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On sharpness in Local Converse Theorems for classical groups and $G_2$

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For symplectic, orthogonal, and $G_2$ groups with large residual characteristic, the optimal local converse theorem requires twisting through half the dimension of the standard representation of the dual group.

desk verdict A strong sharpness paper for local converse theorems in classical groups and G2, with one overreach in the GL4/Ramakrishnan section that should be reframed. read the letter →

arxiv 2509.22390 v2 pith:M6IQJWYY submitted 2025-09-26 math.RT

Moshe Adrian , Shaun Stevens This is my paper · ORCID
classification math.RT MSC 11S7022E5011F8522E55
keywords localconverseproblemtwistedgammafactorsadmissiblepairssupercuspidalrepresentationsclassicalgroupsG2sharpnessLanglandsparameters
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 determines, for large residual characteristic, the exact number of $GL_r$-twists needed to distinguish generic representations of split classical groups and of the exceptional group $G_2$ by their twisted $\gamma$-factors. It shows that for $G_N=Sp_{2N},SO_{2N},SO_{2N+1}$ with $p>N$, the standard local converse theorem is optimal: twisting by $GL_r$ through $r=N$ (half the dimension of the standard representation of the dual group) is necessary, already among generic tempered non-cuspidal representations. For generic supercuspidal representations the required level drops by one when $G=SO_{2N}$ with $N$ odd, and the paper conjectures the same drop for symplectic and odd orthogonal groups. For $G_2$ with $p>3$, level 3 is necessary and sufficient, and level 2 fails even if both fundamental representations of the dual group are used. A separate construction for $GL_4$ disproves the idea that character twists together with all exterior-power $\gamma$-factors can separate supercuspidal representations.

What carries the argument

The engine is Proposition 2.10, a $\gamma$-factor comparison criterion for direct sums $\bigoplus_i \rho_{\chi_i}$ of Weil-group representations induced from totally ramified admissible pairs. Admissible pairs $(E/F,\chi)$ are tame objects: a tamely ramified extension $E/F$ together with a quasi-character $\chi$ of $E^\times$ not inflated from a proper subfield, and their induced representations exhaust the tame irreducible Weil-group representations (Theorem 2.2). Proposition 2.10 says that equality of all twisted $\gamma$-factors against representations of dimension less than $r$ follows from three conditions: the characters $\chi_i,\chi_i'$ agree on the filtration subgroup $U^{t_i}_{E_i}$; the products $\prod_i \chi_i(\beta_i)$ and $\prod_i \chi_i'(\beta_i)$ agree; and the products $\prod_i \chi_i$ and $\prod_i \chi_i'$ agree on $F^\times$. The proof reduces to a product formula for $\gamma$-factors of tensor products (Proposition 2.7) and a Gauss-sum computation; the sharpness examples are precisely choices of admissible pairs that satisfy these conditions while remaining inequivalent. A second mechanism is Lemma 2.12, which produces at least $p$ self-dual admissible characters of either parity, needed for the supercuspidal counterexamples.

What would settle it

Check the explicit pairs of Theorem 3.8 at level $M-1$: if any irreducible twist $\tau$ of dimension $M-1$ produced unequal gamma-factors, the criterion of Proposition 2.10 would be violated and the sharpness claim would collapse. For the conjectural odd-$N$ improvement, find self-dual supercuspidal representations of $GL_{2N}(F)$ of the same parity that are $\gamma$-equivalent to level $N-1$ but not isomorphic; such a pair would refute Conjecture 3.10 and the conjectured converse for symplectic and odd orthogonal groups.

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Extended reading notes

Core claim

The central discovery is a family of sharpness counterexamples built from minimal totally ramified admissible pairs $(E/F,\chi)$ and $(E/F,\chi')$ that agree on $U_E$ but differ by a sign on a uniformizer. Under the condition $p>N$, Proposition 2.10 gives equality of all standard twisted $\gamma$-factors against every irreducible twist of dimension below $N$, while the two parameters are not outer conjugate. These pairs produce generic tempered counterexamples at level $N-1$ and, after passing through self-dual pairs constructed by a fixed-point correspondence for pro-$p$ characters, supercuspidal counterexamples at level $2\lfloor N/2\rfloor-1$ (Corollaries 3.6 and 3.9). For $SO_{2N}$ with odd $N$, a determinant obstruction rules out the last potential irreducible $2N$-dimensional orthogonal component, yielding the unconditional level $N-1$ converse for supercuspidals (Theorem 3.16). For $G_2$, cubic admissible pairs pulled back through $SL_3\subset G_2$ give inequivalent discrete parameters that agree at levels 1 and 2 against both the 7-dimensional standard and 14-dimensional adjoint representations, while Theorem 4.2 makes level 3 sufficient. The $GL_4$ counterexample takes a totally ramified quartic admissible character and its twist by the unramified quadratic character; these are inequivalent, yet every exterior-power twist by every character agrees.

