Category O for p-adic rational Cherednik algebras

TL;DR

The paper develops a unified analytic framework for category O in the p-adic setting by introducing triangular decompositions for Banach and Fréchet-Stein algebras and constructing the associated analytic highest-weight category $\wideparen{\mathcal{O}}$. This framework is then specialized to p-adic rational Cherednik algebras, yielding a Fréchet-Stein presentation and an analytic category $\wideparen{\mathcal{O}}_c$ that is equivalent to the algebraic category $\mathcal{O}_c$ via base change, with simples parameterized by irreducible $G$-representations. The authors also develop the necessary analytic tools, including weight-space decompositions in Banach spaces, GAGA-type results for rigid-analytic quotients, and a precise Arens-Michael envelope theory linking algebraic and analytic Cherednik algebras. Overall, the work bridges algebraic representation theory of Cherednik algebras with p-adic analytic harmonic analysis, establishing a robust analytic category O with Verma and simple objects in correspondence with the finite group representations.

Abstract

We introduce the concept of a triangular decomposition for Banach and Fréchet-Stein algebras over $p$-adic fields, which allows us to define a category $\mathcal{O}$ for a wide array of topological algebras. In particular, we apply this concept to $p$-adic rational Cherednik algebras, which allows us to obtain an analytic version of the category $\mathcal{O}$ developed by Ginzburg, Guay, Opdam and Rouquier. Along the way, we study the global sections of $p$-adic Cherednik algebras on smooth Stein spaces, and determine their behavior with respect to the rigid analytic GAGA functor.

Theorems & Definitions (127)

• Theorem A
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• Theorem B
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• Theorem C
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• Theorem D
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• Theorem E
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