Single conjugacy classes of isometries in orthogonal groups over local fields

TL;DR

This paper classifies certain isometries in quadratic spaces over non-archimedean local fields, extending previous results to more general cases, and provides a detailed understanding of their conjugacy properties.

Contribution

It determines all isometries with unique conjugacy classes in the orthogonal group over local fields, generalizing prior theorems to broader settings.

Findings

Complete classification of isometries with unique conjugacy classes

Extension of Milgram's theorem to arbitrary cases

Enhanced understanding of conjugacy in orthogonal groups

Abstract

All isometries $\sigma$ in a quadratic space over a non-archimedean local field of characteristic not 2 satisfying that any isometry $\tau$ which is conjugate to $\sigma$ in the general linear group is conjugate to $\sigma$ in the orthogonal group are determined. This extends \cite[Theorem 2.1]{Mil} to arbitary cases.