# Victor GinzburgUniversity of Chicago | UC · Department of Mathematics

Victor Ginzburg

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103

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Introduction

**Skills and Expertise**

## Publications

Publications (103)

We study the natural $GL_2$-action on the Hilbert scheme of points in the
plane, resp. $SL_2$-action on the Calogero-Moser space. We describe the closure
of the $GL_2$-orbit, resp. $SL_2$-orbit, of each point fixed by the
corresponding diagonal torus. We also find the character of the representation
of the group $GL_2$ in the fiber of the Procesi b...

We study the natural Gieseker and Uhlenbeck compactifications of the rational
Calogero-Moser phase space. The Gieseker compactification is smooth and
provides a small resolution of the Uhlenbeck compactification. This allows
computing the IC stalks of the Uhlenbeck compactification.

We study holonomic D-modules on SLn(C)×Cn, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cheredni...

We apply the technique of formal geometry to give a necessary and sufficient
condition for a line bundle supported on a smooth Lagrangian subvariety to
deform to a sheaf of modules over a fixed deformation quantization of the
structure sheaf of an algebraic symplectic variety.

We describe the equivariant cohomology of cofibers of spherical perverse
sheaves on the affine Grassmannian of a reductive algebraic group in terms of
the geometry of the Langlands dual group. In fact we give two equivalent
descriptions: one in terms of D-modules of the basic affine space, and one in
terms of intertwining operators for universal Ve...

Let g be a complex reductive Lie algebra with Cartan algebra t: Hotta and Kashiwara defined a holonomic D-module M, on g × t, called the Harish-Chandra module. We relate gr M, an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g × g × t × t which is a ramified cover o...

We study holonomic D-modules on SL_n(C) x C^n, called mirabolic modules,
analogous to Lusztig's character sheaves. We prove that a mirabolic module is
killed by the functor of Hamiltonian reduction from the category of mirabolic
modules to the category of representations of the trigonometric Cherednik
algebra if and only if the characteristic varie...

Based on the ideas of Cuntz and Quillen, we give a simple construction of
cyclic homology of unital algebras in terms of the noncommutative de Rham
complex and a certain differential similar to the equivariant de Rham
differential. We describe the Connes exact sequence in this setting.
We define equivariant Deligne cohomology and construct, for eac...

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and
Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra
module. We relate gr(M), an associated graded module with respect to a
canonical Hodge filtration on M, to the isospectral commuting variety, a
subvariety of g x g x h x h which is a ramified cover of t...

In this article we explore some of the combinatorial consequences of recent
results relating the isospectral commuting variety and the Hilbert scheme of
points in the plane.

One might ask whether the work of producing representations of Weyl groups by geometric means, carried out in the previous chapter, was worth-while. Our point is that absolutely the same machinery can be applied to construct representations of sln(C) and perhaps other semisimple Lie algebras, cf. [Na2]. Many of the objects we use for studying the s...

Let R ⊂ P be a reduced (not necessarily finite) root system as defined, e.g., in 3.1.22. There is a slight difference with 3.1.22,
since now we are working with lattices instead of vector spaces. This makes axiom 3.1.22(3) superfluous. Thus it is assumed
only that, in addition to the above data, a subset R
v ⊂ P
v, called the dual root system, and...

Our primary goal in this chapter is to obtain a classification of simple modules over the affine Hecke algebra H although
the techniques we develop works in much greater generality (we will indicate this on several occasions). In §8.1 we introduce
a class of “standard” H-modules. In the same section we define simple H-modules in terms of a certain...

In this chapter we study some further properties of general complex semisimple groups. Most of the results of the result of
the chapter play a crucial role in the representation theory of semisimple groups and Lie algebras. We have tried to assemble
and give complete proofs for all those results that are, on the one hand, considered “too advanced”...

This chapter is devoted to the fundamentals of equivariant algebraic K–theory. The reader interested mostly in the applications to representation theory may skip this chapter and use it only as a reference for later chapters. As has been explained in the introduction, most of the results here were proved by Thomason [Th1]–[Th4], sometimes in much g...

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We give an explicit description of gr(M), the associated graded module with respect to a canonical Hodge filtration on M. The description involves the isospectral commuting variety, a subvariety...

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (𝒟, G) where 𝒟 is a sheaf of twisted differential operators on X, we form a left ideal generated by the Lie algebra 𝔤 = LieG. Then, 𝒟/𝒟 𝔤 is a holonomic 𝒟-module, and its restriction to a unique Zariski open dense G-orbit in X i...

We begin this section by reviewing some basic facts about semisimple groups and Lie algebras which we will need in the rest
of this book. For further information the reader is referred to [Bour], [Bo3], [Hum], [Se1], and [Di].

Preface.- Chapter 0. Introduction.- Chapter 1. Symplectic Geometry.- Chapter 2. Mosaic.- Chapter 3. Complex Semisimple Groups.- Chapter 4. Springer Theory.- Chapter 5. Equivariant K-Theory.- Chapter 6. Flag Varieties, K-Theory, and Harmonic Polynomials.- Chapter 7. Hecke Algebras and K-Theory.- Chapter 8. Representations of Convolution Algebras.- B...

