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Parabolic geometric Eisenstein series and constant term functors

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Pith's one-line read The paper proves that, under the geometric Casselman-Shalika equivalence, the parabolic Jacquet functor on Whittaker sheaves corresponds exactly to restriction of representations, and its Koszul dual to Lie algebra cohomology along the…

desk verdict Fills a real gap in parabolic geometric Langlands, but the central vanishing is imported from Raskin without checking the identification; referee should push on that seam. read the letter →

arxiv 2507.13930 v2 pith:NUZ35LBS submitted 2025-07-18 math.RT math.AG

Joakim Faergeman , Andreas Hayash This is my paper
classification math.RTmath.AG MSC 14D2422E6714F10
keywords parabolicgeometricEisensteinseriesconstanttermfunctorsCasselman-Shalikaequivalencesemi-infiniteintersectioncohomologysheafWhittakercategoriesJacquetKoszuldualityDrinfeldcompactification
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

Using a factorizable semi-infinite intersection cohomology sheaf as the kernel, this paper proves that the two parabolic Jacquet functors on spherical Whittaker categories are the geometric shadows of two algebraic operations on representations of the Langlands dual group: the !*-Jacquet functor becomes restriction along the Levi subgroup, and the !-Jacquet functor becomes Lie algebra cohomology with respect to the unipotent radical. The argument constructs Hecke and Drinfeld-Plücker structures on this sheaf, compares it to the intersection cohomology sheaf of Drinfeld's compactification, and reduces the main compatibility to a single vanishing result in the Borel case. If correct, the compatibility supplies exactly the diagram needed to show that the spectral-to-automorphic geometric Langlands functor commutes with constant term functors.

What carries the argument

The machine is the factorizable parabolic semi-infinite intersection cohomology sheaf, defined by imposing a Hecke structure on the delta sheaf at the unit section of the affine Grassmannian; acting with this sheaf on Whittaker categories produces the Jacquet functors. The geometric Casselman-Shalika equivalence turns the Hecke structure into representation-theoretic restriction, while Koszul duality between the coalgebra O(N_P) and the Chevalley complex C^•(n_P) turns the O(N_P)-invariants of the same sheaf into Lie algebra cohomology. The local-to-global bridge is a map from a relative version of the compactified Grassmannian to Drinfeld's compactification, which is universally homologically contractible and identifies the semi-infinite IC sheaf with the pullback of the IC sheaf of the compactification.

What would settle it

Compute the principal-case vanishing (6.4.4) for a rank-one group, such as SL_2 with the Borel parabolic: if for some λ ≠ 0 the compactly supported cohomology of the translated semi-infinite IC sheaf tensored with the Whittaker sheaf on the intersection S^λ_x ∩ S'^{-,0}_x is nonzero, the counit (6.4.2) fails to be an isomorphism and Theorem 6.4.1.1 collapses. Alternatively, write out the fiber of the bundle in Lemma 6.4.3.1 explicitly; a counterexample there would already break the identification of the vacuum Whittaker sheaf.

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Extended reading notes

Core claim

The central discovery is that under the factorizable geometric Casselman-Shalika equivalences CS_G and CS_M, the functor Jac^M_!* corresponds to restriction of representations along Mˇ → Gˇ, while Jac^M_! corresponds to derived Lie algebra cohomology C^•(n_P, –). Equivalently, parabolic restriction of Whittaker sheaves is exactly the geometric counterpart of representation-theoretic restriction, and its Koszul dual computes invariants against the Lie algebra of the unipotent radical. Along the way the paper establishes that the semi-infinite IC sheaf carries a canonical Hecke structure, that its invariants with respect to the factorization coalgebra O(N_P) identify with the !-extension of the dualizing sheaf of the open semi-infinite orbit, and that these local sheaves match the IC sheaf of Drinfeld's compactification up to shifts. From this the authors also derive strata-by-strata descriptions of the compactified Eisenstein kernel in terms of universal enveloping algebra and function-algebra factorization algebras, and sketch that the spectral-to-automorphic Langlands functor commutes with constant term functors.

