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Representations of infinite dimension orthogonal groups of quadratic forms with finite index

Bruno Duchesne

Abstract

We study representations $G\to H$ where $G$ is either a simple Lie group with real rank at least 2 or an infinite dimensional orthogonal group of some quadratic form of finite index at least 2 and $H$ is such an orthogonal group as well. The real, complex and quaternionic cases are considered. Contrarily to the rank one case, we show that there is no exotic such representations and we classify these representations. On the way, we make a detour and prove that the projective orthogonal groups $\mathop{PO}_\mathbf{K}(p,\infty)$ or their orthochronous component (where $\mathbf{K}$ denotes the real, complex or quaternionic numbers) are Polish groups that are topologically simple but not abstractly simple.

Representations of infinite dimension orthogonal groups of quadratic forms with finite index

Abstract

We study representations where is either a simple Lie group with real rank at least 2 or an infinite dimensional orthogonal group of some quadratic form of finite index at least 2 and is such an orthogonal group as well. The real, complex and quaternionic cases are considered. Contrarily to the rank one case, we show that there is no exotic such representations and we classify these representations. On the way, we make a detour and prove that the projective orthogonal groups or their orthochronous component (where denotes the real, complex or quaternionic numbers) are Polish groups that are topologically simple but not abstractly simple.

Paper Structure

This paper contains 8 sections, 25 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Symmetric spaces of infinite dimension
  3. Isometry groups of infinite dimensional Riemannian symmetric spaces of finite rank
  4. Full isometry group
  5. Topological simplicity
  6. Actions on infinite dimensional symmetric spaces
  7. Representations of finite dimensional simple Lie groups of rank at least 2
  8. Representations of $\mathop{\mathrm{PO}}\nolimits_\mathbf{K}(p,\kappa)$ for $p\geq2$.

Key Result

Theorem 1.1

Let $G$ be a connected simple non-compact Lie group with trivial center. Let $G\to\mathop{\mathrm{PO}}\nolimits_\mathbf{K}(p,\infty)$ be a continuous representation without totally isotropic invariant subspace. If the real rank of $G$ is at least 2 then the underlying Hilbert space $\mathcal{H}$ spl

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 42 more