Cohomological Mackey formula for quotient stacks

Lucien Hennecart

REVIEW 3 major objections 4 minor 3 references

Cohomological Mackey formula for quotient stacks

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Induction and restriction on quotient stacks obey a Mackey formula

desk verdict A credible and valuable Mackey-formula framework for critical cohomology of quotient stacks, but the proof of the main theorem currently asks the referee to take the key normal-bundle Euler class identity (7.1) on faith. read the letter →

arxiv 2505.09483 v1 pith:XXS2J7TE submitted 2025-05-14 math.RT math.AG

Lucien Hennecart This is my paper · ORCID

classification math.RTmath.AG MSC 14L3014F4355N91

keywords MackeyformulacriticalcohomologyequivariantLandau-Ginzburgmodelquotientstackinduction-restrictionsystemCoxetercomplexvanishingcyclescohomologicalDonaldson-Thomastheory

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 commutation rule between two families of maps — parabolic induction and restriction — acting on the critical cohomology of a quotient stack $V/G$ equipped with a $G$-invariant function $f$. The rule is a cohomological Mackey formula: restricting after inducing is the same as summing, over Weyl-group double cosets, inductions conjugated by braiding isomorphisms and Weyl translations. If correct, this makes the critical cohomology spaces into a localized induction-restriction system, and supplies the geometrically correct restriction counterpart to the induction maps used in cohomological Donaldson–Thomas theory. The paper also relates this to 2d induction systems through cohomological dimensional reduction, and checks the formula explicitly when $f$ vanishes.

What carries the argument

The machinery is the Coxeter complex of $(G,V)$: the real Cartan space $\mathfrak{h}_{\mathbb{R}}$ cut by hyperplanes $\alpha=0$ for weights $\alpha$ of $V$ and of the adjoint representation, whose cells $C$ and flats $F$ are ordered by reverse inclusion and carry a Tits product $C\circ C'$. Around it, the paper organizes induction morphisms $\operatorname{Ind}_C^F$ built from an induction diagram $V_{\langle C\rangle} \leftarrow V_{C\geqslant 0,F} \rightarrow V_F$, restriction morphisms $\operatorname{Res}_C^F$ defined after localizing by Euler classes $\mathrm{Eu}_{V,C,F}$, and braiding isomorphisms multiplying by kernels $k_{C,F}/k_{C',F}$. The Mackey formula is proved by rewriting both sides in torus-equivariant cohomology, averaging over Weyl groups, and reducing to the normal-bundle Euler-class identity (7.1).

What would settle it

Carry out the normal-bundle Euler-class computation in identity (7.1) for a configuration with more than one double coset, for instance $\mathrm{GL}_3$ acting on $\mathbf{C}^3$ with a cubic invariant; a single mismatch between the asserted and computed Euler factors would refute Theorem 7.6.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.6: for cells $C, C' \preceq C''$ in the Coxeter complex of $(G,V)$, the composition $\operatorname{Res}_{\langle C''\rangle}^{C'} \circ \operatorname{Ind}_{C}^{\langle C''\rangle}$ equals a sum over double cosets of inductions, braidings, restrictions, and Weyl translations. Concretely, $$ \operatorname{Res}_{\langle C''\rangle}^{C'} \circ \operatorname{Ind}_{C}^{\langle C''\rangle} = \sum_{w \in W_{\langle C'\rangle}\backslash W_{\langle C''\rangle}/W_{\langle C\rangle}} \operatorname{Ind}_{C'\circ(\dot w\cdot C)}^{C'} \circ \tau_{\dot w\cdot C\circ C'}^{C'\circ(\dot w\cdot C)} \circ \operatorname{Res}_{\langle \dot w\cdot C\rangle}^{\dot w\cdot C\circ C'} \circ (\dot w\cdot -), $$ where $\tau$ is a braiding isomorphism given by multiplication by a ratio of Euler-class kernels. The formula is proved through torus-equivariant induction and restriction, using a double-coset bijection and the Euler-class identity (7.1), and it upgrades the critical cohomology spaces into a localized induction-restriction system with associative induction and coassociative restriction.

