Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields \@slowromancapiii@

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 3

Abstract

Let be a nonperfect separably closed field. Let be a (possibly non-connected) reductive group defined over . We study rationality problems for Serre’s notion of complete reducibility of subgroups of . In our previous work, we constructed examples of subgroups of that are -completely reducible but not -completely reducible over (and vice versa). In this paper, we give a theoretical underpinning of those constructions. To illustrate our result, we present a new such example in a non-connected reductive group of type in characteristic . Then using Geometric Invariant Theory, we generalize the theoretical result above obtaining a new result on the structure of -(and -) orbits in an arbitrary affine -variety. We translate our result into the language of spherical buildings to give a new topological view. A problem on centralizers of completely reducible subgroups and a problem concerning the number of conjugacy classes are also considered.

Keywords: algebraic groups, geometric invariant theory, complete reducibility, rationality, spherical buildings

1 Introduction

Let be a field. We write for an algebraic closure of . Let be a (possibly non-connected) affine algebraic -group: we regard as a -defined algebraic group together with a choice of -structure in the sense of Borel [9, AG. 11]. We say that is reductive if the unipotent radical of is trivial. Throughout, is always a (possibly non-connected) reductive -group. In this paper, we continue our study of rationality problems for complete reducibility of subgroups of  [37][34]. By a subgroup of we mean a (possibly non--defined) closed subgroup of . If a subgroup of needs to be -defined (or needs to be connected) in some statement, we explicit say so. Recall [26, Sec. 3]

Definition 1.1.

A subgroup of is called -completely reducible over (-cr over for short) if whenever is contained in a -defined -parabolic subgroup of , then is contained in a -defined -Levi subgroup of . In particular if is not contained in any proper -defined -parabolic subgroup of , is called -irreducible over (-ir over for short).

We define -parabolic subgroups and -Levi subgroups in the next section (Definition 2.2). These concepts are essential to extend the notion of complete reducibility (initially defined only for subgroups of connected  [26, Sec. 3]) to subgroups of non-connected  [3][4, Sec. 6]. We defined complete reducibility for a possibly non--defined subgroup of . This is because for a subgroup of , some closely related important subgroups of are not necessarily -defined even if is -defined. For example, centralizers or normalizers of -subgroups of are not necessarily -defined; see [37, Thm. 1.2 and Thm. 1.7] for such examples. If is connected and is a subgroup of , our notion of complete reducibility agrees with the usual one of Serre.

Definition 1.2.

Let be an algebraic extension of . We say that a subgroup of is -completely reducible over (-cr over for short) if whenever is contained in a -defined -parabolic subgroup of , is contained in a -defined -Levi subgroup of . In particular, if is not contained in any proper -defined -parabolic subgroup of , is -irreducible over (-ir over for short). We simply say that is -cr (-ir for short) if is -cr over (-ir over ).

So far, most studies on complete reducibility is for complete reducibility over only; see [19][29][30] for example. Not much is known on complete reducibility over (especially for nonperfect ) except a few theoretical results and important examples in [4, Sec. 5][1][37][34]. In particular, in [36, Thm. 1.10] [35, Thm. 1.8][37, Thm. 1.2], we found several examples of -subgroups of that are -cr over but not -cr (and vice versa). The main result of this paper is to give a theoretical underpinning for our (possibly somewhat mysterious) construction of those examples. For an algebraic extension of and an affine group , we denote the set of -points of by . We write for where .

Theorem 1.3.

Let be a nonperfect separably closed field. Suppose that a subgroup of is -cr but not -cr over . Let be a minimal -defined -parabolic subgroup of containing . Then there exists a unipotent element such that is -cr over .

Theorem 1.4.

Let be a nonperfect separably closed field. Suppose that a subgroup of is -cr over but not -cr. Let be a minimal -defined -parabolic subgroup of containing . Then

  1. is -ir over for some -defined -Levi subgroup of .

  2. Moreover, there exists an element such that is not -cr over .

To illustrate our theoretical results (Theorems 1.31.4) and ideas in the proofs, we present a new example of a -subgroup of that is -cr over but not -cr (and vice versa).

Theorem 1.5.

Let be a nonperfect separably closed field of characteristic . Let be a simple -group of type . Let be a non-trivial element in the graph automorphism of . Let . Then there exists a -subgroup of that is -cr over but not -cr (and vice versa).