Load-bearing premise

The proof of the main compatibility rests on an unproved, imported vanishing theorem: the compactly supported cohomology of the translated semi-infinite IC sheaf tensored with the Whittaker sheaf on the intersection S^λ_x ∩ S'^{-,0}_x must vanish for every λ ≠ 0, and the fiber computation in Lemma 6.4.3.1, used to identify the vacuum Whittaker sheaf, is asserted rather than fully expanded.

Editorial extensions

If this is right

  • The !*-Jacquet functor is linear over the representation category of the dual group, so any Hecke eigen-property of an object in the Whittaker category is automatically inherited by its parabolic restriction.
  • The !-Jacquet functor, being C^•(n_P, –) under the equivalence, computes derived n_P-coinvariants; in particular it detects whether the corresponding representation has n_P-cohomology in the relevant weights.
  • Combining the commuting diagrams with known commutativity for Poincaré and localization functors yields that the spectral-to-automorphic Langlands functor commutes with constant term functors, hence that the automorphic-to-spectral functor interchanges Eisenstein series.
  • The Hecke structure on geometric Eisenstein series for an arbitrary parabolic now supports singular-support statements for Eisenstein series, since Hecke functors preserve nilpotent singular support.
  • The local-to-global identification gives a self-contained route to the strata restrictions of the compactified Eisenstein kernel: !-restrictions are universal-enveloping-algebra factorization algebras and *-restrictions are function-algebra factorization algebras.

Reading between the lines

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

  • A likely extension is that the same compatibility holds on the full D-module category of Bun_G rather than only on the Whittaker category, provided one uses the appropriate averaging functors; this would make the constant-term diagram commute at the level of categories rather than just objects.
  • Because the semi-infinite IC sheaf is defined by the Hecke condition rather than by a perverse t-structure, the construction is portable to settings where intermediate extensions are not well behaved, such as sheaves on the Fargues-Fontaine curve; the authors hint at this possibility but do not develop it.
  • The sketched alternative proof of the principal-case vanishing suggests that the one imported vanishing result could be replaced by an explicit construction of a family of objects exhibiting each weight space of the dual torus; carrying out that argument would remove the only non-self-contained step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper develops a factorization-algebraic formalism for parabolic geometric Eisenstein series. The central construction is a factorizable parabolic semi-infinite IC sheaf IC^{8/2}_{P,Ran}, defined via a Hecke structure induced from a Drinfeld-Plücker structure on the delta sheaf. The authors establish local-to-global comparisons between this sheaf and the IC sheaf of Drinfeld's compactification, prove Hecke and Drinfeld-Plücker structures on the relevant Eisenstein series functors, and state as the main result a compatibility: under the geometric Casselman-Shalika equivalences, the Jacquet functor Jac_M^{!*} corresponds to restriction Res_{Mˇ}^{Gˇ}, and Jac_M^! corresponds to Lie algebra cohomology C^•(n_P, -). They also sketch the compatibility of the spectral-to-automorphic Langlands functor with constant term functors.

Significance. If the main results are correct, they are substantial. The paper generalizes Braverman-Gaitsgory Hecke structures from the Borel case to arbitrary parabolics, gives a clean definition of the parabolic semi-infinite IC sheaf via Hecke structures, and extends Raskin's principal-case Casselman-Shalika compatibility to general Levi subgroups. The Koszul-dual statement identifying Jac_M^! with Lie algebra cohomology is a useful and nontrivial consequence. The paper is also commendable for separating theorems from explicitly labelled sketches and for providing detailed proofs of the structural results in Sections 4 and 5. However, the central theorem currently rests on an external vanishing theorem whose applicability is asserted rather than verified, so the significance is conditional on that verification being supplied.

