Endoscopy for representations of disconnected reductive groups over finite fields

Maarten Solleveld

REVIEW 2 major objections 4 minor 1 cited by

Endoscopy for representations of disconnected reductive groups over finite fields

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes canonical equivalences of categories that make the Jordan decomposition for finite reductive groups canonical and extend it to disconnected reductive groups.

desk verdict A transparent and mostly well-executed package: the connected part is a clean repackaging of Lusztig-Yun, the disconnected extension is the real novelty but depends on a coherence step that is asserted rather than proved. read the letter →

arxiv 2507.16304 v1 pith:QSD77JNI submitted 2025-07-22 math.RT

Maarten Solleveld This is my paper

classification math.RT MSC 20C3320G40

keywords JordandecompositionLusztigseriesendoscopydisconnectedreductivegroupscharactersheavesunipotentrepresentationscanonicalequivalencesfiniteofLietype

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 proves that the Jordan decomposition for representations of finite reductive groups can be made canonical: rather than a bijection between irreducible representations with a fixed semisimple parameter and unipotent representations, there is an equivalence of whole categories. The main theorems extend this from connected groups to smooth group schemes whose neutral component is reductive, allowing the group of connected components to be infinite. If the theorems are correct, every irreducible representation of such a group is canonically described by a unipotent representation of a smaller endoscopic group together with an equivariant structure for a component group. This upgrades a classical classification into a functorial statement, and the author notes that the canonical form is what applications to depth-zero supercuspidal representations of $p$-adic groups require.

What carries the argument

The carrying object is the cell-by-cell categorical endoscopy equivalence from [LuYu2, Corollary 12.7], which identifies each cell subcategory of a geometric series with a direct sum of unipotent representation categories of forms of an endoscopic group, equipped with equivariant structures for stabilizer groups. The paper's work is to assemble these cell-wise equivalences into whole-series equivalences, to pass from geometric to rational series by parametrizing rational conjugacy classes through the quotient $B^\circ_L/\mathrm{Ad}^\epsilon(\Omega_L)$ using Lang's theorem and Steinberg's description of component groups, and to handle disconnectedness through the equivalence $\mathrm{Rep}(G^\epsilon) \cong \mathrm{Rep}(G^{\circ,\epsilon})^{\pi_0(G)}$ of Lemma 3.2, which trades a representation of the disconnected group for a representation of its neutral component with a $\pi_0(G)$-equivariant structure. The canonical nature of the final equivalences depends on all these identifications being compatible up to canonical isomorphisms.

What would settle it

Compute, for a small disconnected group satisfying Condition 3.1—for example $G = O_2(\mathbb{F}_q)$ or the normalizer of a maximal torus in $\mathrm{PGL}_2$—the category $\mathrm{Rep}_s(G)$ and the claimed endoscopic category $\mathrm{Rep}_1(H^\epsilon)^{\Omega_L^\epsilon}$ for every semisimple parameter $s$, and compare the number and internal structure of irreducible objects; any mismatch would disprove Theorem 3.5, while verifying the hypotheses of [LuYu2, Corollary 12.7] for those groups would test the load-bearing input.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is the existence of canonical equivalences of categories between representation series of (possibly disconnected) reductive groups over finite fields and unipotent representation categories of endoscopic groups. For a connected reductive group $G^\epsilon$, Theorem 2.4 states that for every stable pair $(L,T)$ corresponding to a semisimple element $s$ in the dual group, $\mathrm{Rep}_s(G^\epsilon)$ is canonically equivalent to $\mathrm{Rep}_1(H^\epsilon)^{\Omega_L^\epsilon}$, where $H$ is the endoscopic group determined by $L$ and the superscript denotes enrichment by $\Omega_L^\epsilon$-equivariant structures. For a smooth group scheme with reductive neutral component satisfying Condition 3.1, Theorem 3.4 gives the analogous equivalence for geometric series, $\mathrm{Rep}_{W(G,T)L}(G) \cong (\bigoplus_{\beta \in B^\circ_L} \mathrm{Rep}_1(H^{\sigma_\beta \epsilon}))^{\Omega_L}$, and Theorem 3.5 gives the rational-series version $\mathrm{Rep}_s(G) \cong \mathrm{Rep}_1(H^\epsilon)^{\Omega_L^\epsilon}$. These results make Lusztig's Jordan decomposition, previously a bijection, into a canonical equivalence of categories and extend it to the disconnected setting.