Load-bearing premise

The load-bearing premise is identity (7.1), an asserted normal-bundle Euler-class computation: if those Euler factors do not match, the double-coset expansion in the Mackey formula collapses.

Editorial extensions

If this is right

When the potential vanishes, induction and restriction have explicit shuffle formulas in Weyl-group invariants; Section 9 verifies the Mackey formula on concrete $\mathrm{GL}_2$ examples.
The newly defined restriction morphisms repair the earlier restriction used for cohomological integrality: induction and restriction now form a compatible pair satisfying the Mackey formula.
The critical cohomological system is a localized induction-restriction system, with associative induction, coassociative restriction, and the Mackey formula controlling their interaction.
Cohomological dimensional reduction relates the 2d Borel–Moore induction system to the critical 3d system, so the same Mackey formalism applies in both settings up to explicit signs.
For quiver representation spaces, the restriction morphisms realize the localized coproduct of the cohomological Hall algebra, so the Mackey formula supplies the product-coproduct compatibility.

Reading between the lines

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

This suggests the Mackey formula should extend from quotient stacks to global stacks, with component lattices replacing the Coxeter complex; the paper points to this as a further direction but does not prove it.
The Euler-class localization used to form the restriction is a fixed-point localization principle in disguise: critical cohomology on a flat should be recoverable from one-parameter-subgroup fixed loci, with braiding operators encoding wall-crossing between chambers.
In the quiver-representation setting, the formula should make the cohomological Hall product and the localized coproduct compatible; verifying that compatibility directly would give an independent check of Theorem 1.6.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things you should know. First, the paper is real: it constructs restriction morphisms and proves a Mackey formula for critical cohomology of V/G with invariant potential, in a general reductive-group setting. The Coxeter-complex formalism and the braiding operators are new at this level, and the GL2 examples show the shuffle formula working. Second, the load-bearing step is not yet demonstrated. Theorem 7.6 reduces the entire Mackey double-coset expansion to identity (7.1), and the text says it “follows from this diagram and the calculation of the Euler classes of the normal bundles” without displaying the calculation. The stress-test note is right: any sign or w-dependent factor in that Euler class would change the coefficients of the double-coset terms. This is internal to the proof, not imported from elsewhere.

What I like: the author is explicit about what reduces to Davison’s localized coproduct and what is new. The comparison theorem with 2d induction systems is substantial. The examples in Section 9 are concrete and honest, though they are all vanishing-potential cases, so they do not exercise the nontrivial-f version of (7.1). I also appreciate the self-caveat on Conjecture 8.13; not many papers flag their own conjecture as likely false as stated.

Soft spots, in proportion. The gap in (7.1) is the main issue, and it is not minor. A referee needs a full derivation or a precise reference. The sheafified identification in Section 4.4 imports approachability by proper maps via [Kin24, Hen24a]; I would like the hypotheses stated explicitly so the reader knows exactly when the induction morphism computes what is claimed. This is a dependency issue more than a flaw. The self-citations to [Hen24b, Hen24a] are legitimate; the new result genuinely builds on them.

Bottom line: the framework and statement are worth serious referee time. I would not desk-reject, and I would send it out with a request to expand §7.4. I would not cite the theorem as established until (7.1) is written out; as written, the result is conditional.

Referee Report

3 major / 4 minor

Summary. The paper develops an induction–restriction formalism for the critical cohomology of equivariant Landau–Ginzburg models attached to a representation V of a reductive group G and an invariant function f. It introduces the Coxeter complex of (G,V), defines critical cohomological systems H^T'_{G,V,f,F} for flats F, constructs induction morphisms Ind^F_C and localized restriction morphisms Res^C_F, and states a Mackey-type formula expressing Res^{C'}_{⟨C''⟩} ∘ Ind^{⟨C''⟩}_C as a sum over double cosets of compositions of induction, braiding, restriction, and Weyl action. A parallel 2d induction system is defined using Borel–Moore homology of zero loci and compared with the critical system via dimensional reduction. The paper ends with explicit GL2 examples with vanishing potential and with a partial description of the image of the 2d-to-shuffle comparison morphism.

