Dennis Gaitsgory | Harvard University (original) (raw)

Papers by Dennis Gaitsgory

Research paper thumbnail of Local geometric Langlands correspondence and affine Kac-Moody algebras

Progress in Mathematics, 2006

1 2 EDWARD FRENKEL AND DENNIS GAITSGORY 22. Convolution 136 23. Categories over topological commu... more 1 2 EDWARD FRENKEL AND DENNIS GAITSGORY 22. Convolution 136 23. Categories over topological commutative algebras 148 References 154 X of the canonical bundle gives a map of functors Op g (X) → Op G (X).

Research paper thumbnail of Algebraic groups over a 2-dimensional local field: Some further constructions

Progress in Mathematics, 2006

... We can consider ker(ρ) and coker(ρ) as functors on Vect: ker(ρ)(U) = {φ : U → V | ρ ◦φ = 0} a... more ... We can consider ker(ρ) and coker(ρ) as functors on Vect: ker(ρ)(U) = {φ : U → V | ρ ◦φ = 0} andcoker(ρ)(U) = {ψ : W → U | ψ ◦ρ = 0}. As in [GK], Proposition 2.8, one shows that coker(ρ) is always representable, and ker(ρ) is representable if condition (**) is satisfied. ...

Research paper thumbnail of Geometric Eisenstein series

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework ... more The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic,

Research paper thumbnail of Relation between two geometrically defined bases in representations of 𝐺𝐿_{𝑛}

Contemporary Mathematics, 2007

To the memory of Iosif Donin 1. The two bases 1.1. Introduction. Following [Gi] and [BG], one can... more To the memory of Iosif Donin 1. The two bases 1.1. Introduction. Following [Gi] and [BG], one can realize the irreducible finite-dimensional representation of GL n , corresponding to a certain Young diagram, in the top cohomology of (the union of) Springer fibers over a nilpotent matrix, whose Jordan decomposition corresponds to this diagram. We will review this construction in Sect. 1.3 below. In particular, the set of irreducible components of these Springer fibers provides a basis for this representation. We call it the Springer basis. 1 On the other hand, we have the theory of geometric Langlands duality, which realizes the category of finite-dimensional representations of any reductive groupǦ in terms of spherical perverse sheaves on the affine Grassmannian of the Langlands dual group, Gr G . In particular, by taking the top (and, in fact, the only non-zero) cohomology with compact supports of a given irreducible spherical perverse sheaf IC λ , corresponding to a dominant coweight λ, along a semi-infinite orbit S(µ) ⊂ Gr G , corresponding to a coweight µ, we obtain a vector space, which is canonically identified with the weight space V λ (µ), where V λ is the irreducible representation ofǦ with highest weight λ. Therefore, the set of irreducible components of the intersection of S(µ) with the support of IC λ , provides a basis for V λ (µ). We call it the Mirković-Vilonen (MV for short) basis.

Research paper thumbnail of Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and ... more We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higherdimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of [BD1] on chiral enveloping algebras of AE-Lie algebras.

Research paper thumbnail of Uhlenbeck Spaces via Affine Lie Algebras

Progress in Mathematics, 2007

Let G be an almost simple simply connected group over ℂ, and let Bun G a (ℙ2, ℙ1) be the moduli s... more Let G be an almost simple simply connected group over ℂ, and let Bun G a (ℙ2, ℙ1) be the moduli scheme of principalG-bundles on the projective plane ℙ2, of second Chern class a, trivialized along a line ℙ1 ⊂ ℙ2. We define the Uhlenbeck compactification \mathfrakUGa \mathfrak{U}_G^a of Bun G a (ℙ2, ℙ1), which classifies, roughly, pairs (ℱG, D),

Research paper thumbnail of Representations of algebraic groups over a 2-dimensional local field

Geometric and Functional Analysis - GEOM FUNCT ANAL, 2004

We introduce a categorical framework for the study of representations of G( F), where G is a redu... more We introduce a categorical framework for the study of representations of G( F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G( F), which is an object of a suitable category of representations of G( F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G( F), when we consider the action by right translations.

