Pith. sign in

REVIEW 4 major objections 5 minor 61 references

The universal monodromic Arkhipov--Bezrukavnikov equivalence

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves a canonical equivalence of categories between universal monodromic Iwahori–Whittaker sheaves on the enhanced affine flag variety and quasicoherent sheaves on the Grothendieck alteration, plus a monoidal…

desk verdict A serious, well-written proof of the universal monodromic Arkhipov–Bezrukavnikov equivalence; the main theorems are new and the proof strategy is genuinely different from AB09, with conditionality stemming from reliance on prior tilting theory rather than internal gaps. read the letter →

arxiv 2501.14156 v1 pith:X2JR74I6 submitted 2025-01-24 math.RT math.AG

Gurbir Dhillon , Jeremy Taylor This is my paper · ORCID
classification math.RTmath.AG MSC 14D2420C0814F0522E67
keywords universalmonodromicsheavesIwahori–WhittakercategoryGrothendieckalterationaffineHecketamelocalBettigeometricLanglandsArkhipov–Bezukavnikovequivalencetiltingduality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a universal-monodromic version of the Arkhipov–Bezukavnikov equivalence: it identifies, for a reductive group $G$ and its Langlands dual $\mathsf{G}$, the category of universal monodromic Iwahori–Whittaker sheaves on the enhanced affine flag variety of $G$ with quasicoherent sheaves on the Grothendieck alteration of $\mathsf{G}$. It also identifies, as monoidal categories, the universal bi-Iwahori–Whittaker category on the loop group of $G$ with quasicoherent sheaves on the adjoint quotient stack of $\mathsf{G}$. This is the first step toward a proof of the tame local Betti geometric Langlands conjecture of Ben-Zvi and Nadler, completed in the sequel. The proof replaces the nilpotent localization of the original argument with localization in semisimple directions, reducing full faithfulness to an order-of-vanishing calculation in semisimple rank one.

What carries the argument

The load-bearing mechanism is a pair of matching order-of-vanishing calculations. On the spectral side, restriction from $B/B$ to the torus sends the global sections of $V_\lambda(i)\otimes \mathcal{O}(B)$ onto the ideal $(e^\alpha-1)^iR$; on the automorphic side, the associated-graded map from $\mathrm{Hom}(W_{\lambda-(n-i)\alpha},Z_\lambda)$ has exactly the same image, with $n=\langle\check\alpha,\lambda\rangle$. Because both sides are free $R$-modules concentrated in degree zero, Hartogs' lemma lets the proof localize away from the wall intersections and compare the two images one wall at a time. The auxiliary constructions—universal monodromic Wakimoto sheaves $W_\lambda$, central sheaves $Z_\lambda$, the big tilting sheaf $\Xi$ with $\mathrm{End}(\Xi)\simeq \mathcal{O}(T\times_C T)$, and the induced Whittaker averaging functor—transport the calculation from coherent sheaves on the spectral side to Iwahori–Whittaker sheaves on the automorphic side.

What would settle it

Specialize the equivalence to a single simple root $\alpha$ and a weight $\lambda$ with $n=\langle\check\alpha,\lambda\rangle$: both sides must have image exactly $(e^\alpha-1)^iR$ for every $0\le i\le n$. A direct computation exhibiting any Hom-space between Whittaker averaged central sheaves that is not free over $R$, or a mismatch between the spectral and automorphic ideals at some $i$, would refute Theorem 1.3.1.

Watch

Extended reading notes

Core claim

The central claim is that the universal monodromic Iwahori–Whittaker category $\mathrm{Shv}_{(I,\chi)}(\mathrm{Fl})$ is canonically equivalent to $\mathrm{QCoh}(B/B)$, where $\mathrm{Fl}$ is the enhanced affine flag variety of $G$ and $B/B$ is the Grothendieck alteration of the dual group $\mathsf{G}$; and that the universal bi-Iwahori–Whittaker category $\chi H_\chi$ is monoidally equivalent to $\mathrm{QCoh}(\mathsf{G}/\mathsf{G})$, compatibly with the actions on both sides of the first equivalence. In concrete terms, the theorem handles all semisimple monodromies at once rather than only unipotent monodromy. The authors construct the functor by a universal-monodromic version of Gaitsgory's nearby-cycles central sheaves together with Wakimoto sheaves, and prove full faithfulness by localizing away from root hyperplanes, reducing to semisimple rank one where both sides are controlled by the same ideal $(e^\alpha-1)^i$ in the Laurent polynomial ring.

