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Faithfulness of highest-weight modules for Iwasawa algebras in type D

Stephen Mann

Abstract

We prove that infinite-dimensional highest-weight modules are faithful for Iwasawa algebras corresponding to a simple Lie algebra of type D. We use this to prove that all non-zero two-sided ideals of the Iwasawa algebra have finite codimension in this case.

Faithfulness of highest-weight modules for Iwasawa algebras in type D

Abstract

We prove that infinite-dimensional highest-weight modules are faithful for Iwasawa algebras corresponding to a simple Lie algebra of type D. We use this to prove that all non-zero two-sided ideals of the Iwasawa algebra have finite codimension in this case.

Paper Structure

This paper contains 17 sections, 61 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Main approach
  3. Notation
  4. Background
  5. Preliminaries about Lie algebras
  6. Completions
  7. Induced modules
  8. Proving Theorem \ref{['maintheorem']}
  9. Outline
  10. Use of simple reflections
  11. Translation functors
  12. Inductive approach
  13. Inductive approach in type D
  14. Highest-weight vectors in generalised Verma modules
  15. The case with an abelian nilpotent subalgebra
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $G = \ker \left( SO_{2m}(\mathcal{O}_{F}) \rightarrow SO_{2m}(\mathcal{O}_{F} / p^{n+1} \mathcal{O}_{F}) \right)$ for some integers $n \geq 0$ and $m \geq 4$, and suppose $p \geq 5$. Then any infinite-dimensional affinoid highest-weight module for $KG$ is a faithful $KG$-module.

Theorems & Definitions (136)

  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 1
  • Theorem 2
  • Theorem 1.0.3
  • Definition 2.1.1
  • Definition 2.1.2
  • proof
  • Definition 2.2.1
  • Definition 2.2.2
  • ...and 126 more