Significance. If the main theorem is correct, the paper provides a genuine structural result: the critical cohomology spaces H^T'_{G,V,f,F} become the stalks of a localized induction–restriction system, with a Mackey formula that generalizes the localized coproduct of Davison and connects to Langlands-type constant-term formulae and to perverse-sheaf descriptions of hyperplane arrangements. The formal framework is careful: the Coxeter complex, Tits product, Euler-class localizations, braiding operators, and double-coset combinatorics are set up precisely, and the GL2 examples give concrete, checkable instantiations of the formula in the f=0 case. There are no fitted parameters and no ad hoc numerical inputs; the claimed identities are functorial. The main caveat is that the central proof is conditional on a normal-bundle Euler class identity that is not derived in the manuscript, and on external structural results whose precise hypotheses are not stated. These gaps are local in nature but they concern the core theorem, so they block acceptance in the current form.

major comments (3)

[§7.4, identity (7.1)] The proof of Theorem 7.6 reduces the Mackey formula to the identity Eu_{V,w·C,F} Ind^{C'}_{C'∘w·C} i^* f = i'^* Eu_{V,C'∘w·C,⟨C'⟩} Ind^F_{w·C} f, and then states that (7.1) follows from diagram (7.2) and the calculation of Euler classes of normal bundles of p_{w·C,F} and p_{C'∘w·C,⟨C'⟩}. No such calculation is actually shown. This identity is load-bearing because it is exactly what matches the Euler-class denominators in the shuffle expansion and fixes the coefficient of each double-coset term; an omitted sign or a w-dependent factor would change the formula. The Section 9 examples do not test it, since they restrict to f=0 and involve no nontrivial vanishing-cycle normal-bundle computation. The authors should provide a complete derivation of (7.1), including signs and the w-dependence, or give a precise reference that contains the calculation.
[§4.4, sheafified induction identification] The definition of the induction morphism Ind^F_C relies on the sentence that, using smoothness of q, properness of p, finiteness of ı, and the fact that π_F is APM (approachable by proper maps), one can canonically identify (ı_{⟨C⟩,F})_*(π_{⟨C⟩})_* φ(q)_* Q with (π_F)_* φ(p)_* Q. The precise hypotheses under which π_F is APM for the stacks V_F/(G_F×T') are not stated; the manuscript cites [Hen24a, Proposition 5.12] and [Kin24], but does not verify their hypotheses for the generality claimed here. This identification is what makes the sheafified induction compute the intended critical cohomology, so the authors should either state the needed structural result as an explicit assumption, prove it in this setting, or restrict the main theorem to the cases where it applies.
[§5.5, torus equivariant 2d induction] The definition of the torus equivariant 2d induction morphism contains the incomplete formula gInd^F_C := ?? · Ind^F_C, where the multiplier is left undefined. This is not a harmless typo: the subsequent average Ind'^F_C and the comparison in Proposition 5.6 depend on this multiplier, and the reader cannot check whether the claimed compatibility with the non-torus induction holds. The missing factor should be supplied explicitly and the comparison proof adapted accordingly.

minor comments (4)