Research paper thumbnail of Weyl Modules and Opers without Monodromy

We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to th... more We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier

Research paper thumbnail of Quantum integrable systems and differential Galois theory

Transformation Groups, 1997

This paper is devoted to a systematic study of quantum completely integrable systems (i.e. comple... more This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding D-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e. its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.

Research paper thumbnail of The Cohen-Macaulay property of the category of $ ({\frak g}, K) $-modules

Selecta Mathematica, 1997

Let (g, K) be a Harish-Chandra pair. In this paper we prove that if P and P are two projective (g... more Let (g, K) be a Harish-Chandra pair. In this paper we prove that if P and P are two projective (g, K)-modules, then Hom(P, P ) is a Cohen-Macaulay module over the algebra Z(g, K) of K-invariant elements in the center of U (g). This fact implies that the category of (g, K)-modules is locally equivalent to the category of modules over a Cohen-Macaulay algebra, where by a Cohen-Macaulay algebra we mean an associative algebra that is a free finitely generated module over a polynomial subalgebra of its center.

Research paper thumbnail of A corollary of the b-function lemma

Selecta Mathematica, 2012

Research paper thumbnail of Algebraic groups over a 2-dimensional local field: Irreducibility of certain induced representations

Journal of Differential Geometry - J DIFFEREN GEOM, 2005

Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop g... more Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop group. In [1], we have introduced the notion of a representation of (the group of K-points) of G((t)) on a pro-vector space. In addition, we have defined an induction procedure, which produced G((t))-representations from usual smooth representations of G. We have conjectured that the induction of a cuspidal irreducible representation of G is irreducible. In this paper, we prove this conjecture for G=SL<sub>2</sub>.

Research paper thumbnail of Poincaré–Birkhoff–Witt Theorem for Quadratic Algebras of Koszul Type

Journal of Algebra, 1996

Let V be a vector space over some field k and let T (V ) = ⊕ T i be its tensor algebra over k. Fi... more Let V be a vector space over some field k and let T (V ) = ⊕ T i be its tensor algebra over k. Fix a subspace R ⊂ T 2 = V ⊗ V , consider the two-sided ideal J(R) in T (V ) generated by R and denote by Q(V, R) the quotient algebra T (V )/J(R). This is what is known as (a homogeneous) quadratic algebra. 0.2. Nonhomogeneous quadratic algebras. In a similar way we define nonhomogeneous quadratic algebras, the main objects of our study. They are filtered analogs of graded homogeneous quadratic algebras.

Research paper thumbnail of Geometric Eisenstein series

Inventiones mathematicae, 2002

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework ... more The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic,

Research paper thumbnail of Deformations of Local Systems and Eisenstein Series

Geometric and Functional Analysis, 2008

Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To expla... more Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To explain it let us recall the general framework of the geometric Langlands correspondence.

Research paper thumbnail of D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Duke Mathematical Journal, 2004

Research paper thumbnail of Crystals via the affine Grassmannian

Duke Mathematical Journal, 2001

Let G be a connected reductive group over C and let g ∨ be the Langlands dual Lie algebra. Crysta... more Let G be a connected reductive group over C and let g ∨ be the Langlands dual Lie algebra. Crystals for g ∨ are combinatorial objects, that were introduced by Kashiwara (cf. for example [5]) as certain "combinatorial skeletons" of finite-dimensional representations of g ∨ . For every dominant weight λ of g ∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq(g ∨ ) and then specializing it to q = 0. Other (independent) constructions of B(λ) were given by Lusztig (cf. [8]) using the combinatorics of root systems and by Littelmann (cf.

Research paper thumbnail of Localization of g-modules on the affine Grassmannian

Annals of Mathematics, 2009

We consider the category of modules over the affine Kac-Moody algebra b g of critical level with ... more We consider the category of modules over the affine Kac-Moody algebra b g of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G..t //=GOEOEt . This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I 0 equivariant categories, where I 0 is the radical of the Iwahori subgroup of G..t //. Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G=B, due to Beilinson-Bernstein and Brylinski-Kashiwara.

