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Compact induction from an open compact-mod-centre subgroup yields topologically irreducible locally analytic representations precisely when the dual module is uniformly simple and has no non-trivial intertwiners.

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Compact induction of a uniformly simple admissible locally analytic representation from an open compact-mod-centre subgroup is topologically irreducible when there are no nontrivial intertwiners with its conjugates.

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load-bearing objection Solid, usable Mackey criterion for compactly induced locally analytic representations; the uniform-semisimplicity hypothesis is explicit and the proofs check out.

arxiv 2607.02976 v1 pith:LDP7HK2K submitted 2026-07-03 math.NT math.RT

A Mackey criterion for locally analytic representations

Andreas Bode This is my paper
classification math.NT math.RT MSC 22E5011F70
keywords locally analytic representationsMackey criterioncompact inductiondistribution algebrauniform semisimplicityind-admissiblep-adic groups
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper extends classical Mackey theory to locally analytic representations of p-adic groups. It shows that if you compactly induce an admissible representation V of an open subgroup H that is compact modulo the centre, then the resulting G-representation is topologically irreducible whenever the dual of V is uniformly simple over the distribution algebra of a compact open subgroup and admits no non-zero intertwiners with its conjugates outside H. The argument proceeds by establishing a Mackey decomposition into H-representations, proving that uniform semisimplicity is preserved under restriction, conjugation and induction, and then applying a standard generation argument. Concrete examples for the Heisenberg group and the Borel of SL2(Qp) illustrate that the criterion can produce both admissible and non-admissible irreducible representations, giving a practical test for irreducibility in the p-adic Langlands setting.

Core claim

If M = V' is a uniformly simple D(H0)-module such that Hom_D(H ∩ gHg^{-1})(gM, M) equals K when g lies in H and equals 0 otherwise, then the compact induction c-Ind_H^G V is a topologically irreducible, ind-admissible G-representation.

What carries the argument

Uniform semisimplicity of coadmissible modules over a Fréchet–Stein distribution algebra: the module remains simple (or a finite direct sum of simples) on all sufficiently large Banach levels of a chosen presentation. This property is preserved under the Mackey operations and supplies the complements needed for topological irreducibility.

Load-bearing premise

The dual module must be uniformly simple relative to a fixed Fréchet–Stein presentation of the distribution algebra; mere topological simplicity is not enough for the argument.

What would settle it

Exhibit a concrete uniformly simple dual module that satisfies the intertwiner vanishing condition yet whose compact induction admits a proper closed G-invariant subspace, or show that a module which is only topologically simple (not uniformly simple) still produces an irreducible induction.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Compact induction becomes a systematic source of topologically irreducible locally analytic representations for non-compact p-adic groups.
  • The associated Hecke algebra End_G(c-Ind V) is reduced to End_H(V), simplifying the study of irreducible constituents after fixing a central character.
  • Admissibility of the induced representation can be decided by checking vanishing of only finitely many Banach-level summands for each radius.
  • The same criterion applies verbatim in the solid-module setting over the distribution algebra.

Where Pith is reading between the lines

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

  • The dependence on a presentation suggests that uniform simplicity may be a temporary technical device; a presentation-independent reformulation would enlarge the class of usable representations.
  • The Heisenberg examples produce admissible irreducibles while the Borel examples never do, hinting that the relative position of the centre controls admissibility of compact inductions.
  • Once the intertwiner condition is verified for a family of characters or finite-dimensional representations, one obtains infinite families of topologically irreducible objects ready for use in p-adic Langlands correspondences.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper establishes a Mackey-type irreducibility criterion for compactly induced locally analytic representations of p-adic groups. For G a locally L-analytic group and H = H0 Z_G an open subgroup compact mod centre, and V an admissible locally analytic H-representation whose dual M = V' is uniformly simple over D(H0), the compact induction c-Ind_H^G V is topologically irreducible (and ind-admissible) provided the intertwining spaces Hom_{D(H ∩ gHg^{-1})}(gM, M) vanish for g otin H (Theorems 4.1–4.2; Banach-level variant Theorem 4.3). The argument proceeds by recalling the Mackey decomposition for ind-admissible representations (Proposition 2.4), introducing uniform semisimplicity of coadmissible modules over Fréchet–Stein distribution algebras (Definitions 3.2–3.3), proving that this property is preserved under restriction, conjugation and induction (Propositions 3.1, 3.7, 3.8), and then running a closed-submodule generation argument. Two families of examples (Heisenberg group characters and finite-dimensional representations of a congruence subgroup of the Borel of SL2(Qp)) illustrate both the criterion and the distinction between admissible and non-admissible outcomes.