[§1.3.6] The sentence fixing representatives of double cosets writes the quotient as W_{⟨C′⟩}\W_{⟨C′′⟩}/W_{⟨C′⟩} in both occurrences; the second factor should be W_{⟨C⟩} to match Theorem 1.6.
[§8, Conjecture 8.13] The conjecture is immediately followed by the admission that it 'does not hold as stated' and likely needs refinement. Since Corollary 8.12 only proves containment, the conjecture should be labeled as tentative and its status made explicit in the main text, so that it is not mistaken for a proved structural result.
[§9] The examples only treat the case of vanishing potential f=0, where the induction and restriction morphisms reduce to shuffle formulas for Weyl-group invariants. A short sentence clarifying that these examples do not exercise the vanishing-cycle and normal-bundle Euler-class computations in the proof of Theorem 7.6 would help calibrate what is verified.
[§1.8 and throughout] There are occasional grammatical slips and notational inconsistencies, e.g. 'The restriction morphism are defined' in §1.2.2 and the repeated 'opposite cell' phrasing in §6; these should be corrected in a final polish.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Mackey formula is an identity of independently defined morphisms, and the unexpanded normal-bundle computation in (7.1) is an incompleteness, not a reduction of the theorem to its inputs.

full rationale

The paper's central claim, Theorem 1.6/7.6, is an equality of morphisms between cohomology spaces, and neither side is fitted to the other. Induction morphisms are defined in §4.4 directly from pullback/pushforward in vanishing-cycle cohomology, and restriction morphisms are defined in §6.2 from the opposite-cell induction diagram and Euler-class denominators; these definitions do not presuppose the Mackey formula. The proof replaces the parabolic induction and restriction by their torus-equivariant shuffle descriptions (Propositions 4.7 and 6.2), and the reduction to identity (7.1) is a genuine normal-bundle Euler-class comparison: the formula is asserted to follow from the Cartesian diagram (7.2) and Euler classes of the closed immersions there. Even though the calculation is not displayed, this is a missing derivation step rather than a circularity, because the identity is not identical to the definition of Res, Ind, or the braiding operators. The self-citations [Hen24b, Hen24a, Hen25] and the citations [Kin24, Dav17, Kap+24] supply structural tools such as the approachability-by-proper-maps property and dimensional reduction; these are prior results with stated hypotheses and are not used to assume Theorem 1.6. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors to force a choice, and the examples in §9 are independent checks in the vanishing-potential case. The omission of the Euler-class calculation in (7.1) is a legitimate correctness risk, but it does not make the derivation circular.

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

The central theorem does not rest on fitted constants or invented physical entities; it rests on a body of external geometric formalism and on the author's earlier induction constructions. The entries above are the background results that carry the load: the monodromic mixed Hodge module machine, dimensional reduction, good moduli space decomposition, and equivariant localization. The Coxeter complex and braiding operators are new mathematical objects introduced in the paper and are self-contained. No free parameters are fitted to data.

assumptions (5)

domain assumption Monodromic mixed Hodge module formalism, including the shifted vanishing cycle functor φ_f and the square root of the Tate twist L^{1/2}, as developed in [DM20].
The entire cohomology theory and the functoriality of vanishing cycles in Section 2 are phrased in this framework; the paper cites [DM20] instead of reproving it.
domain assumption Cohomological dimensional reduction isomorphisms of [Dav17, Theorem A.1] and Corollary 2.6.
Used as a black box in Theorem 1.3, Proposition 5.7, and Section 8 to identify 2d and 3d induction systems.
domain assumption The decomposition theorem and approachability of good moduli space maps from [Kin24] and [Hen24a, Proposition 5.12].
Needed in Section 4.4 for the canonical identification that produces the sheafified induction morphism (4.4) to (4.6).
domain assumption Equivariant localization in Borel-Moore homology [EG98b, Theorem 1], GKM localization [GKM98, Theorem 6.2], and purity-implies-formality [GKM98, Theorem 14.1] / [Dav23, Theorem 9.6].
Used in Section 8 for injectivity, decomposition, and purity arguments; these are external to the paper.
domain assumption The Weyl group invariant description H*_{G×T'}(V, φ_f Q) ≅ (H*_{T×T'}(V, φ_f Q))^W of Lemma 4.4, whose proof uses the classical cohomology of flag varieties in Proposition 2.1.
This identification is the bridge between parabolic and torus equivariant definitions and underlies both Proposition 4.7 and the proof of Theorem 7.6.

