Fréchet–Stein Algebras in p-adic Analysis

• Fréchet–Stein algebras are complete, locally convex non-Archimedean K-algebras constructed as projective limits of Banach algebras with strict flatness and density conditions.
• They provide a robust framework for analytic representation theory via coadmissible modules and the analytic Category O, paralleling classical highest weight theory.
• Their structure supports triangular decomposition and GAGA techniques, exemplified by the p-adic rational Cherednik algebra serving as a universal Fréchet–Stein completion.

A Fréchet–Stein algebra is a complete, locally convex non-Archimedean $K$-algebra that admits a presentation as a projective limit of Banach $K$-algebras with suitable flatness and density conditions. This concept enables a systematic approach to analytic representation theory over $p$-adic fields, in particular through the development of analytic analogues of algebraic structures such as Category $\mathcal{O}$. The framework naturally encompasses important examples, including $p$-adic rational Cherednik algebras, and generalizes powerful techniques familiar from complex and algebraic settings to rigid analytic geometry and non-Archimedean functional analysis (Vázquez, 23 Apr 2025).

1. Definition and Structure of Fréchet–Stein Algebras

A $K$-algebra $A$ with a Fréchet topology is a Fréchet–Stein algebra if there exists an inverse system $\{A_n\}_{n\geq 0}$ of two-sided Noetherian Banach $K$-algebras and continuous homomorphisms $\rho_{n+1,n}\colon A_{n+1}\to A_n$ such that:

• $K$0 as Fréchet topological $K$1-algebras.
• Each transition map $K$2 is two-sided flat, has dense image, and is strict as a map of Banach spaces.
• Each $K$3 is Noetherian on both sides and finitely generated over its center.

In practice, $K$4 is required to be a nuclear Fréchet space, owing to the “compact-type” property of the transition maps (in the sense of Schneider–Teitelbaum).

2. Coadmissible Modules and Their Properties

Given a Fréchet–Stein algebra $K$5, a left $K$6-module $K$7 is coadmissible if there exist finitely generated $K$8-modules $K$9 and topological $p$0-linear isomorphisms

$p$1

such that $p$2. The subcategory of coadmissible $p$3-modules, denoted $p$4, is independent of the choice of presentation $p$5 and is abelian.

This construction ensures compatibility with the non-Archimedean analytic context and is robust under passage to limits, providing an appropriate categorical setting for analytic representation theory.

3. Triangular Decomposition and Analytic Category $p$6

A triangular decomposition is an additional structure reflecting the direct-sum and grading properties critical to highest weight theory. For a Banach $p$7-algebra $p$8, a triangular decomposition is a tuple $p$9 with:

• A dense graded subalgebra $\mathcal{O}$0 graded by $\mathcal{O}$1 via $\mathcal{O}$2.
• Graded subalgebras $\mathcal{O}$3, with $\mathcal{O}$4 (finite-dimensional semisimple), and decompositions $\mathcal{O}$5, $\mathcal{O}$6.
• An isomorphism of Banach spaces $\mathcal{O}$7.
• $\mathcal{O}$8 and $\mathcal{O}$9 admit finite-type, semisimple weight-space decompositions for $p$0 and are two-sided Noetherian Banach algebras, stable under $p$1.

A Fréchet–Stein algebra $p$2 admits a triangular decomposition if each $p$3 does, compatible with the transition maps. This structure enables definition of an analytic Category $p$4 as the full subcategory of $p$5 with modules finitely generated over the positive subalgebra and admitting finite-type weight decompositions for $p$6.

4. The $p$7-adic Rational Cherednik Algebra as a Fréchet–Stein Algebra

Given a finite-dimensional $p$8-vector space $p$9 and $K$0, let $K$1 denote the set of reflections and $K$2 a $K$3-invariant function. The algebraic rational Cherednik algebra $K$4 is constructed as: $K$5 with associated Dunkl–Opdam filtration satisfying $K$6.

Transitioning to the non-Archimedean analytic framework, one defines the analytic rational Cherednik algebra $K$7 as the closure of $K$8 in $K$9, with the explicit presentation: $A$0 where each Banach algebra $A$1 is a $A$2-adic completion corresponding to bounded disks, and

$A$3

with $A$4 and $A$5 (Tate algebras). Hence, $A$6 is itself a Fréchet–Stein algebra admitting a triangular decomposition (Vázquez, 23 Apr 2025).

5. The Analytic Category $A$7 and Its Properties

For $A$8 with a triangular decomposition $A$9, the analytic Category $\{A_n\}_{n\geq 0}$0 is defined as: $\{A_n\}_{n\geq 0}$1 The main properties include:

• $\{A_n\}_{n\geq 0}$2 is an abelian Serre subcategory of $\{A_n\}_{n\geq 0}$3, closed under closed subobjects and finite direct sums.
• It forms a highest-weight category, with standard (Verma) objects $\{A_n\}_{n\geq 0}$4 for $\{A_n\}_{n\geq 0}$5, simple heads $\{A_n\}_{n\geq 0}$6, and block decomposition by “$\{A_n\}_{n\geq 0}$7-eigenvalues”.
• Each Verma module $\{A_n\}_{n\geq 0}$8 has a unique maximal closed submodule, leading to unique simple quotients with a bijection to $\{A_n\}_{n\geq 0}$9.
• Objects in $K$0 possess analytic Verma filtrations and Jordan–Hölder series, analogously to the algebraic case (Vázquez, 23 Apr 2025).

6. GAGA, Arens–Michael Envelopes, and Algebra–Analytic Correspondence

A comparison arises between the algebraic rational Cherednik algebra $K$1 and its analytic counterpart $K$2:

• The canonical homomorphism

$K$3

is faithfully flat with dense image.

• Under rigid-analytic GAGA, $K$4, and $K$5 intertwines global sections for algebraic and analytic sheaves of Cherednik algebras.
• The analytic algebra $K$6 is the Arens–Michael envelope of $K$7—the universal Fréchet–Stein completion.
• The analytic Category $K$8 is equivalent to the completion of the algebraic Category $K$9: $\rho_{n+1,n}\colon A_{n+1}\to A_n$0 These results collectively yield a $\rho_{n+1,n}\colon A_{n+1}\to A_n$1-adic analogue of the familiar triangular decomposition and Category $\rho_{n+1,n}\colon A_{n+1}\to A_n$2 machinery from the complex-analytic and algebraic contexts, with the rational Cherednik algebra as the prototypical example (Vázquez, 23 Apr 2025).