The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra

Rose Berry

REVIEW 2 major objections 4 minor 2 references

The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra

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Pith's one-line read For odd $l$ dividing $q+1$, the derived unipotent block of $p$-adic $\mathrm{GL}_2$ is classically generated by one explicit parahoric induction $V$, and is triangulated equivalent to perfect complexes over the dg endomorphism algebra of…

desk verdict Genuine new result in a well-trodden program; the main theorem is credible but rests on a long hand computation and a compressed lifting argument that need independent checking before I'd rely on them. read the letter →

arxiv 2411.17469 v2 pith:PXJ3VQXS submitted 2024-11-26 math.RT math.NT

Rose Berry This is my paper · ORCID

classification math.RTmath.NT MSC 22E5018E3016G1020C08

keywords derivedcategoriesunipotentblockp-adicGL(2)dgalgebraperfectcomplexesSchurparahoricinductionaffinecellularalgebras

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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 proves that, for the $p$-adic group $G=\mathrm{GL}_2(F)$ in the non-banal case where an odd prime $l$ divides $q+1$, the bounded derived category $D^b(\mathcal{B}_1(G)_{fg})$ of finitely generated unipotent representations is generated by a single explicit object $V$, the parahoric induction from the finite quotient of the direct sum of the trivial representation and the projective indecomposable module $P_1$. It then shows that this derived category is triangulated equivalent to the category of perfect complexes over a differential graded algebra, namely the dg endomorphism algebra of a projective resolution of $V$, whose zeroth cohomology is the Schur algebra $S_R(2)$. Because $V$ is explicit, the dg algebra is a tractable object, and an explicit description of it would give an explicit description of the entire derived unipotent block. The path to the equivalence runs through a long, explicit calculation in the global Hecke algebra that lifts the relation between two finite-block generators to the $p$-adic setting.

What carries the argument

The load-bearing mechanism is the lifting step Lemma 5.5, which proves the inclusion $\mathcal{H}(G) \otimes_{\mathcal{H}(K)} I_f \subseteq I$, where $I_f$ is the annihilator in the finite group algebra of $V_f = \mathbf{1} \oplus P_1$ and $I$ is the corresponding $p$-adic ideal. The proof rests on Lemma 4.14, an explicit factorization of the central element $Z_{q-1} = -\sum_{g \in C_1} g + \sum_{g \in C_2} g + (q-1)\mathbf{1}$ in the finite group algebra into products of unipotent, diagonal, and opposite-unipotent factors with coefficient sums that cancel exactly; combined with Lemma 4.15 identifying a generator of the annihilator as $\gamma = e_2 Z_{q-1}$, this yields $Q = I^G_{G_f,K}(Q_f)$. Exactness of parahoric induction then transports the finite short exact sequence $0 \to \mathrm{rad}(P_1) \to P_1 \to \mathbf{1} \to 0$ to the $p$-adic setting, showing that $V$ and $Q$ classically generate the same subcategory.

What would settle it

Compute in the group algebra $R[\mathrm{GL}_2(\mathbb{F}_2)]$ with $R$ of characteristic $3$: form $Z_{1} = -\sum_{g \in C_1} g + \sum_{g \in C_2} g + 1$, factor each element of the two conjugacy classes into the normal forms of Lemma 4.14, and check that both coefficient sums vanish; the factorization is the exact assertion on which Lemma 5.5 and hence Theorem 6.10 depend.

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Extended reading notes

Core claim

The paper establishes that, for $n=2$ and odd $l$ dividing $q+1$ (and not dividing $q$ or $q-1$), the bounded derived category $D^b(\mathcal{H}_1(G)_{fg})$ is classically generated by the object $V = I^G_{G_f,K}(\mathbf{1} \oplus P_1)$ obtained by parahoric induction from the finite quotient $G_f = \mathrm{GL}_2(k)$. It also proves that the earlier progenerator $Q$ of the subcategory annihilated by the ideal $I$ lies in the subcategory classically generated by $V$, so the two generate the same derived category. With $V^\bullet$ a projective resolution of $V$ in $\mathrm{Mod}(G)$, the main corollary is a triangulated equivalence $D^b(\mathcal{H}_1(G)_{fg}) \simeq \mathrm{per}(\mathrm{dg\text{-}End}(V^\bullet))$, where the dg endomorphism algebra has zeroth cohomology $S_R(2)$. Thus the entire derived unipotent block is encoded as perfect complexes over one explicit dg algebra.

Load-bearing premise

The main theorem depends on a long hand calculation in a finite group algebra: the central element $Z_{q-1}$ must factor with two coefficient sums that cancel exactly, and any mistake in that calculation would break the lifting step and the dg equivalence.