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Cite this review

Pith. "Pith review of Cohomological Mackey formula for quotient stacks." pith.science (2026). https://pith.science/paper/XXS2J7TE

@misc{pith2026250509483,
  author       = {Pith},
  title        = {Pith review of: Cohomological Mackey formula for quotient stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXS2J7TE}},
  note         = {Machine review of arXiv:2505.09483}
}

read the original abstract

In this paper, we construct a restriction morphism on the critical cohomology of an equivariant Landau-Ginzburg model associated to a representation of a reductive group equipped with an invariant function. We show a compatibility formula between the restriction and induction maps as a Mackey-type formula, thereby giving the critical cohomology the structure of a localized induction-restriction system.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 2 linked inside Pith

[1]

Equivariant mixed Hodge modules

[Ach13] Pramod Achar. “Equivariant mixed Hodge modules”. In: Lecture notes to accompany talk “Mixed Hodge modules and their applications (2013). [BD23] Tommaso Maria Botta and Ben Davison. “Okounkov’s conjecture via BPS Lie algebras”. In: arXiv preprint arXiv:2312.14008 (2023). [Bu+25a] Chenjing Bu, Ben Davison, Andr´ es Ib´ a˜ nez N´ u˜ nez, Tasuki Kinjo...

arXiv 2013
[2018]

Equivariant cohomology, Koszul duality, and the localization theorem

[GKM98] Mark Goresky, Robert Kottwitz, and Robert MacPherson. “Equivariant cohomology, Koszul duality, and the localization theorem”. In: Inventiones mathematicae 131.1 (1998), pp. 25–84. [Gun18] Sam Gunningham. “Generalized Springer theory for D-modules on a reductive Lie algebra”. In: Selecta Mathematica 24.5 (2018), pp. 4223–4277. [Hen24a] Lucien Henne...

arXiv 1998
[2025]

The Langlands formula and perverse sheaves

[Kap+24] Mikhail Kapranov, Vadim Schechtman, Olivier Schiffmann, and Jiangfan Yuan. “The Langlands formula and perverse sheaves”. In: arXiv preprint arXiv:2412.01638 (2024). [Kin22] Tasuki Kinjo. “Dimensional reduction in cohomological Donaldson–Thomas theory”. In: Compositio Mathematica 158.1 (2022), pp. 123–167. [Kin24] Tasuki Kinjo. “Decomposition theo...

arXiv 2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Equivariant mixed Hodge modules

[Ach13] Pramod Achar. “Equivariant mixed Hodge modules”. In: Lecture notes to accompany talk “Mixed Hodge modules and their applications (2013). [BD23] Tommaso Maria Botta and Ben Davison. “Okounkov’s conjecture via BPS Lie algebras”. In: arXiv preprint arXiv:2312.14008 (2023). [Bu+25a] Chenjing Bu, Ben Davison, Andr´ es Ib´ a˜ nez N´ u˜ nez, Tasuki Kinjo...

arXiv 2013

[2] [2018]

Equivariant cohomology, Koszul duality, and the localization theorem

[GKM98] Mark Goresky, Robert Kottwitz, and Robert MacPherson. “Equivariant cohomology, Koszul duality, and the localization theorem”. In: Inventiones mathematicae 131.1 (1998), pp. 25–84. [Gun18] Sam Gunningham. “Generalized Springer theory for D-modules on a reductive Lie algebra”. In: Selecta Mathematica 24.5 (2018), pp. 4223–4277. [Hen24a] Lucien Henne...

arXiv 1998

[3] [2025]

The Langlands formula and perverse sheaves

[Kap+24] Mikhail Kapranov, Vadim Schechtman, Olivier Schiffmann, and Jiangfan Yuan. “The Langlands formula and perverse sheaves”. In: arXiv preprint arXiv:2412.01638 (2024). [Kin22] Tasuki Kinjo. “Dimensional reduction in cohomological Donaldson–Thomas theory”. In: Compositio Mathematica 158.1 (2022), pp. 123–167. [Kin24] Tasuki Kinjo. “Decomposition theo...

arXiv 2024