major comments (3)
  1. [§6.4.6, first proof of Theorem 6.4.4.1] The vanishing assertion (6.4.4) is the load-bearing input for the principal case. The text says that after [Gai21, Prop. 3.8.3] this is 'exactly' [Ras21, Thm. 3.4.1], but no verification is provided for the three points needed: (i) the sheaf IC^{8/2}_{B,x} constructed in §4.3.7 via Hecke induction agrees with the semi-infinite IC sheaf considered in [Gai21] under the identification of [Gai21, Prop. 3.8.3]; (ii) the cohomological shift in [Gai21, Prop. 3.8.3] is correctly matched with the shift appearing in (6.4.4); (iii) the intersection S^λ_x ∩ S'^-,0_x is exactly the closed subscheme of the Zastava space to which [Ras21, Thm. 3.4.1] applies. Since Theorem 6.4.4.1 is used to prove Theorem 6.4.1.1 and hence the first diagram of Theorem 1.4.3.1, this gap affects the central claim. The second proof in §6.4.6 is explicitly a sketch and does not remove the dependence on the first proof.
  2. [§6.4.5, reduction to the principal case] The reduction to the principal case is incomplete as written. The argument correctly notes that conservativity of Jac_T^M^{!*} would reduce the question to checking the counit after applying this functor, but the subsequent chain Jac_T^M^{!*}(Jac_M^{!*}(ψ_G)) ≅ Jac_T^{!*}(ψ_G) ≅ δ_{Gr_T} ≅ Jac_T^M^{!*}(ψ_M) only identifies the two objects. It does not show that the image of the specific counit map (6.4.2) under Jac_T^M^{!*} is an isomorphism under these identifications. One needs a naturality or compatibility statement identifying that image with the corresponding counit for the Borel Jacquet functor, via Theorem 6.3.1.1; this is not stated or proved. The gap is likely fixable but is required for the logical chain.
  3. [§6.4.3, Lemma 6.4.3.1] Lemma 6.4.3.1 concludes from transitivity of the LRanN^-_M action that the projection pr_{B_M} is an fppf locally trivial fiber bundle, and then computes the typical fiber as Ran. Transitivity of an ind-group action on a prestack does not by itself imply fppf local triviality, and the displayed fiber computation is abbreviated. Since Corollary 6.4.3.2 and the counit (6.4.2) depend on this lemma, the local-triviality statement needs a proof or a precise reference.
minor comments (5)
  1. [§1.1.2] The phrase 'For or a prestack' should read 'For a prestack'; this typo appears in the sentence introducing D(Y).
  2. [Appendix B] The higher coherence data for the Drinfeld-Plücker structure on δ_{Gr_G,Ran} is only justified in footnote 27 via the relative perverse t-structure. Since this structure is the bootstrap for the Hecke induction defining IC^{8/2}_{P,Ran}, the authors should spell out the compatibility checks or give a precise citation to [HS23] for the exact statement used.
  3. [§6.4.5–§6.4.6] The notation Jac_T^M^{!*} and the chain of equivalences in the reduction to the principal case would be easier to follow with a short definition of this functor and a displayed statement of the compatibility between Theorem 6.3.1.1 and the counit map.
  4. [§6.4.6, first proof of Theorem 6.4.4.1] The Zastava space is not named or described in the proof; since the appeal to [Gai21, Prop. 3.8.3] and [Ras21, Thm. 3.4.1] is central, the authors should specify the scheme and the embedding of S^λ_x ∩ S'^-,0_x into it.
  5. [§6.4.4, Lemma 6.4.4.2] The Hom-space computation (6.4.3) is stated without proof. A reference or a two-line justification via the Casselman-Shalika equivalence and compact generation would make the reduction to the vacuum object ψ_G more transparent.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central Casselman–Shalika compatibility is proven against external results (geometric Casselman–Shalika and Raskin's vanishing theorem), with self-citations that are not load-bearing.

