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arxiv: 2606.28091 · v1 · pith:6A7COGLGnew · submitted 2026-06-26 · 🧮 math.RT · math.NT

Stability of the exterior cube γ-factors for GL(6)

Taiwang Deng , Dongming She This is my paper

Pith reviewed 2026-06-29 01:50 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords Langlands-Shahidi methodexterior cubegamma factorGL(6)stabilityE6 groupBessel integralslocal factors
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The pith

If two generic representations of GL(6) share the same central character, their exterior cube γ-factors become identical after twisting by any sufficiently ramified character.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes stability of the Langlands-Shahidi local γ-factor attached to the exterior cube representation of GL(6). For irreducible admissible generic representations π1 and π2 of GL6(F) with identical central characters, the equality γ(s, π1 ⊗ χ, ∧³, ψ) = γ(s, π2 ⊗ χ, ∧³, ψ) holds once χ is sufficiently ramified. The argument embeds the exterior cube into the Langlands-Shahidi setup via a maximal parabolic subgroup of the simply-connected E6 group. This reduces the γ-factor to partial Bessel integrals on the Levi subgroup. Asymptotic expansion of those integrals together with vanishing of highly ramified Mellin transforms then forces the stability.

Core claim

We prove that the Langlands-Shahidi γ-factor for the exterior cube representation of GL6 is stable: if π1 and π2 are irreducible admissible generic representations of GL6(F) with the same central character, then γ(s,π1⊗χ,∧³,ψ) equals γ(s,π2⊗χ,∧³,ψ) for all sufficiently ramified characters χ of F×. The argument realizes the exterior cube via the maximal parabolic of the E6 group, describes the geometric quotient U_M ackslash N', computes its invariant measure, relates Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup, and deduces stability from an asymptotic expansion together with the vanishing of highly ramified Mellin transforms.

What carries the argument

Realization of the exterior cube representation via the maximal parabolic subgroup of the simply-connected E6 group, which reduces the γ-factor to partial Bessel integrals on the Levi subgroup whose asymptotics and Mellin vanishing properties establish stability.

If this is right

  • The exterior cube γ-factor depends only on the central character once the twisting character is sufficiently ramified.
  • The γ-factor can be recovered from its values on representations with fixed central character in the stable range.
  • Stability supplies an invariance property that is compatible with the expected local Langlands correspondence for the exterior cube.
  • The method extends the list of representations for which Shahidi's γ-factors are known to be stable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction to partial Bessel integrals on a Levi subgroup may apply to other exceptional-group realizations of classical representations.
  • Global automorphic L-functions built from these local γ-factors should inherit multiplicity-one or uniqueness properties when the local factors are stable.
  • The explicit description of the quotient U_M ackslash N' and its measure could be reused to study other integral representations attached to the E6 parabolic.

Load-bearing premise

The exterior cube representation of GL(6) arises as the adjoint action on the unipotent radical of a maximal parabolic subgroup inside the simply-connected E6 group.

What would settle it

Explicit calculation of the exterior cube γ-factor for two non-isomorphic generic representations of GL(6) that share the same central character, twisted by one fixed highly ramified χ, yielding unequal values.

read the original abstract

We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the stability of the Langlands-Shahidi local γ-factor attached to the exterior cube representation of GL_6. Precisely, if π1 and π2 are irreducible admissible generic representations of GL_6(F) with the same central character, then γ(s, π1 ⊗ χ, ∧³, ψ) = γ(s, π2 ⊗ χ, ∧³, ψ) for every sufficiently ramified character χ of F^× (viewed via det). The argument realizes the exterior cube via the maximal parabolic subgroup of the simply-connected E6 group, gives an explicit description of the geometric quotient U_M ackslash N', computes the invariant measure, relates Shahidi's partial Bessel functions to partial Bessel integrals on the Levi, and obtains the stability statement from an asymptotic expansion of those integrals together with the vanishing of highly ramified Mellin transforms.

Significance. If the central computations hold, the result supplies the missing stability statement for the ∧³ γ-factor on GL_6, a necessary ingredient for the local Langlands correspondence, the exterior-cube functoriality, and the comparison of L-functions in this case. The explicit geometric realization via E6 and the reduction to partial Bessel integrals on the Levi constitute a concrete, checkable implementation of the standard Langlands-Shahidi template.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the main theorem should explicitly record the non-archimedean local field F and the additive character ψ at the outset, rather than deferring these to later sections.
  2. The description of the quotient U_M ackslash N' and the invariant measure (mentioned in the abstract and presumably in §3) would benefit from a short table or diagram summarizing the root-space decomposition used to compute the measure.
  3. Notation for the partial Bessel integrals on the Levi (introduced after the geometric quotient) should be cross-referenced to the corresponding objects in Shahidi's earlier papers to aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments are provided in the report, so we have no points to address point-by-point. The manuscript stands as submitted, with the stability result for the exterior cube γ-factor on GL(6) obtained via the E6 parabolic realization, explicit quotient, and asymptotic analysis of partial Bessel integrals.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external E6 embedding and analytic properties

full rationale

The paper's chain proceeds from the external geometric realization of the exterior cube via the maximal parabolic in the simply-connected E6 group, followed by explicit computation of the quotient U_M \ N', invariant measure, reduction to partial Bessel integrals on the Levi, asymptotic expansion of those integrals, and vanishing of highly ramified Mellin transforms. None of these steps is defined in terms of the target γ-factor, fitted to it, or justified solely by self-citation. The stability statement for sufficiently ramified χ follows directly from these independent properties of integrals and representations, without reduction by construction to the input data or prior results by the same authors. This matches the default case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of Langlands-Shahidi γ-factors and the domain-specific assumption that the exterior cube embeds into E6; no free parameters or new entities are introduced.

axioms (2)
  • standard math Langlands-Shahidi method defines local γ-factors via constant terms of Eisenstein series or intertwining operators for generic representations.
    Invoked as the source of the γ-factor whose stability is proved.
  • domain assumption The exterior cube representation of GL6 arises from the maximal parabolic subgroup of the simply connected E6 group.
    Central technical step stated in the abstract as the starting point of the proof.

pith-pipeline@v0.9.1-grok · 5733 in / 1494 out tokens · 55131 ms · 2026-06-29T01:50:31.741455+00:00 · methodology

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Reference graph

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