About

Victor Ginzburg is a professor in the Department of Mathematics at The University of Chicago. His research primarily focuses on geometric representation theory and noncommutative geometry. In geometric representation theory, he applies methods of algebraic geometry to study representations of various algebras, including the classification of irreducible representations of Hecke algebras, applications of D-modules and perverse sheaves to representations of complex or real reductive groups and semisimple Lie algebras, the study of integrable representations of quantum groups via the geometry of quiver varieties, and the geometric Langlands program. In recent years, Ginzburg has also developed interests in noncommutative geometry, drawing inspiration from the theory of quivers, mirror symmetry, Calabi-Yau categories, and the mathematics appearing in string theory. He has authored lectures and survey articles on these topics and has contributed to the understanding of noncommutative symplectic geometry, symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Ginzburg supervises graduate students, encouraging them to pursue research topics of their choosing, and collaborates on joint projects with some of his students.

Research topics

• Mathematics
• Pure mathematics
• Mathematical analysis
• Physics

Selected publications

• The coordinate ring of the universal centralizer via Demazure operators

  ArXiv.org · 2026-04-28

  articleOpen accessSenior author

  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.

  PublisherOA PDF

• The coordinate ring of the universal centralizer via Demazure operators

  arXiv (Cornell University) · 2026-04-28

  preprintOpen accessSenior author

  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.

  PublisherDOI

• Lagrangian Subvarieties of Hyperspherical Varieties

  Geometric and Functional Analysis · 2025-01-22 · 1 citations

  articleOpen access

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

  PublisherOA PDFDOI

• Quantization of the universal centralizer and central D-modules

  arXiv (Cornell University) · 2024-09-26

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Lagrangian subvarieties of hyperspherical varieties

  arXiv (Cornell University) · 2023-10-30 · 1 citations

  preprintOpen access

  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.

  PublisherOA PDFDOI

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

  Advances in Mathematics · 2022 · 8 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Mathematical analysis
  PublisherDOI

• Parabolic induction and the Harish-Chandra 𝒟-module

  Representation Theory of the American Mathematical Society · 2022-03-24

  articleOpen access1st authorCorresponding

  Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a reductive group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A d"> <mml:semantics> <mml:mi>Ad</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {Ad}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant derived categories of either constructible sheaves or (not necessarily holonomic) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathscr {D}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals upper T"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L=T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathscr {D}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G times upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>G</mml:mi> <mml:mo> × </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{G\times T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that this module is flat over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathscr {D}}(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathscr {D}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules.

  PublisherDOI

• Parabolic induction and the Harish-Chandra D-module

  arXiv (Cornell University) · 2021 · 1 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Physics

  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.

  PublisherOA PDF

• COM volume 157 issue 8 Cover and Back matter

  Compositio Mathematica · 2021-08-01

  paratextOpen access

  An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2018-04-15

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

Recent grants

• Moduli Spaces, Quivers, and Duality

  NSF · $230k · 2016–2020

• Symplectic algebraic geometry and representation theory

  NSF · $351k · 2013–2017

• Symplectic Reflection Algebras and their Generalizations

  NSF · $171k · 2006–2009

• Quantization, Noncommutative Geometry, and Applications

  NSF · $237k · 2010–2013

Frequent coauthors

• Pavel Etingof

  36 shared

• Michael Finkelberg

  21 shared

• Simon Riche

  Université Clermont Auvergne

  17 shared

• Yuri Berest

  15 shared

• Vladimir Baranovsky

  University of California, Irvine

  15 shared

• Travis Schedler

  11 shared

• Gwyn Bellamy

  11 shared

• Wee Liang Gan

  University of California, Riverside

  10 shared