Load-bearing premise

The paper rests on the black-box cell-by-cell canonical endoscopy equivalence of [LuYu2, Corollary 12.7], with the second fragile point being that the compatibility of the $\Omega_L$ and $\pi_0(G)$ actions is only described 'up to canonical isomorphisms'.

Editorial extensions

If this is right

For any fixed semisimple parameter, the full category of representations is canonically equivalent to a unipotent category of an endoscopic group, so categorical invariants such as characters, cohomology, and natural transformations can be transferred between series.
The geometric series of a disconnected group splits into rational series parametrized by $B^\circ_L/\mathrm{Ad}^\epsilon(\Omega_L)$, one for each rational form $H^{\sigma_\beta \epsilon}$ of the endoscopic group.
Representations of groups with reductive neutral component, including infinite component groups, are reduced to unipotent representations of connected endoscopic groups plus equivariant structures.
The canonical form of the equivalence is what makes the depth-zero supercuspidal application viable, since the construction is now functorial rather than a mere parametrization.

Reading between the lines

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

The categorical, rather than bijective, nature suggests that natural operations on representation series, such as Deligne-Lusztig induction or restriction, can be made compatible with the Jordan decomposition, something a parametrization cannot express.
The equivariant-structure viewpoint may give a uniform language for representation rings of disconnected groups with nontrivial component-group actions, possibly connecting to cohomological induction for such groups.
A concrete numerical consequence of the main theorem would be identities between sizes of Lusztig series and sums of sizes of unipotent series; checking these for small $q$ would be an easy consistency test.
Because the construction inherits its canonicity from the cell-by-cell input, making that input more explicit (or replacing it) would directly control how constructive the whole equivalence is.

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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

The short version: this paper does something useful and labels its debts honestly. The connected-group part is a cell-free reformulation of [LuYu2, Cor. 12.7]; the genuinely new content is the canonical rational-series statement (Theorem 2.4) and the extension to disconnected reductive groups (Theorems 3.4 and 3.5). The exposition is clear, the citation pattern is clean, and the paper is open about the fact that the geometric connected case largely restates Lusztig-Yun. The reader's circularity score of 1 is right: there is no parameter-fitting or conclusion-normalizing here.

The soft spot is exactly where the stress-test note lands. In the proof of Theorems 3.4 and 3.5, Eq. (3.7) passes from an Omega_L^0-equivariant structure plus a pi_0(G)W_L^0-equivariant structure to a single Omega_L-equivariant structure. The text says these carry 'precisely the same information' because pi_0(G)W_L^0 is isomorphic to Omega_L/Omega_L^0. As a statement about groups that is true, but as a statement about categories it is not automatic. Combining the two actions requires that the chosen lifts of Omega_L act coherently on the direct sum of unipotent categories. The paper defines the Omega_L-action only 'up to canonical isomorphisms' and does not verify the required associativity or pentagon-type coherence. Without that check, the right side of (3.7) could define a different equivariant category, and the main theorems would not follow. I suspect the coherence can be extracted from the explicit morphisms in Section 2, but the paper needs to show it rather than assert it.

Two smaller issues. The proof of Theorem 2.4 says 'from [LuYu2, Section 12.10] one can see' how the restriction functor behaves; that is a handwave in the central connected statement. And the whole edifice rests on the black box [LuYu2, Cor. 12.7]; if that result carries hypotheses not satisfied by Condition 3.1 groups, the conclusions would not follow. I do not think either of these is fatal; they are exactly what a referee should ask to be pinned down.

The audience is representation theorists working on finite groups of Lie type and on depth-zero p-adic supercuspidals. This paper deserves a serious referee. I would not desk-reject; I would send it out with the explicit request to prove the coherence claim in Theorems 3.4 and 3.5. I personally would not cite it in my own work until that step is settled.

Referee Report

2 major / 4 minor

Summary. The paper proves canonical categorical analogues of Lusztig's Jordan decomposition for finite groups of Lie type. For a connected reductive group G over a finite field, Theorem 2.1 assembles the cell-wise endoscopy equivalences of Lusztig and Yun into a canonical equivalence between the geometric series Rep_{WL}(G) and a direct sum of unipotent representation categories of endoscopic forms enriched with Ω_L-equivariant structures. Theorem 2.4 uses a regular embedding to obtain a canonical equivalence for rational series, adding canonicity to Lusztig's Jordan decomposition. The main new results are Theorems 3.4 and 3.5, which extend these categorical equivalences to smooth group schemes G with reductive neutral component G^∘, allowing π_0(G) to be infinite. The proof in the disconnected case proceeds by restricting to G^∘, applying the connected results, and then combining a π_0(G)W^∘_L-action with an Ω^∘_L-action into an Ω_L-action. The paper also provides a parametrization of rational series by pairs (L, β) and discusses applications to depth-zero supercuspidal representations of p-adic groups.

