Generic weights for finite reductive groups

Gunter Malle; Jiping Zhang; Zhicheng Feng

REVIEW 4 major objections 6 minor 57 references

Generic weights for finite reductive groups

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Generic weights can replace Alperin weights in the inductive verification of Alperin's weight conjecture.

desk verdict A genuinely new framework for the Alperin weight conjecture, with a load-bearing dependency on the first author's unpublished preprint [21]; the right call is conditional, not acceptance in current form. read the letter →

arxiv 2505.22064 v1 pith:OJGC3J6Q submitted 2025-05-28 math.RT math.GR

Zhicheng Feng , Gunter Malle , Jiping Zhang This is my paper

classification math.RTmath.GR MSC 20C3320C2020G40

keywords Alperinweightconjecturegenericweightse-Harish-ChandratheoryfinitereductivegroupsinductiveconditionblocksofcharactertriplesrelativeWeyl

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

The paper's goal is to make Alperin's weight conjecture, an open conjecture predicting a numerical equality between certain ordinary and modular characters of any finite group, approachable for finite groups of Lie type by replacing Alperin's weights with new generic weights built from generalized Harish-Chandra theory. It defines $(e,\ell)$-generalized-cuspidal characters, a widening of $e$-cuspidality in which Deligne--Lusztig induction along $E_{e,\ell}$-split Levi subgroups replaces ordinary Harish-Chandra induction, and uses them to define generic weights $(T,\eta)$ for $G^F$ in non-defining characteristic $\ell$. Its main theorem asserts that, for odd good primes satisfying Condition 6.1, there is a blockwise equivariant bijection between generic weights and Alperin weights that preserves the relevant stabilizers and block character-triple isomorphisms. From this the paper derives criteria under which the inductive Alperin weight condition, respectively its blockwise version, follows from a single equivariant bijection between ordinary characters in a block and its generic weights. If correct, this would reduce the hard verification for simple groups of Lie type from an analysis of many radical subgroups to the construction of character correspondences in generalized Harish-Chandra theory.

What carries the argument

The central object is the generic weight $(T,\eta)$: $T$ is an $e$-torus of $G$ with $T=Z^\circ(C_G(T))_{\varphi_e}$, and $\eta$ is an irreducible character of $N_{G^F}(T)$ lying over an $(e,\ell)$-Jordan-generalized-cuspidal character of $C_{G^F}(T)$ inside the set $\mathrm{rdz}(N_{G^F}(T)\mid\lambda)$, whose members have the defect-zero character ratio $\chi(1)_\ell/\theta(1)_\ell=|N_{G^F}(T)/C_{G^F}(T)|_\ell$. The proof has three main parts: the classification of $(e,\ell)$-generalized-cuspidal characters (hook partitions for $SL_n$ and $SU_n$ at the relevant primes, and explicit Harish-Chandra series for bad primes $\ell\ge3$ in Table 1); the reduction of generic weights to quasi-isolated blocks through the Bonnaf\'e--Dat--Rouquier correspondence in Theorem 3.29; and the block isomorphism of character triples that allows stabilizers and block inductions to be compared between generic weights and Alperin weights. In the proof of Theorem 6.2, the maps for individual $e$-tori are constructed by passing to the Levi subgroup $L=C_G(T)$, decomposing $[L,L]$ into simple factors, and invoking the type-$A$ bijection of Theorem 5.1 together with the character-triple theorems of the preprint [21].

What would settle it

Concretely, compute $|W(B)|$ and $|\mathrm{Alp}(B)|$ for every $\ell$-block of a small group of Lie type satisfying Condition 6.1, for example a principal block of an exceptional group with non-abelian Sylow $\ell$-subgroup at $\ell=5$ or $7$; if the sizes differ, or if no bijection with the stabilizer inclusions and block character-triple isomorphisms of Theorem 6.2 exists, the central claim is false. Independently, disproving the uni-triangular basic-set property for any group of Lie type at a good prime would remove the standing hypotheses of Corollary 6.7 and Theorem 6.8.

