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Irreducibility results for equivariant $\mathcal{D}$-modules on rigid analytic spaces

Konstantin Ardakov, Tobias Schmidt

TL;DR

The paper develops a robust geometric framework for irreducibility of equivariant D-modules on rigid analytic spaces and applies it to locally analytic representations. It introduces and analyzes induction, duality, and side-changing functors, proving that under suitable geometric and homological hypotheses, induced modules are simple and often self-dual. By leveraging Beilinson–Bernstein localisation, analytification, and the Orlik–Strauch correspondence, the authors produce geometric proofs of irreducibility for a broad class of locally analytic representations, including Schubert-variety–derived objects in both projective and full flag settings. The results unify representations-theoretic constructions with rigid-analytic D-module techniques, enabling new irreducibility criteria that extend beyond split groups to non-split cases and arbitrary primes, with significant implications for geometric representation theory in the p-adic setting.

Abstract

We prove a general irreducibility result for geometrically induced coadmissible equivariant $\mathcal{D}$-modules on rigid analytic spaces. As an application, we geometrically reprove the irreducibility of certain locally analytic representations previously constructed by Orlik-Strauch.

Irreducibility results for equivariant $\mathcal{D}$-modules on rigid analytic spaces

TL;DR

The paper develops a robust geometric framework for irreducibility of equivariant D-modules on rigid analytic spaces and applies it to locally analytic representations. It introduces and analyzes induction, duality, and side-changing functors, proving that under suitable geometric and homological hypotheses, induced modules are simple and often self-dual. By leveraging Beilinson–Bernstein localisation, analytification, and the Orlik–Strauch correspondence, the authors produce geometric proofs of irreducibility for a broad class of locally analytic representations, including Schubert-variety–derived objects in both projective and full flag settings. The results unify representations-theoretic constructions with rigid-analytic D-module techniques, enabling new irreducibility criteria that extend beyond split groups to non-split cases and arbitrary primes, with significant implications for geometric representation theory in the p-adic setting.

Abstract

We prove a general irreducibility result for geometrically induced coadmissible equivariant -modules on rigid analytic spaces. As an application, we geometrically reprove the irreducibility of certain locally analytic representations previously constructed by Orlik-Strauch.
Paper Structure (35 sections, 79 theorems, 231 equations)

This paper contains 35 sections, 79 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Induction and side-changing
  3. Induction and duality
  4. The main irreducibility result.
  5. The set of self-intersections
  6. Schubert varieties.
  7. Application to locally analytic representations.
  8. Some complements on $D$-modules
  9. Sheaves and supports
  10. Analytification
  11. Some complements to localization, induction and duality
  12. Intersection obstructions and inverse images
  13. Induction and restriction
  14. Induction and side-changing
  15. Induction and duality
  16. ...and 20 more sections

Key Result

Theorem A

Let $P\subseteq G$ be a closed cocompact subgroup. Let $\mathcal{N}\in \mathcal{C}_{\mathbf{X} /P}$. There is a natural isomorphism in ${^r\mathcal{C}}_{\mathbf{X} /G}$

Theorems & Definitions (157)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Proposition 2.1.1
  • proof
  • Proposition 2.2.1
  • proof
  • ...and 147 more