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Single conjugacy classes of isometries in orthogonal groups over local fields

Fei Xu, Bo Zhang

Abstract

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

Single conjugacy classes of isometries in orthogonal groups over local fields

Abstract

All isometries in a quadratic space over a non-archimedean local field of characteristic not 2 satisfying that any isometry which is conjugate to in the general linear group is conjugate to in the orthogonal group are determined. This extends \cite[Theorem 2.1]{Mil} to arbitary cases.
Paper Structure (4 sections, 13 theorems, 179 equations)

This paper contains 4 sections, 13 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Hermitian structures for non-reciprocal cases
  3. More complements about Milnor's classification of conjugacy classes of orthogonal groups
  4. Proof of main theorem

Key Result

Theorem 1.2

Let $V$ be a non-degenerated quadratic space over a non-archimedean local field $k$ with $char(k)\neq 2$. Suppose $\sigma\in {\rm O}(V)$ and $f(x)$ is the characteristic polynomial of $\sigma$ with the factorization of (factor). Write Then any isometry in ${\rm O}(V)$ which is conjugate to $\sigma$ in ${\rm GL}(V)$ is conjugate to $\sigma$ in ${\rm O}(V)$ if and only if one of the following condi

Theorems & Definitions (26)

  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 16 more