Load-bearing premise

Everything rests on the existence and monoidality of the universal monodromic tilting sheaf $\Xi$ from the prior tilting theory: if that sheaf, with its endomorphism ring $\mathcal{O}(T\times_C T)$ and strict monoidal Soergel functor, did not exist in the unbounded analytic setting, the Whittaker categories would not be well-defined and both main theorems would collapse.

Editorial extensions

If this is right

  • For every reductive group, universal Iwahori–Whittaker sheaves and coherent sheaves on the Grothendieck alteration form the same category, so homological invariants on either side can be read on the other.
  • The monoidal equivalence $\chi H_\chi \simeq \mathrm{QCoh}(\mathsf{G}/\mathsf{G})$ upgrades Bezrukavnikov's unipotent result to all monodromies, making the bi-Whittaker category a spectral stack in the adjoint quotient.
  • The proof gives a new route to full faithfulness—semisimple localization instead of unipotent localization—which avoids the regular-centralizer machinery for classical groups.
  • Together with the sequel, these theorems imply the tame local Betti geometric Langlands conjecture of Ben-Zvi and Nadler.
  • The freeness and vanishing results imply that Hom-spaces between Whittaker averaged central sheaves are flat families over the torus, so no jumping occurs as semisimple monodromy varies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the comparison is performed one wall at a time, the same order-of-vanishing format should extend to other coefficient systems or to families of reductive groups whenever universal monodromic tilting theory is available.
  • Editorial inference: the reduction to semisimple rank one suggests that the universal monodromic equivalence could be proved by a finite list of rank-one checks plus sheaf-theoretic gluing, a format adaptable to other Hecke-theoretic settings such as higher-depth Bernstein blocks.
  • Editorial inference: the paper's description of the affinization of the Grothendieck–Springer variety supplies a structural fact that may be useful independently in geometric representation theory.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proves a universal monodromic deformation of the Arkhipov--Bezukavnikov equivalence. Its two main theorems are Theorem 1.3.1, a canonical equivalence between the universal monodromic Iwahori--Whittaker category Shv(I,chi)(Fl) and QCoh(B/B), and Theorem 1.3.3, a monoidal equivalence between the universal bi-Iwahori--Whittaker category chi H_chi and QCoh(G/G). The proof constructs the automorphic-to-spectral functor in Part I via Gaitsgory's central sheaves, Wakimoto sheaves, Pluecker relations, and a Tannakian identification of the associated graded functor. Part II proves the equivalence by localizing away from all but one wall, reducing fully faithfulness to an order-of-vanishing calculation in semi-simple rank one. The paper also states that these results are used in a sequel to prove the tame local Betti geometric Langlands conjecture.

Significance. If the main theorems are correct, they are significant: they provide the universal-monodromic generalization of the Arkhipov--Bezukavnikov equivalence and the bi-Iwahori--Whittaker monoidal equivalence, which are key inputs for the tame local Betti geometric Langlands program. The proof strategy is original in localizing in semi-simple directions rather than unipotent directions, and the order-of-vanishing calculations appear genuinely new. For classical groups the paper is largely self-contained up to the cited universal monodromic tilting theory, and the authors are transparent about the remaining dependencies. The paper also contains useful structural results, such as the affinization of the Grothendieck alteration and compact-generation statements for the Whittaker categories.

