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arxiv: 2605.05201 · v2 · submitted 2026-05-06 · 🧮 math.RT

A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

Manish Mishra This is my paper

Pith reviewed 2026-05-15 07:23 UTC · model grok-4.3

classification 🧮 math.RT MSC 22E5011S37
keywords local Langlands correspondencedepth-zero supercuspidal representationspinned splittingenhanced parametersreductive groupsparahoric quotientsnon-archimedean local fields
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The pith

Fixing a pinned splitting of the quasi-split inner form produces a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters.

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

This paper constructs an explicit local Langlands correspondence for the depth-zero supercuspidal representations of connected reductive groups over non-archimedean local fields. It decomposes each representation into its tame toral piece, finite cuspidal piece on a parahoric quotient, and unipotent piece, then matches them separately to the parameter side using torus Langlands maps, pinned Jordan decompositions that retain Clifford data, and the Feng-Opdam-Solleveld unipotent correspondence. A sympathetic reader cares because the result supplies a normalization-dependent, reversible bijection that respects natural compatibilities such as weakly unramified twists and central characters. Under the DeBacker-Reeder logarithm hypothesis the attached packets have stable dimension-weighted distributions.

Core claim

After fixing a pinned splitting of the quasi-split inner form, there is a canonical bijection between the irreducible depth-zero supercuspidal representations of a connected reductive group over a non-archimedean local field and the relevant cuspidal enhanced depth-zero Langlands parameters. The forward map attaches to a depth-zero type its enhanced parameter by combining the local Langlands correspondence for maximally unramified elliptic tori, a pinned Jordan decomposition for the finite cuspidal data on possibly disconnected quotients, and the known correspondence for supercuspidal unipotent representations; the inverse map reverses the same steps.

What carries the argument

The pinned splitting of the quasi-split inner form, which normalizes L-embeddings for the toral part and supplies the pinning for the Jordan decomposition on finite reductive quotients.

Load-bearing premise

The construction requires a fixed pinned splitting of the quasi-split inner form; without it the normalization that makes the bijection canonical is undefined.

What would settle it

A concrete depth-zero supercuspidal representation of a group such as GL(2) or a classical group over a p-adic field whose forward image under the construction fails to recover the original representation when the inverse map is applied would disprove the claimed bijection.

read the original abstract

We construct a pinning-normalized local Langlands correspondence for depth-zero supercuspidal representations of a connected reductive group over a non-archimedean local field. After fixing a pinned splitting of the quasi-split inner form, we obtain a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction is organized around the two pieces naturally present in a depth-zero type: a tame toral part and a finite cuspidal representation of a parahoric quotient. The toral part is matched using the local Langlands correspondence for maximally unramified elliptic tori and normalized \(L\)-embeddings. The finite cuspidal part is compared with the parameter side by a pinned Jordan decomposition for the relevant finite reductive quotients. Since these quotients may be disconnected, the finite comparison must retain the Clifford-theoretic data that records the possible extension ambiguity. On the connected unipotent part we use the correspondence of Feng--Opdam--Solleveld for supercuspidal unipotent representations. Combining the toral, unipotent, and Clifford-theoretic pieces gives the enhanced parameter attached to a depth-zero supercuspidal representation, and the inverse map is obtained by reversing the same construction. The correspondence is canonical relative to the fixed pinned normalization. It is compatible with the tame inertial parameter attached to the depth-zero character, with weakly unramified twists, and with central characters via the torus correspondence. Under the DeBacker--Reeder logarithm hypothesis, the dimension-weighted packet distributions attached to the resulting packets are stable.

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 constructs a pinning-normalized local Langlands correspondence for depth-zero supercuspidal representations of connected reductive groups over non-archimedean local fields. After fixing a pinned splitting of the quasi-split inner form, it obtains a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction decomposes each such representation into a tame toral character, a finite cuspidal representation of the parahoric quotient (retaining Clifford data), and an unipotent supercuspidal factor; these are matched separately via the LLC for unramified elliptic tori, pinned Jordan decomposition on finite quotients, and the Feng-Opdam-Solleveld correspondence, with the inverse obtained by reversing the steps. Compatibilities with inertial parameters, unramified twists, and central characters are stated to follow from the component maps, while stability of packet distributions is conditional on the DeBacker-Reeder logarithm hypothesis.

Significance. If the bijection is rigorously established, the result supplies a canonical, pinning-normalized matching in the depth-zero supercuspidal case by coherently assembling independent prior correspondences for tori and unipotent representations while retaining Clifford-theoretic data for possibly disconnected parahoric quotients. The explicit construction and stated compatibilities advance the local Langlands program in this setting; the conditional stability statement is clearly separated from the bijection itself.

minor comments (3)
  1. [Abstract] The abstract introduces 'relevant cuspidal enhanced depth-zero Langlands parameters' without a self-contained definition or forward reference; a one-sentence gloss in the introduction would aid readers.
  2. [§2 (Toral matching)] The precise normalization of the L-embeddings for the toral part (mentioned in the abstract) should be stated explicitly in the first section where the toral matching is defined, rather than deferred.
  3. [§4 (Finite cuspidal comparison)] Notation for the Clifford-theoretic extension data on the finite quotient side is introduced but its compatibility with the enhanced parameter on the Langlands side could be recorded in a short table or diagram for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of the pinning-normalized construction and the recognition that the conditional stability statement is clearly separated from the bijection. We are pleased that the significance for the local Langlands program is acknowledged. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central bijection is assembled by composing three independent prior results: the LLC for unramified elliptic tori, the pinned Jordan decomposition on finite reductive quotients (with Clifford data retained for disconnectedness), and the Feng-Opdam-Solleveld correspondence for unipotent supercuspidals. These are applied componentwise to the tame toral part, finite cuspidal part, and unipotent part of a depth-zero type, then inverted by reversing the same steps. The fixed pinned splitting supplies only a normalization convention and does not enter the definition of the maps themselves. No equation or construction inside the paper reduces the new correspondence to a fitted quantity or to a self-citation chain; the DeBacker-Reeder hypothesis is invoked only for the optional stability statement and is not required for the bijection. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central construction rests on three established prior correspondences used as black boxes; no new free parameters or invented entities are introduced.

axioms (3)
  • domain assumption Local Langlands correspondence for maximally unramified elliptic tori with normalized L-embeddings
    Invoked to match the tame toral part of the depth-zero type.
  • domain assumption Feng-Opdam-Solleveld correspondence for supercuspidal unipotent representations
    Used for the connected unipotent part of the finite reductive quotient.
  • domain assumption Pinned Jordan decomposition for finite reductive quotients that may be disconnected
    Required to compare the finite cuspidal representation while retaining Clifford-theoretic extension data.

pith-pipeline@v0.9.0 · 5591 in / 1485 out tokens · 78687 ms · 2026-05-15T07:23:20.501450+00:00 · methodology

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