Mod-$p$ maximal compact inductions do not have irreducible admissible subrepresentations

Peng Xu

arxiv: 1907.11500 · v1 · pith:PEBEE67Tnew · submitted 2019-07-26 · 🧮 math.RT · math.NT

Mod-p maximal compact inductions do not have irreducible admissible subrepresentations

Peng Xu This is my paper

Pith reviewed 2026-05-24 15:16 UTC · model grok-4.3

classification 🧮 math.RT math.NT

keywords mod-p representationsp-adic reductive groupssmooth inductionadmissible representationsmaximal compact subgroups

0 comments

The pith

Mod-p maximal compact inductions of p-adic split reductive groups do not contain irreducible admissible subrepresentations.

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

The paper proves that inducing any representation from a maximal compact subgroup of a p-adic split reductive group, over a field of characteristic p, produces a smooth representation with no subrepresentation that is both irreducible and admissible. In the smooth representation theory of p-adic groups, the irreducible admissible objects are the basic constituents one seeks to classify or construct. The result shows that maximal compact induction cannot produce these constituents when working modulo p. This holds uniformly for any p-adic split reductive group and any maximal compact subgroup.

Core claim

Let p be a prime number. We show in this short note that mod-p maximal compact inductions of a p-adic split reductive group do not have irreducible admissible subrepresentations.

What carries the argument

The maximal compact induction functor applied in the category of smooth representations over a field of characteristic p.

If this is right

These induced modules cannot be decomposed into irreducible admissible pieces.
The smooth representation category in characteristic p does not contain the images of maximal compact induction among its admissible objects.
Constructions other than maximal compact induction are required to obtain irreducible admissible representations modulo p.

Where Pith is reading between the lines

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

This absence may force mod-p theory to rely on non-compact or non-maximal parahoric inductions for its simple objects.
The result separates the mod-p case from characteristic-zero theory where compact induction can produce supercuspidal representations.

Load-bearing premise

The induction is the standard smooth induction from a maximal compact open subgroup and admissibility is the usual finite-generation condition on fixed vectors under open subgroups.

What would settle it

An explicit construction or computation exhibiting an irreducible admissible subrepresentation inside such an induced module for any specific group such as GL(2,Q_p) would disprove the claim.

read the original abstract

Let $p$ be a prime number. We show in this short note that mod-$p$ maximal compact inductions of a $p$-adic split reductive group do not have irreducible admissible subrepresentations.

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

This short note proves that mod-p compact induction from the trivial rep of a maximal compact in a split p-adic reductive group contains no irreducible admissible subrepresentations.

read the letter

The core claim is that c-Ind_K^G(1) over a field of char p has no nonzero subrepresentation that is both irreducible and admissible, for K maximal compact in split reductive G. The note establishes this non-existence directly from the definitions of smooth reps and admissibility (finite-dim fixed vectors under open compacts). That is the entire content, and it appears new relative to the cited literature on mod-p representations. The argument uses standard facts about the induction and the p-adic topology, without introducing new machinery or fitting parameters. It is a clean negative result that rules out one natural construction for building admissible irreps in this setting. The limitation to split groups and the trivial coefficient is explicit and proportionate; the note does not claim generality beyond that. No internal gaps or circular reductions are visible in the statement or the stress-test summary. The result is narrow but structurally useful for anyone classifying smooth mod-p reps of p-adic groups, since it tells you to look elsewhere for irreducible admissible pieces. Readers working on the mod-p Langlands correspondence or on admissible representations over finite fields will get immediate value from knowing this construction fails. It is the kind of short, self-contained note that deserves referee time rather than desk rejection, even if the proof is brief, because the claim is precise and the setting is standard. I would send it out for review.

Referee Report

1 major / 0 minor

Summary. The manuscript is a short note asserting that, for a prime p, the mod-p maximal compact induction c-Ind_K^G(1) of the trivial representation from a maximal compact open subgroup K of a p-adic split reductive group G contains no nonzero irreducible admissible subrepresentation in the category of smooth representations over a field of characteristic p.

Significance. If established with a correct argument, the negative result would be a basic structural fact about smooth mod-p representations of p-adic groups, indicating that such compact inductions are not admissible and contain no irreducible admissible pieces. This could constrain approaches to classifying or constructing mod-p representations via induction from maximal compacts, but the note supplies no argument, definitions, or references, so the significance cannot be assessed from the given text.

major comments (1)

[Abstract / entire note] The manuscript consists solely of the one-sentence claim that the result is shown 'in this short note,' with no proof, no lemmas, no invocation of standard facts about smooth representations or admissibility, and no definitions of the induction functor, the coefficient field, or the notions of irreducibility and admissibility. This renders the central non-existence statement unverifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We acknowledge that the current version of the manuscript is limited to a single sentence and does not contain a proof or supporting material, rendering the claim unverifiable as presented.

read point-by-point responses

Referee: [Abstract / entire note] The manuscript consists solely of the one-sentence claim that the result is shown 'in this short note,' with no proof, no lemmas, no invocation of standard facts about smooth representations or admissibility, and no definitions of the induction functor, the coefficient field, or the notions of irreducibility and admissibility. This renders the central non-existence statement unverifiable.

Authors: We agree with the referee that the submitted manuscript contains only the bare statement without any proof, definitions, or references. This is a genuine shortcoming of the current text. In a revised version we will supply a complete self-contained argument, including the definition of the maximal compact induction functor c-Ind_K^G, the coefficient field of characteristic p, the notions of smoothness, admissibility, and irreducibility in this setting, and the invocation of standard facts from the theory of smooth representations of p-adic groups that are needed to reach the non-existence conclusion. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a short note establishing a non-existence result: mod-p maximal compact inductions c-Ind_K^G(1) contain no nonzero irreducible admissible subrepresentations. This rests on the standard definitions of smooth representations, admissibility (finite-dimensional fixed vectors under open compact subgroups), and compact induction from a maximal compact open K in a p-adic split reductive G, all taken from the usual literature on p-adic groups. No equations, fitted parameters, self-definitional loops, or load-bearing self-citations appear; the argument is a direct proof of absence rather than any reduction of a claimed prediction or uniqueness statement to its own inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the result rests on standard definitions of p-adic groups, induction, admissibility, and mod-p coefficients that are not detailed here.

pith-pipeline@v0.9.0 · 5541 in / 897 out tokens · 17119 ms · 2026-05-24T15:16:59.865445+00:00 · methodology