There are two essential differences between symplectic and Riemannian geometries. First, the Riemannian geometry is “rigid” in the sense that two Riemannian manifolds chosen at random are most likely to be locally nonisometric. On the contrary, any two symplectic manifolds are locally isometric in the sense that the symplectic 2-form on any symplec...

We prove a criterion stating when a line bundle on a smooth coisotropic subvariety Y of a smooth variety X with an algebraic Poisson structure, admits a first order deformation quantization. Comment: 16 pages

Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf O_X to a sheaf of noncommutative algebras and of the sheaves O_Y and O_Z to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkovisk...

This is an expanded version of lectures given at a Summer School "Geometric methods in Representation Theory" (Grenoble, 2008). Comment: 40pp

We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall–Littlewood basis of symmetric functions, introduced by Shoji.

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice.
In this paper, we define and study Harish-Chandra bimodules over Premet’s algebras. We apply the technique of Harish-Chand...

We present a new approach to cyclic homology that does not involve the Connes
differential and is based on a `noncommutative equivariant de Rham complex' of
an associative algebra. The differential in that complex is a sum of the
Karoubi-de Rham differential, which replaces the Connes differential, and
another operation analogous to contraction wit...

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view...

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra, and let L
G be a complex semisimple group with trivial center whose root system is dual to that of \( \mathfrak{g} \). We establish a graded algebra isomorphism \( H^{\raise0.145em\hbox{${\scriptscriptstyle \bullet}$}} {\left( {X_{\lambda } ,\mathbb{C}} \right)} \cong {{\text{S}}\mathfrak{...

This is a copy of the article by the same authors published in Duke Math. J. (1994).

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DerA, the bimodule of double derivations. Our differential operators act not on the algebra A itself...

The hypersurface in ℂ3 with an isolated quasi-homogeneous elliptic singularity of type Ēr, r = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type Er provides a semiuniversal Poisson deformation of that Poisson structure.
We also construct a deformation-quantization of the coordinate ri...

To George Lusztig with admirationFor any algebraic curve C and n≥1, Etingof introduced a “global” Cherednik algebra as a natural deformation of the cross product \(\mathcal{D}({C}^{n}) \rtimes {\mathbb{S}}_{n}\) of the algebra of differential operators on C
n
and the symmetric group. We provide a construction of the global Cherednik algebra in term...

We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T∗B, which is a basic example of noncommutative symplectic man...

We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau 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 C...

We study a class of associative algebras associated to finite groups acting on a vector space. These algebras are non-homogeneous N-Koszul algebra generalizations of symplectic reflection algebras [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002) 243–34...

We compute the stable homology of necklace Lie algebras associated with quivers and give a construction of stable homology classes from certain $A_\infty$-categories. Our construction is a generalization of the construction of homology classes of moduli spaces of curves due to M. Kontsevich. In the second part of the paper we produce a Moyal-type q...

We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep_n(A), of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, inclu...

We prove a localization theorem for the type A n- 1 rational Cherednik algebra H c = H 1,c (A n-1) over double-struck F sign̄ p. an algebraic closure of the finite field. In the most interesting special case where c ε double-struck F sign p, we construct an Azumaya algebra ℋ c on Hilb n double-struck A sign 2, the Hubert scheme of n points in the p...

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in...

We deduce a special case of a theorem of M. Haiman concerning alternating polynomials in 2n variables from our results about almost commuting variety, obtained earlier in a joint work with W.-L. Gan.

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncom...

From symplectic reflection algebras, some algebras are naturally introduced. We show that these algebras are non-homogeneous N-Koszul algebras, through a PBW theorem.

We give a new construction of cyclic homology of an associative algebra A that does not involve Connes' differential. Our approach is based on an extended version of the complex \Omega A, of noncommutative differential forms on A, and is similar in spirit to the de Rham approach to equivariant cohomology. Indeed, our extended complex maps naturally...

We use Moyal-type formulas to construct a Hopf algebra quantization of the necklace Lie bialgebra associated with a quiver.

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to commutativity of the cup-product on Hochschild cohomology, a morphism from the graded algebra Sym(k^n) to Ext^*(...

We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T^*B, which is a basic example of noncommutative symplectic ma...

The theory of PBW properties of quadratic algebras, to which this
paper aims to be a modest contribution, originates from the
pioneering work of Drinfeld (see [Dr1]). In particular, as we
learned after publication of [EG] (to the embarrassment of
two of us!), symplectic reflection algebras, as well as PBW theorems for
them, were discovered by Drinf...

We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding sympl...

We study a scheme M closely related to the set of pairs of n by 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 Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. S...

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero–Moser deformation of the orbifold V/G is, in an appro...

We prove a localization theorem for the type A rational Cherednik algebra H_c=H_{1,c} over an algebraic closure of the finite field F_p. In the most interesting special case where the parameter c takes values in F_p, we construct an Azumaya algebra A_c on Hilb^n, the Hilbert scheme of n points in the plane, such that the algebra of global sections...

We study the category ?? of representations of the rational Cherednik algebra A W attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: ???H W-mod, where H W is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between...

