On relative cuspidality

Nadir Matringe

arxiv: 2506.08393 · v2 · pith:3JPFQPZAnew · submitted 2025-06-10 · 🧮 math.RT

On relative cuspidality

Nadir Matringe This is my paper

Pith reviewed 2026-05-22 01:23 UTC · model grok-4.3

classification 🧮 math.RT

keywords relative cuspidalitysymmetric pairsreductive groupsp-adic fieldscuspidal representationsθ-split torusBeuzart-Plessis proof

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The pith

Under a torus anisotropy assumption, G(F) admits strongly relatively cuspidal representations.

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

The paper establishes that symmetric pairs of reductive groups over a p-adic field with p not equal to 2 admit strongly relatively cuspidal representations. It reaches this by extending Beuzart-Plessis' earlier proof of ordinary cuspidal representation existence, under the given assumption on the torus. The result confirms expectations previously stated by Kato and Takano. A reader would care because the existence provides a concrete relative counterpart to classical cuspidality, which is used in harmonic analysis and the study of automorphic forms on these groups.

Core claim

Let (G, H) be a symmetric pair of reductive groups over a p-adic field F with p ≠ 2, attached to an involution θ. Under the assumption that there exists a maximally θ-split torus in G, which is anisotropic modulo its intersection with the split component of G, the group G(F) admits strongly relatively cuspidal representations. This is shown by extending Beuzart-Plessis' proof of existence of cuspidal representations to the relative setting.

What carries the argument

Extension of Beuzart-Plessis' proof of existence of cuspidal representations applied to strongly relatively cuspidal representations in the symmetric pair setting.

If this is right

Confirms expectations of Kato and Takano on relative cuspidality.
Gives a relative version of the existence theorem for cuspidal representations.
Applies to symmetric pairs attached to an involution θ over p-adic fields with odd residue characteristic.

Where Pith is reading between the lines

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

The result may support constructions of relative discrete series in broader symmetric pair contexts.
Similar existence statements could be explored when the anisotropy condition is relaxed or replaced by other tori properties.

Load-bearing premise

There exists a maximally θ-split torus in G which is anisotropic modulo its intersection with the split component of G.

What would settle it

An explicit symmetric pair satisfying the torus assumption for which G(F) has no strongly relatively cuspidal representations would disprove the existence claim.

read the original abstract

Let $(\mathbb{G},\mathbb{H})$ be a symmetric pair of reductive groups over a $p$-adic field with $p\neq 2$, attached to the involution $\theta$. Under the assumption that there exists a maximally $\theta$-split torus in $\mathbb{G}$, which is anisotropic modulo its intersection with the split component of $\mathbb{G}$, we extend Beuzart-Plessis' proof of existence of cuspidal representations, and prove that $\mathbb{G}(F)$ admits strongly relatively cuspidal representations. This confirms expectations of Kato and Takano.

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

Matringe extends Beuzart-Plessis to prove existence of strongly relatively cuspidal representations under the standard torus assumption.

read the letter

Matringe extends Beuzart-Plessis to prove existence of strongly relatively cuspidal representations under the standard torus assumption. The paper takes the known argument for ordinary cuspidality and adapts it to the relative case for a symmetric pair (G, H) over a p-adic field with p odd. Under the assumption of a maximally θ-split torus that is anisotropic modulo its intersection with the split component, it concludes that G(F) admits strongly relatively cuspidal representations. This matches the expectations noted by Kato and Takano and is presented as a direct extension rather than a new method from scratch.

Referee Report

0 major / 3 minor

Summary. The manuscript considers symmetric pairs (𝔾, 𝔥) of reductive groups over a p-adic field F with p ≠ 2, arising from an involution θ. Under the assumption that there exists a maximally θ-split torus in 𝔾 which is anisotropic modulo its intersection with the split component of 𝔾, the authors extend Beuzart-Plessis' proof to show that 𝔾(F) admits strongly relatively cuspidal representations. This result confirms expectations of Kato and Takano regarding relative cuspidality.

Significance. If the result holds, it advances the understanding of the relative cuspidal spectrum for symmetric pairs in p-adic groups by providing a conditional existence theorem. The extension of Beuzart-Plessis' established argument to the relative setting is a strength, as it builds directly on prior techniques without introducing circularity or ad-hoc parameters. This contributes to the development of relative harmonic analysis and aligns with expectations in the literature on symmetric varieties.

minor comments (3)

§1: The definition of 'strongly relatively cuspidal' is referenced but not restated; including a brief self-contained definition or precise citation to the standard reference would improve readability for readers unfamiliar with the relative setting.
Introduction: The comparison with Beuzart-Plessis' original argument could be expanded with one or two sentences outlining the key adaptations (e.g., handling of matrix coefficients or test functions in the relative case).
References: Ensure the works of Kato-Takano and Beuzart-Plessis are cited with full bibliographic details; the current citation pattern appears incomplete for the expectations mentioned in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. We are pleased that the extension of Beuzart-Plessis' techniques to the relative setting, under the stated assumption on the maximally θ-split torus, is viewed as advancing the understanding of the relative cuspidal spectrum.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an existence result for strongly relatively cuspidal representations that is explicitly conditional on the given assumption about a maximally θ-split torus and proceeds by extending Beuzart-Plessis' prior independent argument for ordinary cuspidality. No load-bearing step reduces by the paper's own equations or self-citation to its inputs; the central claim remains an extension of external prior work rather than a self-definitional or fitted-input construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on a domain assumption about the existence of a specific torus and on extending a known proof method from prior literature; no free parameters or new entities are introduced.

axioms (1)

domain assumption There exists a maximally θ-split torus in G which is anisotropic modulo its intersection with the split component of G
Explicitly required in the abstract as the condition under which the proof extension holds.

pith-pipeline@v0.9.0 · 5604 in / 1051 out tokens · 67093 ms · 2026-05-22T01:23:51.270945+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Under the assumption that there exists a maximally θ-split torus in G, which is anisotropic modulo its intersection with the split component of G, we extend Beuzart-Plessis' proof of existence of cuspidal representations, and prove that G(F) admits strongly relatively cuspidal representations.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The main ingredient to show that the space of relatively cuspidal functions on hzg is not reduced to zero is our assumption that G'-elliptic tori exist in G. More precisely we use that the regular elements of their Lie algebra form an open subset of hzg, which we prove in Theorem 2.5

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.