Maximal connected k-subgroups of maximal rank in connected reductive algebraic k-groups

TL;DR

This paper classifies maximal rank connected reductive subgroups within connected reductive algebraic groups over various fields, introducing invariants and equivalence relations to understand their conjugacy classes and embeddings.

Contribution

It introduces the embedding of indices as an invariant for classifying maximal rank subgroups and provides a classification for exceptional types over different fields.

Findings

Classification of embeddings of indices for exceptional groups.

Introduction of index-conjugacy as an invariant for subgroup conjugacy classes.

Existence results for embeddings over fields with specific cohomological properties.

Abstract

Let $k$ be any field and let $G$ be a connected reductive algebraic $k$-group. Associated to $G$ is an invariant first studied by Satake and Tits that is called the index of $G$ (a Dynkin diagram along with some additional combinatorial information). Tits showed that the $k$-isogeny class of $G$ is uniquely determined by its index and the $k$-isogeny class of its anisotropic kernel $G_a$. For the cases where $G$ is absolutely simple, Satake and Tits classified all possibilities for the index of $G$. Let $H$ be a connected reductive $k$-subgroup of maximal rank in $G$. We introduce an invariant of the $G(k)$-conjugacy class of $H$ in $G$ called the embedding of indices of $H$ in $G$. This consists of the index of $H$ and the index of $G$ along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of $k$-subgroups of $G$, and observe that the $G(k)$-conjugacy class of $H$ in $G$ is determined by its index-conjugacy class and the $G(k)$-conjugacy class of $H_a$ in $G$. We show that the index-conjugacy class of $H$ in $G$ is uniquely determined by its embedding of indices. For the cases where $G$ is absolutely simple of exceptional type and $H$ is maximal connected in $G$, we classify all possibilities for the embedding of indices of $H$ in $G$. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when $k$ has cohomological dimension $1$ (resp. $k=R$, $k$ is $p$-adic).