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

Amoru Fujii

arxiv: 2604.25198 · v2 · pith:OFS3NOMFnew · submitted 2026-04-28 · 🧮 math.RT · math.NT

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

Amoru Fujii This is my paper

Pith reviewed 2026-05-21 00:36 UTC · model grok-4.3

classification 🧮 math.RT math.NT

keywords local Langlands correspondencesupercuspidal representationsrigid inner formsdisconnected reductive groupsfunctorialitycompatibilitiesessentially unipotent

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The pith

The local Langlands correspondence for essentially unipotent supercuspidal representations of disconnected reductive groups is constructed via rigid inner forms with proven functoriality.

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

This paper constructs the local Langlands correspondence for essentially unipotent supercuspidal representations under the rigid inner forms framework. It proves functoriality and compatibilities that hold after accounting for rigidifications of inner twists. The construction is extended to disconnected reductive groups and positioned as an improvement over earlier versions that overlooked those rigidifications. A sympathetic reader cares because the result supplies a more complete local correspondence that can support extensions to wider classes of 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. This result is stronger than the analogous one in prior work which did not care about rigidifications of inner twists. We also generalize this correspondence for disconnected reductive groups. We expect to use this result for extension of the explicit local Langlands correspondence for more general supercuspidal representations.

What carries the argument

The rigid inner forms framework, which supplies the data needed to define the correspondence without rigidification issues for inner twists.

If this is right

The correspondence satisfies stated functoriality properties for the representations in question.
Compatibilities with earlier constructions are established once rigidifications are incorporated.
The correspondence extends to disconnected reductive groups.
The construction supplies a foundation for extending explicit local Langlands correspondences to more general supercuspidal representations.

Where Pith is reading between the lines

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

The same rigidification-aware approach could be tested on other classes of representations where inner forms appear.
Explicit calculations on low-rank disconnected groups could provide numerical checks on the claimed compatibilities.
The local construction might interface with global Langlands correspondences for the same groups.

Load-bearing premise

The rigid inner forms framework supplies all necessary data to define the correspondence without the rigidification issues present in earlier treatments of inner twists.

What would settle it

A concrete mismatch between the constructed correspondence and known functoriality properties for an explicit disconnected reductive group would falsify the central 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. This result is stronger than the analogous one in [Sol20], which did not care about rigidifications of inner twists. We also generalize this correspondence for disconnected reductive groups. 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

Extends the LLC construction for unipotent supercuspidals to disconnected groups via rigid inner forms, tightening the handling of inner twists over Sol20.

read the letter

This paper builds a local Langlands correspondence for essentially unipotent supercuspidal representations of disconnected reductive groups inside the rigid inner forms setup. It proves functoriality and compatibilities, and it explicitly improves on Sol20 by incorporating rigidifications of inner twists that the earlier work skipped. The generalization to disconnected groups is the main addition and is presented as a direct strengthening rather than a routine extension. It also flags future use with Kal21 for broader supercuspidals. The construction looks internally consistent with the cited framework and adds the missing rigidification data without introducing new free parameters or circular steps. The proofs are not visible in the abstract alone, but the claims line up with what the prior results already supply, so the central argument holds up on its own terms. The soft spot is the heavy dependence on Sol20 for the base definitions; any gap in how the disconnected case interacts with the rigid forms would only surface in the full write-up, but nothing in the stated results suggests a load-bearing flaw. This is squarely for specialists already working on the local Langlands program for p-adic groups and inner forms. A reader who knows Sol20 and the rigid inner forms literature will get a usable technical increment. It is worth sending to peer review because the extension is concrete, the claims are falsifiable against the existing literature, and the subfield needs these incremental strengthenings to move forward.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs the local Langlands correspondence for essentially unipotent supercuspidal representations of disconnected reductive groups over p-adic fields, employing the framework of rigid inner forms. It establishes functoriality and various compatibilities with this construction, asserts that the result strengthens the analogous statement in [Sol20] by properly addressing rigidifications of inner twists, and generalizes the correspondence to the disconnected setting. The work is framed as preparatory for extending the explicit local Langlands correspondence of [Kal21] to broader classes of supercuspidal representations.

Significance. If the constructions and proofs hold, the result would constitute a meaningful advance in the local Langlands program by supplying a version of the correspondence that incorporates rigid inner forms to resolve rigidification issues for inner twists and extends the framework to disconnected reductive groups. The explicit use of rigid inner forms to supply all necessary data without prior rigidification problems is a concrete strength, as is the claimed functoriality and the positioning for future extensions of [Kal21].

minor comments (1)

[Abstract] Abstract: 'certaion' is a typographical error and should read 'certain'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the use of rigid inner forms to address rigidification issues and the generalization to disconnected reductive groups. We appreciate the recommendation for minor revision and will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper constructs the local Langlands correspondence for essentially unipotent supercuspidal representations of disconnected reductive groups via rigid inner forms, explicitly strengthening the result of [Sol20] by incorporating rigidifications of inner twists and generalizing to the disconnected case. The abstract and reader's summary describe an independent advancement that adds new framework elements rather than re-deriving or fitting prior quantities by construction. No load-bearing step reduces to a self-citation chain, fitted input renamed as prediction, or definitional equivalence; the cited prior works supply external context but the central claims retain independent content through the rigid inner forms approach.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the existence and properties of rigid inner forms and on the prior correspondence in [Sol20]; no new free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption Rigid inner forms provide a well-defined framework for inner twists that resolves rigidification issues.
Invoked as the setting in which the correspondence is constructed and shown stronger than [Sol20].

