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p-adic Cherednik algebras on rigid analytic spaces

Fernando Peña Vázquez

Abstract

Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X/G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small étale site of the quotient $X/G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fréchet $K$-algebras on $X/G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof. Furthermore, their sections on small enough $G$-invariant affinoid spaces are canonically Fréchet-Stein algebras. Along the way, we construct sheaves of infinite order twisted differential operators on $X$, we give a $G$-equivariant classification of the Atiyah algebras (Picard algebroids) on $X$, and study the category of co-admissible modules over a sheaf of infinite order twisted differential operators.

p-adic Cherednik algebras on rigid analytic spaces

Abstract

Let be a smooth rigid space with an action of a finite group satisfying that is represented by a rigid space. We construct sheaves of -adic Cherednik algebras on the small étale site of the quotient , and study some of their properties. The sheaves of -adic Cherednik algebras are sheaves of Fréchet -algebras on , which can be regarded as -adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof. Furthermore, their sections on small enough -invariant affinoid spaces are canonically Fréchet-Stein algebras. Along the way, we construct sheaves of infinite order twisted differential operators on , we give a -equivariant classification of the Atiyah algebras (Picard algebroids) on , and study the category of co-admissible modules over a sheaf of infinite order twisted differential operators.

Paper Structure

This paper contains 18 sections, 56 theorems, 414 equations.

Table of Contents

  1. Introduction
  2. Twisted differential operators on rigid analytic spaces
  3. Basic constructions
  4. Classification of Atiyah algebras on smooth rigid analytic spaces
  5. Sheaves of twisted differential operators on smooth rigid analytic spaces
  6. Group actions on twisted differential operators I
  7. Group actions on twisted differential operators II
  8. Extension to the étale site
  9. Cherednik algebras on rigid analytic spaces
  10. Cherednik algebras on rigid analytic spaces
  11. The presheaf of Cherednik algebras
  12. Rational Cherednik algebras
  13. Complete localization of Cherednik algebras
  14. The PBW Theorem for Cherednik algebras
  15. Basics on Adic spaces
  16. ...and 3 more sections

Key Result

Theorem A

Let $U\subset X^{\operatorname{reg}}$, and assume $U$ contains the Shilov boundary of $X$. Then $\mathcal{H}_{t,c,\omega}(X,G)_U$ is a Fréchet-Stein algebra, and is independent of $U$.

Theorems & Definitions (195)

  • Theorem A
  • proof
  • Theorem B
  • proof
  • Remark 2.1.4
  • Lemma 2.1.5
  • proof
  • Lemma 2.1.6
  • proof
  • Lemma 2.1.7
  • ...and 185 more