major comments (4)
  1. [Proposition 11.2.1 and footnote 16] Proposition 11.2.1 asserts p_*O_~G is isomorphic to O_{G x_C T} for every reductive G, but the proof uses the normality of G x_C T, while footnote 16 immediately below states that if the derived subgroup of G is not simply connected, then G x_{G//G} T is neither smooth in codimension 1 nor normal. These two statements are contradictory. Since Proposition 11.2.1 is explicitly used in the proof of Theorem 17.6.1 to conclude that p^* is fully faithful, the proof of Theorem 1.3.3 does not currently cover all reductive G as stated. The authors need either to prove Proposition 11.2.1 without relying on normality of the target, or to restrict the statement to the simply-connected-derived-subgroup case and handle the remaining groups by the alternative route sketched in Remark 17.6.2.
  2. [Definitions 12.1.1-12.1.3, Eq. (54), Lemma 17.2.1] The definition of the finite Whittaker category Shv(B,chi)(G/U) and the construction of Shv(I,chi)(Fl) rest on the universal monodromic tilting theory of [T23], specifically the existence of the big tilting sheaf Xi, the isomorphism End(Xi) = O(T x_C T) in Eq. (54), and the strict monoidality of Soergel's functor V = Hom(Xi,-). These are asserted by reference but not proved or even precisely stated in the present paper, and it is not discussed whether they hold in the unbounded analytic sheaf category of Section 2.3, which imposes no finiteness conditions on stalks. These inputs are load-bearing: without them the Whittaker categories are not well-defined and Theorem 1.3.3 does not follow. The authors should state the needed results from [T23] as explicit hypotheses with precise references, or include proofs in the present framework.
  3. [Proposition 4.2.3] Proposition 4.2.3 states that conditions (a) and (b) are equivalent for universal perversity, but the proof only establishes the implication (a) implies (b). The converse is needed later: Lemma 4.2.4 verifies that Delta_w * Nabla_v lies in the intersection <Delta[>=0]> and <Nabla[<=0]> (condition (b)) and concludes that it is universally perverse, and this is used to show that Wakimoto sheaves are universally perverse and to justify an injectivity statement in Lemma 4.3.1(b). Please supply the missing direction or a reference for it.
  4. [Proposition 13.2.1] For exceptional groups, Proposition 13.2.1 is imported from [BFO09, Section 2.6], and the paper itself notes in Section 1.4 that [BFO09] depends on the regular centralizer arguments of [AB09]. Since Proposition 13.2.1 is used in Proposition 13.3.1 to establish the freeness and vanishing properties that feed into Proposition 16.1.1 and Theorem 16.2.1, the proof of the Whittaker equivalence for exceptional groups is contingent on an external theorem that uses the circle of ideas being generalized. This is not circular, but it should be transparently declared as a dependency, and ideally the necessary statement from [BFO09] should be reproduced in the paper.
minor comments (5)
  1. [Section 2.1] The sentence 'Let U subset B the unipotent radical of the negative Borel' appears to contain a typo or a missing symbol, and the notation is confusing because U was already used for the unipotent radical of the positive Borel. Please disambiguate the two Borels and their unipotent radicals.
  2. [Section 2.3] In the definition Shv(H)(Y) = lim! Shv(H)(Y_i), the direction of the limit is not immediately clear; please specify the indexing category and the convention for limits under !-pullback.
  3. [Section 6.1] The curve C used in the definition of Fl_C is not introduced in this section; it is presumably the projective line fixed by the choice of coordinate mentioned in Section 5.1. Please make this explicit.
  4. [Section 17.6, Theorem 17.6.1] The phrase 'phi is fully faithful by Equation (54)' is terse; a sentence explaining how the Endomorphismensatz (54) implies full faithfulness of phi would improve readability.
  5. [References] The citation [T23] is given as an arXiv preprint (2305.03033); if this paper has been published, please update the reference.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Whittaker and bi-Whittaker equivalences are proved from independent prior theorems and new order-of-vanishing calculations; the main self-citation [T23] is load-bearing but not circular.

full rationale

After walking the derivation chain, I find no circular step. The main theorem 1.3.1 is obtained by constructing F in Part I from independent ingredients (Gaitsgory's central functor, Wakimoto sheaves, Plücker relations, and the Tannakian identification of grF), and then proving fully faithfulness in Part II by comparing two order-of-vanishing calculations, Proposition 11.6.1 on the spectral side and Proposition 15.4.1 on the automorphic side, after localization to semisimple rank one. Theorem 1.3.3 is proved from the center of the affine Hecke category, Lemma 17.2.1, and faithfully flat descent; it does not reduce to Theorem 1.3.1 by construction, and Remark 17.6.2 notes that a direct proof is possible. The paper does rely heavily on the second author's prior theorem [T23] for the big tilting sheaf Ξ, strict monoidality of Soergel's functor V=Hom(Ξ,−), and the formula End(Ξ)≃O(T×_C T) (Eq. 54), and on [BFO09] for exceptional groups. These are load-bearing self- or predecessor citations, but each is an independent theorem with stated hypotheses that do not include the present equivalence; they are not restatements of the target result. The paper explicitly acknowledges its lack of logical self-containment for exceptional groups in Section 1.4. I therefore record the dependence as a correctness/independence caveat, not as a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical parameters are fitted and no new physical or categorical entities are postulated from thin air. The universal monodromic Iwahori-Whittaker category Shv(I,χ)(Fl) and the notion of universal perversity are new mathematical definitions, but they are constructed explicitly from prior sheaf-theoretic ingredients rather than introduced as unexplained inputs. The main unproved inputs are prior results from [T23], [BFO09], [G01], and standard categorical and nearby-cycles formalism.

