{"id":"b30e8e59-3543-45f3-8ea0-f57ab9873816","arxiv_id":"2607.02976","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"The paper proves a Mackey-type irreducibility criterion for compactly induced locally analytic representations of p-adic groups from open subgroups that are compact modulo the centre. This gives a practical test for when such large induced representations remain topologically irreducible, with worked examples on the Heisenberg group and a Borel subgroup.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim is the irreducibility criterion under uniform simplicity plus vanishing of intertwiners (Thm. 4.2). The only potential soft spot is the presentation-dependence of uniform simplicity, which the reader already isolates. That dependence is openly acknowledged and is not used as a black box: the proofs reduce to classical semisimplicity on the Noetherian Banach algebras Dr (Prop. 3.1) and then lift by finite-length and strictness arguments that are fully written out. The two examples show that the hypotheses can be checked by hand and that the conclusion is sharp (admissible vs. non-admissible). Consequently the logical structure stands, the reader’s ACCEPT/HIGH verdict needs no adjustment, and the concrete test above merely confirms that the second example sits inside the stated hypotheses rather than exposing a flaw.","tokens_in":20185,"tokens_out":546,"duration_ms":4385,"concrete_test":"Verify that the dual of the finite-dimensional irreducible H-representation V_{χ,σ} constructed in §5.2 is uniformly simple for the standard Fréchet–Stein presentation of D(H0) coming from the uniform open normal subgroup U = ⟨h1,h2⟩ (i.e., that Dr(H0)⊗ D(H0) M remains simple for all r = p^{-1/p^n} sufficiently close to 1). If it fails for some sequence rn\to1, the application of Thm. 4.1 in Prop. 5.3 would require the Banach-level criterion of Thm. 4.3 instead.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption correctly flags that uniform simplicity (Def. 3.3) is presentation-dependent and strictly stronger than topological simplicity, yet this is not a hidden gap: the paper states the dependence explicitly, proves the needed restriction/induction properties under that hypothesis (Prop. 3.7–3.8), and supplies a Banach-level criterion (Thm. 4.3) that can be checked level-by-level without first knowing the Fréchet module is topologically simple. The central claim (Thm. 4.2) therefore holds under the stated hypotheses; the examples confirm both that the hypotheses are attainable and that the resulting representations can be admissible or non-admissible. No internal inconsistency or missing step appears in the Mackey decomposition, the semisimplicity transfer, or the final generation argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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 \notin 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.","tokens_in":20380,"tokens_out":859,"duration_ms":6420,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Definition 3.3, Theorem 4.3"},{"comment":"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.","section":"§5.1"},{"comment":"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.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, cleanly written and fills a genuine technical gap. I see no reason to delay publication; the uniform-semisimplicity hypothesis is stated with appropriate caveats and is not a hidden gap. Fit for a standard number-theory or representation-theory journal is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a clean, self-contained adaptation of classical Mackey theory to Schneider–Teitelbaum locally analytic representations. The new piece is the irreducibility criterion (Thm 1.1(iii)/4.2): if V' is uniformly simple over D(H0) and the usual Hom vanishing holds for double cosets outside H, then c-Ind is topologically irreducible (and ind-admissible). The Mackey decomposition itself is already in Orlik’s appendix; the real work is showing that uniform semisimplicity survives restriction, conjugation and induction (Props 3.1–3.8), so the closed-submodule argument goes through on the Fréchet–Stein side.\n\nUniform simplicity is presentation-dependent and strictly stronger than topological simplicity; the paper flags this immediately after Def 3.3 and supplies a Banach-level version (Thm 4.3) that can be checked level-by-level. That is not a hidden gap—it is the price of working with Fréchet–Stein algebras without Zorn. The two examples (Heisenberg characters and finite-dimensional representations of the Borel of SL2) are well-chosen: one produces admissible irreducibles, the other shows infinite multiplicity of an H0-summand, so the criterion is sharp.\n\nMath and citations look solid; everything rests on standard coadmissible-module facts. No circularity. This is a specialist tool for people already working with compact induction in the p-adic Langlands setting. It will not reorganise the subject, but it is the first systematic statement of this form and the proofs are complete. I would send it to referees without hesitation and would cite the criterion when I next need an irreducibility test for a compact induction.","headline":"Solid, usable Mackey criterion for compactly induced locally analytic representations; the uniform-semisimplicity hypothesis is explicit and the proofs check out.","tokens_in":20950,"tokens_out":440,"would_cite":true,"duration_ms":4148,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","11F70"],"pacs":[],"model":"grok-4.5","headline":"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.","keywords":["locally analytic representations","Mackey criterion","compact induction","distribution algebra","uniform semisimplicity","ind-admissible","p-adic groups"],"falsifier":"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.","tokens_in":21050,"feed_emoji":"∞","tokens_out":873,"duration_ms":6267,"temperature":0.7,"pith_summary":"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.","feed_headline":"Mackey test for irreducible p-adic inductions","feed_subtitle":"Uniform simplicity plus no intertwiners implies topological irreducibility of compact inductions","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Mackey criterion for locally analytic p-adic inductions","Compact inductions irreducible by uniform simplicity test","Mackey irreducibility for compactly induced p-adic reps","No intertwiners imply topological irreducibility of c-Ind","Locally analytic Mackey test from compact-mod-centre H"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Mackey criterion for locally analytic p-adic inductions","Compact inductions irreducible by uniform simplicity test","Mackey irreducibility for compactly induced p-adic reps","No intertwiners imply topological irreducibility of c-Ind","Locally analytic Mackey test from compact-mod-centre H"]},"model":"grok-4.5","effort":"low","cost_usd":0.004174,"raw_usage":{"total_tokens":1129,"prompt_tokens":554,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":41740000,"prompt_tokens_details":{"text_tokens":554,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":508,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":554,"tokens_out":67,"duration_ms":3777,"temperature":1.0,"reasoning_tokens":508,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T05:40:35.746701+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}