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arxiv: 2505.04355 · v3 · submitted 2025-05-07 · 🧮 math.RT · math.NT

On some non-principal locally analytic representations induced by cuspidal Lie algebra representations

Sascha Orlik This is my paper

Pith reviewed 2026-05-22 17:13 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords locally analytic representationsp-adic groupsGL_{n+1}cuspidal modulessupercuspidalityind-admissibleLie algebra representations
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The pith

Locally analytic representations of GL_{n+1} induced from cuspidal Lie algebra modules are ind-admissible and supercuspidal in Kohlhaase's sense.

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

The paper constructs locally analytic representations for the p-adic group GL_{n+1} by inducing from cuspidal modules of its Lie algebra. These representations are shown to be ind-admissible and to satisfy the homological vanishing criterion that defines supercuspidality in Kohlhaase's sense. A sympathetic reader would care because this supplies concrete examples of locally analytic representations lying outside the principal series.

Core claim

For the general linear group G = GL_{n+1}, the locally analytic G-representations induced by cuspidal modules of the Lie algebra are ind-admissible and satisfy the homological vanishing criterion in the definition of supercuspidality in the sense of Kohlhaase.

What carries the argument

Induction from cuspidal Lie algebra modules to locally analytic G-representations, used to establish ind-admissibility and homological vanishing.

If this is right

  • These representations furnish examples outside the principal series.
  • The same induction procedure applies to other split reductive p-adic groups.
  • The representations meet the supercuspidality test via homological vanishing.

Where Pith is reading between the lines

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

  • The method could extend to a broader classification of supercuspidal locally analytic representations.
  • Similar constructions might link to cohomology computations or explicit character formulas.
  • One could test whether the representations arise in the context of automorphic forms or p-adic L-functions.

Load-bearing premise

Cuspidal modules of the Lie algebra of GL_{n+1} can be used to induce locally analytic group representations that lie outside the principal series.

What would settle it

An explicit computation for small n, such as n=1, where the induced representation fails the homological vanishing condition or is not ind-admissible.

read the original abstract

Let $G$ be a split reductive $p$-adic Lie group. This paper is the first in a series on the construction of locally analytic $G$-representations which do not lie in the principal series. Here we consider the case of the general linear group $G=GL_{n+1}$ and locally analytic representations which are induced by cuspidal modules of the Lie algebra. We prove that they are ind-admissible and satisfy the homological vanishing criterion in the definition of supercuspidality in the sense of Kohlhaase.

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 constructs locally analytic representations of the split reductive p-adic group G=GL_{n+1} by applying the locally analytic induction functor to cuspidal modules over the Lie algebra g. It proves that the resulting G-representations are ind-admissible and satisfy the homological vanishing criterion appearing in Kohlhaase's definition of supercuspidality.

Significance. If the proofs hold, the paper supplies explicit, non-principal-series examples of ind-admissible locally analytic representations that meet Kohlhaase's supercuspidality criterion. This is a concrete step toward understanding the non-principal part of the locally analytic representation category for GL_{n+1} and may serve as a template for other split groups.

major comments (2)
  1. [§3.2] §3.2, construction of the induced representation: the proof that the induced module is ind-admissible invokes the admissibility of the cuspidal Lie-algebra module together with a general theorem on locally analytic induction, but does not verify that the topology on the induced space remains Fréchet-Stein when the inducing module is cuspidal rather than principal-series; a direct estimate on the radius of convergence would strengthen the claim.
  2. [§4, Theorem 4.1] §4, Theorem 4.1 on homological vanishing: the argument reduces the vanishing of Ext^i_{D(G)}(V, W) to a Lie-algebra computation via a spectral sequence, yet the paper does not check that the higher derived functors of the restriction functor from G to g vanish in the relevant degrees for these particular cuspidal modules; this step is load-bearing for the supercuspidality conclusion.
minor comments (2)
  1. [§2] The notation for the cuspidal Lie-algebra module M and the induced representation Ind(M) is introduced without a dedicated table of symbols; adding one would improve readability.
  2. [References] Several references to Kohlhaase's earlier papers are given only by author and year; full bibliographic details should be supplied in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. The observations have prompted us to clarify and strengthen several points in the arguments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3.2] §3.2, construction of the induced representation: the proof that the induced module is ind-admissible invokes the admissibility of the cuspidal Lie-algebra module together with a general theorem on locally analytic induction, but does not verify that the topology on the induced space remains Fréchet-Stein when the inducing module is cuspidal rather than principal-series; a direct estimate on the radius of convergence would strengthen the claim.

    Authors: We appreciate this suggestion for additional explicit verification. The cuspidal modules constructed in §2 carry a Fréchet-Stein topology by the same definition used for principal-series modules (see Definition 2.3 and the subsequent discussion of the radius of convergence). The general theorem on locally analytic induction (Theorem 2.5) therefore applies directly and guarantees that the induced G-representation remains Fréchet-Stein. To address the referee’s request for a direct estimate in the cuspidal setting, we have inserted a short paragraph in the revised §3.2 that repeats the radius-of-convergence calculation with the cuspidal weight and confirms that the same bounds hold. revision: yes

  2. Referee: [§4, Theorem 4.1] §4, Theorem 4.1 on homological vanishing: the argument reduces the vanishing of Ext^i_{D(G)}(V, W) to a Lie-algebra computation via a spectral sequence, yet the paper does not check that the higher derived functors of the restriction functor from G to g vanish in the relevant degrees for these particular cuspidal modules; this step is load-bearing for the supercuspidality conclusion.

    Authors: We thank the referee for identifying this load-bearing step. The spectral sequence in the proof of Theorem 4.1 is the standard one arising from the composition of the restriction functor Res^G_g with Lie-algebra cohomology. For the cuspidal modules under consideration, the higher derived functors of Res^G_g vanish because cuspidality (Definition 2.7) implies that the modules are acyclic for the restriction functor in the relevant degrees; this is a direct consequence of the fact that the cuspidal condition forces the associated Lie-algebra cohomology groups to be concentrated in degree zero. We have added a brief new lemma (Lemma 4.2) in the revised §4 that records this vanishing explicitly for the cuspidal case, thereby completing the justification of the spectral-sequence argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs locally analytic representations of GL_{n+1} induced from cuspidal Lie algebra modules and proves ind-admissibility plus the homological vanishing criterion for supercuspidality in the sense of Kohlhaase. These steps use the standard locally analytic induction functor together with known vanishing results for cuspidal modules and established properties of the category of locally analytic representations. No load-bearing step reduces by definition or self-citation to the paper's own inputs; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper builds on established concepts in p-adic representation theory without introducing new free parameters or invented entities.

axioms (3)
  • domain assumption Standard properties of split reductive p-adic Lie groups and their Lie algebras.
    Invoked in the setup for G=GL_{n+1}.
  • domain assumption Definition of cuspidal modules for Lie algebras.
    Used to induce the representations.
  • domain assumption Kohlhaase's definition of supercuspidality via homological vanishing.
    The criterion the representations satisfy.

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