Significance. If the disconnected-group theorems are correct, the paper delivers a canonical, functorial Jordan decomposition for representations of finite groups of Lie type, including disconnected reductive groups, which is a natural and useful extension of the bijections of Lusztig and the categorical results of Lusztig and Yun. The connected case is a clean and useful repackaging of [LuYu2], and the parametrization of rational series in Lemma 2.3 is a valuable contribution. The promised applications to depth-zero representations of p-adic groups give the results clear potential impact. However, the genuinely new step for disconnected groups — the coherence of the combined Ω_L-action — is asserted rather than proved, so the significance of the main theorems is currently conditional on a gap being filled.

major comments (2)

[§3, proof of Theorem 3.4, Eq. (3.7)] The proof of Theorem 3.4 contains a load-bearing gap. After establishing the chain of equivalences Rep_{WL}(G_ϵ) ≅ Rep_{W^∘L}(G^∘_ϵ)^{π_0(G)W^∘_L} ≅ ((⊕_{β∈B^∘_L} Rep_1(H^{σ_β ϵ}))^{Ω^∘_L})^{π_0(G)W^∘_L}, the text asserts that 'a π_0(G)W^∘_L-equivariant structure on top of a Ω^∘_L-equivariant structure contains precisely the same information as an Ω_L-equivariant structure' because π_0(G)W^∘_L ≅ Ω_L/Ω^∘_L (3.5). This is not automatic for weak actions on categories: the actions are defined only up to canonical isomorphisms, and one must prove that the π_0(G)W^∘_L-action on the Ω^∘_L-equivariant category and the Ω^∘_L-action combine into a coherent Ω_L-action satisfying the relevant cocycle and associativity (pentagon) conditions. A nontrivial 2-cocycle would make the iterated equivariant category inequivalent to the Ω_L-equivariant category. The proof does not construct the comparison functor or verify coherence, so Theorem 3.4 is not established as written. Since Theorem 3.5 invokes the same 'combine' step with the same justification, it inherits the same gap.
[§3, definition of the Ω_L-action before Theorem 3.4] The definition of the Ω_L-action on ⊕_{β∈B^∘_L} Rep_1(H^{σ_β ϵ}) is not sufficiently precise to support the equivalence claimed in Theorem 3.4. The text says that Ω_L acts 'by the same constructions as in the case of a connected G' and that the action is 'well-defined up to canonical isomorphisms,' but it does not prove that this action is compatible with the Ω^∘_L-action and the transferred π_0(G)W^∘_L-action. In particular, one needs to know whether the Ω_L-action restricts to the Ω^∘_L-action on each summand, whether the transferred π_0(G)W^∘_L-action agrees with the conjugation action induced by Ω_L on the category of Ω^∘_L-equivariant objects, and whether the resulting combined action satisfies the coherence conditions needed for a well-defined category of equivariant objects. These compatibilities are the substance of the passage from (3.7) to the theorem's conclusion, and they cannot be assumed without proof.

minor comments (4)

[§2, Eq. (2.1)] The formula for the action of Ω_L on B_L is hard to parse: 'Ad ϵ(ω)w = ωwϵ(ω)−1' should be typeset with explicit parentheses and a clarification of whether ϵ(ω) denotes the image of the representative of ω under the Frobenius action, and where the inverse is taken.
[§2, Theorem 2.1] Theorem 2.1 invokes [LuYu2, Corollary 12.7] as a black box without stating its hypotheses. Please state the precise conditions on the rank-one character sheaf L under which the cell-wise equivalences of [LuYu2, Corollary 12.7] hold, and confirm that those conditions are satisfied for the L considered in Theorem 2.1.
[§3, Theorem 3.4 statement] In Theorem 3.4 the hypothesis is written as 'W^∘L is ϵ-stable,' but since W^∘ is a group, what is presumably meant is that the W^∘-orbit of L is ϵ-stable. The wording should be corrected for clarity.
[§3, rational series for disconnected groups] The definition of Rep_{W(L,β)}(G_ϵ) is followed by the remark that it 'actually corresponds to a set of semisimple elements in G^{∘∨}_ϵ larger than the rational conjugacy class of s_β.' This is vague; please give a precise description of the parameter set, perhaps using the action of π_0(G) on the pairs (L, β) as in (3.8).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper transparently derives its results from the external Lusztig–Yun theorem and formal equivariance transfers, without fitting parameters or redefining its conclusions into its premises.