full rationale

The paper's central claim (Theorem 1.4.3.1 = Theorem 6.4.1.1 + Corollary 6.4.1.2) is an internal comparison between the Jacquet functor Jac^M_!* and representation-theoretic restriction Res^G_M under the geometric Casselman–Shalika equivalence. The proof does not assume the target statement. It reduces the general case to the principal (Borel) case using the composition theorem 6.3.1.1, and in the principal case invokes Raskin's vanishing theorem [Ras21, Thm 3.4.1] after identifying the restricted semi-infinite IC sheaf via [Gai21, Prop. 3.8.3]. These are external, non-circular inputs; the paper explicitly states that the proof is self-contained except for this vanishing result (§1.4.4). The definition of IC^{8/2}_{P,Ran} via Hecke structures (§4.3.7, Remark 1.5.5.2) is a definition, not a prediction deduced from the theorem. Self-citations such as [Hay25] and [FR22] are contextual or used for alternate constructions, and they are not load-bearing for the main compatibility. The second proof sketch in §6.4.6 does contain phrases like 'by construction, under geometric Casselman–Shalika, the functor Res^G_T corresponds to the same named functor', but it is explicitly a sketch and is not the rigorous route used to establish Theorem 6.4.4.1. The identified risk—that the application of Raskin's vanishing theorem requires careful verification of identifications and shifts—is a correctness concern about an external input, not circularity. Overall, the derivation chain is independent of the conclusions it claims to prove.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters. The proofs rest on standard geometric Langlands theorems (Casselman-Shalika equivalence, Raskin vanishing) and on structural assumptions (simply-connected derived subgroup, D-module formalism in characteristic zero). The semi-infinite IC sheaf is a constructed object, not a postulated entity; Theorem 5.3.8.1 identifies it with the IC sheaf of Drinfeld's compactification.

assumptions (6)
  • domain assumption Geometric Casselman-Shalika equivalence CS_G: Whit(Gr_G,Ran) ≅ Rep(Gˇ)_Ran
    Cited as Theorem 6.1.4.1 from [FGV01] and [Ras18]; the paper's main theorem expresses Jacquet functors only after transporting along this equivalence.
  • domain assumption Raskin's vanishing theorem [Ras21, Theorem 3.4.1]
    Used in §6.4.6 to prove the principal case of Theorem 6.4.4.1; the paper states its proof is self-contained except for this vanishing result.
  • domain assumption The maps π_P,pol and π_P are universally homologically contractible (Prop 5.2.8.2, Cor 5.2.8.3)
    Proven in the text using [Gai11, Lemma 3.1.2] and [Bar12]; this contractibility underpins the local-to-global comparison Theorem 5.3.8.1.
  • domain assumption G has simply-connected derived subgroup
    Stated in §2.3.1; it simplifies Drinfeld compactifications and Zastava spaces, and the authors note it can be removed via [ABB'05, §4.1] or [Sch15, §7].
  • standard math D-module formalism for prestacks and higher algebra of [Lur09], [Lur17], [GR17], [GR19]
    The categorical and geometric framework used throughout, including six-functor formalism and factorization categories.
  • domain assumption t-exactness of the forgetful functor Sph^+_{M,Conf} → D(Gr^+_{M,Conf}) (Lemma 3.4.17.1, [Ber21, Lemma 2.1.15])
    Used to compute perverse degrees of the factorization algebras and of the strata restrictions of the IC sheaf in Section 5.3.

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Pith. "Pith review of Parabolic geometric Eisenstein series and constant term functors." pith.science (2026). https://pith.science/paper/NUZ35LBS

@misc{pith2026250713930,
  author       = {Pith},
  title        = {Pith review of: Parabolic geometric Eisenstein series and constant term functors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUZ35LBS}},
  note         = {Machine review of arXiv:2507.13930}
}
read the original abstract

We prove a compatibility between parabolic restriction of Whittaker sheaves and restriction of representations under the geometric Casselman-Shalika equivalence. To do this, we establish various Hecke structures on geometric Eisenstein series functors, generalizing results of Braverman-Gaitsgory in the case of a principal parabolic. Moreover, we relate compactified and non-compactified geometric Eisenstein series functors via Koszul duality. We sketch a proof that the spectral-to-automorphic geometric Langlands functor commutes with constant term functors.

