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arxiv: 2606.06243 · v1 · pith:PMZ6YJ57new · submitted 2026-06-04 · 🧮 math.RT · math.NT

Stability of the smooth Casselman-Jacquet functor

Kei Yuen Chan , Kaidi Wu , Jun Yu , Hongfeng Zhang This is my paper

Pith reviewed 2026-06-27 22:57 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords Casselman-Jacquet functorJacquet subspacesBernstein-Zelevinsky filtrationreal reductive groupssmooth Fréchet representationsmoderate growthrepresentation theory
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The pith

The smooth Casselman-Jacquet functor stabilizes on intersections of Jacquet subspaces for real reductive groups.

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

The paper proves several properties of the smooth Casselman-Jacquet submodule and quotient functors, including exactness, surjectivity, and globalization. Its central new result is a stability statement for the functors applied to intersections of Jacquet subspaces. This stability yields a complete form of the real Bernstein-Zelevinsky filtrations on smooth Fréchet representations of moderate growth. A reader would care because the filtrations give an organized description of how these representations decompose, extending the known p-adic picture to the real case. The work assumes standard definitions for the functors on real reductive groups.

Core claim

We establish a stability on the intersection of Jacquet subspaces for the smooth Casselman-Jacquet submodule and quotient functors. As an application, we establish a full version of the real Bernstein-Zelevinsky filtrations for smooth Fréchet representations of moderate growth rate.

What carries the argument

Stability on the intersection of Jacquet subspaces under the smooth Casselman-Jacquet functors.

Load-bearing premise

The groups are real reductive and the representations are smooth Fréchet representations of moderate growth rate, with the standard definitions and properties of the smooth Casselman-Jacquet functors holding.

What would settle it

An explicit counterexample on a low-rank group such as SL(2,R) where the image of the intersection of two Jacquet subspaces under the smooth Casselman-Jacquet functor fails to equal the intersection of the individual images.

read the original abstract

We establish and prove several results for the smooth Casselman-Jacquet submodule and quotient functors for real reductive groups. Other than exactness, surjectivity and globalization results, we establish a stability on the intersection of Jacquet subspaces. As an application, we establish a full version of the real Bernstein-Zelevinsky filtrations for smooth Fr\'echet representations of moderate growth rate.

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

0 major / 1 minor

Summary. The paper establishes exactness, surjectivity, and globalization properties for the smooth Casselman-Jacquet submodule and quotient functors on real reductive groups. It proves a stability result on intersections of Jacquet subspaces and applies this to obtain a full version of the real Bernstein-Zelevinsky filtrations for smooth Fréchet representations of moderate growth rate.

Significance. If the stability result holds, it supplies a useful technical tool for controlling filtrations in the smooth Fréchet category, extending prior Bernstein-Zelevinsky work to the real moderate-growth setting. The exactness and globalization statements would also be of independent interest for functorial constructions on real reductive groups.

minor comments (1)
  1. The abstract states the main results but does not indicate where the stability theorem is proved or how the moderate-growth hypothesis is used in the filtration construction; adding a brief outline or theorem numbering would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the paper and for recognizing the potential utility of the stability result for controlling filtrations in the smooth Fréchet category, as well as the independent interest of the exactness and globalization statements. The recommendation is listed as uncertain, but the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives stability of intersections of Jacquet subspaces and a full real Bernstein-Zelevinsky filtration from the standard exactness, surjectivity, and globalization properties of the smooth Casselman-Jacquet functor on real reductive groups acting on smooth Fréchet representations of moderate growth. These are taken as given from the literature on the functor (with the paper stating it verifies or uses them), without any reduction of the central claims to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain remains self-contained against external benchmarks in representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from the abstract regarding free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5583 in / 1016 out tokens · 24343 ms · 2026-06-27T22:57:50.291779+00:00 · methodology

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Reference graph

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