A few comments are in order. First, the non-perfectness of is (almost) essential in Theorems 1.31.4, and 1.5 in view of the following [5, Thm. 1.1]:

Proposition 1.6.

Let be connected. Let be a subgroup of . Then is -cr over if and only if is -cr over .

So in particular if is perfect and is connected, a subgroup of is -cr over if and only if it is -cr. The forward direction of Proposition 1.6 holds for non-connected . The reverse direction depends on the recently proved center conjecture of Tits [26][31][21] in spherical buildings, but this method does not work for non-connected ; the set of -parabolic subgroups does not form a simplicial complex in the usual sense of Tits [32] as we have shown in [34, Thm. 1.12]. In the following we assume that is separably closed. So every maximal -torus of splits over , thus is -split. This simplifies arguments in many places. For the theory of complete reducibility over arbitrary , see [1][2].

Second, note that the -definedness of in Theorem 1.5 is important. Actually it is not difficult to find a -subgroup with the desired property. For our construction to work, it is essential for to be nonseparable in . We write or for the Lie algebra of . Recall [7, Def. 1.1]

Definition 1.7.

A subgroup of is nonseparable if the dimension of is strictly smaller than the dimension of (where acts on via the adjoint action). In other words, the scheme-theoretic centralizer of in (in the sense of [13, Def. A.1.9]) is not smooth.

We exhibit the importance of nonseparability of in the proof of Theorem 1.5. Nonseparable -subgroups of are hard to find, and only a handful examples are known [7, Sec. 7][36, Thm. 1.10] [35, Thm. 1.8][37, Thm. 1.2]. Note that if characteristic of is very good for connected , every subgroup of is separable [7, Thm. 1.2]. Thus, to find a nonseparable subgroup we are forced to work in small (at least for connected ). See [7][16] for more on separability.

Our second main result is a generalization of Theorems 1.3 and 1.4 using the language of Geometric Invariant Theory (GIT for short) [22]. Let be a (possibly non-connected) affine -variety. When acts on -morphically, we say that is a -variety. One of the main themes of GIT is to study the structure of -orbits (and -orbits) in  [18][8][1]. Recently studies on completely reducibility (over ) via GIT have been very fruitful; GIT gives a very short and uniform proof for many results [4][8][1]. This makes a striking contrast to traditional representation theoretic methods (which depend on long case-by-case analyses) [19][29][30].

We recall the following algebro-geometric characterization for complete reducibility (over ) via GIT ([4, Prop. 2.16, Thm. 3.1] and [1, Thm. 9.3]). This turns problems on complete reducibility into problems on the structure of -(or -) orbits. Let be a subgroup of such that for some and . Suppose that (and ) acts on via simultaneous conjugation.

Proposition 1.8.

is -cr if and only if is Zariski closed in . Moreover, is -cr over if and only if is cocharacter closed over .

The definition of a cocharacter closed orbit is given in the next section (Definition 2.7). Using Proposition 1.8 and various techniques from GIT we can sometimes generalize results on complete reducibility (over ) to obtain new results on GIT where (or ) acts on an arbitrary affine -variety rather than on some tuple of ; see [8][1] for example. We follow the same philosophy here and generalize Theorems 1.3 and 1.4.

Theorem 1.9.

Let be nonperfect. Suppose that there exists such that is Zariski closed but is not cocharacter closed over . Let be the set of -cocharacters of destabilizing over . Pick such that is minimal among -parabolic subgroups for . Then there exists a unipotent element such that is cocharacter closed over .

Theorem 1.10.

Let be nonperfect. Suppose that there exists such that is cocharacter closed over but is not Zariski closed. Let be the set of -cocharacters of destabilizing over . Pick such that is minimal among -parabolic subgroups for . Then

  1. There exists such that and fixes .

  2. Any -defined cocharacter of destabilizing over is central in .

  3. Moreover, there exists an element such that is not cocharacter closed over .

Roughly speaking, we say that is destabilized over by a -cocharacter of if is taken outside of by taking a limit of along in the sense of GIT [18][22]; see Definition 2.8 for the precise definition. Note that if is perfect, Theorems 1.9 and 1.10 have no content: in that case is cocharacter closed if and only if is Zariski closed [1, Cor. 7.2, Prop. 7.4].

To complement the paper we also investigate the structure of centralizers of completely reducible subgroups of . In particular we ask [34, Open Problem 1.4]:

Open Problem 1.11.

Suppose that a -subgroup of is -cr over . Is -cr over ?