full rationale

The paper's derivation chain is not circular. The connected-group results are explicitly presented as consequences of [LuYu2, Corollary 12.7]; Theorem 2.1 is called 'a version of [LuYu2, Corollary 12.7] without cells' and its proof invokes that corollary at equation (2.5), while Theorem 2.4 uses the functorial properties of the same external work. This is a transparent import of an independent prior result, not a self-citation or a hidden use of the target statement. The disconnected-group theorems are obtained by the natural formal route: Lemma 3.2 identifies representations of a disconnected group with G^0-representations carrying a pi0(G)-equivariant structure, equation (3.7) transfers this along Theorem 2.1, and the final step identifying a pi0(G)W^0_L-equivariant structure on top of an Omega^0_L-equivariant structure with an Omega_L-equivariant structure rests on the group-theoretic isomorphisms (3.5) and (3.9). That identification is a structural consequence of the split exact sequence (3.1), not a restatement of the theorem being proved. There is no fitted parameter being renamed as a prediction, no uniqueness theorem imported from the present author's prior work, and no ansatz smuggled in by citation. The skeptical concern about coherence of the transferred Omega_L-action is a substantive technical worry about whether the asserted 'canonical isomorphisms' satisfy the necessary higher coherence conditions, but that is a potential gap or correctness risk, not circularity: nothing in the paper defines the conclusion to be equivalent to its own input. Accordingly the correct circularity finding is a non-finding, with score 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No fitted constants and no new physical or algebraic entities are introduced. The central claims rest on standard background (Deligne-Lusztig theory, Lang's theorem, Steinberg's centralizer theorem, regular embeddings) and, crucially, on [LuYu2, Corollary 12.7], which is used as a black box. Condition 3.1 defines the class of disconnected groups treated.

assumptions (8)

standard math Deligne-Lusztig theory provides geometric and rational series and disjointness results for irreducible representations.
Invoked in Section 1 and Section 2 to define Rep_{WL}(G_epsilon) and to justify the rational/geometric series parametrization.
domain assumption [LuYu2, Corollary 12.7] gives cell-by-cell canonical categorical endoscopy equivalences.
Used as a black box at Eq. (2.5) to prove Theorem 2.1; all later theorems inherit it.
standard math Lang's theorem: for connected Fq-groups the morphism g maps to g^{-1}epsilon(g) is surjective, and H^1(Fr, G^vee) is trivial.
Used in Lemma 2.2 and Lemma 2.3 to transfer Frobenius twists and parametrize rational conjugacy classes.
standard math Steinberg's description of centralizers of semisimple elements: pi_0(Z_{G^vee}(s)) is isomorphic to W^o_s \ W_s.
Used in Lemma 2.3 and Theorem 2.4 to identify the component group of the centralizer with Omega_L.
standard math The natural map N_{G_epsilon}(T_0^epsilon, B_0^epsilon) / T_0^epsilon to pi_0(G) is a group isomorphism (Lemma 3.3).
This structural fact is needed to split the exact sequences in (3.1) and to define the pi_0(G)-action.
domain assumption The pi_0(G)-action on the direct summands in (3.2) is transitive; W^o \ W / W_L is a single orbit for W modulo W^o.
Stated after (3.2) with only a brief justification; used to obtain (3.4) and hence Theorem 3.4.
domain assumption Condition 3.1: G is a smooth Fq-group scheme with reductive neutral component and all connected components Fr-stable.
Defines the class of groups for Theorems 3.4 and 3.5; the Fr-stability clause is harmless because other components contribute no Fq-points.
standard math Regular embedding G into G_c with connected center exists and preserves rational and geometric series.
Used in the proof of Theorem 2.4 to reduce to the connected-center case via [GeMa, Section 1.7] and Eq. (2.14).

