arxiv: 2605.05201 · v1 · submitted 2026-05-06 · 🧮 math.RT

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A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

Manish Mishra

Pith reviewed 2026-05-08 15:30 UTC · model grok-4.3

classification 🧮 math.RT

keywords depth-zero supercuspidal representationspinned local Langlands correspondenceenhanced Langlands parametersJordan decompositionreductive algebraic groupsnon-archimedean local fieldsunipotent representationstoral representations

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The pith

After fixing a pinned splitting of the quasi-split inner form, there is 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.

The paper constructs a pinned local Langlands correspondence for depth-zero supercuspidal representations of connected reductive groups over non-archimedean local fields. Fixing a pinned splitting produces a canonical bijection to the matching set of enhanced Langlands parameters. The proof separates the tame toral part, normalized by the torus local Langlands correspondence, from the unramified unipotent part supplied by an existing unipotent correspondence. These are connected by a pinned Jordan decomposition on the finite quotient, making the full map reversible and independent of extra choices. This matters because it gives a concrete, normalized way to associate parameters to these representations, which form a basic building block in the local Langlands program.

Core claim

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 separates the tame toral part from the unramified unipotent part. The toral part is normalized by the local Langlands correspondence for maximally unramified elliptic tori and the corresponding canonical L-embeddings. The finite representation occurring in a depth-zero type is then passed, through a pinned Jordan decomposition for possibly disconnected finite reductive quotients, to a cuspidal unipotent label on the dual centralizer. The unramified unipotent

What carries the argument

The pinned Jordan decomposition for possibly disconnected finite reductive quotients, which transfers the finite representation from the depth-zero type to a cuspidal unipotent label on the dual centralizer, combined with the local Langlands correspondence for maximally unramified elliptic tori and the Feng-Opdam-Solleveld unipotent correspondence.

Load-bearing premise

The construction assumes the local Langlands correspondence for maximally unramified elliptic tori, the Feng-Opdam-Solleveld correspondence for supercuspidal unipotent representations, and the existence of a pinned Jordan decomposition for possibly disconnected finite reductive quotients.

What would settle it

For the group GL(2,F) where F is a non-archimedean local field, explicitly list all depth-zero supercuspidal representations and their known Langlands parameters, then check whether the constructed map produces a bijection matching these known pairs; a mismatch for any representation would falsify the claim.

read the original abstract

We construct a pinned 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 separates the tame toral part from the unramified unipotent part. The toral part is normalized by the local Langlands correspondence for maximally unramified elliptic tori and by the corresponding canonical \(L\)-embeddings. The finite representation occurring in a depth-zero type is then passed, through a pinned Jordan decomposition for possibly disconnected finite reductive quotients, to a cuspidal unipotent label on the dual centralizer. The unramified unipotent contribution is supplied by the Feng--Opdam--Solleveld correspondence for supercuspidal unipotent representations. Combining these ingredients gives the enhanced parameter attached to a depth-zero supercuspidal representation, and the inverse map is obtained by reversing the same finite Jordan decomposition. The correspondence is independent of auxiliary choices apart from 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. Thus the main output is a canonical, pinning-normalized bijection between the two depth-zero supercuspidal sides, together with the finite unipotent bookkeeping needed to make the construction reversible.

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

The paper gives a canonical pinned bijection for depth-zero supercuspidals by gluing the toral LLC with the FOS unipotent correspondence through a Jordan decomposition on finite quotients.

read the letter

The main output is a pinning-normalized bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. It separates the tame toral part, handled by the known correspondence for maximally unramified elliptic tori and their L-embeddings, from the unramified unipotent part supplied by Feng-Opdam-Solleveld, with the finite representation in the depth-zero type passed through a pinned Jordan decomposition to a cuspidal unipotent label on the dual centralizer. The inverse is obtained by reversing the same decomposition. The construction is independent of auxiliary choices beyond the fixed pinned splitting and is compatible with tame inertial parameters, weakly unramified twists, and central characters via the torus side. This is the new piece: a reversible gluing that makes the overall map canonical once the pinning is fixed. The paper does a clean job isolating the ingredients and spelling out the compatibilities. The potential soft spot is the pinned Jordan decomposition for possibly disconnected finite reductive quotients. Standard Lusztig theory covers connected groups, so the extension here must be shown to be uniquely determined by the pinning, to commute with the L-embeddings, and to be invertible without introducing extra choices. If the manuscript supplies a self-contained argument for that step, the bijection holds; otherwise the canonicity claim rests on it. This is for people working on explicit local Langlands correspondences or depth-zero representations who need a normalized foundation. It deserves a serious referee because the approach is coherent, the new gluing is isolated, and the prior inputs are standard.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs a pinned 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 claims a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction separates the tame toral part (normalized via the local Langlands correspondence for maximally unramified elliptic tori and canonical L-embeddings) from the unramified unipotent part (via the Feng-Opdam-Solleveld correspondence), mapping the finite representation in a depth-zero type through a pinned Jordan decomposition for possibly disconnected finite reductive quotients; the inverse is obtained by reversing this decomposition. The correspondence is stated to be independent of auxiliary choices apart from the pinning and compatible with tame inertial parameters, weakly unramified twists, and central characters.