Watch

Extended reading notes

Core claim

The paper's central claim is that, under Condition 6.1, Alperin's weights for a finite reductive group $G^F$ in non-defining characteristic can be systematically replaced by generic weights without changing the block theory. A generic weight is a pair $(T,\eta)$ in which $T$ is an $e$-torus of $G$ with $T=Z^\circ(C_G(T))_{\varphi_e}$, and $\eta\in\mathrm{rdz}(N_{G^F}(T)\mid\lambda)$ lies over an $(e,\ell)$-Jordan-generalized-cuspidal character $\lambda$ of $C_{G^F}(T)$. Theorem 6.2 asserts that for every $\ell$-block $B$ there is a bijection $\Omega:W(B)\to\mathrm{Alp}(B)$ that is equivariant for the stabilizer of $B$ inside $\tilde G^F\rtimes B$ (where the second $B$ denotes the group generated by field and graph automorphisms), and every image $(R,\varphi)$ of $(T,\eta)$ satisfies the torus identity $T=Z^\circ(C^\circ_G(Z(R)))_{\varphi_e}$, the block equation $\mathrm{bl}(\varphi)^{N_{G^F}(T)}=\mathrm{bl}(\eta)$, and a block isomorphism of character triples between $((\tilde G^F\rtimes B)_{T,\eta},N_{G^F}(T),\eta)$ and $((\tilde G^F\rtimes B)_{R,\varphi},N_{G^F}(R),\varphi)$. The paper then shows that if $\mathrm{E}(G^F,\ell')$ is a uni-triangular basic set for $G^F$, such an equivariant bijection between ordinary characters and generic weights is all that is needed to trigger the inductive Alperin weight condition, and analogously its blockwise version, for the corresponding simple group.

Load-bearing premise

The argument depends on the unproved assumption that the set $\mathrm{E}(G^F,\ell')$ is a uni-triangular basic set for $G^F$, a property that is open for general groups of Lie type, and on the correctness of the three theorems taken as black boxes from the unpublished preprint [21].

Editorial extensions

If this is right

In type $A$ (Theorem 5.1), the generic weights of $SL_n(\epsilon q)$ are in explicit equivariant bijection with Alperin weights of $GL_n(\epsilon q)$ modulo the $\ell$-part of the diagonal group, with stabilizer inclusions and character-triple block isomorphisms; this is the model case for the general replacement.
Under Condition 6.1, verification of the inductive Alperin weight condition for $S=G^F/Z(G^F)$ reduces to exhibiting an equivariant bijection $\mathrm{E}(G^F,\ell')\to W(G^F)$ whose character-triple block isomorphism holds, assuming $\mathrm{E}(G^F,\ell')$ is a uni-triangular basic set.
The blockwise version reduces the inductive blockwise Alperin weight condition for a block $B$ to an equivariant bijection from $\mathrm{Irr}(B)\cap\mathrm{E}(G^F,\ell')$ to $W(B)$, again modulo the uni-triangular basic-set hypothesis.
The correspondence proposed in Question 7.1, partitioning irreducible characters of one relative Weyl group into defect-zero characters of other relative Weyl groups, holds for all unipotent blocks and for quasi-isolated blocks of exceptional groups, giving explicit numerical checks via Tables 2 and 3.
The classification results show that, outside type $A$ at primes dividing $q-\epsilon$ or $\ell=2$ with $4\mid q+\epsilon$, the new $e$-generalized-cuspidal characters coincide with the classical $e$-cuspidal characters, so the generic-weight framework genuinely differs from earlier theory only in restricted situations.

Reading between the lines

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

If the uni-triangular basic-set assumption is eventually proved for all good primes, the criteria in Corollary 6.7 and Theorem 6.8 would turn any equivariant bijection from ordinary characters in a block to its generic weights into a complete proof of the inductive AW and BAW conditions; the remaining hard work would then be supplying such bijections via Deligne--Lusztig and Jordan decomposition r
Question 7.1 suggests a concrete combinatorial refinement: irreducible characters of relative Weyl groups should split into defect-zero characters of smaller relative Weyl groups, and in type $A$ the unipotent side is labeled by hook partitions, so one can test the claimed partition numerically with explicit character tables of small Weyl groups.
Because Theorem 6.2 invokes three results from the unpublished preprint [21] as black boxes, a reader wanting to apply the replacement theorem should first check [21]: if any of those three theorems fails, the generic-weight replacement is not yet established even though the bijection statements of Section 5 may still hold.