We show that solving the Maurer-Cartan equations is, essentially, the same thing as performing the Hamiltonian reduction construction. In particular, any differential graded Lie algebra equipped with an even nondegenerate invariant bilinear form gives rise to modular stacks with symplectic structures.

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expre...

We study the category of representations of the rational Cherednik algebra AW attached to a complex reflection group W . We construct an exact functor, called KnizhnikZamolodchikov functor: -mod, where is the (finite) Iwahori-Hecke algebra associated to W . We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between tor , the qu...

We study a BGG-type category of infinite-dimensional representations of H[W ], a semidirect product of the quantum torus with parameter q, built on the root lattice of a semisimple group G, and the Weyl group of G. Irreducible objects of our category turn out to be parametrized by semistable G-bundles on the elliptic curve C * /q Z .

To any finite group Gamma subset of SL2(C) and each element tau in the center of the group algebra of Gamma, we associate a category, Coh(P-Gamma,tau(2), P-1). It is defined as a suitable quotient of the category of graded modules over ( a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category C...

A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

We classify the rational Cherednik algebras H_c(W) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when W is the symmetric group and `c' is a generic parameter value.

Let the group μm of m th roots of unity act on the complex line by multiplication. This gives a μm-action on Diff, the algebra of polynomial differential operators on the line. Following Crawley-Boevey and Holland (1998),
we introduce a multiparameter deformation Dτ of the smash product Diff #μm. Our main result provides natural bijections between...

We extend two well-known results on primitive ideals in enveloping algebras
of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem',
to much wider classes of algebras. Our general version of Irreducibility
theorem says that if A is a positively filtered associative algebra such that
gr(A) is a commutative Poisson algebra with f...

Let W be a finite Coxeter group in a Euclidean vector space V, and let m be a W-invariant Z(+)-valued function on the set of reflections in W. Chalykh and Veselov introduced an interesting algebra Q(m), called the algebra of m-quasi-invariants for W, such that C[V](W) subset of Q(m) subset of C[V], Q(0) = C[V], and Q(m) superset of Q(m') whenever m...

We consider the canonical map from the Calogero-Moser space to symmetric powers of the affine line, sending conjugacy classes of pairs of n by n matrices to their eigenvalues. We show that the character of a natural C^*-action on the scheme-theoretic zero fiber of this map is given by Kostka polynomials.

We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. We further stud...

Let W be a finite Coxeter group in a Euclidean vector space V, and m a W-invariant Z_+-valued function on the set of reflections in W. Chalyh and Veselov introduced in an interesting algebra Q_m, called the algebra of m-quasiinvariants for W. This is the algebra of quantum integrals of the rational Calogero-Moser system with coupling constants m. I...

We give a direct proof of (a slight generalization of) the recent result of
A. Premet related to generalized Gelfand-Graev representations and of an
equivalence due to Skryabin.

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh_t should be thought of as the category...

To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra, is related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/G. If G is...

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint...

We study a BGG-type category of infinite dimensional representations of H[W], a semi-direct product of the quantum torus with parameter `q' built on the root lattice of a semisimple group G, and the Weyl group of G. Irreducible objects of our category turn out to be parameterized by semistable G-bundles on the elliptic curve C^*/q^Z. In the second...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. Each principal nilpote...

Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be th...

These lectures are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum...

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applicatio...

Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology classes of a configuration space. The main observation of the present paper is that the screening operators co...

The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program, due to Beilinson-Drinfeld, s...

Hecke algebras are usually defined algebraically, via generators and relations. We give a new algebra-geometric construction of affine and double-affine Hecke algebras (the former is known as the Iwahori-Hecke algebra, and the latter was introduced by Cherednik) in terms of residues. More generally, to any generalized Cartan matrixAand a pointqin a...

We added an additional result (theorem 1.6) that strengthenns our main theorem in the G=GL-case by establishing an equivalence of tensor categories.

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certai...

An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "a...

In this paper we explain the parallelism in the classification of three different kinds of mathematical objects: (i) Classical r-matrices. (ii) Generalized cohomology theories that have Chern classes for complex vector bundles. (iii) 1-dimensional formal groups. The main point of the paper is a construction of the elliptic algebra associated to Bel...

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra generated by the Hecke operators turns out to be a homomorphic image of the {\it quantum toroidal algebra}. T...

This paper is a continuation of [BG]. In that paper, for any smooth complex curve X and n > 1, we constructed a canonical completion of the configuration space of all ordered n-tuples of distinct points of X. The completion is called Resolution of Diagonals. There is a natural stratification of the resolution of diagonals with the set of strata bei...

A survey is given of the method of orbits which makes it possible to construct irreducible unitary representations of an arbitrary Lie group proceeding from mechanical considerations. After a brief introduction to symplectic geometry, a construction of a representation associated with an orbit of a group in the dual space of its Lie algebra is give...

We establish a link between two geometric approaches to the rep- resentation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction (GS1); the other involving quantum hamil- tonian reduction of an algebra of dierential operators (GG). In this paper, we combine these two points of view by showing that the pr...