Victor Ginzburg

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Representation theory. Let us try to study a given object X through vector spaces V associated naturally to it. The symmetries S of X can form a group or more generally some kind of an algebra usually related to a group (Lie algebra, quantum group,...). The Vector space V inherits the symmetries of X, and they act on V by linear operators. This is what one(More)
To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of P, where r =number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic(More)
We study the category O of representations of the rational Cherednik algebra AW attached to a complex reflection group W . We construct an exact functor, called KnizhnikZamolodchikov functor: O → HW -mod, where HW is the (finite) Iwahori-Hecke algebra associated to W . We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/Otor,(More)
on the set of F-rational points of X given by the alternating sum of traces of Fr, the Frobenius action on stalks of the cohomology sheaves HiF . He then went on to initiate an ambitious program of giving geometric (= sheaf theoretic) meaning to various classical algebraic formulas via the above “functions-faisceaux” correspondence F 7→ χ F . This program(More)
Fix a finite Coxeter group W in a complex vector space h. Thus, h is the complexification of a real Euclidean vector space and W is generated by a finite set S ⊂ W of reflections s ∈ S with respect to certain hyperplanes {Hs}s∈S in that Euclidean space. For each s ∈ S, we choose a nonzero linear function αs ∈ h∗ which vanishes on Hs (the positive root(More)
We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately(More)
From symplectic reflection algebras [12], some algebras are naturally introduced. We show that these algebras are non-homogeneous N Koszul algebras. The Koszul property was generalized to homogeneous algebras of degree N > 2 in [6]. In the present paper, the extension of the Koszul property to non-homogeneous algebras is realized through a PBW theorem. This(More)
We study a scheme M closely related to the set of pairs of n × n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n + 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil ⊂ M . We introduce a category, C , of D-modules whose characteristic variety is contained in Mnil. Simple(More)