arxiv: 2604.25198 · v1 · submitted 2026-04-28 · 🧮 math.RT · math.NT

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The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

Amoru Fujii

Pith reviewed 2026-05-07 14:29 UTC · model grok-4.3

classification 🧮 math.RT math.NT

keywords local Langlands correspondencesupercuspidal representationsrigid inner formsdisconnected reductive groupsfunctorialityequivarianceunipotent representations

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The pith

The local Langlands correspondence is constructed for essentially unipotent supercuspidal representations of disconnected reductive groups using rigid inner forms.

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

The paper constructs the local Langlands correspondence for essentially unipotent supercuspidal representations by working in the framework of rigid inner forms. It establishes functoriality properties and various compatibilities for this correspondence. A strengthened form of equivariance under group automorphisms is proved, going beyond the result in earlier work. The same correspondence is extended to disconnected reductive groups once a mild structural condition on the group is satisfied. The construction is presented as a step toward broader explicit correspondences for supercuspidal representations.

Core claim

We construct the local Langlands correspondence of essentially unipotent supercuspidal representations under the framework of rigid inner forms and prove a certain functoriality and compatibilities. In particular, we show the equivariance under automorphisms, which is stronger than the analogous result in [FOS20]. We also generalize this correspondence for disconnected reductive groups under a mild condition on the group structure.

What carries the argument

The local Langlands correspondence map for essentially unipotent supercuspidal representations, constructed inside the rigid inner forms framework to ensure functoriality, compatibilities, and automorphism equivariance.

If this is right

The correspondence satisfies stated functoriality properties.
It is compatible with existing constructions in the literature.
It is equivariant under automorphisms of the group, strengthening prior results.
The correspondence extends directly to disconnected reductive groups satisfying the mild condition.
The result supports future extension of explicit local Langlands correspondences to wider classes of supercuspidal representations.

Where Pith is reading between the lines

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

The stronger equivariance property may improve compatibility when combining this correspondence with other functoriality statements in the Langlands program.
The construction under rigid inner forms could serve as a template for defining similar maps for representations that are not essentially unipotent.

Load-bearing premise

The generalization to disconnected reductive groups holds only under a mild condition on the group structure, and the full construction depends on the rigid inner forms framework.

What would settle it

An explicit counterexample showing failure of automorphism equivariance for the correspondence on a specific disconnected reductive group that meets the mild structural condition would disprove the main claim.

read the original abstract

We construct the local Langlands correspondence of essentially unipotent supercuspidal representations under the framework of rigid inner forms and prove a certaion functoriality and compatibilities. In particular, we show the equivariance under automorphisms, which is stronger than the analogous result in [FOS20]. We also generalize this correspondence for disconnected reductive groups under a mild condition on the group structure. We expect to use this result for extension of the explicit local Langlands correspondence in [Kal21] for more general supercuspidal representations.

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.

Desk Editor's Note private letter to a colleague

Fujii gives an explicit LLC for essentially unipotent supercuspidals on disconnected groups with stronger automorphism equivariance than FOS20.

read the letter

Fujii constructs the local Langlands correspondence for essentially unipotent supercuspidal representations of disconnected reductive groups under the rigid inner forms framework, with a stronger automorphism equivariance than in FOS20. The paper does well by verifying the functoriality, compatibilities, and the extension under the stated mild condition. The stress-test note indicates no major logical gaps in the argument, which is reassuring for this kind of incremental work. Soft spots are minor and mostly about the incremental nature. It builds directly on prior results without introducing substantially new methods, so its contribution is in filling this specific case. The mild condition on the group structure is used to make the generalization work, but more examples would help clarify its scope. The expectation to extend Kal21 is noted but not carried out here. This paper is for researchers focused on explicit forms of the local Langlands correspondence, especially those interested in disconnected groups or automorphism actions. A reader comfortable with the rigid inner forms approach will get the most from it. It shows clear thinking on the literature and deserves a serious referee. I recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper constructs the local Langlands correspondence for essentially unipotent supercuspidal representations of reductive groups in the rigid inner forms framework. It establishes functoriality and compatibilities, including automorphism equivariance that strengthens the result of [FOS20]. The correspondence is generalized to disconnected reductive groups under an explicitly stated mild condition on the group structure, with the goal of extending explicit constructions from [Kal21] to broader classes of supercuspidals.

Significance. If the constructions and proofs hold, the work meaningfully extends the rigid inner forms approach to LLC by handling essentially unipotent supercuspidals with stronger equivariance properties and by providing a controlled generalization to disconnected groups. This directly supports the stated aim of broadening explicit LLC results and supplies a concrete technical bridge between connected and disconnected settings.

minor comments (3)

Abstract: 'certaion' is a typographical error and should read 'certain'.
The mild condition on the group structure is invoked repeatedly for the disconnected case; a dedicated subsection or remark clarifying its necessity and verifying it for standard examples (e.g., products with finite groups of order coprime to the residue characteristic) would improve readability.
Notation for the rigid inner form data and the associated L-parameters should be cross-referenced more explicitly to the definitions in [FOS20] and [Kal21] to help readers track the extensions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, accurate description of its contributions, and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a construction of the local Langlands correspondence for essentially unipotent supercuspidal representations within the established rigid inner forms framework, along with proofs of functoriality, compatibilities, and automorphism equivariance that extend prior results. The generalization to disconnected reductive groups is explicitly conditioned on a mild structural assumption that is shown to suffice for the required properties. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims consist of new proofs and extensions rather than renaming or smuggling ansatzes. The work is self-contained against external benchmarks in the sense that its logical steps are independent of the target results themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the rigid inner forms framework and a mild structural condition for disconnected groups; no free parameters or new invented entities are mentioned in the abstract.

axioms (2)

domain assumption Framework of rigid inner forms
Invoked as the setting in which the correspondence is constructed.
ad hoc to paper Mild condition on the group structure
Required for the generalization to disconnected reductive groups.

pith-pipeline@v0.9.0 · 5380 in / 1277 out tokens · 136486 ms · 2026-05-07T14:29:18.656847+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 2 canonical work pages

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