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Victor Ginzburg

Publications and source records attributed to Victor Ginzburg.

At least 19 recordsLinked to original sources
arXiv2026

The coordinate ring of the universal centralizer via Demazure operators

We give a simple description of the coordinate ring of the universal centralizer associated to a simply connected semisimple group. To this end, we prove a general result on Weil restriction of affine schemes $X$ over the Cartan subalgebra $\mathfrak{t}$ equipped with a compatible action of the Weyl group $W$. Specifically, we show that the coordinate ring of the scheme $\mathrm{Res}^W(X)$ of $W$-fixed points of Weil restriction of $X$ to the categorical quotient $\mathfrak{t}//W$ can be obtained from the coordinate ring of $X$ by applying Demazure operators if and only if the scheme $\mathrm{Res}^W(X)$ is integral.

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arXiv2026

Mirabolic affine Grassmannian and character sheaves

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.

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arXiv2025

Quantization of the universal centralizer and central D-modules

The group scheme of universal centralizers of a complex reductive group $G$ has a quantization called the spherical nil-DAHA. The category of modules over this ring is equivalent, as a symmetric monoidal category, to the category of bi-Whittaker $D$-modules on $G$. We construct a braided monoidal equivalence, called the Knop-Ngô functor, of this category with a full monoidal subcategory of the abelian category of $\mathrm{Ad}(G)$-equivariant $D$-modules, establishing a $D$-module abelian counterpart of an equivalence established by Bezrukavnikov and Deshpande, in a different way. As an application of our methods, we prove conjectures of Ben-Zvi and Gunningham by relating this equivalence to parabolic induction and prove a conjecture of Braverman and Kazhdan in the $D$-module setting.

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arXiv2025

Pointwise purity, derived Satake, and Symplectic duality

It has been known for a long time that Ext's between IC-sheaves may often be expressed in terms of Hom's between cohomology groups. We prove a more general result under weaker assumptions. The result is used to describe the action of the derived Satake equivalence on !-pure objects and show that the equivalence enjoys a new kind of functoriality with respect to morphisms of reductive groups. We find and prove normality of the symplectic dual X^! for many smooth affine Hamiltonian G-varieties X, including X=T^*(G/H) for all connected reductive subgroups H of G. We also describe the symplectic duals M^! in the case of Coulomb branches and prove that M^! has symplectic singularities.

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arXiv2025

Lagrangian subvarieties of hyperspherical varieties

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $Λ_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $Λ_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}Λ_{\mathscr X}$ and $\mathrm{Irr}Λ_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

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arXiv2022

Differential operators on G/U and the Gelfand-Graev action

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent bundle. A long time ago, Gelfand and Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of D(G/U). Our main result has an interpretation, via geometric Satake, in terms of spherical perverse sheaves on the affine Grassmanian for the Langlands dual group.

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arXiv2022

Parabolic induction and the Harish-Chandra D-module

Let G be a reductive group and L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L=T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D-module on G x T. We show that this module is flat over D(T), which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of D-modules.

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arXiv2021

Cherednik algebras and Hilbert schemes in characteristic p (with an appendix by Pavel Etingof)

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 of A_c is isomorphic to H_c. Our localisation theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent A_c-modules on the Hilbert scheme, respectively. Furthermore, we show that the Azumaya algebra splits on the formal completion of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman.

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arXiv2021

Mirabolic Satake equivalence and supergroups

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $GL_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

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arXiv2018

Gaiotto's Lagrangian subvarieties via derived symplectic geometry

Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, D. Gaiotto associated to any symplectic representation of G a Lagrangian subvariety of the cotangent bundle of Bun_G. We give a simple interpretation of (a generalization of) Gaiotto's construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.

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arXiv2018

Nil Hecke algebras and Whittaker D-modules

Given a reductive group G, Kostant and Kumar defined a nil Hecke algebra that may be viewed as a degenerate version of the double affine nil Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on G. This result has an interpretation in terms of the geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homology of the affine flag variety (due to Kostant and Kumar) and of the affine Grassmannian (due to Bezrukavnikov and Finkelberg), respectively. It follows from our result that the category of Whittaker D-modules on G considered by Drinfeld is equivalent to the category of holonomic modules over the nil Hecke algebra, and it is also equivalent to a certain subcategory of the category of Weyl group equivariant holonomic D-modules on the maximal torus.

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arXiv2016

Moduli spaces, indecomposable objects and potentials over a finite field

Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give formulas for an exponential sum over the set of absolutely indecomposable objects and a stacky sum over the set of all objects of the category, respectively, in terms of the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible etale local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

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arXiv2015

$SL_2$-action on Hilbert schemes and Calogero-Moser spaces

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 bundle, and its Calogero-Moser analogue, over the $SL_2$-fixed point.

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arXiv2015

Quantization of line bundles on Lagrangian subvarieties

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.

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arXiv2014

Differential operators on G/U and the affine Grassmannian

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 Verma modules. We also construct natural collections of isomorphisms parametrized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman-Finkelberg.

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arXiv2013

Hamiltonian reduction and nearby cycles for Mirabolic D-modules

We study holonomic D-modules on SL_n(C)xC^n, 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 Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of the Verdier specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

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arXiv2013

Quantization of Slodowy slices

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.

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