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

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).