assumptions (5)
  • domain assumption Universal monodromic tilting theory of [T23]: existence of a big tilting sheaf Ξ with End(Ξ) ≃ O(T ×_C T), and strict monoidality of Soergel's functor V = Hom(Ξ,-).
    Invoked in Definition 12.1.1, Lemma 12.1.5, and Section 17.2 to define the Whittaker categories and to identify the finite bi-Whittaker category; without it the automorphic side of the main theorems cannot be constructed.
  • domain assumption For exceptional G, the tilting property of Ξ*Z_λ at each closed monodromy point (Section 2.6 of [BFO09]).
    Used in Proposition 13.2.1 to prove universal tilting-ness of Whittaker averaged central sheaves for exceptional groups; this is the only place the paper is not logically self-contained for G exceptional.
  • standard math Nearby cycles formalism in the unbounded weakly constructible derived category, with commutation with pushforward along the convolution map via polar decomposition.
    Used throughout Part I, especially Section 6.3, following [G01] and [KS].
  • standard math Tannakian formalism identifying monoidal functors from Rep(G×T) to Free_T(T) with maps of stacks, and uniqueness of B-torsors on T/T with prescribed associated graded T-torsor.
    Used in Section 8.2 and in the proof of Proposition 9.3.1(c).
  • standard math Hartogs' lemma: a vector bundle over T whose sections are free R-modules can be checked on the complement of codimension-two intersections of root hyperplanes.
    Used in Proposition 16.1.1 to reduce fully faithfulness to rank one after localizing away from all but one wall.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The universal monodromic Arkhipov--Bezrukavnikov equivalence." pith.science (2026). https://pith.science/paper/X2JR74I6

@misc{pith2026250114156,
  author       = {Pith},
  title        = {Pith review of: The universal monodromic Arkhipov--Bezrukavnikov equivalence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2JR74I6}},
  note         = {Machine review of arXiv:2501.14156}
}
abstract

We identify equivariant quasicoherent sheaves on the Grothendieck alteration of a reductive group $\mathsf{G}$ with universal monodromic Iwahori--Whittaker sheaves on the enhanced affine flag variety of the Langlands dual group $G$. This extends a similar result for equivariant quasicoherent sheaves on the Springer resolution due to Arkhipov--Bezrukavnikov. We further give a monoidal identification between adjoint equivariant coherent sheaves on the group $\mathsf{G}$ itself and bi-Iwahori--Whittaker sheaves on the loop group of $G$. 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 52 canonical work pages

  1. [1]

    Perverse coherent sheaves

    Arinkin, Dmitry, and Roman Bezrukavnikov. Perverse coherent sheaves. (2009)

    work page 2009

  2. [2]

    Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic

    Johannes, Anschütz, João Lourenço, Zhiyou Wu, and Jize Yu. Gaitsgory's central functor and the Arkhipov--Bezrukavnikov equivalence in mixed characteristic. arXiv:2311.04043 (2023)

    work page Pith review arXiv 2023

  3. [3]

    Geometric Satake, Springer correspondence, and small representations II

    Achar, Pramod, Anthony Henderson, and Simon Riche. Geometric Satake, Springer correspondence, and small representations II. Representation Theory of the American Mathematical Society 19.6 (2015): 94-166

    work page 2015

  4. [4]

    Central sheaves on affine flag varieties

    Achar, Pramod, and Simon Riche. Central sheaves on affine flag varieties. Book in preparation, (2024)

    work page 2024

  5. [5]

    Proof of the geometric Langlands conjecture I-V, arXiv:2405.03599, 2405.03648, 2409.07051, 2409.08670, 2409.09856 (2024)

    Arinkin, Dima, Dario Beraldo, Justin Campbell, Lin Chen, Joakim F rgeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, and Nick Rozenblyum. Proof of the geometric Langlands conjecture I-V, arXiv:2405.03599, 2405.03648, 2409.07051, 2409.08670, 2409.09856 (2024)

    arXiv 2024

  6. [6]

