Sam Raskin
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View article: Proof of the geometric Langlands conjecture V: the multiplicity one theorem
Proof of the geometric Langlands conjecture V: the multiplicity one theorem Open
This is the final paper in the series of five, in which we prove the geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing that there exists a unique (up to tensoring up by a vector space) Hecke eigensheaf correspon…
View article: Proof of the geometric Langlands conjecture IV: ambidexterity
Proof of the geometric Langlands conjecture IV: ambidexterity Open
This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is ambidext…
View article: Proof of the geometric Langlands conjecture III: compatibility with parabolic induction
Proof of the geometric Langlands conjecture III: compatibility with parabolic induction Open
We establish the compatibility of the Langlands functor with the operations of Eisenstein series constant term, and deduce that the Langlands functor induces an equivalence on Eisenstein-generated subcategories.
View article: Proof of the geometric Langlands conjecture I: construction of the functor
Proof of the geometric Langlands conjecture I: construction of the functor Open
We construct the geometric Langlands functor in one direction (from the automorphic to the spectral side) in characteristic zero settings (i.e., de Rham and Betti). We prove that various forms of the conjecture (de Rham vs Betti, restricte…
View article: Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE
Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE Open
This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction b…
View article: Langlands duality on the Beilinson-Drinfeld Grassmannian
Langlands duality on the Beilinson-Drinfeld Grassmannian Open
We calculate various categories of equivariant sheaves on the Beilinson-Drinfeld Grassmannian in Langlands dual terms. For one, we obtain the factorizable derived geometric Satake theorem. More generally, we calculate the categorical analo…
View article: Non-vanishing of geometric Whittaker coefficients for reductive groups
Non-vanishing of geometric Whittaker coefficients for reductive groups Open
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whi…
View article: Affine Beilinson-Bernstein localization at the critical level
Affine Beilinson-Bernstein localization at the critical level Open
We prove the Frenkel-Gaitsgory localization conjecture describing regular Kac-Moody representations at critical level via eigensheaves on the affine Grassmannian using categorical Moy-Prasad theory. This extends previous work of the author…
View article: Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)
Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd) Open
We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying certain an…
View article: Fundamental local equivalences in quantum geometric Langlands
Fundamental local equivalences in quantum geometric Langlands Open
In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence . With a few exceptions,…
View article: COM volume 157 issue 12 Cover and Back matter
COM volume 157 issue 12 Cover and Back matter Open
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View article: A generalization of the <i>b</i>-function lemma
A generalization of the <i>b</i>-function lemma Open
We establish some cohomological bounds in $D$ -module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $b$ -function lemma for non-holonomic $D$ -modules.
View article: The Arinkin-Gaitsgory temperedness conjecture
The Arinkin-Gaitsgory temperedness conjecture Open
Arinkin and Gaitsgory defined a category of tempered D-modules on Bun_G that is conjecturally equivalent to the category of quasi-coherent (not ind-coherent!) sheaves on LocSys_{\check{G}}. However, their definition depends on the auxiliar…
View article: Tate's thesis in the de Rham Setting
Tate's thesis in the de Rham Setting Open
We calculate the category of D-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham l…
View article: Exceptional loci in Lefschetz theory
Exceptional loci in Lefschetz theory Open
Let $ϕ:X\rightarrow \mathbb{P}^n$ be a morphism of varieties. Given a hyperplane $H$ in $\mathbb{P}^n$, there is a Gysin map from the compactly supported cohomology of $ϕ^{-1}(H)$ to that of $X$. We give conditions on the degree of the coh…
View article: PROJECTIVE GENERATION FOR EQUIVARIANT -MODULES
PROJECTIVE GENERATION FOR EQUIVARIANT -MODULES Open
We investigate compact projective generators in the category of equivariant "Image missing"-modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X , there is a natural countable set of compact proje…
View article: Automorphic functions as the trace of Frobenius
Automorphic functions as the trace of Frobenius Open
We prove that the trace of the Frobenius endofunctor of the category of automorphic sheaves with nilpotent singular support maps isomorphically to the space of unramified automorphic functions, settling a conjecture from [AGKRRV1]. More ge…
View article: Duality for automorphic sheaves with nilpotent singular support
Duality for automorphic sheaves with nilpotent singular support Open
We identify the category Shv_{Nilp}(Bun_G) of automorphic sheaves with nilpotent singular support with its own dual, and relate this structure to the Serre functor on Shv_{Nilp}(Bun_G) and miraculous duality.
View article: Localization for affine $W$-algebras
Localization for affine $W$-algebras Open
We prove a localization theorem for affine $W$-algebras in the spirit of Beilinson--Bernstein and Kashiwara--Tanisaki. More precisely, for any non-critical regular weight $λ$, we identify $λ$-monodromic Whittaker $D$-modules on the enhance…
View article: The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support
The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support Open
We define a new geometric object--the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as l-adic sheaves). …
View article: Affine Beilinson-Bernstein localization at the critical level for $GL_2$
Affine Beilinson-Bernstein localization at the critical level for $GL_2$ Open
We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the …
View article: Homological methods in semi-infinite contexts
Homological methods in semi-infinite contexts Open
Actions of algebraic groups on DG categories provide a convenient, unifying framework in some parts of geometric representation theory, especially the representation theory of reductive Lie algebras. We extend this theory to loop groups an…
View article: Fundamental local equivalences in quantum geometric Langlands
Fundamental local equivalences in quantum geometric Langlands Open
In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory--Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we …
View article: Projective generation for equivariant $D$-modules
Projective generation for equivariant $D$-modules Open
We investigate compact projective generators in the category of equivariant $D$-modules on a smooth affine variety. For a reductive group $G$ acting on a smooth affine variety $X$, there is a natural countable set of compact projective gen…
View article: On the Dundas-Goodwillie-McCarthy theorem
On the Dundas-Goodwillie-McCarthy theorem Open
We give a modern presentation of the Dundas-Goodwillie-McCarthy theorem identifying relative K-theory and topological cyclic homology for nilpotent ring extensions.
View article: A generalization of the b-function lemma
A generalization of the b-function lemma Open
We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.
View article: W-algebras and Whittaker categories
W-algebras and Whittaker categories Open
Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of at…
View article: Acyclic complexes and 1-affineness
Acyclic complexes and 1-affineness Open
This short note is an erratum to arXiv:1306.4304, correcting the proof of one of its main results. It includes some counterexamples regarding infinite-dimensional unipotent groups and affine spaces that may be of independent interest.
View article: On the notion of spectral decomposition in local geometric Langlands
On the notion of spectral decomposition in local geometric Langlands Open
The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with non-ab…