Overgroups of regular unipotent elements in simple algebraic groups

By Gunter Malle and Donna M. Testerman

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:

Theorem 1.

Let be a simple linear algebraic group over an algebraically closed field, a closed reductive subgroup containing a regular unipotent element of . If , then lies in no proper parabolic subgroup of .

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 -subgroups 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 (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 -modules are rational, as are all extensions, and cohomology groups are those associated to rational cocycles.

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.

Remark 2.1.

In the situation of Theorem 1, assume that . As is finite, some power of a regular unipotent element will lie in . In characteristic 0 any power of a regular unipotent element is again regular unipotent, so here we are thus reduced to studying the connected reductive subgroup satisfying the same assumptions. In that case, the conclusion of Theorem 1 was established in Reference 22, Thm 1.2. Hence, in proving Theorem 1 we may assume whenever convenient. Furthermore, we will assume without loss of generality that .

Remark 2.2.

Let be a reductive subgroup of a connected reductive group such that is regular unipotent in and . Let be one of the simple components of and set . Then , so acts transitively on the set of simple components of , and if lies in a proper parabolic subgroup of , then so does . Thus, when proving Theorem 1 we may as well assume that the simple components of are permuted transitively by .

2.1. Jordan forms and tensor products

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

Lemma 2.3.

Assume that and let be unipotent with a single Jordan block. Write with . Then has Jordan blocks, of size  and the other of size .

Lemma 2.4.

Let be a unipotent element with a single Jordan block of size , or with two Jordan blocks of sizes or . If preserves the factors in a non-trivial tensor product decomposition of then and has two Jordan blocks on . If these are of sizes .

Proof.

Using the description of Jordan block sizes of unipotent elements in tensor products given in Reference 17, Lemma 1.5 we see that necessarily and either has Jordan block sizes , or and has Jordan block sizes , as claimed.

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:

Lemma 2.5.

Let be groups with finite cyclic and be a finite-dimensional irreducible -module. Then , where is any irreducible -submodule of and runs over a system of coset representatives of the stabiliser of in . Moreover, the are pairwise non-isomorphic -modules.

We now show the desired corollary of Lemma 2.4:

Lemma 2.6.

Let be connected reductive with non-trivial derived subgroup and assume that is completely reducible and homogeneous. If is normalised by a unipotent element with a single Jordan block of size , or with two Jordan blocks of sizes or then either is an irreducible -module, or , preserves a non-trivial tensor product decomposition of , and has two Jordan blocks on , of sizes if .

Proof.

Let be the unipotent element normalising as in the assumption. Let be an irreducible -submodule of . As is homogeneous as an -module, Lemma 2.5 shows that is irreducible. Since , we have . Then with we have as an -module, and this decomposition is stabilised by (see e.g. Reference 15, Prop. 18.1). Applying Lemma 2.4, this implies that either , whence is irreducible for , or we are in the exceptional case of that result, as in the conclusion.

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

Lemma 2.7.

Assume and let be a reductive subgroup and a unipotent element with a single Jordan block of size , or with two Jordan blocks of sizes or . If acts irreducibly on , then either is simple, or one of the following holds:

(1)

, preserves a non-trivial tensor decomposition of  and has two Jordan blocks on . If these are of size ;

(2)

, as an -module with , , permutes both sets of factors transitively and has a single Jordan block on . Moreover, has a single Jordan block on each ; or

(3)

, as an -module with , has Jordan blocks of sizes on and has a single Jordan block on each .

Here, in and , does not preserve the stated tensor product decomposition of .

Proof.

Note that as and acts irreducibly. Write with simple algebraic groups , so with non-trivial irreducible -modules . Now permutes the factors and their corresponding tensor factors . Assume that . If has at least two orbits on the set of , this yields a corresponding -invariant tensor decomposition of . By Lemma 2.4 we reach case (1).

Henceforth, we may assume that permutes the , and thus the , transitively. In particular all have the same dimension , that is, , and is a power of . Let be minimal with . Now, stabilises all , so is a tensor product of matrices of size and thus of order at most . Hence divides . On the other hand,

The above conditions imply that either , and , or , and .

In the first case, , , and as all simple factors of must have type , as in (2). The statement about the Jordan form of follows from Lemma 2.3.

In the second case we have , and our inequalities force that and hence has Jordan blocks of sizes or and by Lemma 2.3, has Jordan blocks of sizes , respectively . The latter cannot arise as the block sizes of a tensor product of two unipotent matrices by Reference 17, Lemma 1.5, so we are in the former case and has a single Jordan block on each , as in (3).

2.2. On subgroups containing regular unipotent elements

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

Lemma 2.8.

Let be connected reductive, a parabolic subgroup with Levi complement and assume that is regular unipotent in . Then the image of is regular unipotent in and hence in each simple factor of .

Lemma 2.9.

Let be simple and be a connected reductive subgroup normalised by a regular unipotent element of . Assume that is a central product with such that acts by an inner automorphism on . Then contains a regular unipotent element of .