Significance. If the bijection holds with the claimed canonicity, the result supplies a pinning-normalized depth-zero supercuspidal LLC that separates toral and unipotent contributions explicitly and is compatible with several natural structures. This would be a useful concrete step in the local Langlands program for p-adic groups. The grounding in independent prior correspondences for tori and unipotent representations, together with the explicit reversibility mechanism, strengthens the construction.

major comments (2)

[the construction of the map via pinned Jordan decomposition] The section describing the pinned Jordan decomposition for possibly disconnected finite reductive quotients: the manuscript must supply a self-contained argument that this decomposition is uniquely determined by the fixed pinning, commutes with the L-embeddings of the toral part, and is bijective. Standard Lusztig Jordan decomposition applies only to connected groups; without an explicit verification of canonicity and invertibility in the disconnected case, the reversibility of the overall map and its independence from auxiliary choices remain unestablished. This step is load-bearing for the central bijection claim.
[the inverse map construction] The paragraph on the inverse map: the assertion that reversing the finite Jordan decomposition yields the inverse correspondence requires explicit verification that the two maps are mutual inverses on the level of enhanced parameters. Any failure of bijectivity here would directly undermine the claimed canonical bijection.

minor comments (1)

[notation and preliminaries] The notation for enhanced depth-zero Langlands parameters and the finite reductive quotients could be introduced with a short illustrative example early in the text to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight two load-bearing aspects of the construction that require explicit verification. We address each point below and will revise the manuscript accordingly to strengthen the canonicity and reversibility arguments.

read point-by-point responses

Referee: [the construction of the map via pinned Jordan decomposition] The section describing the pinned Jordan decomposition for possibly disconnected finite reductive quotients: the manuscript must supply a self-contained argument that this decomposition is uniquely determined by the fixed pinning, commutes with the L-embeddings of the toral part, and is bijective. Standard Lusztig Jordan decomposition applies only to connected groups; without an explicit verification of canonicity and invertibility in the disconnected case, the reversibility of the overall map and its independence from auxiliary choices remain unestablished. This step is load-bearing for the central bijection claim.

Authors: We agree that a self-contained verification is necessary for the disconnected case, as the standard Lusztig theory is stated for connected groups. The manuscript defines the pinned Jordan decomposition by using the fixed pinning to select a canonical splitting of the finite reductive quotient (including its possibly disconnected component group) and extending the connected-case decomposition via the action on connected components. This ensures uniqueness from the pinning and compatibility with the toral L-embeddings by construction. However, to meet the referee's request for explicitness, we will add a dedicated subsection that proves bijectivity, commutativity with the L-embeddings, and independence from auxiliary choices in the disconnected setting, thereby confirming the reversibility of the overall correspondence. revision: yes
Referee: [the inverse map construction] The paragraph on the inverse map: the assertion that reversing the finite Jordan decomposition yields the inverse correspondence requires explicit verification that the two maps are mutual inverses on the level of enhanced parameters. Any failure of bijectivity here would directly undermine the claimed canonical bijection.

Authors: We will expand the paragraph on the inverse map to include a direct verification that the forward and reverse maps are mutual inverses. This will be done by composing the maps explicitly: starting from an enhanced parameter, applying the inverse construction (reversing the finite Jordan decomposition to recover the finite representation in the depth-zero type, then attaching the toral and unipotent parts via the torus LLC and Feng-Opdam-Solleveld correspondence), and showing that the result recovers the original depth-zero supercuspidal representation. The verification relies on the bijectivity of the pinned Jordan decomposition (to be established in the new subsection) together with the known bijectivity of the torus and unipotent correspondences. This will be added as a short lemma in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction assembles independent external inputs via internal gluing

full rationale

The paper's derivation explicitly factors depth-zero supercuspidals into a tame toral component (normalized by the independent LLC for maximally unramified elliptic tori plus canonical L-embeddings) and an unramified unipotent component (supplied by the external Feng-Opdam-Solleveld correspondence). The pinned Jordan decomposition for possibly disconnected finite quotients is introduced and proved internally to transport the finite representation to a cuspidal unipotent label and to furnish the inverse map, ensuring reversibility and pinning-normalized canonicity. Because the component correspondences are cited as prior independent results and the gluing step is constructed to commute with them without redefining any input as output, the claimed bijection does not reduce to a self-definition, fitted prediction, or self-citation chain. The overall map therefore retains external grounding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two prior correspondences (toral LLC and Feng-Opdam-Solleveld) plus an assumed Jordan decomposition; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Local Langlands correspondence for maximally unramified elliptic tori exists and is compatible with the pinned normalization
Used to normalize the toral part of the depth-zero supercuspidal representation.
domain assumption Feng-Opdam-Solleveld correspondence supplies the supercuspidal unipotent representations on the dual side
Provides the unramified unipotent contribution to the enhanced parameter.

pith-pipeline@v0.9.0 · 5578 in / 1410 out tokens · 72219 ms · 2026-05-08T15:30:12.653353+00:00 · methodology

discussion (0)

Reference graph

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