    Perverse sheaves on affine flags and Langlands dual group

    Arkhipov, Sergey, and Roman Bezrukavnikov. Perverse sheaves on affine flags and Langlands dual group. Israel Journal of Mathematics 170 (2009): 135-183

    work page 2009

  7. [7]

    Integral transforms and Drinfeld centers in derived algebraic geometry

    Ben-Zvi, David, Francis, John, and Nadler, David. Integral transforms and Drinfeld centers in derived algebraic geometry. Journal of the American Mathematical Society 23.4 (2010): 909-966

    work page 2010

  8. [8]

    Morita equivalence for convolution categories: appendix to arXiv:0805.0157, arXiv:1209.0193 (2012)

    Ben-Zvi, David, Francis, John, and Nadler, David. Morita equivalence for convolution categories: appendix to arXiv:0805.0157, arXiv:1209.0193 (2012)

    arXiv 2012

Show all 61 references
  1. [9]

    Loop spaces and Langlands parameters, arXiv:0706.0322 (2007)

    Ben-Zvi, David, and David Nadler. Loop spaces and Langlands parameters, arXiv:0706.0322 (2007)

    2007 arXiv

  2. [10]

    Betti geometric Langlands

    Ben-Zvi, David, and David Nadler. Betti geometric Langlands. Algebraic geometry: Salt Lake City 2015 97 (2018): 3-41

    2018

  3. [11]

    On tensor categories attached to cells in affine Weyl groups

    Bezrukavnikov, Roman. On tensor categories attached to cells in affine Weyl groups. Representation theory of algebraic groups and quantum groups 40 (2004): 69-90

    2004

  4. [12]

    Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group

    Bezrukavnikov, Roman. Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group. Israel Journal of Mathematics. 170, 185–206 (2009)

    2009

  5. [13]

    On two geometric realizations of an affine Hecke algebra

    Bezrukavnikov, Roman. On two geometric realizations of an affine Hecke algebra. Publications mathématiques de l'IHÉS 123 (2016): 1-67

    2016

  6. [14]

    Beilinson

    Bernstein, J., and A. Beilinson. A proof of Jantzen’s conjecture. Advances in Soviet Mathematics 16 (1993): 1-50

    1993

  7. [15]

    Vanishing sheaves and the geometric Whittaker model

    Bezrukavnikov, Roman, and Tanmay Deshpande. Vanishing sheaves and the geometric Whittaker model. arXiv:2310.14834 (2023)

    2023 arXiv

  8. [16]

    Equivariant Satake category and Kostant-Whittaker reduction

    Bezrukavnikov, Roman, and Michael Finkelberg. Equivariant Satake category and Kostant-Whittaker reduction. Mosc. Math. J. 8 (2008), no. 1, 39–72, 183

    2008

  9. [17]

    On tensor categories attached to cells in affine Weyl groups, III

    Bezrukavnikov, Roman, Michael Finkelberg, and Victor Ostrik. On tensor categories attached to cells in affine Weyl groups, III. Israel Journal of Mathematics 170 (2009): 207-234

    2009

  10. [18]

    Localization of modules for a semi-simple Lie algebra in prime characteristic

    Bezrukavnikov, Roman, Ivan Mirković, and Dmitriy Rumynin. Localization of modules for a semi-simple Lie algebra in prime characteristic. Annals of Mathematics (2008): 945-991

    2008

  11. [19]

    A topological approach to Soergel theory

    Bezrukavnikov, Roman, and Simon Riche. A topological approach to Soergel theory. Representation Theory and Algebraic Geometry: A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg, (2021): 267-343

    2021

  12. [20]

    Modular affine Hecke category and regular centralizer

    Bezrukavnikov, Roman, and Simon Riche. Modular affine Hecke category and regular centralizer. (2022)

    2022

  13. [21]

    Modular affine Hecke category and regular unipotent centralizer, I

    Bezrukavnikov, Roman, Simon Riche, and Laura Rider. Modular affine Hecke category and regular unipotent centralizer, I. arXiv:2005.05583 (2020)

    2020 arXiv

  14. [22]

    On Koszul duality for Kac-Moody groups

    Bezrukavnikov, Roman, and Zhiwei Yun. On Koszul duality for Kac-Moody groups. Representation Theory of the American Mathematical Society 17.1 (2013): 1-98