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Works this paper leans on

16 extracted references · 9 canonical work pages

  1. [7]

    Langlands duality on the Beilinson-Drinfeld Grassmannian

    [CR23] Justin Campbell and Sam Raskin. Langlands duality on the Beilinson-Drinfeld Grassmannian. arXiv preprint arXiv:2310.19734,

    work page Pith review arXiv

  2. [10]

    Semiinfinite Flags. I. Case of global curve $P^1$

    [FM97] Michael Finkelberg and Ivan Mirković. Semiinfinite Flags. I. Case of global curve P1. arXiv preprint alg-geom/9707010,

    work page Pith review arXiv

  3. [15]

    Geometric Eisenstein series I: finiteness theorems.arXiv preprint arXiv:2409.07363,

    [HHS24] Linus Hamann, David Hansen, and Peter Scholze. Geometric Eisenstein series I: finiteness theorems.arXiv preprint arXiv:2409.07363,

    arXiv

  4. [16]

    The Harder–Narasimhan stratification of the moduli stack of G-bundles via Drinfeld’s compactifications

    [Sch15] Simon Schieder. The Harder–Narasimhan stratification of the moduli stack of G-bundles via Drinfeld’s compactifications. Selecta Mathematica, 21(3):763–831, 2015

    work page 2015

  5. [1997]

    Non-vanishing of geometric Whittaker coefficients for reductive groups

    [FR22] Joakim Faergeman and Sam Raskin. Non-vanishing of geometric Whittaker coefficients for reductive groups. arXiv preprint arXiv:2207.02955 ,

    arXiv

  6. [2001]

    Deformations of local systems and Eisenstein series

    [BG06] Alexander Braverman and Dennis Gaitsgory. Deformations of local systems and Eisenstein series. arXiv preprint math/0605139,

    work page Pith review arXiv

  7. [2002]

    Geometric Eisenstein series

    [BG99] Alexander Braverman and Dennis Gaitsgory. Geometric Eisenstein series. arXiv preprint math/9912097 ,

    work page Pith review arXiv

  8. [2005]

    Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE

    [ABC`24] Dima Arinkin, Dario Beraldo, J Campbell, L Chen, J Faergeman, Dennis Gaitsgory, K Lin, S Raskin, and N Rozenblyum. Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE. arXiv preprint arXiv:2405.03648 ,

    arXiv

Show all 16 references
  1. [2006]

    Proof of the geometric Langlands conjecture III: compatibility with parabolic induction

    [CCF`24] Justin Campbell, Lin Chen, Joakim Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, and Nick Rozenblyum. Proof of the geometric Langlands conjecture III: compatibility with parabolic induction. arXiv preprint arXiv:2409.07051 ,

    arXiv

  2. [2010]

    [Gai11] Dennis Gaitsgory

    Available at https://people.mpim-bonn.mpg.de/gaitsgde/GL/GenVan.pdf. [Gai11] Dennis Gaitsgory. Contractibility of the space of rational maps. arXiv preprint arXiv:1108.1741 ,

    arXiv

  3. [2012]

    On the geometric Ramanujan conjecture

    [Ber21] Dario Beraldo. On the geometric Ramanujan conjecture. arXiv preprint arXiv:2103.17211 ,

    arXiv

  4. [2019]

    Semi-infinite flags and Zastava spaces

    [Hay25] Andreas Hayash. Semi-infinite flags and Zastava spaces. arXiv preprint arXiv:2311.13067 ,

  5. [2021]

    Metaplectic Whittaker category and quantum groups: the "small" FLE

    [GL19] Dennis Gaitsgory and Sergey Lysenko. Metaplectic Whittaker category and quantum groups: the "small" FLE. arxiv preprint arXiv:1903.02279 ,

    1903 arXiv

  6. [2023]

    Semi-infinite parabolic IC sheaf

    [DL25] Gurbir Dhillon and Sergey Lysenko. Semi-infinite parabolic IC sheaf. arXiv preprint arXiv:2310.06386 ,

    arXiv

  7. [2024]

    ThestackoflocalsystemswithrestrictedvariationandgeometricLanglandstheorywithnilpotentsingular support

    [AGK`20] Dima Arinkin, Dennis Gaitsgory, David Kazhdan, Sam Raskin, Nick Rozenblyum, and Yasha Varshavsky. ThestackoflocalsystemswithrestrictedvariationandgeometricLanglandstheorywithnilpotentsingular support. arXiv preprint arXiv:2010.01906 ,

    2010 arXiv

  8. [2025]

    Semiinfinite Flags

    [FFKM97] Boris Feigin, Michael Finkelberg, Alexander Kuznetsov, and Ivan Mirković. Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps Spaces.arXiv preprint alg-geom/9711009,

    arXiv

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