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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, this is a serious contribution to the Alperin weight program, not a superficial repackaging: the generic-weight formalism is new, it reduces the inductive condition to fewer local subgroups, and it yields a fresh combinatorial problem (Question 7.1) with nontrivial positive results. Second, the central reduction in Theorem 6.2 depends on the first author's unpublished preprint [21] for the decisive equivariant bijections; if any of those black-boxed statements is wrong, the claimed replacement of Alperin weights by generic weights fails. The paper deserves a careful referee, but in its current form it is conditional.

What is actually new and good: Definition 3.5 of (e,ℓ)-generalized-cuspidal characters, the classification in Proposition 3.12 for type A (hook partitions), the systematic reduction to quasi-isolated blocks in Section 3.V, the partition of weights by centers of radical subgroups in Section 4, the equivariant bijection for type A in Theorem 5.1, and the reformulation of the inductive conditions in Corollaries 6.7 and 6.10. The proof of Theorem 7.5 for unipotent blocks is a concrete, checkable result that goes beyond previous statements. The paper is carefully written: many lemmas are proved in detail, and the authors flag the known open cases (Assumption 3.19, the basic-set hypothesis) rather than hiding them.

Soft spots, in proportion. The load-bearing issue: Theorem 6.2 is stated as a theorem, but its proof passes through [21, Thm. 4.3, 5.8] and Theorem 5.1 of the present paper uses [21, Thm. 5.1] as a black box. Reference [21] is an unpublished arXiv preprint; none of the key statements are proved or even stated in enough detail here for independent verification. This is the single most important risk. Secondary: Theorem 6.5 and Corollary 6.7 assume E(G^F,ℓ′) is a uni-triangular basic set, which is open for general groups of Lie type. The paper flags this, but it is a genuine limitation. The tables in Propositions 7.6 and 7.7 are asserted with minimal derivation; they look plausible and are consistent with the cited fusion-system data, but a referee will want the calculations shown or independently checkable. The intrinsic Assumption 3.19 is open in general, though Späth has proved the key cases.

Bottom line: I found no load-bearing error on the published side; the architecture is coherent and the type A analysis is genuinely useful. The dependency on [21] is the difference between conditional and accepted. I would send it to a serious referee and ask them to verify the black-box statements or obtain an independent check of [21], and I would not cite Theorem 6.2 in my own work until that dependency is resolved.

Referee Report

4 major / 6 minor

Summary. The paper introduces a generalization of e-cuspidality, called (e,ℓ)-generalized-cuspidality, and defines 'generic weights' for finite reductive groups in non-defining characteristic. It studies a bijection between generic weights and Alperin weights: Theorem 6.2 asserts a blockwise equivariant bijection W(B) → Alp(B) under Condition 6.1, and Theorems 6.5–6.8 give criteria under which such a bijection implies the inductive Alperin weight (AW) or blockwise Alperin weight (BAW) conditions. Sections 5 and 7 contain type A results, explicit comparisons with Alperin–Fong and An classifications, and a new combinatorial Question 7.1 about character correspondences for relative Weyl groups, proved for unipotent blocks and quasi-isolated blocks of exceptional groups.

Significance. If the central bijection is correct, the paper offers a substantial conceptual simplification: it reduces the inductive AW/BAW verification for groups of Lie type to finding a character-to-generic-weight bijection with good equivariance properties, thereby avoiding the full local analysis of radical subgroups. The paper is carefully structured and contains several independently valuable contributions: Corollary 4.4 gives a canonical bijection in the abelian Sylow case; the type A analysis in Section 5 is checked against the independent Alperin–Fong and An classifications; and Question 7.1 is a genuinely new combinatorial statement that the authors verify for all unipotent blocks and for quasi-isolated blocks of exceptional groups. These strengths are substantial. The main caveats are the reliance on the unpublished preprint [21] for the decisive bijections and the unproved unitriangular basic-set assumption in the inductive criteria; both are load-bearing for the paper's central claim.

major comments (4)