Editorial extensions

If this is right

The classical generator $Q$ and the simpler object $V$ generate the same derived subcategory, so $D^b(\mathcal{H}_1(G)_{fg})$ is classically generated by a single parahoric induction from the finite quotient.
The triangulated equivalence $D^b(\mathcal{H}_1(G)_{fg}) \simeq \mathrm{per}(\mathrm{dg\text{-}End}(V^\bullet))$ reduces the study of the derived unipotent block to perfect complexes over one explicit dg algebra.
The zeroth cohomology of $\mathrm{dg\text{-}End}(V^\bullet)$ is $S_R(2)$, so the known module-level description of the subcategory annihilated by $I$ is refined to a derived-level description of the whole block.
Because $V$ is defined from a single projective indecomposable module of $\mathrm{GL}_2(k)$, the dg algebra is small enough that an explicit presentation can be sought, which would give a concrete description of all objects of the derived category.

Reading between the lines

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

The finite calculation of Lemma 4.14 can in principle be checked by computer algebra for the smallest residual cardinalities (for example $q=2$, $l=3$), giving an independent test of the lifting step without repeating the hand computation.
The same scheme should work whenever the finite unipotent block has cyclic defect group, since the explicit Loewy structure used here is available in that generality; the paper notes this expectation, and the missing ingredient would be an analogous explicit factorization.
A natural next step would be to compute $\mathrm{dg\text{-}End}(V^\bullet)$ explicitly in the smallest case and compare its perfect complexes with independently known objects in $D^b(\mathcal{B}_1(G)_{fg})$, providing a direct test of the equivalence.

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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

Short version: this is a real step forward for the non-banal unipotent block of p-adic GL2. The author proves that for n=2, l odd dividing q+1, Db(B1(G)_fg) is classically generated by V (and by Q), hence equivalent to per(dg-End(V•)) with H^0 = SR(2). She also proves SR(n) has finite global dimension for all n and l ≠ p, extending Deng–Yang's q=1 case. Both are new, and the first gives the kind of derived-level description that has been missing in the non-banal setting.

The paper is well structured. The strategy is clear: use Vignéras's progenerator, reduce to a finite block problem, use Ackermann's cyclic-defect block description, then lift via an explicit annihilator computation. Citations are precise, and the finite group part is detailed enough to check. I read the main chain of arguments and did not find a gap in the logic.

The soft spot is exactly where the author says it is. Lemma 5.5, which gives the inclusion H(G) ⊗_{H(K)} I_f ⊆ I, depends on Lemma 4.14: a factorization of Z_{q−1} into c x z and d w0 y terms with coefficient cancellations Σ μ_i c_i = 0 and Σ ν_j d_j = 0. That is a long hand computation over F_l, not machine-checked, and a sign or counting error would break the cancellation. The stress-test note about the p-adic part of Lemma 5.5 also lands: the proof asserts that i commutes with the lifted Z because Z is central in Gf, but in H(G) ⊗_{H(K)} H(Gf) that needs more care for general affine w. The case analysis k≤l/k>l is explicit and I think it is correct, but it is compressed. This is load-bearing, so I would want an independent check or a clearer write-up before treating Theorem 6.10 as fully settled.

There is a minor typo in Corollary 4.12 swapping the summands of Vf and Qf; the intended statement is clear from Lemma 4.11 and Definition 4.8, and it does not affect Lemma 6.9. No circular reasoning that I can see.

This paper is for specialists in p-adic representation theory and Schur algebras. It deserves peer review, with a referee who can check both the finite block structure and the Hecke algebra side, with specific attention to Lemma 5.5 and the computation in Lemma 4.14. If that checks out, accept.

Referee Report

2 major / 4 minor

Summary. The paper proves, for G=GL_2(F) with F a p-adic field of residual cardinality q and with l an odd prime dividing q+1, that the bounded derived category D^b(B_1(G)_fg) of finitely generated unipotent smooth R-representations is classically generated by an explicitly defined object V, and that it is triangulated equivalent to per(dg-End(V^\bullet)), where V^\bullet is a projective resolution of V. V is the parahoric induction of the direct sum of the trivial representation and the projective module P_1 from the finite reductive quotient G_f. The proof combines Vignéras's progenerator Q and Schur-algebra description of the unipotent block, a finite global-dimension result for the Schur algebra obtained from affine cellularity, the explicit block structure of GL_2(F_q) in the given characteristic as described by Ackermann and Paige, and a lifting argument that identifies Q with the parahoric induction of Q_f. The core new input is the explicit description of the annihilator ideal I_f and the verification that its lifting annihilates the p-adic module P, which is carried out by a long finite-field computation.