Proof.

By assumption, acts as an inner automorphism on , say by . Thus, and are contained in and so . But then Reference 22, Prop. 2.3 implies that . Now is regular unipotent. Replacing and by their unipotent parts respectively, we may assume both to be unipotent and lying in a common Borel subgroup of . As centralises and thus isn’t regular, must be regular by Reference 22, Lemma 2.4.

Lemma 2.10.

Let be simple and a connected reductive subgroup containing a regular unipotent element of . Then is regular in .

Proof.

Let be a Borel subgroup of containing . Assume is not regular unipotent in . By Reference 19, Ch. III, 1.13 it may be written as a product of root elements

where the first product runs over a proper subset of the simple roots of the root system of with respect to the pair where is some maximal torus, and the second one over the roots in of height at least 2. Thus, lies in the unipotent radical of the parabolic subgroup of whose Levi factor is generated by the root subgroups for the simple roots not occurring in the representation of and their negatives, which thus is not a torus.

By Borel–Tits, then also lies in the unipotent radical of a proper parabolic subgroup of with non-toral Levi factor. But then, when writing as a product of root elements for with respect to a Borel subgroup contained in , not all simple roots can occur, whence is not regular in . This contradiction achieves the proof.

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:

Proposition 2.11.

Let be a reductive subgroup of the simple classical group , or with simple and irreducible on , where when is of orthogonal type. If contains a regular unipotent element of , then acts tensor indecomposably on .

Furthermore, either , or and the highest weight of on are as in Table 1 (up to Frobenius twists and taking duals) and has a single Jordan block.

Proof.

Assume that for non-trivial irreducible -modules . If , then Lemma 2.4 gives that and , but this is not simple. If some power acts as an inner element on , then centralises , hence, as acts irreducibly, we must have that equals the unipotent part of and so lies in . So now we may assume that and hence (), () or , with , or and . Unipotent elements in have order at most 32, but there is no faithful representation of of dimension less than 54 (the 27-dimensional modules for are not invariant under the graph automorphism). Thus is of classical type. Let denote the dimension of its natural module. Then, e.g., by Reference 12, Tab. 2 we have , so , but unipotent elements of have order less than . Thus . Note that with cannot occur, as here unipotent elements have order at most  (see Lemma 2.12). This only leaves the possibility , , and . But there is no 9-dimensional irreducible orthogonal module in characteristic 2, so in fact must be tensor indecomposable for . The remaining assertions are now shown in 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 -conjugacy class 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.

Lemma 2.12.

Let be almost simple of classical type with regular unipotent in . Then in the natural representation of :

(a)

has a single Jordan block for , (when ), , and for when ;

(b)

has two Jordan blocks of sizes when for ;

(c)

has two Jordan blocks of sizes when , respectively of sizes when for ;

(d)

has a single Jordan block of size when for with odd; and

(e)

has two Jordan blocks of sizes when for with even.

Proof.

Only (d) and (e) are not shown in Reference 17, Lemma 1.2. Spaltenstein Reference 18, I.2.7, I.2.8(c) gives a description of the unipotent classes in in terms of the Jordan normal form of the square of the elements on the natural -module, and a formula for the centraliser dimension. From this it can be seen that elements with minimal centraliser dimension are those for which has one Jordan block of size if is odd, and two blocks of sizes if is even. Thus, in the natural -dimensional orthogonal representation of , the element has two Jordan blocks of size , respectively four of sizes . Given the possible Jordan block shapes of unipotent elements in in its natural representation Reference 18, I.2.6, the claim for follows with Lemma 2.3.

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).

Lemma 2.13.

Let be a group, and with , , be -modules. Then

We thank Jacques Thévenaz for pointing out the following result:

Lemma 2.14.

Let be a field, be groups and two finite-dimensional -modules on which acts trivially. Assume that . Then

Proof.

We use the (exact) inflation-restriction sequence for cohomology (see Reference 24, 6.8.3) for a -module :

which by Reference 24, 6.1.2 can be interpreted as the -sequence

Applying this with and using (see Reference 1, Cor. 1) the previous sequence becomes

As acts trivially on and , the first term equals , while the third is by our hypothesis on and Lemma 2.13, whence exactness of the sequence implies our claim.

Lemma 2.15.
(a)

Let be a semisimple algebraic group. Then there are no non-trivial self-extensions between irreducible -modules.

(b)

Let be a semisimple group acting on , with acting trivially on . Then

Proof.

Part (a) is Reference 4, II.2.12(1). In (b) by Lemma 2.13 we have

As acts trivially on , the first summand is isomorphic to

by two applications of Lemma 2.14, and similarly for the second summand.

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 -invariant subspaces, then has a single Jordan block on .

3.1. The completely reducible case

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

Lemma 3.1.

Let be a reductive subgroup of the form for a regular unipotent element of , such that is not a torus. Assume that