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

TL;DR

This paper classifies representations of certain infinite-dimensional orthogonal groups and simple Lie groups, showing the absence of exotic representations in higher rank cases and analyzing their topological simplicity.

Contribution

It provides a classification of representations of infinite-dimensional orthogonal groups with finite index and explores their topological properties, contrasting with rank one cases.

Findings

No exotic representations exist for higher rank cases.

Projective orthogonal groups are topologically simple but not abstractly simple.

Classification results extend understanding of infinite-dimensional orthogonal group representations.

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.