Maximal connected subgroups of algebraic k-groups
File(s)
Author(s)
Sercombe, Damian John
Type
Thesis
Abstract
Let $k$ be any field. Let $G$ be a connected reductive algebraic $k$-group. Associated to $G$ is an invariant that is called the index of $G$. For the cases where $G$ is absolutely simple, 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 pair $H<G$ that is called the \textit{embedding of indices of $H<G$}. This consists of the index of $H$ and the index of $G$ along with a map that satisfies some compatibility conditions. For the cases where $G$ is absolutely simple of exceptional type and $H$ is $k$-isotropic and maximal connected in $G$, we classify all possibilities for the embedding of indices of $H<G$. Furthermore, we consider which of these possibilities exist when $k$ has cohomological dimension $1$ (resp. $k=\R$, $k$ is $\mathfrak{p}$-adic).
We now specialise to the case where $k=\R$ and relax the requirement that $G$ is reductive. An unrefinable chain of $G$ is a chain of subgroups $G=G_0>G_1>...>G_t=1$ where each $G_i$ is a maximal connected real subgroup of $G_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $G$. We give a precise formula for the length of $G$, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If $G$ is simple then we bound the depth of $G$ above and below, and in many cases we compute the exact value. In particular, the depth of any simple $G$ is at most $9$.
We now specialise to the case where $k=\R$ and relax the requirement that $G$ is reductive. An unrefinable chain of $G$ is a chain of subgroups $G=G_0>G_1>...>G_t=1$ where each $G_i$ is a maximal connected real subgroup of $G_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $G$. We give a precise formula for the length of $G$, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If $G$ is simple then we bound the depth of $G$ above and below, and in many cases we compute the exact value. In particular, the depth of any simple $G$ is at most $9$.
Version
Open Access
Date Issued
2019-10
Date Awarded
2020-03
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Liebeck, Martin
Sponsor
Imperial College London
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)