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Gurbir Dhillon

Publications and source records attributed to Gurbir Dhillon.

At least 19 recordsLinked to original sources
arXiv2026

The universal monodromic Arkhipov-Bezrukavnikov equivalence

Working with integral coefficients, we identify universal monodromic Iwahori-Whittaker sheaves on the enhanced affine flag variety with ind-perfect complexes on the Grothendieck-Springer stack of the Langlands dual group. This extends a similar result for quasi-coherent sheaves on the Springer stack due to Arkhipov-Bezrukavnikov. We further prove a monoidal equivalence between bi-Iwahori-Whittaker sheaves and ind-perfect complexes on the adjoint quotient of the Langlands dual group. These results are used in the sequel to this paper to prove the tame local Betti geometric Langlands conjecture of Ben-Zvi-Nadler. Our proof of fully faithfulness provides an alternative to the argument of Arkhipov-Bezrukavnikov. Namely, while they localize in unipotent directions, we localize in semi-simple directions, thereby reducing fully faithfulness to an order of vanishing calculation in semi-simple rank one and characteristic zero.

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arXiv2026

Tame local Betti geometric Langlands

Working over the integers, we prove that the universal monodromic affine Hecke category is monoidally equivalent to ind-coherent sheaves on the Steinberg stack, confirming a conjecture of Ben-Zvi and Nadler. Specializing to unipotent monodromy, this provides another argument for a fundamental theorem of Bezrukavnikov.

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arXiv2026

Categorical representations of algebraic groups in positive characteristic

In this paper we present a formalism of categorical representation theory for affine group schemes of finite type over fields (both in the weak and strong settings) which applies to arbitrary base fields. This is based on the development of an appropriate $\infty$-categorical setting for the study of representations and Harish-Chandra (bi)modules of such group schemes, which constitute the building blocks and basic examples of such structures. We also study categorical traces in this context, and show in particular that the categorical trace of the identity morphism, resp.~Frobenius morphism, on the appropriate $\infty$-category of representations of a connected reductive algebraic group $G$ over an algebraically closed field $k$ of positive characteristic identifies (under suitable assumptions) with the $\infty$-category of Ind-coherent sheaves on the adjoint quotient $G/G$, resp.~of the fixed points $G^{\mathrm{F}}$ of the Frobenius.

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arXiv2026

Affine Harish-Chandra center in positive characteristic

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish--Chandra centers of the associated Kac--Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $κ$ for $G$, consider the associated Kac--Moody vertex algebra $V_κ(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish--Chandra center of arc group invariants $$V_κ(\mathfrak{g})^{\mathscr{J}G} \subset V_κ(\mathfrak{g}).$$ Write $\checkκ$ for the dual level for $\check{G}$, $\checkκ^p - \checkκ$ for its image under the Artin--Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}$ for the moduli space of $(\checkκ^p - \checkκ)$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_κ(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}.$$ There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level $κ$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish--Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish--Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.

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arXiv2026

Semiinfinite sheaves on affine flag varieties

We study a category of semiinfinite sheaves on the affine flag variety of a connected reductive algebraic group, with coefficients in a field (of arbitrary characteristic different from that of the base field), generalizing some results of Gaitsgory and showing that this category behaves like familiar categories of sheaves on flag varieties in many respects. Our interest is motivated by expected relations with representation theory of the Lie algebra of the Langlands dual reductive group.

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arXiv2026

The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f \in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^κ(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $κ\in \mathbb{C}$ is obtained from the Kazhdan-Lusztig category of the affine vertex algebra $V^κ(\mathfrak{g})$ via the quantum Hamiltonian reduction associated with $f$. We show that this is a braided tensor equivalence for any $f$ and any irrational level $κ\in \mathbb{C} \backslash \mathbb{Q}$.

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arXiv2025

Semi-infinite parabolic IC-sheaf II: the Ran space version

This paper is a sequel to arXiv:2310.0638. Let G be a split reductive group. We define the parabolic semi-infinite category on the Ran version of the affine Grassmanian Gr_{G,Ran}. We study the semi-infinite parabolic IC-sheaf in this category. We provide several descriptions of this object, one as certain colimit, another as an intermediate extension in certain category. We relate the global and local semi-infinite categories of sheaves. We also relate the intersection cohomology sheaf of the Drinfeld compactification of Bun_P with the semi-infinite parabolic IC-sheaf.

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arXiv2025

Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)

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 anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our construction goes through certain "universal" ring object in the twisted derived Satake category of the symplectic group $Sp(2n)$. The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf introduced by S.Lysenko on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We also discuss applications of our construction to a potential mathematical construction of $S$-duality for super-symmetric boundary conditions in 4-dimensional gauge theory and to (some extension of) the conjectures of D.Ben-Zvi, Y.Sakellaridis and A.Venkatesh.

