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 .

Abstract. Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P 2 . More precisely, it predicts the existence of an action of the corresponding W -algebra on the above cohomology, satisfying certain properties.We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P 1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W -algebra U (g, e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P ; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = GL(N ), using the works of Brundan and Kleshchev interpreting the algebra U (g, e) in terms of certain shifted Yangians.

Abstract. We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U (g) of a semisimple Lie algebra g . This family is parameterized by collections µ; z1, . . . , zn , where µ ∈ g * , and z1, . . . , zn are pairwise distinct complex numbers. The construction presented here generalizes the famous construction of the higher Gaudin hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1 , our construction gives a quantization of the family of maximal Poissoncommutative subalgebras of S(g) obtained by the argument shift method. Next, we describe natural representations of commutative algebras of our family in tensor products of finite-dimensional g -modules as certain degenerations of the Gaudin model. In the case of g = slr we prove that our commutative subalgebras have simple spectrum in tensor products of finite-dimensional g -modules for generic µ and zi . This implies simplicity of spectrum in the "generic" slr Gaudin model.