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arxiv: 2506.17907 · v3 · submitted 2025-06-22 · 🧮 math.RT

The endoscopic character identity for even special orthogonal groups

Hao Peng This is my paper

Pith reviewed 2026-05-19 08:37 UTC · model grok-4.3

classification 🧮 math.RT
keywords endoscopic character identityA-packetseven special orthogonal groupsnon-quasisplit groupselliptic endoscopic tripleslocal Langlands correspondencemultiplicity formula
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The pith

The endoscopic character identity holds for bounded A-packets of non-quasisplit even special orthogonal groups relative to elliptic endoscopic triples.

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

The paper aims to prove the endoscopic character identity for the bounded A-packets of non-quasisplit even special orthogonal groups in the context of elliptic endoscopic triples. It achieves this by reducing the problem to the quasisplit case and certain real cases through the local-global compatibility principle and an existing multiplicity formula. A reader would care because this identity is crucial for linking different formulations of the local Langlands correspondence for these groups and advancing the theory of automorphic representations.

Core claim

We establish the endoscopic character identity for bounded A-packets of non-quasisplit even special orthogonal groups, with respect to elliptic endoscopic triples. The proof reduces the non-quasisplit case to the quasisplit case and the real Adam-Johnson case by combining local-global compatibility principle with Arthur's multiplicity formula for non-quasisplit even special orthogonal groups.

What carries the argument

The endoscopic character identity, which equates the characters of representations in A-packets under the transfer maps induced by elliptic endoscopic triples.

If this is right

  • This identity is a key step in proving the compatibility between the Fargues-Scholze local Langlands correspondence and the classical local Langlands correspondence for even special orthogonal groups.
  • It extends the endoscopic character identities previously known for quasisplit groups to the non-quasisplit setting.
  • The result allows for the application of endoscopic methods to study automorphic forms on non-quasisplit even special orthogonal groups.

Where Pith is reading between the lines

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

  • The reduction technique could potentially be adapted to establish similar identities for other non-quasisplit classical groups where multiplicity formulas are known.
  • Explicit checks in low-dimensional cases might provide numerical evidence supporting the identity.
  • This work suggests that endoscopic transfers preserve the boundedness of A-packets across different forms of the groups.

Load-bearing premise

The argument relies on the local-global compatibility principle together with the Arthur multiplicity formula for non-quasisplit even special orthogonal groups.

What would settle it

An explicit calculation of the endoscopic transfer of characters for a specific bounded A-packet of a non-quasisplit even special orthogonal group over a global field, checking if it matches the predicted identity.

read the original abstract

We establish the endoscopic character identity for bounded $A$-packets of non-quasisplit even special orthogonal groups, with respect to elliptic endoscopic triples. The proof reduces the non-quasisplit case to the quasisplit case and the real Adam--Johnson case by combining local-global compatibility principle with Arthur's multiplicity formula for non-quasisplit even special orthogonal groups established by Chen and Zou in arXiv:2103.07956. This result plays a key role in the author's work arXiv:2503.04623 on the compatibility between the Fargues--Scholze local Langlands correspondence and classical local Langlands correspondence for even special orthogonal 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

2 major / 2 minor

Summary. The manuscript establishes the endoscopic character identity for bounded A-packets of non-quasisplit even special orthogonal groups with respect to elliptic endoscopic triples. The proof reduces the non-quasisplit case to the quasisplit case and the real Adam-Johnson case by combining the local-global compatibility principle with Arthur's multiplicity formula for non-quasisplit even special orthogonal groups from Chen and Zou (arXiv:2103.07956).

Significance. If the reduction is valid and the cited results apply directly, this completes the endoscopic character identity in the non-quasisplit setting and serves as a key step toward compatibility between the Fargues-Scholze and classical local Langlands correspondences, as noted in the abstract and the author's related work arXiv:2503.04623. The reduction approach efficiently leverages existing multiplicity formulas rather than re-deriving them.

major comments (2)
  1. [Proof of Theorem 1.1 / reduction argument] The central reduction (described in the proof of the main result) invokes the multiplicity formula of Chen and Zou (arXiv:2103.07956) for non-quasisplit even special orthogonal groups. Explicit verification is needed that the bounded A-packets and elliptic endoscopic triples treated here satisfy all hypotheses of that formula, including any restrictions on the groups, packets, and global fields.
  2. [Section on local-global compatibility] The local-global compatibility principle is used to complete the reduction. The precise statement of this principle (including the version of the principle and the precise matching of local and global data for the endoscopic triples) should be stated, together with confirmation that no additional assumptions are required for the non-quasisplit even special orthogonal groups.
minor comments (2)
  1. [Introduction] Ensure the terminology 'bounded A-packets' is defined or clearly referenced when first used in the introduction.
  2. [References] The references to arXiv:2103.07956 and arXiv:2503.04623 should include full bibliographic details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will make the indicated revisions to improve the clarity of the arguments.

read point-by-point responses

  1. Referee: The central reduction (described in the proof of the main result) invokes the multiplicity formula of Chen and Zou (arXiv:2103.07956) for non-quasisplit even special orthogonal groups. Explicit verification is needed that the bounded A-packets and elliptic endoscopic triples treated here satisfy all hypotheses of that formula, including any restrictions on the groups, packets, and global fields.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript, we will add a dedicated paragraph in the proof of Theorem 1.1 that checks the hypotheses of the Chen-Zou multiplicity formula against our bounded A-packets, elliptic endoscopic triples, and the number-field setting, confirming that all stated restrictions on groups and packets are met. revision: yes

  2. Referee: The local-global compatibility principle is used to complete the reduction. The precise statement of this principle (including the version of the principle and the precise matching of local and global data for the endoscopic triples) should be stated, together with confirmation that no additional assumptions are required for the non-quasisplit even special orthogonal groups.

    Authors: We will revise the section discussing local-global compatibility to include the exact statement of the principle as invoked, together with the matching of local and global endoscopic data. We will also add a short confirmation that the cited version applies directly to non-quasisplit even special orthogonal groups without extra hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity; reduction relies on independent external multiplicity formula

full rationale

The paper's derivation is a reduction of the endoscopic character identity for non-quasisplit even special orthogonal groups to the quasisplit case and real Adam-Johnson case. This reduction explicitly invokes the local-global compatibility principle together with Arthur's multiplicity formula as established in the independent prior work of Chen and Zou (arXiv:2103.07956). The citation to the author's own arXiv:2503.04623 appears only as a downstream application of the present result, not as a load-bearing input or justification for the proof. No self-definitional relations, fitted parameters renamed as predictions, or uniqueness theorems imported from overlapping authors are used. The central claim therefore stands on external, independently established results rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on two external results: Arthur's multiplicity formula from Chen-Zou and the local-global compatibility principle. No new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Arthur's multiplicity formula for non-quasisplit even special orthogonal groups
    Invoked to reduce the non-quasisplit case; cited from Chen and Zou arXiv:2103.07956
  • domain assumption local-global compatibility principle
    Used to combine local and global information in the reduction proof

pith-pipeline@v0.9.0 · 5623 in / 1363 out tokens · 49257 ms · 2026-05-19T08:37:05.697883+00:00 · methodology

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