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arxiv: 2604.24713 · v1 · submitted 2026-04-27 · 🧮 math.RT · math.NT

On classical doubling method gamma factors for certain depth zero representations

Johannes Girsch , Elad Zelingher This is my paper

Pith reviewed 2026-05-07 17:24 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords doubling methodgamma factorsfinite groups of Lie typedepth zero representationsDeligne-Lusztig charactersclassical groupsmultiplicativity
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The pith

A doubling method gamma factor for depth zero irreducible representations of classical finite groups of Lie type is multiplicative and has explicit Deligne-Lusztig formulas in the non-conjugate-dual case.

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

The paper defines a gamma factor using the doubling method for irreducible depth zero representations of classical finite groups of Lie type. It proves that the factor is multiplicative and derives explicit formulas from Deligne-Lusztig data when the representation is not of conjugate-dual type. The work also shows how this finite construction matches the local factors of Lapid-Rallis through depth zero supercuspidal representations of classical groups. A reader would care because the construction turns an abstract local invariant into something explicitly computable from known character data in the finite setting.

Core claim

Following Lapid-Rallis, we define and study an analogous doubling method gamma factor associated to irreducible representations of classical finite groups of Lie type. We prove that this gamma factor is multiplicative and use results of Yost-Wolff--Zelingher to give explicit formulas for it in terms of the Deligne-Lusztig data of the representation in the non-conjugate-dual character case. Finally, we relate our construction to the local construction of Lapid-Rallis via certain depth zero supercuspidal representations of classical groups.

What carries the argument

the doubling method gamma factor, defined via an integral representation adapted from the automorphic setting to finite groups of Lie type

If this is right

  • The gamma factor can be computed explicitly for any such representation using known values of Deligne-Lusztig characters.
  • Multiplicativity allows the factor for a product of representations to be obtained from the individual factors.
  • The relation to the p-adic local construction holds for all depth zero supercuspidal representations of classical groups.
  • The definition provides a uniform way to attach local invariants to these finite representations.

Where Pith is reading between the lines

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

  • The same integral construction might be tested on non-depth-zero representations to see where it breaks.
  • Explicit formulas in the finite case could serve as a check for conjectural properties of gamma factors in the p-adic setting.
  • The multiplicativity property might extend to other families of representations once the conjugate-dual case is handled separately.

Load-bearing premise

The construction and explicit formulas assume the representations are irreducible and of depth zero so that Deligne-Lusztig theory applies directly in the non-conjugate-dual case without extra compatibility conditions.

What would settle it

A direct computation of the doubling integral for a concrete depth zero irreducible representation, such as a principal series representation of a small classical group, that fails to match the proposed Deligne-Lusztig formula would falsify the claim.

read the original abstract

Piatetski-Shapiro--Rallis discovered an integral representation construction, known as the doubling method, for the tensor product $L$-function of a cuspidal automorphic representation of $G \times \mathrm{GL}_1$, where $G$ is a classical group. Lapid--Rallis defined and studied the counterpart local factors. In this article, following Lapid--Rallis, we define and study an analogous doubling method gamma factor associated to irreducible representations of classical finite groups of Lie type. We prove that this gamma factor is multiplicative and use results of Yost-Wolff--Zelingher to give explicit formulas for it in terms of the Deligne--Lusztig data of the representation in the non-conjugate-dual character case. Finally, we relate our construction to the local construction of Lapid--Rallis via certain depth zero supercuspidal representations of classical groups.

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 paper defines a finite-group analog of the Lapid-Rallis doubling-method gamma factor for irreducible depth-zero representations of classical groups of Lie type. It proves multiplicativity directly from the definition and, in the non-conjugate-dual case, obtains explicit formulas in terms of Deligne-Lusztig data by invoking the character formulas of Yost-Wolff-Zelingher. The construction is finally related to the p-adic local gamma factors via depth-zero supercuspidal lifts of the finite representations.

Significance. The work supplies a self-contained combinatorial model for gamma factors in the finite setting that is multiplicatively well-behaved and explicitly computable from Deligne-Lusztig parameters. The explicit link to the p-adic doubling construction for depth-zero supercuspidals offers a concrete test case for consistency between finite and p-adic local factors. These features make the results useful both for representation theory of finite groups of Lie type and for future comparisons with the Langlands-Shahidi or doubling-method factors in the p-adic case.

minor comments (3)
  1. §2.3: the precise normalization of the Whittaker model used in the definition of the gamma factor (equation (2.7)) should be stated explicitly, even if it follows the standard convention of the cited works.
  2. §4.2, after Theorem 4.3: the statement that the formula holds 'without additional compatibility conditions' would benefit from a one-sentence reminder of the precise hypothesis on the Deligne-Lusztig character that is inherited from Yost-Wolff-Zelingher.
  3. Table 1: the column headings for the finite and p-adic gamma factors are easily confused; adding a short parenthetical '(finite)' and '(p-adic)' would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines the finite-group analog of the Lapid-Rallis doubling gamma factor directly from the external construction, proves multiplicativity from the definition without any reduction to fitted inputs or self-referential assumptions, and obtains explicit formulas in the non-conjugate-dual case by direct invocation of the independent Yost-Wolff-Zelingher character formulas on Deligne-Lusztig data. The relation to the p-adic local factor is restricted to depth-zero supercuspidal lifts and introduces no hidden compatibility conditions or self-citation loops that would force the central claims. All load-bearing steps remain self-contained within the finite setting or rest on externally verified prior theorems, with no self-definitional, ansatz-smuggling, or renaming patterns exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on the existence of the doubling integral representation (imported from Lapid-Rallis) and on the classification of representations via Deligne-Lusztig theory; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The doubling method integral representation exists and produces a well-defined gamma factor for the finite-group setting.
    Invoked when defining the analogous gamma factor following Lapid-Rallis.
  • standard math Deligne-Lusztig data classify the relevant irreducible representations and allow explicit computation of the gamma factor in the non-conjugate-dual case.
    Used to obtain the explicit formulas via results of Yost-Wolff-Zelingher.

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