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Publication# Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Résumé

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$. In this thesis, we investigate closed connected reductive subgroups $X < G$ that contain a given distinguished unipotent element $u$ of $G$. Our main result is the classification of all such $X$ that are maximal among the closed connected subgroups of $G$.

When $G$ is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where $G$ is simple of classical type, say $G = \operatorname{SL}(V)$, $G = \operatorname{Sp}(V)$, or $G = \operatorname{SO}(V)$. We begin by considering the maximal closed connected subgroups $X$ of $G$ which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where $X$ is the stabilizer of a tensor decomposition of $V$. For $p = 2$ and $X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$, we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015).

After the geometric subgroups, the maximal closed connected subgroups that remain are the $X < G$ such that $X$ is simple and $V$ is an irreducible and tensor indecomposable $X$-module. The bulk of this thesis is concerned with this case. We determine all triples $(X, u, \varphi)$ where $X$ is a simple algebraic group, $u \in X$ is a unipotent element, and $\varphi: X \rightarrow G$ is a rational irreducible representation such that $\varphi(u)$ is a distinguished unipotent element of $G$. When $p = 0$, this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015).

In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups $X$ of $G$ that contain a distinguished unipotent element $u$ of $G$. This leads us to consider connected reductive overgroups $X$ of $u$ which are contained in some proper parabolic subgroup of $G$. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when $u$ is a regular unipotent element of $G$, no such $X$ exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where $(G, p) = (C_2, 2)$, a connected reductive overgroup of a distinguished unipotent element of order $p$ cannot be contained in a proper parabolic subgroup of $G$.

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This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

We study (connected) reductive subgroups G of a reductive algebraic group H, where G contains a regular unipotent element of H. The main result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H = SL(n, F), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element.

2013Fix an algebraically closed field $K$ having characteristic $p\geq 0$ and let $Y$ be a simple algebraic group of classical type over $K.$ Also let $X$ be maximal among closed connected subgroups of $Y$ and consider a non-trivial $p$-restricted irreducible rational $KY$-module $V.$ In this thesis, we investigate the triples $(Y,X,V)$ such that $X$ acts with exactly two composition factors on $V$ and see how it generalizes a question initially investigated by Dynkin in the $1950$s and then studied by numerous mathematicians. In particular, we study the natural embeddings of $\mbox{SO}_{2n}(K)$ in both $\mbox{Spin}_{2n+1}(K)$ and $\mbox{SL}_{2n}(K)$ and obtain results on the structure of certain Weyl modules.