Totally Ramified Maximal Tori and Bruhat-Tits theory

TL;DR

This paper uses Bruhat-Tits theory to classify and describe the conjugacy and embedding properties of certain maximal tori in reductive groups over nonarchimedean local fields, providing explicit descriptions of their classes.

Contribution

It offers a complete and explicit classification of rational and stable conjugacy classes and embeddings of maximal tori in reductive groups over local fields, extending Bruhat-Tits theory.

Findings

Classification of stable classes of tori in G-orbits.

Explicit description of rational conjugacy classes of tori.

Detailed analysis of embeddings of tori into G.

Abstract

Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. If $\mathbf{T}$ is a $K$-minisotropic maximal $k$-torus in $\mathbf{G}$, then we use Bruhat-Tits theory to describe the stable classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, the rational classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of $\mathbf{T}$. We also provide, via Bruhat-Tits theory, a complete and explicit description of: the rational conjugacy classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; the stable classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of a $K$-minisotropic maximal tame $k$-torus of $\mathbf{G}$.