# Overgroups of regular unipotent elements in simple algebraic groups

## Abstract

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

## 1. Introduction

Let be a simple linear algebraic group defined over an algebraically closed field. The regular unipotent elements of are those whose centraliser has minimal possible dimension (the rank of and these form a single conjugacy class which is dense in the variety of unipotent elements of ) The main result of our paper is a contribution to the study of positive-dimensional subgroups of . which meet the class of regular unipotent elements. Since any parabolic subgroup must contain representatives from every unipotent conjugacy class, the question arises only for reductive, not necessarily connected subgroups, where we establish the following:

In addition, we show that for many simple groups there exists a closed reductive subgroup , with a torus and such that meets the class of regular unipotent elements of (See Proposition .7.2 and Examples 7.7, 7.11.) Finally, we go on to consider subgroups of non-simple almost simple algebraic groups where there is a well-defined notion of regular unipotent elements in unipotent cosets of We establish the corresponding result in this setting; see Corollary .6.2.

The investigation of the possible overgroups of regular unipotent elements in simple linear algebraic groups has a long history. The *maximal* closed positive-dimensional reductive subgroups of which meet the class of regular unipotent elements were classified by Saxl and Seitz Reference 17 in 1997. In earlier work, see Reference 21, Thm 1.9, Suprunenko obtained a particular case of their result. In order to derive from the Saxl–Seitz classification an inductive description of all closed positive-dimensional reductive subgroups containing regular unipotent elements, one needs to exclude that any of these can lie in proper parabolic subgroups. For connected this was shown by Testerman and Zalesski in Reference 22, Thm 1.2 in 2013. They then went on to determine all connected reductive subgroups of simple algebraic groups which meet the class of regular unipotent elements. Our result generalises Reference 22, Thm 1.2 to the disconnected case and thus makes the inductive approach possible. It is worth pointing out that the analogous result is no longer true even for simple subgroups once one relaxes the condition of positive-dimensionality. For example, there exist reducible indecomposable representations of the group whose image in the corresponding contains a matrix with a single Jordan block, i.e., the image meets the class of regular unipotent elements in In .Reference 3, Burness and Testerman consider of exceptional type simple algebraic groups which meet the class of regular unipotent elements and show that with the exception of two precise configurations, such a subgroup does not lie in a proper parabolic subgroup of -subgroups (see Reference 3, Thms 1 and 2).

Our proof of Theorem 1 relies on the result of Testerman–Zalesski Reference 22 in the connected case, which actually implies our theorem in characteristic 0 (see Remark 2.1) as well as on results of Saxl–Seitz Reference 17 classifying almost simple irreducible and tensor indecomposable subgroups of classical groups containing regular unipotent elements and maximal reductive subgroups in exceptional groups with this property. For the exceptional groups we also use information on centralisers of unipotent elements and detailed knowledge of Jordan block sizes of unipotent elements acting on small modules, as found in Lawther Reference 6. For establishing the existence of positive-dimensional reductive subgroups with , a torus, and meeting the class of regular unipotent elements, we produce subgroups which centralise a non-trivial unipotent element and hence necessarily lie in a proper parabolic subgroup of (See .Reference 15, Thm 17.10, Cor. 17.15.)

After collecting some useful preliminary results we deal with the case of in Section 3, with the orthogonal case in Section 4, and with the simple groups of exceptional type in Section 5. The case of almost simple groups is deduced from the connected case in Corollary 6.2. Finally, in Section 7 we discuss the case when is a torus.