We have some partial answer [34, Thm. 1.5]:

Proposition 1.12.

Let be connected. Suppose that a -subgroup of is -cr over . If is reductive, then it is -cr over .

We need the connectedness assumption in Proposition 1.12 since it depends on the center conjecture of Tits. If (or more generally if is perfect), the answer to Open Problem 1.11 is “yes” by [4, Cor. 3.17] (and Proposition 1.6). A trouble arises for nonperfect since is not necessarily reductive even if is -cr over  [37, Rem. 3.11]. This does not happen if by [4, Prop. 3.12] that depends on a deep result of Richardson [24, Thm. A]. The reductivity of was crucial in the proof of [4, Cor. 3.17] to apply a tool from GIT.

In general, if is nonperfect, even if a -subgroup of is -cr over , it is not necessarily reductive  [37, Prop. 1.10]. This pathology happens because the classical construction of Borel-Tits [11, Prop. 3.1] fails over nonperfect ; see [37, Sec. 3.2]. This does not happen if ; a -cr subgroup is always reductive [26, Prop. 4.1].

Here is our third main result in this paper. Let be connected. Fix a maximal -torus of . We write for an automorphism of which normalizes and induces on (the set of positive roots of ). It is known that for of not type , , or , and for of type , , or where is the longest element of the Weyl group of and is a suitable graph automorphism of (cf. [19, Proof of Thm. 4.1]).

Theorem 1.13.

Let be connected. Suppose that a semisimple -subgroup of is -cr over . Let be a minimal -parabolic subgroup containing , and be a -Levi subgroup of . If the automorphism of extends to an automorphism of (in particular if is not of type , , or ), then is -cr over .

Here is the structure of the paper. In Section 2, we set out the notation and show some preliminary results. Then in Section 3, we prove our first main result (Theorems 1.3 and 1.4). In Section 4, we present the example (Theorem 1.5). In Section 5, we generalize Theorems 1.3 and 1.4 and prove our second main result (Theorems 1.9 and 1.10). In Section 6, we translate Theorems  1.3 and 1.4 into the language of spherical buildings, and prove Theorems 6.4 and 6.5. This gives a new topological perspective for the rationality problems for complete reducibility and GIT. Then in Section 7, we attack Open Problem 1.11, and prove Theorem 1.13 in a purely combinatorial way. In Section 8, we consider a problem on the number of conjugacy classes and prove Theorem 8.1. We note that nonseparability comes into play in a crucial way in the proof of Theorem 8.1.

2 Preliminaries

Throughout, we denote by a separably closed field. Our references for algebraic groups are [9][10][13][17], and [28].

Let be a (possibly non-connected) affine algebraic group. We write for the identity component of . It is clear that if is -defined, is -defined. We write for the derived group of . A reductive group is called simple as an algebraic group if is connected and all proper normal subgroups of are finite. We write and ( and ) for the set of -characters and -cocharacters (-characters and -cocharacters) of respectively. For -characters and -cocharacters we simply say characters and cocharacters of .

Fix a maximal -torus of (such a exists by [9, Cor. 18.8]). Then splits over since is separably closed. Let denote the set of roots of with respect to . We sometimes write for . Let . We write for the corresponding root subgroup of . We define . Let . Let be the coroot corresponding to . Then is a -homomorphism such that for some . Let denote the reflection corresponding to in the Weyl group of . Each acts on the set of roots by the following formula [28, Lem. 7.1.8]: By [12, Prop. 6.4.2, Lem. 7.2.1] we can choose -homomorphisms so that

We recall the notions of -parabolic subgroups and -Levi subgroups from [25, Sec. 2.1–2.3]. These notions are essential to define -complete reducibility for subgroups of non-connected and also to translate results on complete reducibility into the language of GIT; see [3] and [4, Sec. 6].

Definition 2.1.

Let be a affine -variety. Let be a -morphism of affine -varieties. We say that exists if there exists a -morphism (necessarily unique) whose restriction to is . If this limit exists, we set .

Definition 2.2.

Let . Define

We call an -parabolic subgroup of , an -Levi subgroup of . Note that the unipotent radical of . If is -defined, , , and are -defined [25, Sec. 2.1-2.3]. Any -defined parabolic subgroups and -defined Levi subgroups of arise in this way since is separably closed. It is well known that . Note that -defined -Levi subgroups of a -defined -parabolic subgroup of are -conjugate [8, Lem. 2.5(iii)]. Let be a reductive -subgroup of . Then, there is a natural inclusion of -cocharacter groups. Let . We write or just for the -parabolic subgroup of corresponding to , and for the -parabolic subgroup of corresponding to . It is clear that and . If is connected, -parabolic subgroups and -Levi subgroups are parabolic subgroups and Levi subgroups in the usual sense [28, Prop. 8.4.5].

