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

Abstract

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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Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).

Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n).

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