Stability of the smooth Casselman-Jacquet functor

Technical Summary: Stability of the Smooth Casselman-Jacquet Functor

Problem Statement
The paper addresses the development of a comprehensive theory for the Bernstein-Zelevinsky (BZ) filtrations for smooth Fréchet representations of moderate growth rate on real reductive groups, specifically general linear groups $GL_n(\mathbb{R})$ and $GL_n(\mathbb{C})$. While the BZ filtrations are well-established for $p$-adic groups, their extension to the real case faces significant topological obstructions. The classical Casselman-Jacquet functor, originally defined for Harish-Chandra modules, behaves differently in the context of smooth Fréchet representations due to the lack of total disconnectedness in real groups and the necessity of handling topological closures. The primary challenge is to establish exactness, surjectivity, and stability properties for these functors within the category of moderate growth representations ($Rep^{\infty, F}(G)$), which is quasi-abelian but not abelian.

Methodology
The authors employ a combination of functional analysis and representation theory, distinguishing their approach from previous works on Harish-Chandra modules or earlier studies on Jacquet modules (e.g., [AGS15a, AGS15b]). Key methodological components include:

1. Topological Refinements: The authors define the Casselman-Jacquet functors with explicit topological closures to ensure the resulting quotient spaces are Hausdorff. They utilize the quotient topology and projective tensor products of Fréchet spaces to analyze the structure of these functors.
2. Schwartz Algebra Actions: A central technique involves the continuous action of the Schwartz algebra $S(\mathfrak{n})$ (where $\mathfrak{n}$ is the Lie algebra of the unipotent radical) on representations of moderate growth. By leveraging the Dixmier-Malliavin theorem, the authors establish the surjectivity of the action map $S(\mathfrak{n}) \hat{\otimes} \pi \to \pi$.
3. Semi-norms and Stability: The paper constructs specific families of semi-norms on the Casselman-Jacquet modules. These semi-norms are used to prove "stability" properties, specifically that the intersection of subspaces $\bigcap_{k \geq 0} \mathfrak{n}^k \cdot \pi$ is stable under the action of $\mathfrak{n}$ (i.e., $\mathfrak{n} \cdot CJ^\infty_{s,P}(\pi) = CJ^\infty_{s,P}(\pi)$).
4. Artin-Rees Lemma: For the subcategory of Casselman-Wallach representations, the authors prove a topological version of the Artin-Rees lemma. This lemma is crucial for establishing the exactness of the functors, ensuring that intersections of submodules behave predictably with respect to powers of the nilpotent ideal.
5. Imprimitivity Theorem: In the context of mirabolic subgroups, the authors apply the imprimitivity theorem (from [dCl91]) to analyze the structure of Schwartz inductions and the functors $\Phi^+$ and $\Phi^-$.

Key Contributions and Results

• Definition and Properties of Functors: The authors define two functors for a parabolic subgroup $P$ with unipotent radical $N$ and Lie algebra $\mathfrak{n}$:

  • The submodule functor: $CJ^\infty_{s,P}(\pi) = \bigcap_{k \geq 0} \mathfrak{n}^k \cdot \pi$.
  • The quotient functor: $CJ^\infty_P(\pi) = \varprojlim \pi / \mathfrak{n}^k \cdot \pi$.
    They prove that the natural map $\pi \to CJ^\infty_P(\pi)$ is surjective, yielding a short exact sequence $0 \to CJ^\infty_{s,P}(\pi) \to \pi \to CJ^\infty_P(\pi) \to 0$.

• Stability and Epiness: A major result is the stability of the submodule functor: $\mathfrak{n} \cdot CJ^\infty_{s,P}(\pi) = CJ^\infty_{s,P}(\pi)$. Furthermore, the functor $CJ^\infty_{s,P}$ is shown to be "epi" (preserving surjectivity in the quasi-abelian sense), meaning it maps epimorphisms to epimorphisms.

• Exactness for Casselman-Wallach Representations: When restricted to the category of Casselman-Wallach representations (admissible, finite length, moderate growth), both $CJ^\infty_P$ and $CJ^\infty_{s,P}$ are proven to be exact functors. This relies on the topological Artin-Rees lemma and the fact that the categories involved become abelian in this restricted setting.

• Real Bernstein-Zelevinsky Filtrations: As a primary application, the paper constructs a full version of the Bernstein-Zelevinsky filtration for smooth Fréchet representations of $GL_n$. This involves defining functors $\Phi^-$ (a variant of the Jacquet functor) and $\Phi^+$ (a variant of parabolic induction) and proving they are adjoint. The authors establish that any representation $\pi$ admits a filtration $0 = \pi_n \subset \dots \subset \pi_0 = \pi$ where the successive subquotients are related to the images of these functors and are Hausdorff.

• Globalization: The paper establishes a globalization result, showing that the Casselman-Jacquet modules for smooth Fréchet representations are isomorphic to the Casselman-Wallach globalization of the corresponding Harish-Chandra modules.

Significance and Claims
The paper claims to provide the first comprehensive mathematical proof of the Bernstein-Zelevinsky filtration for real general linear groups within the framework of moderate growth representations. The authors emphasize that their results complement and extend previous work (such as [AGS15a, AGS15b]) by addressing topological difficulties inherent to real groups that are absent in the $p$-adic case.

The significance lies in:

1. Bridging the Gap: Successfully adapting the $p$-adic BZ filtration machinery to the real world, overcoming the topological differences (total disconnectedness vs. connectedness).
2. Topological Rigor: Providing a rigorous treatment of the topology of Jacquet modules, ensuring they are Hausdorff and well-behaved under the operations of the category.
3. Foundation for Branching Laws: The work is presented as a foundational step toward a comprehensive proof of branching laws for real general linear groups, following the blueprint established for $p$-adic groups. The authors note that the stability and exactness results are prerequisites for computing higher structures and simple quotients in future work.

The paper remains modest regarding immediate computational applications, noting that while the theoretical framework is established, the actual computation of simple quotients and higher structures will be facilitated by tools from $p$-adic groups in forthcoming works (e.g., [CW26, CP25+]).