    2013

  15. [23]

    Hyperbolic localization of intersection cohomology

    Braden, Tom. Hyperbolic localization of intersection cohomology. Transformation groups 8 (2003): 209-216

    2003

  16. [24]

    Langlands duality on the Beilinson--Drinfeld Grassmannian, arXiv:2310.19734 (2023)

    Campbell, Justin and Sam Raskin. Langlands duality on the Beilinson--Drinfeld Grassmannian, arXiv:2310.19734 (2023)

    2023 arXiv

  17. [25]

    A Langlands dual realization of coherent sheaves on the nilpotent cone, arXiv:2310.10539 (2023)

    Chen, Harrison and Gurbir Dhillon. A Langlands dual realization of coherent sheaves on the nilpotent cone, arXiv:2310.10539 (2023)

    2023 arXiv

  18. [26]

    Tame local Betti geometric Langlands, (2024)

    Dhillon, Gurbir and Jeremy Taylor. Tame local Betti geometric Langlands, (2024)

    2024

  19. [27]

    Endoscopy for affine Hecke categories, in preparation

    Dhillon, Gurbir, Yau-Wing Li, Zhiwei Yun and Xinwen Zhu. Endoscopy for affine Hecke categories, in preparation

  20. [28]

    Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras

    Dodd, Christopher. Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras. arXiv:1108.4028 (2011)

    2011 arXiv

  21. [29]

    Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands, arXiv:2405.18268 (2024)

    F rgeman, Joakim. Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands, arXiv:2405.18268 (2024)

    2024 arXiv

  22. [30]

    Non-vanishing of geometric Whittaker coefficients for reductive groups, arXiv:2207.02955 (2022)

    F rgeman, Joakim and Sam Raskin. Non-vanishing of geometric Whittaker coefficients for reductive groups, arXiv:2207.02955 (2022)

    2022 arXiv

  23. [31]

    D-modules on the affine flag variety and representations of affine Kac--Moody algebras

    Frenkel, Edward and Dennis Gaitsgory. D-modules on the affine flag variety and representations of affine Kac--Moody algebras. Represent. Theory, Vol. 13 (2009): 470-608

    2009

  24. [32]

    Humphreys, James Reflection groups and Coxeter groups. No. 29. Cambridge university press, 1990

    1990

  25. [33]

    Conjugacy classes in semi-simple algebraic groups

    Humphreys, James E. Conjugacy classes in semi-simple algebraic groups. No. 43. American Mathematical Soc., 1995

    1995

  26. [34]

    Tilting sheaves for real groups and Koszul duality

    Ionov, Andrei, and Zhiwei Yun. Tilting sheaves for real groups and Koszul duality. arXiv:2301.05409 (2023)

    2023

  27. [35]

    Construction of central elements in the affine Hecke algebra via nearby cycles

    Gaitsgory, Dennis. Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. math, (2001): 253-280

    2001

  28. [36]

    Braiding compatibilities

    Gaitsgory, Dennis. Braiding compatibilities. Representation theory of algebraic groups and quantum groups. Vol. 40. Mathematical Society of Japan, 2004. 91-101

    2004

  29. [37]

    The local and global versions of the Whittaker category

    Gaitsgory, Dennis. The local and global versions of the Whittaker category. (2018)

    2018

  30. [38]

    Classification of nondegenerate G-categories

    Gannon, Tom. Classification of nondegenerate G-categories. arXiv:2206.11247 (2022)

    2022 arXiv

  31. [39]

    Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale, arXiv:1709.00978 (2017)

    Genestier, Alain, and Vincent Lafforgue. Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale, arXiv:1709.00978 (2017)

    2017 arXiv

  32. [40]

    A study in derived algebraic geometry: Volume I: correspondences and duality

    Gaitsgory, Dennis, and Nick Rozenblyum. A study in derived algebraic geometry: Volume I: correspondences and duality. Vol. 221. American Mathematical Society, 2019

    2019

  33. [41]

    Nil-Hecke Algebras and Whittaker D-Modules

    Ginzburg, Victor. Nil-Hecke Algebras and Whittaker D-Modules. Lie Groups, Geometry, and Representation Theory: A Tribute to the Life and Work of Bertram Kostant (2018): 137-184

    2018

  34. [42]