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Cite this review

Pith. "Pith review of Endoscopy for representations of disconnected reductive groups over finite fields." pith.science (2026). https://pith.science/paper/QSD77JNI

@misc{pith2026250716304,
  author       = {Pith},
  title        = {Pith review of: Endoscopy for representations of disconnected reductive groups over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QSD77JNI}},
  note         = {Machine review of arXiv:2507.16304}
}

read the original abstract

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence between the category of G-representations with a fixed semisimple parameter s and the category of unipotent representations of an endoscopic group of G, enriched with an equivariant structure with respect to the component group of the centralizer of s. Next we generalize these results, replacing the connected reductive group by a smooth group scheme with reductive neutral component. Again we establish canonical equivalences for both the rational and the geometric series of G-representations, in terms of unipotent representations of endoscopic groups.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finite Langlands correspondence
math.NT 2025-08 conditional novelty 6.0 of 10

Every irreducible representation of a connected reductive group over a finite field is matched with a special Langlands parameter, with the fiber over each parameter given by irreducible representations of a finite co...

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

Deligne, G

P. Deligne, G. Lusztig, ``Representations of reductive groups over a finite field", Annals Math. 103 (1976) 103--161

work page 1976
[2]

Digne, J

F. Digne, J. Michel, Representations of Finite Groups of Lie Type 2nd ed., London Mathematical Society Student Texts 95 , Cambridge University Press, 2020

work page 2020
[3]

Parametrization of supercuspidal representations of depth zero for some simple adjoint groups

A. Fujii, ``Parametrization of supercuspidal representations of depth zero for simple adjoint groups", arXiv:2504.17225, 2025

work page Pith review arXiv 2025
[4]

M. Geck, G. Malle, The character theory of finite groups of Lie type. A guided tour, Cambridge studies in advanced mathematics 187 , Cambridge University Press, 2020

work page 2020
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G. Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studes 107 , Princeton University Press, Princeton NJ, 1984

work page 1984
[6]

Lusztig, ``On the representations of reductive groups with disconnected centre", Ast\'erisque 168 (1988), 157--166

G. Lusztig, ``On the representations of reductive groups with disconnected centre", Ast\'erisque 168 (1988), 157--166

work page 1988
[7]

Lusztig, Z

G. Lusztig, Z. Yun, ``Endoscopy for Hecke categories, character sheaves and representations", Forum Math. Pi 8 (2020), doi.org/10.1017/fmp.2020.9

doi:10.1017/fmp.2020.9 2020
[8]

Lusztig, Z

G. Lusztig, Z. Yun ``Corrigendum to ‘Endoscopy for Hecke categories, character sheaves and representations´ " Forum Math. Pi 9 (2021), doi.org/10.1017/fmp.2021.14

work page doi:10.1017/fmp.2021.14 2021

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Deligne, G

P. Deligne, G. Lusztig, ``Representations of reductive groups over a finite field", Annals Math. 103 (1976) 103--161

work page 1976

[2] [2]

Digne, J

F. Digne, J. Michel, Representations of Finite Groups of Lie Type 2nd ed., London Mathematical Society Student Texts 95 , Cambridge University Press, 2020

work page 2020

[3] [3]

Parametrization of supercuspidal representations of depth zero for some simple adjoint groups

A. Fujii, ``Parametrization of supercuspidal representations of depth zero for simple adjoint groups", arXiv:2504.17225, 2025

work page Pith review arXiv 2025

[4] [4]

M. Geck, G. Malle, The character theory of finite groups of Lie type. A guided tour, Cambridge studies in advanced mathematics 187 , Cambridge University Press, 2020

work page 2020

[5] [5]

Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studes 107 , Princeton University Press, Princeton NJ, 1984

G. Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studes 107 , Princeton University Press, Princeton NJ, 1984

work page 1984

[6] [6]

Lusztig, ``On the representations of reductive groups with disconnected centre", Ast\'erisque 168 (1988), 157--166

G. Lusztig, ``On the representations of reductive groups with disconnected centre", Ast\'erisque 168 (1988), 157--166

work page 1988

[7] [7]

Lusztig, Z

G. Lusztig, Z. Yun, ``Endoscopy for Hecke categories, character sheaves and representations", Forum Math. Pi 8 (2020), doi.org/10.1017/fmp.2020.9

doi:10.1017/fmp.2020.9 2020

[8] [8]

Lusztig, Z

G. Lusztig, Z. Yun ``Corrigendum to ‘Endoscopy for Hecke categories, character sheaves and representations´ " Forum Math. Pi 9 (2021), doi.org/10.1017/fmp.2021.14

work page doi:10.1017/fmp.2021.14 2021