Research paper thumbnail of Geometric Realization of Whittaker Functions and the Langlands Conjecture

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic func... more We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GL_n(A) associated to an irreducible l-adic local system of rank n on an algebraic curve X over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following

Research paper thumbnail of Local geometric Langlands correspondence and affine Kac-Moody algebras

Progress in Mathematics, 2006

1 2 EDWARD FRENKEL AND DENNIS GAITSGORY 22. Convolution 136 23. Categories over topological commu... more 1 2 EDWARD FRENKEL AND DENNIS GAITSGORY 22. Convolution 136 23. Categories over topological commutative algebras 148 References 154 X of the canonical bundle gives a map of functors Op g (X) → Op G (X).

Research paper thumbnail of Algebraic groups over a 2-dimensional local field: Some further constructions

Progress in Mathematics, 2006

... We can consider ker(ρ) and coker(ρ) as functors on Vect: ker(ρ)(U) = {φ : U → V | ρ ◦φ = 0} a... more ... We can consider ker(ρ) and coker(ρ) as functors on Vect: ker(ρ)(U) = {φ : U → V | ρ ◦φ = 0} andcoker(ρ)(U) = {ψ : W → U | ψ ◦ρ = 0}. As in [GK], Proposition 2.8, one shows that coker(ρ) is always representable, and ker(ρ) is representable if condition (**) is satisfied. ...

Research paper thumbnail of Geometric Eisenstein series

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework ... more The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic,

Research paper thumbnail of Relation between two geometrically defined bases in representations of 𝐺𝐿_{𝑛}

Contemporary Mathematics, 2007

To the memory of Iosif Donin 1. The two bases 1.1. Introduction. Following [Gi] and [BG], one can... more To the memory of Iosif Donin 1. The two bases 1.1. Introduction. Following [Gi] and [BG], one can realize the irreducible finite-dimensional representation of GL n , corresponding to a certain Young diagram, in the top cohomology of (the union of) Springer fibers over a nilpotent matrix, whose Jordan decomposition corresponds to this diagram. We will review this construction in Sect. 1.3 below. In particular, the set of irreducible components of these Springer fibers provides a basis for this representation. We call it the Springer basis. 1 On the other hand, we have the theory of geometric Langlands duality, which realizes the category of finite-dimensional representations of any reductive groupǦ in terms of spherical perverse sheaves on the affine Grassmannian of the Langlands dual group, Gr G . In particular, by taking the top (and, in fact, the only non-zero) cohomology with compact supports of a given irreducible spherical perverse sheaf IC λ , corresponding to a dominant coweight λ, along a semi-infinite orbit S(µ) ⊂ Gr G , corresponding to a coweight µ, we obtain a vector space, which is canonically identified with the weight space V λ (µ), where V λ is the irreducible representation ofǦ with highest weight λ. Therefore, the set of irreducible components of the intersection of S(µ) with the support of IC λ , provides a basis for V λ (µ). We call it the Mirković-Vilonen (MV for short) basis.

Research paper thumbnail of Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and ... more We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higherdimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of [BD1] on chiral enveloping algebras of AE-Lie algebras.

Research paper thumbnail of Uhlenbeck Spaces via Affine Lie Algebras

Progress in Mathematics, 2007

Let G be an almost simple simply connected group over ℂ, and let Bun G a (ℙ2, ℙ1) be the moduli s... more Let G be an almost simple simply connected group over ℂ, and let Bun G a (ℙ2, ℙ1) be the moduli scheme of principalG-bundles on the projective plane ℙ2, of second Chern class a, trivialized along a line ℙ1 ⊂ ℙ2. We define the Uhlenbeck compactification \mathfrakUGa \mathfrak{U}_G^a of Bun G a (ℙ2, ℙ1), which classifies, roughly, pairs (ℱG, D),

Research paper thumbnail of Representations of algebraic groups over a 2-dimensional local field

Geometric and Functional Analysis - GEOM FUNCT ANAL, 2004

We introduce a categorical framework for the study of representations of G( F), where G is a redu... more We introduce a categorical framework for the study of representations of G( F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G( F), which is an object of a suitable category of representations of G( F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G( F), when we consider the action by right translations.

