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The Metaplectic Representation is Faithful

Christopher Chang, Simeon Hellsten, Mario Marcos Losada, Sergiu Novac

TL;DR

The paper develops a general framework for proving faithfulness of infinite-dimensional modules over Iwasawa algebras of uniform pro-$p$ groups, centering on the metaplectic representation for the symplectic group. It constructs a metaplectic map from the affinoid enveloping algebra to the completed Weyl algebra and shows injectivity of the restricted map on $KG$ by a Gluing Lemma that orchestrates faithfulness from subalgebras and controlled multiplication maps. The work also provides abstract criteria for faithfulness on abelian subalgebras via invariant-ideal analysis under a uniform automorphism group, illustrating that nontrivial invariant ideals have finite codimension. Collectively, these results yield injectivity of the metaplectic representation on $KG$ and demonstrate a robust method potentially extensible to other simple Lie algebras and highest-weight modules.

Abstract

We develop methods to show that infinite-dimensional modules over the Iwasawa algebra $KG$ of a uniform pro-p group are faithful and apply them to show that the metaplectic representation for the symplectic group is faithful.

The Metaplectic Representation is Faithful

TL;DR

The paper develops a general framework for proving faithfulness of infinite-dimensional modules over Iwasawa algebras of uniform pro- groups, centering on the metaplectic representation for the symplectic group. It constructs a metaplectic map from the affinoid enveloping algebra to the completed Weyl algebra and shows injectivity of the restricted map on by a Gluing Lemma that orchestrates faithfulness from subalgebras and controlled multiplication maps. The work also provides abstract criteria for faithfulness on abelian subalgebras via invariant-ideal analysis under a uniform automorphism group, illustrating that nontrivial invariant ideals have finite codimension. Collectively, these results yield injectivity of the metaplectic representation on and demonstrate a robust method potentially extensible to other simple Lie algebras and highest-weight modules.

Abstract

We develop methods to show that infinite-dimensional modules over the Iwasawa algebra of a uniform pro-p group are faithful and apply them to show that the metaplectic representation for the symplectic group is faithful.
Paper Structure (11 sections, 20 theorems, 82 equations, 4 figures)

This paper contains 11 sections, 20 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Constructing the Metaplectic Representation
  4. The Gluing Lemma
  5. Faithfulness of the Metaplectic Representation
  6. The Multiplication Map
  7. The subalgebra KA
  8. Final Gluing
  9. Abelian subalgebras
  10. Invariant ideals in commutative algebras
  11. Abelian subalgebras

Key Result

Theorem 2

The metaplectic representation ${ \left.\nulldelimiterspace \rho \newline \right|_{KG} }:KG\to\widehat{A_{n}(R)}_K$ is injective.

Figures (4)

  • Figure 1: Root space decomposition of $\mathfrak{sp}_4$, and image under $\rho$
  • Figure 2: Root space decomposition of $\mathfrak{sp}_6$, and image under $\rho$
  • Figure :
  • Figure :

Theorems & Definitions (35)

  • Conjecture 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Proposition 5: Metaplectic Representation
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Proposition 8: Gluing Lemma
  • ...and 25 more