Parameterizing Conjugacy Classes of Unramified Tori via Bruhat-Tits Theory

TL;DR

This paper uses Bruhat-Tits theory to parameterize conjugacy classes of unramified tori and related subgroups in reductive groups over nonarchimedean local fields, extending to finite groups of Lie type.

Contribution

It introduces new parameterizations of conjugacy classes of unramified tori and subgroups in reductive groups using Bruhat-Tits theory, including finite groups of Lie type.

Findings

Parameterization of rational conjugacy classes of unramified tori in reductive groups.

Parameterization of conjugacy classes of unramified twisted Levi subgroups.

Extension of parameterizations to finite groups of Lie type.

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. In this paper we provide parameterizations via Bruhat-Tits theory of: the rational conjugacy classes of $k$-tori in $\mathbf{G}$ that split over $K$; the rational and stable conjugacy classes of the $K$-split components of the centers of unramified twisted Levi subgroups of $\mathbf{G}$; and the rational conjugacy classes of unramified twisted generalized Levi subgroups of $\mathbf{G}$. We also provide parameterizations of analogous objects for finite groups of Lie type.