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Victor Ginzburg

Publications and source records attributed to Victor Ginzburg.

At least 73 records · Page 4Linked to original sources
arXiv2002

Wilson's grassmannian and a noncommutative Quadric

Let the group G of m-th roots of unity act on the complex line by multiplication, inducing an action on the algebra, Diff, of polynomial differential operators on the line. Following Crawley-Boevey and Holland, we introduce a multiparameter deformation, D_c, of the smash-product (Diff # G). Our main result provides natural bijections between (roughly speaking) the following spaces: (1) G-equivariant version of Wilson's adelic Grassmannian of rank r; (2) Rank r projective D_c-modules (equipped with generic trivialization); (3) Rank r torsion-free sheaves on a `noncommutative quadric'; (4) Disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between D-modules and sheaves. The bijections between (2), (3) and (4) were motivated by our previous work math.AG/0103068. The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and G=1, to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces, discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to Wilson's result.

math.AG↗
arXiv2002

Finite dimensional representations of rational Cherednik algebras

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.

math.RT↗
arXiv2001

Calogero-Moser space and Kostka polynomials

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.

math.RT↗
arXiv2001

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

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 the Weyl group of a root system in a vector space h and V=h\oplus h^*, then the algebras H_k are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=S_n, the Weyl group of g=gl_n. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g)^g, the algebra of invariant polynomial differential operators on gl_n, to the algebra of S_n-invariant differential operators with rational coefficients on C^n. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator with rational potential. Our crucial idea is to reinterpret the deformed homomorphism as a homomorphism: D(g)^g \to {spherical subalgebra in H_k}, where H_k is the symplectic reflection algebra associated to S_n. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of `quantum' Hamiltonian reduction. In the classical limit k -> \infty, our construction gives an isomorphism between the spherical subalgebra in H_\infty and the coordinate ring of the Calogero-Moser space. We prove that all simple H_\infty-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The algebra H_\infty is isomorphic to the endomorphism algebra of a distinguished rank n! vector bundle on this space.

math.AG↗
arXiv2001

Quiver varieties and a non-commutative P^2

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 of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety arising from G via the McKay correspodence. We apply the above isomorphism to deduce generalized Crawley-Boevey & Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for G=1, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A_1, by points of the Calogero-Moser space, due to Cannings-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on the noncommutative projective 2-space. It is totally different from the one used by Berest-Wilson, involving $τ$-functions.

math.AG↗
arXiv2001

On m-quasiinvariants of Coxeter groups

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. In a recent paper math-ph/0105014, Feigin and Veselov proposed a number of interesting conjectures concerning the structure of Q_m, and verified them for dihedral groups and constant functions m. Our goal is to prove some of these conjectures in the general case.

math.QA↗
arXiv2000

Perverse sheaves on a Loop group and Langlands' duality

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 "automorphic" sheaves on the moduli space of G-bundles.

alg-geom↗
arXiv2000

The global nilpotent variety is Lagrangian

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, since it insures that the D-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic, hence, e.g. have finite length.

alg-geom↗
arXiv2000

Non-commutative Symplectic Geometry, Quiver varieties, and Operads

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 orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.

math.QA↗
arXiv2000

Representations of quantum tori and double-affine Hecke algebras

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 part of the paper we construct a family of algebras depending on a parameter `v' that specializes to H[W] at v=0, and specializes to the double-affine Hecke algebra introduced by Cherednik, at v=1. We propose a Deligne-Langlands-Lusztig type conjecture relating irreducible modules over the double-affine Hecke algebra to Higgs G-bundles on the elliptic curve. The conjecture may be seen as a natural `v-deformation' of the classification of simple H[W]-modules obtained in the first part of the paper. Also, an `operator realization' of the double-affine Hecke algebra, as well as of its Spherical subalgebra, in terms of certain `zero-residue' conditions is given.

math.RT↗
arXiv1999

Principal Nilpotent pairs in a semisimple Lie algebra, I

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. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of sl_n, the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, S_n, are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple S_n-modules in terms of Young's symmetrisers.

math.RT↗
arXiv1998

Wall-Crossing functors and D-modules

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 applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.

alg-geom↗
arXiv1998

Loop Grassmannian cohomology, the principal nilpotent and Kostant theorem

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 the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last section we give a new "topological" proof of Kostant's theorem about the polynomial algebra of a semisimple Lie algebra, based on purity of the equivariant intersection cohomology groups of G[[z]]-orbits on Gr.

math.AG↗
arXiv1998

Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 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 group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a p-adic group, are considered. We give a uniform geometric construction of these algebras in terms of homology of an appropriate "Steinberg-type" variety Z (or its modification, such as K-theory or elliptic cohomology of Z, or an equivariant version thereof). We then explain how to obtain a complete classification of finite dimensional irreducible representations of the algebras in question, using our geometric construction and perverse sheaves methods. Similar techniques can be applied to other algebras, e.g. the Double-affine Hecke algebras, Elliptic algebras, quantum toroidal algebras.

math.AG↗
arXiv1997

Screenings and a universal Lie-de Rham cocycle

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 contain more information. Specifically, at the chain level, the screening operators provide a certain canonical cocycle of the Virasoro (resp. affine Kac-Moody) algebra with coefficients in the de Rham complex of an operator-valued local system on the configuration space. This way we obtain canonical morphisms from higher homology groups of the above local systems to appropriate higher Ext-groups between the Fock space representations. Our construction is motivated by, and in a special case reduces to the construction of Bowknegt et al, see [BMP1], [BMP2].

q-alg↗
arXiv1995

Residue construction of Hecke algebras

Hecke algebras are usually defined algebraically, via generators and relations. We give a new algebro-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 [Ch1]). More generally, to any generalized Cartan matrix A and a point q in a 1-dimensional complex algebraic group \c we associate an associative algebra H. If A is of finite type and \c=C^*, the algebra H is the affine Hecke algebra of the corresponding finite root system. If A is of affine type and \c=C^* then H is, essentially, the Cherednik algebra. The case \c=C corresponds to `degenerate' counterparts of the above objects cosidered by Drinfeld [Dr] and Lusztig [L2]. Finally, taking \c to be an elliptic curve one gets some new elliptic analogues of the affine Hecke algebra.

alg-geom↗
arXiv1995

Elliptic Algebras and Equivariant Elliptic Cohomology

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 Belavin's classical elliptic r-matrix in terms of Equivariant elliptic cohomology of the Steinberg varieties associated to some partial flag manifolds.

q-alg↗