[§6, Theorem 6.2 and proof of Theorem 6.3] Theorem 6.2 is the central replacement claim of the paper. Its proof through Theorem 6.3 invokes [21, Thm. 4.3] and [21, Thm. 5.8] as black boxes for the decisive bijections f_{k_i} for the simple components H_{k_i}, and Theorem 5.1 is proved by applying [21, Thm. 5.1] and [21, Prop. 5.6]. Reference [21] (arXiv:2502.00774) is an unpublished preprint by the first author, and none of the invoked statements is restated or proved in the present paper. Since any gap in [21] directly invalidates Theorem 6.2 and hence the unconditional phrasing in the abstract, the manuscript should either include the statements and proofs of the needed results from [21] or explicitly state Theorem 6.2 and the abstract claim as conditional on [21].
[§6, Theorems 6.5, 6.8 and Corollaries 6.7, 6.10] The inductive AW and BAW criteria depend essentially on the assumption that E(G^F,ℓ′) is a unitriangular basic set for G^F (Theorem 6.5) or that Irr(B)∩E(G^F,ℓ′) is a unitriangular basic set for the block B (Theorem 6.8). This is not proved in the paper and is open in general for groups of Lie type. The paper does flag this assumption, but it is load-bearing for the conclusions of the theorems and for Corollaries 6.7 and 6.10; the scope of the main claim should be stated to make this dependence explicit.
[§7, Propositions 7.6 and 7.7] The proofs of Propositions 7.6 and 7.7 rely on Tables 2 and 3, which list cardinalities such as |W_G^F(L_0,1)| and the right-hand sums for quasi-isolated blocks. The derivation of these entries is only sketched ('Arguing as there' and 'we can now argue as before'), and no reference is given for the table data. Since these tables are used to prove the existence of bijections (7.2) for all unipotent blocks and for quasi-isolated blocks of exceptional groups, the data should be either derived in detail or traced to a published source.
[§3.IV, Assumption 3.19; Lemmas 3.21 and 4.5(c)] Assumption 3.19 (maximal extendibility of characters of e-split Levi subgroups) is used to identify W_0(B,T) with defect-zero characters of relative Weyl groups in Lemma 3.21 and to count W(B) in Proposition 4.5(c) and in Section 7. The paper correctly notes that this assumption is open in general. However, the counting formula |W(B)| = ∑ |dz(W_G^F(L,ζ))| in Section 7 and several bijections in Section 6 hold only modulo Assumption 3.19; this limitation should be stated clearly in the main theorems and in the abstract.

minor comments (6)

[Key words and phrases] The keyword 'gereralized Harish-Chandra theory' contains a typo and should read 'generalized Harish-Chandra theory'.
[Abstract] The abstract says the approach 'will constitute a major step' while the introduction says 'a step'; the formulations should be aligned to avoid overstating the claim.
[Definition 4.6] In Definition 4.6(c), Alp(G^F,T) and Alp(B,T) are defined using N_G^F(T)-conjugacy classes, whereas Alp(G^F) and Alp(B) elsewhere use G^F-conjugacy classes; this difference should be stated explicitly to prevent ambiguity.
[Propositions 7.6–7.7, Tables 2–3] The column heading 'P' in Tables 2 and 3 is unexplained; a caption or a sentence should define it and clarify the meaning of the displayed sums.
[Proof of Proposition 3.14] The final part of the proof says certain claims 'can be checked easily in CHEVIE'; this is not a substitute for a verifiable mathematical argument, and the relevant data or a precise reference should be supplied.
[Section 5, introductory paragraph] The standing assumption '4|(q−ϵ) when ℓ=2' is stated after Theorem 5.1, but Proposition 3.12(b) also treats the case 4|(q+ϵ) with ℓ=2; the scope of Theorem 5.1 with respect to that case should be clarified.

Circularity Check

2 steps flagged · score 4.0 of 10

Central bijection between generic weights and Alperin weights is imported from the first author's unpublished preprint [21] (Thms. 4.3, 5.1, 5.8, Prop. 5.6); this is load-bearing self-citation, but the paper also contains independent content.

self citation load bearing [Section 5.IV, proof of Theorem 5.1]
"We prove this assertion by applying [21, Thm. 5.1] by taking A= eGF ⋊B, eG= eGF, G=G F, E=B, eI=W( eGF, eG) and eA=Alp( eGF, eG)."