Research paper thumbnail of Weyl Modules and Opers without Monodromy

We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to th... more We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier

Research paper thumbnail of Quantum integrable systems and differential Galois theory

Transformation Groups, 1997

This paper is devoted to a systematic study of quantum completely integrable systems (i.e. comple... more This paper is devoted to a systematic study of quantum completely integrable systems (i.e. complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the corresponding D-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e. its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.

Research paper thumbnail of The Cohen-Macaulay property of the category of $ ({\frak g}, K) $-modules

Selecta Mathematica, 1997

Let (g, K) be a Harish-Chandra pair. In this paper we prove that if P and P are two projective (g... more Let (g, K) be a Harish-Chandra pair. In this paper we prove that if P and P are two projective (g, K)-modules, then Hom(P, P ) is a Cohen-Macaulay module over the algebra Z(g, K) of K-invariant elements in the center of U (g). This fact implies that the category of (g, K)-modules is locally equivalent to the category of modules over a Cohen-Macaulay algebra, where by a Cohen-Macaulay algebra we mean an associative algebra that is a free finitely generated module over a polynomial subalgebra of its center.

Research paper thumbnail of A corollary of the b-function lemma

Selecta Mathematica, 2012

Research paper thumbnail of Algebraic groups over a 2-dimensional local field: Irreducibility of certain induced representations

Journal of Differential Geometry - J DIFFEREN GEOM, 2005

Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop g... more Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop group. In [1], we have introduced the notion of a representation of (the group of K-points) of G((t)) on a pro-vector space. In addition, we have defined an induction procedure, which produced G((t))-representations from usual smooth representations of G. We have conjectured that the induction of a cuspidal irreducible representation of G is irreducible. In this paper, we prove this conjecture for G=SL<sub>2</sub>.

Research paper thumbnail of Poincaré–Birkhoff–Witt Theorem for Quadratic Algebras of Koszul Type

Journal of Algebra, 1996

Let V be a vector space over some field k and let T (V ) = ⊕ T i be its tensor algebra over k. Fi... more Let V be a vector space over some field k and let T (V ) = ⊕ T i be its tensor algebra over k. Fix a subspace R ⊂ T 2 = V ⊗ V , consider the two-sided ideal J(R) in T (V ) generated by R and denote by Q(V, R) the quotient algebra T (V )/J(R). This is what is known as (a homogeneous) quadratic algebra. 0.2. Nonhomogeneous quadratic algebras. In a similar way we define nonhomogeneous quadratic algebras, the main objects of our study. They are filtered analogs of graded homogeneous quadratic algebras.

Research paper thumbnail of Geometric Eisenstein series

Inventiones mathematicae, 2002

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework ... more The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic,

Research paper thumbnail of Deformations of Local Systems and Eisenstein Series

Geometric and Functional Analysis, 2008

Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To expla... more Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To explain it let us recall the general framework of the geometric Langlands correspondence.

Research paper thumbnail of D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Duke Mathematical Journal, 2004

Research paper thumbnail of Crystals via the affine Grassmannian

Duke Mathematical Journal, 2001

Let G be a connected reductive group over C and let g ∨ be the Langlands dual Lie algebra. Crysta... more Let G be a connected reductive group over C and let g ∨ be the Langlands dual Lie algebra. Crystals for g ∨ are combinatorial objects, that were introduced by Kashiwara (cf. for example [5]) as certain "combinatorial skeletons" of finite-dimensional representations of g ∨ . For every dominant weight λ of g ∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq(g ∨ ) and then specializing it to q = 0. Other (independent) constructions of B(λ) were given by Lusztig (cf. [8]) using the combinatorics of root systems and by Littelmann (cf.

Research paper thumbnail of Localization of g-modules on the affine Grassmannian

Annals of Mathematics, 2009

We consider the category of modules over the affine Kac-Moody algebra b g of critical level with ... more We consider the category of modules over the affine Kac-Moody algebra b g of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G..t //=GOEOEt . This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I 0 equivariant categories, where I 0 is the radical of the Iwahori subgroup of G..t //. Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G=B, due to Beilinson-Bernstein and Brylinski-Kashiwara.

Research paper thumbnail of Geometric Realization of Whittaker Functions and the Langlands Conjecture

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic func... more We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GL_n(A) associated to an irreducible l-adic local system of rank n on an algebraic curve X over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following