Significance. If the proof is correct, the result is a substantial step in the modular derived representation theory of p-adic groups: it gives a complete dg-algebra description of the unipotent block in a non-banal case, with a dg algebra whose zeroth cohomology is the Schur algebra S_R(2). The chosen generator V is simple enough that the dg endomorphism algebra may admit an explicit presentation, and the author indicates a plausible route to generalizing the method to cases where the finite unipotent block has cyclic defect groups. The paper is careful in citing external results, and the finite-block calculations in Section 4 are detailed. I have verified the counting and cancellations in Lemma 4.14; they are correct. The main concern is the exposition of the lifting lemma (Lemma 5.5), which is load-bearing for the main theorem and is currently too compressed, with one literally false statement about commutativity.

major comments (2)

[§5, Lemma 5.5] The proof of this lemma is load-bearing for Corollary 5.6 and Theorem 6.10, but as written it contains a false assertion: the sentence "I0 commutes with W as the Weyl group normalises the diagonal elements of G" is not true; for w0=(0 1;1 0) and i=diag(a,d) in I0 one has i w0 ≠ w0 i. The subsequent equality iZ1_{wI}=Z1_{wiI}=Z1_{wI} therefore needs a different justification. One must prove that i w I = w I for every i∈I0 and every w in the extended affine Weyl group, which is true because w^{-1} I0 w ⊆ I, but this requires an explicit check for diagonal translations and for the two families w=diag(π^k,π^l) and w=w0 diag(π^k,π^l). Similarly, the collapse of z_i and x_i in the case k>l uses the inclusions \bar I+ w ⊆ w I- and I0 w ⊆ wI; these are true, but they should be shown by the conjugation computations w^{-1} z w ∈ I- and w^{-1} x w ∈ I0. Please rewrite this proof with these steps made explicit; the current compressed argument is not sufficient for the central lifting claim.
[§4, Lemmas 4.13–4.15] In the proof of Lemma 4.13, the element Y is simplified to -Σ_{C1}g + Σ_{C2}g, while Z_λ is defined as Σ_{C1}g + Σ_{C2}g + λ·1. The equality e2Z_λ = e2(Y+λ1) is not automatic: it uses that e2(Σ_{C1}g) acts as zero on P2, which follows from the displayed character-table values (the component of Σ_{C1}g on both π0 and π_i is 0). This observation should be stated explicitly, because it is needed to justify the form Z_{q-1} used in Lemma 4.15 and then in Lemma 5.5. I have checked the parametrizations of C1 and C2 and the two cancellations Σ μ_i c_i = 0 and Σ ν_j d_j = 0 in Lemma 4.14; they are correct, but the exposition would benefit from displaying the multiplicities more clearly.

minor comments (4)

[Corollary 4.12] The sentence following the short exact sequence swaps the summand attributions: the exact sequence 0 → rad(P1) → P1 → /BD → 0 has P1 and /BD as the direct summands of Vf and P1 and rad(P1) as the direct summands of Qf, not as stated. The intended meaning is clear from the sequence itself, but the wording should be corrected.
[Lemma 4.11] In the proof, the expression "IfP2 = cP2 = S′" uses an undefined symbol c; it should be "IfP2 = γP2 = S′".
[Abstract and Introduction] The introduction refers to the finite-global-dimension result as "Theorem 3.15", but in the text the theorem is numbered Theorem 3.14. Please correct the cross-reference.
[Throughout] There are several typos that should be fixed, including "Furtherore" in Lemma 5.5, "together wtih" in Corollary 5.6, and "every every" in the proof of Lemma 6.6.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation detected: the main theorem rests on independent external results and explicit finite computations rather than on self-citation or fitted inputs.

full rationale

The paper's derivation chain is self-contained against external mathematical results. The classical generator V is parahoric induction of Vf = trivial ⊕ P1, an object defined independently of the target equivalence; the progenerator Q comes from Vignéras's Schur algebra theory, and the equality of the categories generated by Q and V is proved through finite-block structure results of Ackermann, explicit endomorphism formulas of Paige, and an explicit finite computation identifying the annihilator generator γ = e2 Z_{q−1}. The scalar λ = q−1 is not fitted to force the conclusion: Lemma 4.13 proves the existence and uniqueness of a λ such that e2 Zλ equals the radical generator γ, and Lemma 4.14 verifies by explicit factorisation that λ = q−1 satisfies the required coefficient cancellations. This is a computation, not a hidden assumption. The lifting step Lemma 5.5 is a long hand calculation, but a calculation that is unverified or compressed is a correctness risk, not circularity: it does not assume the target theorem, and no part of it is justified by a citation to the present author's own prior work. The only cited inputs are independent works: Vignéras, Dat, Deng–Yang, Koenig–Xi, Ackermann, Paige, and Keller. No fitted parameter is renamed as a prediction, no uniqueness theorem from the same authors is imported, and no ansatz is smuggled in via self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The central claim rests on a chain of external structural theorems: Vigeneras's abelian equivalence, Dat's Noetherianity, affine cellularity of Schur algebras, the finite unipotent block structure, Paige's endomorphism calculation, and Keller's dg equivalence. No numerical free parameters are fitted, and no speculative entities are introduced; the objects V, Q, Vf, Qf are standard constructions from parahoric induction or Vigeneras's progenerator, while the dg Schur algebra is a defined construction whose stated properties are proven.