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arXiv2025

Endoscopy for metaplectic affine Hecke categories

For a possibly twisted loop group $LG$, and any character sheaf of its Iwahori subgroup, we identify the associated affine Hecke category with a combinatorial category of Soergel bimodules. In fact, we prove such results for affine Hecke categories arising from central extensions of the loop group $LG$. Our results work for mod $\ell$ or integral $\ell$-adic coefficients. As applications, we obtain endoscopic equivalences between affine Hecke categories, including the derived Satake equivalence for metaplectic groups, and a series of conjectures by Gaitsgory in quantum geometric Langlands.

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arXiv2025

Singular support for G-categories

For a reductive group $G$, we introduce a notion of singular support for cocomplete dualizable DG-categories equipped with a strong $G$-action. This is done by considering the singular support of the sheaves of matrix coefficients arising from the action. We focus particularly on dualizable $G$-categories whose singular support lies in the nilpotent cone of $\mathfrak{g}^*$ and refer to these as nilpotent $G$-categories. For such categories, we give a characterization of the singular support in terms of the vanishing of its generalized Whittaker models. We study parabolic induction and restriction functors of nilpotent $G$-categories and show that they interact with singular support in a desired way. We prove that if an orbit is maximal in the singular support of a nilpotent $G$-category $\mathcal{C}$, the Hochschild homology of the generalized Whittaker model of $\mathcal{C}$ coincides with the microstalk of the character sheaf of $\mathcal{C}$ at that orbit. This should be considered a categorified analogue of a result of Moeglin-Waldspurger that the dimension of the generalized Whittaker model of a smooth admissible representation of a reductive group over a non-Archimedean local field of characteristic zero coincides with the Fourier coefficient in the wave-front set of that orbit. As a consequence, we give another proof of a theorem of Bezrukavnikov-Losev, classifying finite-dimensional modules for $W$-algebras with fixed regular central character. More precisely, we realize the (rationalized) Grothendieck group of this category as a certain subrepresentation of the Springer representation. Along the way, we show that the Springer action of the Weyl group on the twisted Grothendieck--Springer sheaves is the categorical trace of the wall crossing functors, extending an observation of Zhu for integral central characters.

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arXiv2025

Modular intersection cohomology of Drinfeld's compactifications

We compute the dimension of the cohomology of stalks of intersection cohomology complexes on Zastava schemes and Drinfeld compactifications associated with a connected reductive algebraic group $G$, in case the characteristic of the coefficients field $\Bbbk$ is good for $G$. In particular, we show that these dimensions do not depend on the choice of $\Bbbk$.

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arXiv2023

A Langlands dual realization of coherent sheaves on the nilpotent cone

Let $G$ and $\check{G}$ be Langlands dual connected reductive groups. We establish a monoidal equivalence of $\infty$-categories between equivariant quasicoherent sheaves on the formal neighborhood of the nilpotent cone in $G$ and Steinberg-Whittaker D-modules on the loop group of $\check{G}$, as conjectured by Bezrukavnikov. More generally, we establish equivalences between various spectral and automorphic realizations of affine Hecke categories and their modules, confirming conjectures of Bezrukavnikov.

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arXiv2022

The weights of simple modules in Category $\mathscr{O}$ for Kac-Moody algebras

We give the first positive formulas for the weights of every simple highest weight module $L(λ)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we also express the weights of $L(λ)$ as an alternating sum similar to the Weyl-Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding $D$-module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group.

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arXiv2021

The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras

Fix an affine Lie algebra $\widehat{\mathfrak{g}}_κ$ with associated principal affine W-algebra $\mathcal{W}_κ$. A basic conjecture of Frenkel--Kac--Wakimoto asserts that Drinfeld--Sokolov reduction sends admissible $\widehat{\mathfrak{g}}_κ$-modules to zero or cohomological shifts of minimal series $\mathcal{W}_κ$-modules. We prove this conjecture and a natural generalization to the spectrally flowed Drinfeld--Sokolov reduction functors. This extends the previous results of Arakawa for the minus reduction.

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arXiv2021

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the category of Lie algebra representations with a generalized infinitesimal character as the parabolic induction, in the sense of categorical representation theory, of a Steinberg module.

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arXiv2020

Localization for affine $W$-algebras

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 enhanced affine flag variety with a full subcategory of Category $\mathscr{O}$ for the $W$-algebra. To identify the essential image of our functor, we provide a new realization of Category $\mathscr{O}$ for affine $W$-algebras using Iwahori--Whittaker modules for the corresponding Kac--Moody algebra. Using these methods, we also obtain a new proof of Arakawa's character formulae for simple positive energy representations of the $W$-algebra.

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arXiv2020

Fundamental local equivalences in quantum geometric Langlands

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 prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.

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