The next result is used repeatedly to reduce problems on -complete reducibility to those on -complete reducibility where is an -Levi subgroup of .

Proposition 2.3.

Suppose that a subgroup of is contained in a -defined -Levi subgroup of . Then is -cr over if and only if it is -cr over .

Proof.

This follows from Proposition 1.8 and [1, Thm. 5.4(ii)]. ∎

The next result shows how complete reducibility behaves under central isogenies.

Definition 2.4.

Let and be reductive -groups. A -isogeny is central if is central in where is the differential of at the identity of and is the Lie algebra of .

Proposition 2.5.

Let and be reductive -groups. Let and be subgroups of and be subgroups of and respectively. Let be a central -isogeny.

  1. Suppose that and (in particular if is connected). If is -cr over , then is -cr over .

  2. Suppose that and . If is -cr over , then is -cr over .

Proof.

Proposition 1.8 and [1, Cor. 5.3] show that a subgroup of a reductive is -cr over if and only if it is -cr over . Now the result follows from the connected case [37, Prop. 1.12]. ∎

Remark 2.6.

In Proposition 2.5 if we know that a -defined -parabolic subgroup of always arises as the inverse image of a -defined -parabolic subgroup of , then a similar argument as in the proof of [37, Prop. 1.12] goes through and we can omit the assumptions “ and ” in Part 1 and “ and ” in Part 2. We do not know this is the case or not.

Now we recall some terminology form GIT [1, Def. 1.1, Sec. 2.4]. Let be a -variety. Let .

Definition 2.7.

We say that is cocharacter closed over if for every such that exists, is -conjugate to . Moreover, we say that is cocharacter closed if for every cocharacter of such that exists, is -conjugate to .

Note that by the Hilbert-Mumford theorem [18], is cocharacter closed if and only if it is Zariski closed.

Definition 2.8.

Let . We say that destabilize over if exists. Moreover if exists and is not -conjugate to , we say that properly destabilizes over . Similarly, for , if exists, we say that destabilize . If exists for and is not -conjugate to , we say that properly destabilizes .

We use the following very useful results from GIT [8, Thm. 3.3] and [1, Cor. 5.1].

Proposition 2.9.

Let be perfect. Suppose that exists for and is -conjugate to . Then is -conjugate to .

For nonperfect , we do not know whether Proposition 2.9 still holds [1, Question 7.8]. It is known that if the centralizer of in is separable, it holds for nonperfect  [1, Thm. 7.1]. See [1] and [2] for details.

Proposition 2.10.

Suppose that exists for and is -conjugate to . If is cocharacter closed over , then is -conjugate to .

3 -cr over vs -cr

We prove theorems 1.3 and 1.4. Our proof works for both connected and non-connected in a uniform way.

Proof of Theorem 1.3.

Since is not -cr over , there exists a proper -defined -parabolic subgroup of containing . Let be a minimal such -defined -parabolic subgroup where . Since is -cr and -Levi subgroups of are -conjugate by [4, Cor. 6.7], there exists such that is contained in . Then is contained in . Suppose that is not -cr over . Then it is not -cr over by Proposition 2.3. So there exists a proper -defined -parabolic subgroup of containing . Thus is contained in a -defined -parabolic subgroup of . Then is contained in . Note that . Thus and we have . It is clear that is strictly contained in . This contradicts the minimality of . So we conclude that is -cr over . ∎

Proof of Theorem 1.4.

We start with Part 1. Let be a minimal -defined -parabolic subgroup of containing . Since is -cr over , there exists a -defined -Levi subgroup of containing . Since -defined -Levi subgroups of are -conjugate, there exists such that . Set . Suppose that is not -ir over . So there exists a -defined proper -parabolic subgroup of containing . Then we have . Since is a -defined -parabolic subgroup of strictly contained in , this contradicts the minimality of .