Significance. The result supplies a usable irreducibility test in the Schneider–Teitelbaum category of locally analytic representations, a setting central to the p-adic Langlands programme where compact inductions are natural but rarely admissible or irreducible. The introduction of uniform semisimplicity is a clean technical device that lets classical Mackey arguments pass to the Fréchet–Stein level without topological pathologies; the Banach-level criterion (Theorem 4.3) further makes the hypotheses checkable in practice. The examples are concrete and show both that the hypotheses are attainable and that the resulting representations can be admissible (Heisenberg) or systematically non-admissible (Borel). The paper is self-contained once standard facts on coadmissible modules are granted, and the proofs are algebraic once the uniform condition is in place.

minor comments (3)
  1. [Definition 3.3, Theorem 4.3] Definition 3.3 and the warning that follows correctly flag that uniform (semi)simplicity depends on the chosen Fréchet–Stein presentation. A short remark after Proposition 3.7 or Theorem 4.2 noting that the Banach-level criterion of Theorem 4.3 is independent of any global choice of presentation would make the practical status of the hypothesis even clearer.
  2. [§5.1] In the Heisenberg example the injectivity assumption on μ (and the consequent equality ker μ = H0 ∩ Z_G) is used both for the intertwining vanishing and for the radius estimates that give admissibility. A one-sentence clarification that any character with the same kernel works would avoid the impression that a specific system of roots of unity is essential.
  3. A few minor typos and notational inconsistencies appear (e.g., occasional missing spaces around math operators, and the dual identification after Proposition 2.4 could be cross-referenced more explicitly when used in the proof of Theorem 4.1). None affect readability of the arguments.

Circularity Check

0 steps flagged

No circularity: self-contained adaptation of classical Mackey theory to Fréchet–Stein distribution algebras

full rationale

The paper proves an analogue of Mackey's irreducibility criterion for compactly induced locally analytic representations. The Mackey decomposition (Prop. 2.4) is taken from prior independent work (Orlik [15, Lem. A.19]); the new technical content is the introduction of uniform (semi)simplicity (Def. 3.3) relative to a Fréchet–Stein presentation, the verification that this notion is preserved under restriction/conjugation/induction (Props. 3.1, 3.7, 3.8), and the standard generation argument that yields topological irreducibility once the Hom-vanishing condition holds (Thms. 4.1–4.3). None of these steps defines a quantity in terms of the claimed conclusion, fits a parameter to data, or relies on a load-bearing self-citation of an unverified uniqueness theorem by the same author. The two examples independently check the Hom condition and the coadmissibility criterion of Cor. 2.3; they do not feed fitted values back into the general theorem. The derivation is therefore free of the six circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper rests on the standard foundations of locally analytic representation theory (Schneider–Teitelbaum distribution algebras, coadmissible modules, Fréchet–Stein structure) together with the earlier definition of ind-admissible representations. No free parameters are fitted; the only new technical device is the definition of uniform (semi)simplicity, which is an ad-hoc strengthening introduced to make the Banach-level arguments work.

axioms (4)
  • domain assumption Distribution algebras D(G) of compact p-adic analytic groups are nuclear Fréchet–Stein algebras; coadmissible modules are dual to admissible locally analytic representations.
    Taken as standard from Schneider–Teitelbaum (Invent. Math. 2003) and used throughout §§2–4.
  • domain assumption Pro-coadmissible modules form an abelian category closed under countable products, and continuous morphisms between them are strict.
    Cited from the appendix of Orlik (arXiv 2505.04355); used for the dual Mackey decomposition and closed-submodule arguments.
  • standard math For r = p^{-1/p^n} sufficiently large, Dr(G0) is free of finite rank over Dr(H0) when H0 is open in the compact group G0.
    Standard consequence of the crossed-product description of the Banach completions; used in Proposition 3.1.
  • ad hoc to paper Uniform semisimplicity (semisimplicity of all sufficiently large Banach levels) is a well-defined and useful strengthening of topological semisimplicity for coadmissible modules.
    Introduced in Definition 3.3 precisely to obtain Proposition 3.7(ii) and the subsequent irreducibility criterion; the paper notes that the notion depends on the chosen Fréchet–Stein presentation.
invented entities (1)
  • uniform (semi)simplicity of a coadmissible module no independent evidence
    purpose: To guarantee that topological semisimplicity descends under restriction from D(G0) to D(H0) and that the Mackey summands remain topologically semisimple, allowing the classical intertwiner argument to go through on the Fréchet level.
    Defined in §3; independent evidence is limited to the internal consistency of the proofs and the two examples; no external characterisation is given.

reviewed 2026-07-12 · how reviews work

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Cite this review

Pith. "Pith review of A Mackey criterion for locally analytic representations." pith.science (2026). https://pith.science/paper/LDP7HK2K

@misc{pith2026260702976,
  author       = {Pith},
  title        = {Pith review of: A Mackey criterion for locally analytic representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDP7HK2K}},
  note         = {Machine review of arXiv:2607.02976}
}
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abstract

We prove an analogue of Mackey's irreducibility criterion for compactly induced locally analytic representations of $p$-adic groups, where we induce from an open subgroup which is compact mod centre. We also discuss several examples.

discussion (0)

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Reference graph

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This paper was first reviewed by grok-4.5 on July 12, 2026.