## 2. Preliminary results

In this paper we consider almost simple algebraic groups defined over an algebraically closed field of characteristic and investigate closed positive-dimensional subgroups that contain a regular unipotent element. For us, throughout “algebraic group” will mean “linear algebraic group”, and all vector spaces will be finite-dimensional vector spaces over An algebraic group . is called an *almost simple algebraic group* if is simple and embeds into Thus, . is an extension of by a subgroup of its group of graph automorphisms (see, e.g., Reference 15, Thm 11.11). As a matter of convention, a “reductive subgroup” of an algebraic group will always mean a closed subgroup whose unipotent radical is trivial. In particular, a reductive group may be disconnected. For an algebraic group we write , to denote the unipotent radical of Throughout, all . are rational, as are all extensions, and cohomology groups are those associated to rational cocycles. -modules

Let us point out that for the question treated here, the precise isogeny type of the ambient simple algebraic group will not matter, as isogenies preserve parabolic subgroups as well as regular unipotent elements. (If is almost simple and does not divide the order of the fundamental group of the natural map , induces an isogeny of onto its adjoint quotient, preserving regular unipotent elements in in the general case, a reduction to ; of adjoint type is given in Reference 18, I.1.7.) In particular, for a classical type simple algebraic group we will argue for the groups , and and for the groups of type , and defined over of characteristic we may choose to work with whichever group is more convenient under the given circumstances. ,

We start by making two useful observations which will simplify the later analysis.

### 2.1. Jordan forms and tensor products

The following elementary fact will be used throughout (see also Reference 17, Lemma 1.3(i)):

Before establishing a useful consequence of Lemma 2.4, we recall the following well-known Clifford-theoretic result, see, e.g., Reference 2, Prop. 2.6.2:

We now show the desired corollary of Lemma 2.4:

The proof of the next result is modelled after the proof of Reference 17, Prop. 2.1 which treats a more special situation:

### 2.2. On subgroups containing regular unipotent elements

For connected groups, the following result from Reference 22, Lemma 2.6 will be useful:

We will make frequent use of the following result, the second part of which was essentially shown by Saxl and Seitz Reference 17, Prop. 2.2:

We will obtain a similar classification for unipotent elements in with a Jordan block of size when in Proposition 4.1.

### 2.3. Jordan forms and orders of regular unipotent elements

While the notion of regular unipotent element is well known for connected reductive groups, this is much less so for non-connected reductive groups. Still, similar results hold.

Let be a not necessarily connected reductive algebraic group and let be unipotent. Spaltenstein Reference 18, p. 41 and II.10.1 has shown (generalising a result of Steinberg in the connected case) that the coset of the connected component contains a unipotent class -conjugacy that is dense in the variety of unipotent elements of called the class of ,*regular unipotent elements of *. Since this variety is irreducible Reference 18, Cor. I.1.6, is also the unique class of unipotent elements in of maximal dimension.

We now describe the Jordan block structure of regular unipotent elements in classical type almost simple groups on their natural representation. For us, the *natural representation* for the extension of , by its graph automorphism of order 2 is defined by its embedding into the stabiliser in , of a pair of complementary totally singular subspaces, with acting in its natural representation, respectively its dual, on these subspaces. We do not consider or as being of classical type.

Since the natural representations are faithful, the above result also allows one to read off the orders of regular unipotent elements of almost simple classical groups.

### 2.4. Some results on extensions

We conclude this preparatory section by collecting some basic properties on Ext-groups. We state the following well-known result for future reference (see Reference 24, Prop. 3.3.4).

We thank Jacques Thévenaz for pointing out the following result:

## 3. The case of

In this section we prove Theorem 1 for those classical type simple algebraic groups for which regular unipotent elements have a single Jordan block on their natural module (see Lemma 2.12). We are in the following situation: is a (not necessarily connected) reductive subgroup of the form for a regular unipotent element of We also assume that . is not a torus; this case will be considered in Section 7.1. We will show that cannot be contained in a proper parabolic subgroup of that is, , acts irreducibly on For this, we may whenever convenient, assume that . is semisimple, since if is not contained in a proper parabolic subgroup of then neither is As . has a single Jordan block on if , are subspaces, then -invariant has a single Jordan block on .

### 3.1. The completely reducible case

We first deal with the case when is completely reducible for .