega --> T(\omega) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(\omega) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras. These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form \omega the invertible function exp T(\omega).Comment: 23 page