The theorem being proved is exactly a blockwise equivariant bijection between generic weights W( eGF, eG) and Alperin weights Alp( eGF, eG). Rather than proving this bijection, the proof applies [21, Thm. 5.1] from the first author's unpublished preprint; even condition (iv.b) is verified by [21, Prop. 5.6]. The present paper's own Theorem 5.10 only supplies a cardinality/action compatibility statement, so the existence of the equivariant bijection is imported from a self-citation whose proofs are not contained in this paper. If [21, Thm. 5.1 or Prop. 5.6] fails, Theorem 5.1, and hence the type-A pillar of Theorem 6.2, has no support in the present text.
self citation load bearing [Section 6.I, proof of Theorem 6.3]
"for 1≤i≤t, according to [21, Thm. 4.3] and Theorem 5.1, there exists a blockwise Aut(H^{F^{n_i}}_{k_i})-equivariant bijection f_{k_i}:W(H^{F^{n_i}}_{k_i},H_{k_i})→Alp_0(H^{F^{n_i}}_{k_i})/∼ Diagℓ(H^{F^{n_i}}_{k_i})."

The central bijection of Theorem 6.2 is assembled in Theorem 6.3; the decisive maps f_{k_i} for arbitrary simple components and the final lift Ω_T are obtained by invoking [21, Thm. 4.3] and [21, Thm. 5.8]. The paper does not state or prove these results; they are from the first author's unpublished arXiv preprint. Thus the conclusion that Alperin weights can be replaced by generic weights terminates in a self-citation. The surrounding reduction to Levi components and the type-A verification are independent work, but the existence part of the central bijection is not derived in this paper.

full rationale

The derivation is not circular in the definitional sense: W(B) and Alp(B) are defined independently, and Corollary 4.4 is a genuine bijection derived from the known e-cuspidal classification rather than a fitted prediction. The type-A section is anchored in the independent Alperin–Fong/An weight classifications, and Question 7.1 is a new combinatorial statement checked for unipotent and quasi-isolated blocks. The serious issue is that Theorem 6.2, the central 'replacement' claim, is proved only by reducing to [21], an unpublished preprint by the first author: Theorem 5.1 applies [21, Thm. 5.1] and [21, Prop. 5.6], and Theorem 6.3 obtains its simple-component bijections from [21, Thm. 4.3] and the final lift from [21, Thm. 5.8]. Since none of these statements is proved or reproduced in this paper, the central conclusion depends on a self-citation; this is load-bearing self-citation rather than a definitional circle. The uni-triangular basic set assumption in Theorem 6.5 and Corollary 6.7 is explicitly flagged and open in general, so it is a correctness condition rather than a circular step. Overall, the paper contains substantial independent content, but the main theorem's proof chain is not self-contained.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

Pure mathematics paper; no parameters fitted to data. The 'axioms' are the standing hypotheses (good prime conditions, uni-triangular basic set, extendibility) and the dependency on the preprint [21]. New objects are definitions, not empirical postulates.

assumptions (5)

domain assumption Standing hypotheses of Condition 6.1: ℓ odd, good for G, ℓ ∤ |Z(G)^F|, ℓ > 3 for 3D4
Underpins Theorems 6.2 and 6.5; excludes bad primes and some small cases.
domain assumption E(G^F, ℓ') is a uni-triangular basic set for G^F
Assumed in Theorem 6.5 and Corollary 6.7; not proved here and open in general.
domain assumption Extendibility/stabilizer conditions (1c),(1d),(2),(3) in Theorem 6.5
Required to apply the Brough-Späth criterion [12, Thm. 3.3]; partially known, assumed here.
ad hoc to paper Validity of Theorems 5.1, 4.3, 5.8 of Feng's preprint [21]
Used as black boxes in Theorem 6.2 and Theorem 5.1 proof; no independent verification in this paper.
domain assumption Assumption 3.19 (maximal extendibility of characters of e-split Levi subgroups)
Explicitly stated as open in general; used in Lemma 3.21 and Proposition 4.5(c).

invented entities (2)

Generic (e,ℓ)-weights (T,η)
purpose: Serve as a replacement for Alperin weights in the inductive Alperin weight condition
Definitions 3.17-3.18; mathematically well-defined, no external falsifiable prediction.
(e,ℓ)-generalized-cuspidal (e-GC) characters
purpose: Extend e-cuspidality to allow a generalized Harish-Chandra theory and generic weights
Definition 3.5; new structural object, no empirical handle.

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Pith. "Pith review of Generic weights for finite reductive groups." pith.science (2026). https://pith.science/paper/OJGC3J6Q

@misc{pith2026250522064,
  author       = {Pith},
  title        = {Pith review of: Generic weights for finite reductive groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJGC3J6Q}},
  note         = {Machine review of arXiv:2505.22064}
}

abstract

This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of $e$-cuspidality in the $e$-Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.

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