For part 2, let be a minimal -defined -Levi subgroup containing . Since is not -cr, there exists a proper -parabolic subgroup of containing . Let be a minimal such -parabolic subgroup of . Since an -parabolic subgroup of is -conjugate to a -defined -parabolic subgroup of , there exists such that is a -defined -parabolic subgroup of . Then . Suppose that is -cr over . Then is -cr over by Proposition 2.3, so there exist a -defined -Levi subgroup of containing . Note that is not -cr since is not -cr. Then there exists a proper -parabolic subgroup of containing . Thus is an -parabolic subgroup of containing . Then is an -parabolic subgroup of containing . It is clear that is a proper subgroup of . This contradicts the minimality of . Thus is not -cr over . ∎

Remark 3.1.

Although Theorems 1.3 and 1.4 (and ideas in the proofs) explain necessary conditions to have examples of a subgroup of that is -cr over but not -cr (or vice versa), it is still a difficult problem to find concrete such examples with a -defined . In the next section, we use the converse of Theorems 1.3 and 1.4: we start with some subgroup of and conjugate it by (or ) as in the proof of Theorems 1.3 and 1.4 to obtain a subgroup with the desired property. For our construction to work, (or ) needs to be chosen very carefully and the choice is closely related to the nonseparability of . We show all details in the next section. The same idea was used in [7][37][35], and [36].

4 The example

In this section we prove Theorem 1.5. We use the triality of in an essential way.

Let be a simple algebraic group of type defined over a nonperfect field of characteristic . Fix a maximal -torus of and a -defined Borel subgroup of . let be the set of roots corresponding to , and be the set of positive roots of corresponding to and . The following Dynkin diagram defines the set of simple roots of .

Let where is the non-trivial element of the graph automorphism group of (normalizing and ) as the diagram defines; we have , and is fixed by . We label in the following. The corresponding negative roos are defined accordingly. Note that Roots 1, 2, 3, 4 correspond to , , , respectively.

1

0
1 0 0

2

1
0 0 0

3

0
0 0 1

4

0
0 1 0

5

0
1 1 0

6

1
0 1 0

7

0
0 1 1

8

1
1 1 0

9

0
1 1 1

10

1
0 1 1

11

1
1 1 1

12

1
1 2 1

Define . Then

Let . Let . Define

Here is our first main result in this section.

Proposition 4.1.

is -defined. Moreover, is -cr but not -cr over .

Proof.

First, we have Using this and the commutation relations [17, Lem. 32.5 and Prop. 33.3], we obtain

An easy computation shows that commutes with . Now it is clear that is -defined.

Now we show that is -cr. It is sufficient to show that is -cr since it is -conjugate to . Since is contained in , by Proposition 2.3 it is enough to show that is -cr. We actually show that is -ir. Note that . We have

Thus contains , , and . Now it is clear that is -ir.

Next, we show that is not -cr over . Suppose the contrary. Clearly is contained in a -defined -parabolic subgroup . Then there exists a -defined -Levi subgroup of containing . Then by [8, Lem. 2.5(iii)] there exists such that is contained in . Thus . So . By [28, Prop. 8.2.1], we set

We compute how acts . Using the labelling of the positive roots above, we have . We compute how acts on :

(4.1)

Using this and the commutation relations,

Thus if we must have

Set . Then we have . Thus . This is impossible since and . We are done. ∎

Remark 4.2.

From the computations above we see that the curve is not contained in , but the corresponding element in , that is, is contained in . Then the argument in the proof of [36, Prop. 3.3] shows that is strictly smaller than . So is non-separable in .

Now we move on to the second main result in this section. We use the same , , and as above. We also use the same labelling of the roots of . Let . Let

Define

Proposition 4.3.

is -defined. Moreover, is -ir over but not -cr.

Proof.

is clearly -defined. First, we show that is -ir over . Note that

Thus we see that is contained in . So is contained in .

Lemma 4.4.

is the unique proper -parabolic subgroup of containing .

Proof.

Suppose that is a proper -parabolic subgroup containing . In the proof of Proposition 4.1 we have shown that is -cr. Then there exists a -Levi subgroup of containing since is contained in . Since -Levi subgroups of are -conjugate by [8, Lem. 2.5(iii)], without loss, we set . Then , so centralizes . Recall that by [28, Thm. 13.4.2], is an open set of where is the opposite of containing .

Lemma 4.5.

.

Proof.

First of all, from Equation (4.1) we see that is contained in . Since , is also contained in . So is contained in . Set for some . Using Equation 4.1 and the commutation relations, we obtain

So, if we must have . But , so for . Then

So we must have if