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arxiv: 2109.01424 · v3 · pith:VEWMCHIJnew · submitted 2021-09-03 · 🧮 math.AG · math.RT

On a decomposition of p-adic Coxeter orbits

Alexander B. Ivanov This is my paper

Pith reviewed 2026-05-24 12:48 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords p-adic Deligne-Lusztig spacesCoxeter elementsclassical groupsbasic elementsunramified toriSteinberg cross sectionpure inner formslocal fields
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The pith

When the group is classical, b basic and w a Coxeter element, the p-adic Deligne-Lusztig space X_w(b) decomposes as a disjoint union of translates of an integral p-adic Deligne-Lusztig space.

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

The paper proves a decomposition result for the p-adic Deligne-Lusztig spaces X_w(b) attached to an unramified reductive group over a non-archimedean local field. For classical groups, with b basic and w Coxeter, it shows that X_w(b) is a disjoint union of translates of a certain integral version of such a space. The work also extends results on rational conjugacy classes of unramified tori to extended pure inner forms and establishes a loop version of a Frobenius-twisted Steinberg cross section. A sympathetic reader cares because the decomposition reduces the geometry of these spaces to simpler integral pieces whose properties are easier to control.

Core claim

When G is classical, b basic and w Coxeter, X_w(b) decomposes as a disjoint union of translates of a certain integral p-adic Deligne-Lusztig space. The paper extends observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms and proves a loop version of Frobenius-twisted Steinberg's cross section.

What carries the argument

The decomposition of X_w(b) into translates of an integral p-adic Deligne-Lusztig space, for classical groups with basic b and Coxeter w.

Load-bearing premise

The p-adic Deligne-Lusztig spaces X_w(b) are well-defined and satisfy the geometric properties stated in the author's prior work, with G an unramified reductive group over a non-archimedean local field.

What would settle it

An explicit classical group G, basic b, and Coxeter w for which X_w(b) cannot be written as a disjoint union of translates of an integral p-adic Deligne-Lusztig space.

read the original abstract

We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$ introduced in [Iva21] attached to an unramified reductive group ${\bf G}$ over a non-archimedean local field. We prove that when ${\bf G}$ is classical, $b$ basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms, and prove a loop version of Frobenius-twisted Steinberg's cross section.

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 / 2 minor

Summary. The manuscript analyzes the geometry of p-adic Deligne-Lusztig spaces X_w(b) for an unramified reductive group G. It proves that when G is classical, b is basic, and w is Coxeter, X_w(b) decomposes as a disjoint union of translates of a certain integral p-adic Deligne-Lusztig space. Supporting results include an extension of DeBacker-Reeder observations on rational conjugacy classes of unramified tori to extended pure inner forms, and a proof of a loop version of the Frobenius-twisted Steinberg cross-section.

Significance. If the decomposition holds, the result clarifies the structure of these spaces in the classical case and may facilitate explicit computations or further applications in the representation theory of p-adic groups. The two auxiliary results on conjugacy classes and the cross-section appear to be of independent interest and strengthen the geometric toolkit for p-adic Deligne-Lusztig theory.

minor comments (2)
  1. The abstract and introduction cite [Iva21] for the definition of X_w(b); a brief self-contained reminder of the key geometric properties used (e.g., dimension or smoothness statements) would improve readability without lengthening the paper substantially.
  2. Notation for the integral p-adic Deligne-Lusztig space appearing in the decomposition statement should be introduced explicitly in §1 or §2 rather than only in the statement of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the independent interest of the auxiliary results on conjugacy classes and the cross-section, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Minor self-citation for foundational definitions; central decomposition independent

full rationale

The manuscript cites [Iva21] solely to introduce the spaces X_w(b) and their geometric properties, then derives a new decomposition result for classical G, basic b, and Coxeter w, together with extensions of DeBacker-Reeder observations and a loop Steinberg cross-section. These supporting ingredients are presented as original contributions and do not reduce the decomposition statement to a self-referential definition or fitted input. No equation or claim in the derivation chain collapses by construction to prior self-citations; the work remains self-contained beyond standard foundational references.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, specific axioms, or invented entities.

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Reference graph

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