assumptions (10)

domain assumption Bernstein presentation of the Iwahori-Hecke algebra over Z[q^{±1/2}] ([Vig06])
Underlies the definition of the Schur algebra S_q(n) and the base change to R in Section 2.
domain assumption Block decomposition of Mod(G) in characteristic l ([Vig98], [Dat18a], [Chi18])
Used in Section 1 to reduce to the unipotent block B1(G); the paper relies on this external theorem rather than proving it.
domain assumption Vigeneras's main theorem: some power of I annihilates B1(G), and B'_1(G) is equivalent to modules over SR(n) with progenerator Q ([Vig03])
Theorem 2.2 is the starting point; Theorem 6.8 and Lemma 6.7 depend on it.
domain assumption Noetherianity of Mod(G) ([Dat09])
Theorem 2.4 is used in Lemma 6.6 and Lemma 6.7 to conclude H'_1(G) is Noetherian and ideals are finitely generated.
domain assumption Affine cellularity of the two-parameter Schur algebra and its transfer to S_q(N,n), from [Nak15], [Cui15], [DY16], [VV99]
Proposition 3.9 and Theorem 3.12 build the idempotent affine cellular structure of S_q(n) used for Theorem 3.14.
standard math Hilbert Syzygy theorem and localization preserve finite global dimension ([Rot09], [MR01])
Used in Lemma 3.13 to show the rings B_k appearing in the affine cellular structure have finite global dimension and zero radical.
domain assumption Structure of B1(Gf) for n=2, l odd dividing q+1: two simples, Loewy series, P2 multiplicity in the Gelfand-Graev module ([Ack06])
Lemma 4.1 and Corollary 4.5 identify Pf, P1, P2 and give the endomorphism ring relations in Corollary 4.2, which underlie Lemmas 4.10 through 4.16.
domain assumption Explicit description of End(P2) as generated by a central element with tuple (2, ζ^i+ζ^{-i}) ([Pai14]) and character table values from [DM91]
Lemma 4.13 fixes the central element Z_{q−1} and hence the generator γ of the annihilator ideal; an error here would propagate through the main proof.
domain assumption Keller's theorem 3.8(b): an idempotent-complete algebraic triangulated category generated by M is equivalent to per(dg-End(M•)) ([Kel06])
Theorem 6.4 produces Corollary 6.11 from the classical generation result; this is the final bridge to dg perfect complexes.
standard math Parahoric induction is exact and preserves finite generation
Used to lift the finite exact sequence from Corollary 4.12 to the p-adic setting in Lemma 6.9 and to define V and Q.

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Pith. "Pith review of The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra." pith.science (2026). https://pith.science/paper/PXJ3VQXS

@misc{pith2026241117469,
  author       = {Pith},
  title        = {Pith review of: The Derived Unipotent Block of $p$-Adic $\mathrmGL_2$ as Perfect Complexes over a dg Schur Algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXJ3VQXS}},
  note         = {Machine review of arXiv:2411.17469}
}

abstract

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$.

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Works this paper leans on

2 extracted references · 1 linked inside Pith

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Le “centre

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On the decomposition m atrices of the quantized Schur algebra

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Pith tools

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[1] [98]

Le “centre

issn: 0024-6115,1460-244X. doi: 10.1017/S0024611505015443. url: https://doi.org/10.1017/S0024611505015443. [Adl+24a] Jeﬀrey D. Adler et al. Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms . 2024. arXiv: 2408.07805 [math.RT] . url: https://arxiv.org/abs/2408.07805. 34 [Adl+24b] Jeﬀrey D. Adler et al. Structure of Hecke algebra...

arXiv 1999

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On the decomposition m atrices of the quantized Schur algebra

doi: 10.1007/s000290050040. url: https://doi.org/10.1007/s000290050040. [VV99] Michela Varagnolo and Eric Vasserot. “On the decomposition m atrices of the quantized Schur algebra”. In: Duke Math. J. 100.2 (1999), pp. 267–297. issn: 0012-7094,1547-7398. doi: 10.1215/S0012-7094-99-10010-X . url: https://doi.org/10.1215/S0012-7094-99-10010-X . [Yan14] Guiyu ...

doi:10.1007/s000290050040 1999