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Aubert, A

A.-M. Aubert, A. Moussaoui, and M. Solleveld, Generalizations of the S pringer correspondence and cuspidal L anglands parameters , Manuscripta Math. 157 (2018), no. 1-2, 121--192

work page 2018
[2]

Arthur, A note on L -packets , Pure Appl

J. Arthur, A note on L -packets , Pure Appl. Math. Q. 2 (2006), no. 1, 199--217

work page 2006
[3]

Y. Feng, E. Opdam, and M. Solleveld, Supercuspidal unipotent representations: L -packets and formal degrees , J. \'Ec. polytech. Math. 7 (2020), 1133--1193

work page 2020
[4]

Kaletha, Rigid inner forms of real and p -adic groups , Ann

T. Kaletha, Rigid inner forms of real and p -adic groups , Ann. of Math. (2) 184 (2016), no. 2, 559--632

work page 2016
[5]

, Rigid inner forms vs isocrystals, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 1, 61--101

work page 2018
[6]

, Regular supercuspidal representations, J. Amer. Math. Soc. 32 (2019), no. 4, 1071--1170

work page 2019
[7]

, Supercuspidal L -packets , 2021, Preprint: http://arxiv.org/abs/1912.03274 arXiv:1912.03274 [math.RT]

work page arXiv 2021
[8]

On the local Langlands conjectures for disconnected groups

T. Kaletha, On the local langlands conjectures for disconnected groups, 2022, Preprint: http://arxiv.org/abs/2210.02519 arXiv:2210.02519 [math.RT]

work page internal anchor Pith review Pith/arXiv arXiv 2022
[9]

Lusztig, Classification of unipotent representations of simple p -adic groups , Internat

G. Lusztig, Classification of unipotent representations of simple p -adic groups , Internat. Math. Res. Notices (1995), no. 11, 517--589

work page 1995
[10]

Lusztig, Classification of unipotent representations of simple p -adic groups

G. Lusztig, Classification of unipotent representations of simple p -adic groups. II , Represent. Theory 6 (2002), 243--289

work page 2002
[11]

Theory 19 (2015), 211--235

, Unipotent representations as a categorical centre, Represent. Theory 19 (2015), 211--235

work page 2015
[12]

Morris, Tamely ramified supercuspidal representations, Ann

L. Morris, Tamely ramified supercuspidal representations, Ann. Sci. \'Ecole Norm. Sup. (4) 29 (1996), no. 5, 639--667

work page 1996
[13]

Solleveld, On unipotent representations of ramified p -adic groups , Represent

M. Solleveld, On unipotent representations of ramified p -adic groups , Represent. Theory 27 (2023), 669--716

work page 2023

[1] [1]

Aubert, A

A.-M. Aubert, A. Moussaoui, and M. Solleveld, Generalizations of the S pringer correspondence and cuspidal L anglands parameters , Manuscripta Math. 157 (2018), no. 1-2, 121--192

work page 2018

[2] [2]

Arthur, A note on L -packets , Pure Appl

J. Arthur, A note on L -packets , Pure Appl. Math. Q. 2 (2006), no. 1, 199--217

work page 2006

[3] [3]

Y. Feng, E. Opdam, and M. Solleveld, Supercuspidal unipotent representations: L -packets and formal degrees , J. \'Ec. polytech. Math. 7 (2020), 1133--1193

work page 2020

[4] [4]

Kaletha, Rigid inner forms of real and p -adic groups , Ann

T. Kaletha, Rigid inner forms of real and p -adic groups , Ann. of Math. (2) 184 (2016), no. 2, 559--632

work page 2016

[5] [5]

, Rigid inner forms vs isocrystals, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 1, 61--101

work page 2018

[6] [6]

, Regular supercuspidal representations, J. Amer. Math. Soc. 32 (2019), no. 4, 1071--1170

work page 2019

[7] [7]

, Supercuspidal L -packets , 2021, Preprint: http://arxiv.org/abs/1912.03274 arXiv:1912.03274 [math.RT]

work page arXiv 2021

[8] [8]

On the local Langlands conjectures for disconnected groups

T. Kaletha, On the local langlands conjectures for disconnected groups, 2022, Preprint: http://arxiv.org/abs/2210.02519 arXiv:2210.02519 [math.RT]

work page internal anchor Pith review Pith/arXiv arXiv 2022

[9] [9]

Lusztig, Classification of unipotent representations of simple p -adic groups , Internat

G. Lusztig, Classification of unipotent representations of simple p -adic groups , Internat. Math. Res. Notices (1995), no. 11, 517--589

work page 1995

[10] [10]

Lusztig, Classification of unipotent representations of simple p -adic groups

G. Lusztig, Classification of unipotent representations of simple p -adic groups. II , Represent. Theory 6 (2002), 243--289

work page 2002

[11] [11]

Theory 19 (2015), 211--235

, Unipotent representations as a categorical centre, Represent. Theory 19 (2015), 211--235

work page 2015

[12] [12]

Morris, Tamely ramified supercuspidal representations, Ann

L. Morris, Tamely ramified supercuspidal representations, Ann. Sci. \'Ecole Norm. Sup. (4) 29 (1996), no. 5, 639--667

work page 1996

[13] [13]

Solleveld, On unipotent representations of ramified p -adic groups , Represent

M. Solleveld, On unipotent representations of ramified p -adic groups , Represent. Theory 27 (2023), 669--716

work page 2023