    Representations of algebraic groups

    Jantzen, Jens Carsten. Representations of algebraic groups. American Mathematical Society, 2003

    2003

  35. [43]

    Proof of the Deligne-Langlands conjecture for Hecke algebras

    Kazhdan, David, and George Lusztig. Proof of the Deligne-Langlands conjecture for Hecke algebras. Inventiones mathematicae 87 (1987): 153-215

    1987

  36. [44]

    Frobenius splitting of cotangent bundles of flag varieties

    Kumar, Shrawan, Niels Lauritzen, and Jesper Funch Thomsen. Frobenius splitting of cotangent bundles of flag varieties. Inventiones mathematicae 136.3 (1999): 603-621

    1999

  37. [45]

    Sheaves on Manifolds, volume 292 of Grundlehren der mathematischen Wissenschaften

    Kashiwara, Masaki, and Pierre Schapira. Sheaves on Manifolds, volume 292 of Grundlehren der mathematischen Wissenschaften. Springer, 1990

    1990

  38. [46]

    Functions on the commuting stack via Langlands duality

    Li, Penghui, David Nadler, and Zhiwei Yun. Functions on the commuting stack via Langlands duality. (2023)

    2023

  39. [47]

    A Fourier transform for the quantum Toda lattice

    Lonergan, Gus. A Fourier transform for the quantum Toda lattice. Selecta Mathematica 24 (2018): 4577-4615

    2018

  40. [48]

    Singularities, character formulas, and a q-analog of weight multiplicities

    Lusztig, George. Singularities, character formulas, and a q-analog of weight multiplicities. Astérisque 101.102 (1983): 208-229

    1983

  41. [49]

    Characters of reductive groups over a finite field

    Lusztig, George. Characters of reductive groups over a finite field. No. 107. Princeton University Press, 1984

    1984

  42. [50]

    Cells in affine Weyl groups, IV, J

    Lusztig, George. Cells in affine Weyl groups, IV, J. Fac. Sci. Univ. Tokyo 36 (1989), 297–328

    1989

  43. [51]

    Endoscopy for Hecke categories, character sheaves and representations

    Lusztig, George, and Zhiwei Yun. Endoscopy for Hecke categories, character sheaves and representations. Forum of Mathematics, Pi. Vol. 8. Cambridge University Press, 2020

    2020

  44. [52]

    Geometric Langlands duality and representations of algebraic groups over commutative rings

    Mirković, Ivan, and Kari Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Annals of mathematics (2007): 95-143

    2007

  45. [53]

    Morse theory and tilting sheaves

    Nadler, David. Morse theory and tilting sheaves. Pure Appl. Math. Quart. 2 (2006), 719–744

    2006

  46. [54]

    The Whittaker Functional Is a Shifted Microstalk

    Nadler, David and Jeremy Taylor. The Whittaker Functional Is a Shifted Microstalk. Transformation Groups (2024)

    2024

  47. [55]

    The connection between the K-theory localization Theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel

    Neeman, Amnon. The connection between the K-theory localization Theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Annales scientifiques de l'Ecole normale supérieure. Vol. 25. No. 5. 1992

    1992

  48. [56]

    Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe

    Soergel, Wolfgang. Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. Journal of the American Mathematical Society 3.2 (1990): 421-445

    1990

  49. [57]

    Regular elements of semi-simple algebraic groups

    Steinberg, Robert. Regular elements of semi-simple algebraic groups. Publications Mathématiques de l'IHÉS 25 (1965): 49-80

    1965

  50. [58]

    Projective modules over Laurent polynomial rings

    Swan, Richard. Projective modules over Laurent polynomial rings. Transactions of the American Mathematical Society 237 (1978): 111-120

    1978

  51. [59]

    Universal monodromic tilting sheaves

    Taylor, Jeremy. Universal monodromic tilting sheaves. arxiv:2305.03033 (2023)

    2023 arXiv

  52. [60]

    Higher algebraic K-theory of schemes and of derived categories

    Thomason, Robert, and Thomas Trobaugh. Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift (1990): 347-435

    1990

  53. [61]

    Associated varieties and unipotent representations, Harmonic Analysis on Reductive Groups, ed

    Vogan, David. Associated varieties and unipotent representations, Harmonic Analysis on Reductive Groups, ed. W. Barker and P. Sally, New York: Springer Science+Business Media (1991), 315–388

    1991

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.