Load-bearing premise

Everything rests on having residual characteristic $p$ larger than the twisting level ($p>N$, or $p>3$ for $G_2$), so that every irreducible representation of dimension below $r$ is induced from an admissible pair; if that tameness hypothesis fails, the constructed counterexamples are not known to exist.

Editorial extensions

If this is right

  • For $p>N$, the standard local converse theorem for $Sp_{2N}$, $SO_{2N}$, and $SO_{2N+1}$ cannot be improved: the bound $r\le N$ is necessary, so the existing theorems are optimal in the setting of generic tempered representations.
  • For generic supercuspidal representations, the $SO_{2N}$ case with odd $N$ is settled at level $N-1$; if Conjecture 3.10 holds, all classical groups attain the same improved bound and Corollary 3.9 becomes optimal.
  • For $G_2$ with $p>3$, a local converse theorem must twist by $GL_3$; the counterexample at level 2 works with both fundamental representations, so including the adjoint representation does not rescue a level-2 theorem.
  • The $SO_6$ construction makes $SO_{2N}(C)$ generically $WD_F$-unacceptable for $q\equiv 3\pmod 4$: some outer-equivalent parameters have identical twisted $\gamma$-factors for every algebraic representation of the dual group, so for even orthogonal groups no local converse theorem based on all dual-group representations can distinguish such parameters.
  • The $GL_4$ example rules out the proposed shortcut of using only character twists with all exterior-power $\gamma$-factors to separate supercuspidal representations, even before the same question is considered for $GL_N$ with $N\ge 5$.

Reading between the lines

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

  • One could test whether the same unramified-quadratic-twist construction produces $GL_{2m}$ counterexamples for every even $2m$, not just $GL_4$; if so, character-twist-only converse theorems fail in all even ranks.
  • The determinant obstruction behind Theorem 3.16 suggests a general principle: whenever the relevant self-dual Weil-group representations are forced to have nontrivial determinant, the odd-$N$ improvement should hold; this may transfer to other groups whose duals have similar parity restrictions.
  • The sharpness examples are constructed at the level of Langlands parameters; with currently available local Langlands correspondences for other groups, the same admissible-pair technique could plausibly produce analogous counterexamples for classical groups in positive characteristic.
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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

2 major / 3 minor

Summary. The paper studies sharpness in local converse theorems for split classical groups and G2 over non-archimedean local fields of characteristic 0. The main technical engine is a pair of propositions (Propositions 2.7 and 2.10), proved from Moy's formulas, giving criteria for two direct sums of tame supercuspidal Weil-group representations to have equal twisted gamma factors against all representations of dimension below a given bound. Using these criteria, the authors construct generic tempered non-cuspidal representations of Sp_{2N}, SO_{2N}, SO_{2N+1} that are gamma-equivalent to level N-1 but not outer conjugate, and supercuspidal examples at level M-1. For odd N they prove an unconditional improved bound for SO_{2N} and reduce the symplectic and odd orthogonal cases to Conjecture 3.10. For G2 they prove a local converse theorem at level 3 and show sharpness at level 2 even when both fundamental representations of the dual group are used. The final section gives Galois-side parameters for GL_4 whose exterior-power gamma factors agree under all character twists, and the authors interpret this as disproving a conjecture of Ramakrishnan and Ye--Zelingher.

Significance. If the main results stand, the paper gives a substantial and essentially complete answer to the sharpness question for standard local converse theorems in the tame setting, including the first such results for G2. The strengths of the paper are the self-contained proof of the key technical propositions from Moy's Gauss-sum formulas, the explicit and checkable constructions, and the honest separation of proved results from Conjecture 3.10. The authors also correctly note the conditional nature of the improved bound for symplectic and odd orthogonal groups. However, the headline claim in Section 5 goes beyond what is logically established, and the proof of Corollary 3.18 has a gap concerning local conjugacy. These issues do not appear to affect the central classical-group and G2 sharpness theorems, but they do require correction before the paper can be accepted.

major comments (2)
  1. [§5, Theorem 1.7 and Eq. (5.1)] Theorem 1.7 is presented as a disproof of the Ramakrishnan/Ye–Zelingher conjecture, but the equality proved in (5.1) is for Galois-side exterior-power gamma factors γ(s,(∧^i φ_j)⊗η,ψ). The conjecture concerns automorphic exterior-power gamma factors for representations of GL_N(F). The paper does not cite or prove a compatibility theorem asserting that the local Langlands correspondence for GL_4 preserves these exterior-power gamma factors for i≥2; hence the constructed parameters need not contradict the automorphic conjecture as stated. The caveat in §5 that the factors are interpreted on the Galois side is not carried into Theorem 1.7 or the abstract. Please restrict the theorem to the Galois-side/parameter-level statement, or supply the missing compatibility argument.
  2. [§3.4, Corollary 3.18] The proof of Corollary 3.18 shows that the inflated parameters φ'_1 and φ'_2 are outer equivalent and not conjugate, but it does not verify the defining condition of local conjugacy, namely equality after composing with every algebraic representation of SO_{2N}(C). In particular, the half-spin representations V_N^± are not discussed. Example 3.17 supplies the needed half-spin equality only for SO_6, and the inflation step from SO_6 to SO_{2N} is not shown to preserve it. Please add the missing argument, or cite a lemma from [Yu22] that establishes local conjugacy directly.
minor comments (3)
  1. [§4.1, Definition 4.1] The phrase 'all irreducible irreducibler-dimensional representations' should read 'all irreducible r-dimensional representations'.
  2. [§5, first paragraph] The sentence 'for N≥6 the requisite γ-functions have not yet been defined on the automorphic side' is confusing immediately before a theorem about N=4; please clarify that the Galois-side interpretation is needed for the exterior powers under consideration even when N=4.
  3. [References] The reference [KT] is listed as a URL without a year; please add publication details if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sharpness machinery is proved in-paper from Moy's formulas and external LLC inputs.

full rationale

The derivation chain is not circular. The engine, Proposition 2.10, is proved internally: it reduces gamma-factor equality to the conditions in the proposition via Proposition 2.7, whose proof uses Moy's Proposition 2.4 and an explicit double-coset calculation; the citations to [Adr23] and [ALST18] are for techniques and identities, not for the target conclusions. The sharpness examples (Corollaries 3.6, 3.9, Theorem 4.6) verify the hypotheses of Proposition 2.10 and then use the known LLC (Arthur, Gan-Savin, etc.) to pass from parameters to representations; the LLC and external converse theorems are not being replaced by the paper's own conclusions. Theorem 3.13 is explicitly conditional on Conjecture 3.10, which the paper flags as having no evidence. Section 5 carefully changes the interpretation of exterior-power gamma factors to the Galois side for N≥6 and computes at the parameter level; this is a scope caveat, not a circular definition or a fitted-input prediction. No parameter is fitted and renamed as a prediction, and no uniqueness conclusion is imported solely from the authors' prior work. The strongest caveat is the Section 5 Galois/automorphic matching, which is a correctness and justification issue outside the scope of circularity.

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

The paper's main results rest on prior structural results (LLC, transfer, tame parametrization) and on the technical Proposition 2.10, which is proved in the text. The only unproved input is Conjecture 3.10, which is explicitly labeled and used only for a conditional result. No new entities are introduced beyond constructed examples.

assumptions (5)
  • domain assumption Moy's tame parametrization (Theorem 2.2): for p not dividing N, every irreducible N-dimensional representation of W_F is induced from an admissible pair (E/F, chi).
    Used in Proposition 2.10 and in the construction of all sharpness examples; it limits the results to large residual characteristic.
  • domain assumption Arthur's local Langlands correspondence for split classical groups, with unicity of generic representations in tempered L-packets (Varma).
    Invoked in Section 3 (before Theorem 3.2 and in Lemma 3.1) to map representations of G_N(F) to parameters and to transfer to GL_{N*}(F).
  • domain assumption Gan-Savin local Langlands correspondence and gamma-factor definition for G2.
    Section 4 defines gamma factors via Langlands parameters because no direct automorphic construction exists yet; used in Theorems 4.2 and 4.6.
  • ad hoc to paper Conjecture 3.10: for p > N, N odd, self-dual supercuspidal representations of GL_{2N}(F) of the same parity that are gamma-equivalent to level N-1 are isomorphic.
    Unproved; needed for Theorem 3.13, the conjectural improved bound for Sp and SO_{2N+1} supercuspidals. Remark 3.11 explicitly says there is no evidence.
  • domain assumption The CKPSS transfer to GL_{N*}(F) for generic supercuspidal representations of G_N, coinciding with Arthur transfer (Henniart), preserves twisted gamma factors.
    Used in Theorem 3.2 and Lemma 3.1 to reduce to the local converse theorem for GL_{N*}(F).

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Pith. "Pith review of On sharpness in Local Converse Theorems for classical groups and $G_2$." pith.science (2026). https://pith.science/paper/M6IQJWYY

@misc{pith2026250922390,
  author       = {Pith},
  title        = {Pith review of: On sharpness in Local Converse Theorems for classical groups and $G_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6IQJWYY}},
  note         = {Machine review of arXiv:2509.22390}
}
abstract

We prove various results about the Local Converse Problem for split reductive groups $G$ over a non-archimedean local field~$F$ of characteristic $0$ and residual characteristic $p$. In particular, we prove that when $G$ is a symplectic or special orthogonal group, or the exceptional group $G_2$, and $p$ is large enough, then the optimal standard Local Converse Theorem for $G(F)$ requires twisting by representations of $GL_r(F)$ with $r$ up to half the dimension of the standard representation of the dual group of $G$. However, if we restrict to generic supercuspidal representations of $G(F)$ then it can be improved when $G=SO_{2N}$; we conjecture that the same is true for symplectic and odd special orthogonal groups. We also consider the possibility of using non-standard representations of the dual group to distinguish representations, giving counterexamples to possible improvements for general linear groups, $G_2$ and $SO_{2N}$.

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