Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example

Falk Bannuscher; Alastair Litterick; Tomohiro Uchiyama

doi:10.1515/jgth-2020-0191

Article Publicly Available

Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹₄ example

Falk Bannuscher , Alastair Litterick and Tomohiro Uchiyama

Published/Copyright: November 30, 2021

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From the journal Journal of Group Theory Volume 25 Issue 3

Abstract

Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ only.

1 Introduction

Let 𝑘 be a field. Let k ¯ be an algebraic closure of 𝑘. Let 𝐺 be a connected affine algebraic 𝑘-group: we regard 𝐺 as a k ¯ -defined algebraic group together with a choice of 𝑘-structure in the sense of Borel [8, AG. 11]. We say that 𝐺 is reductive if the unipotent radical R u ⁢ ( G ) of 𝐺 is trivial. Throughout, 𝐺 is always a connected reductive 𝑘-group (unless stated otherwise). In this paper, we continue the investigation of rationality problems for complete reducibility of subgroups of 𝐺; see [30, 31, 33]. By a subgroup of 𝐺, we mean a (possibly non-𝑘-defined) closed subgroup of 𝐺. Following Serre [24, Section 3], we make the following definition.

Definition 1.1

A subgroup 𝐻 of 𝐺 is said to be 𝐺-completely reducible over 𝑘 (𝐺-cr over 𝑘 for short) if, whenever 𝐻 is contained in a 𝑘-defined parabolic subgroup 𝑃 of 𝐺, 𝐻 is contained in a 𝑘-defined Levi subgroup of 𝑃. In particular, if 𝐻 is not contained in any proper 𝑘-defined parabolic subgroup of 𝐺, 𝐻 is said to be 𝐺-irreducible over 𝑘 (𝐺-ir over 𝑘 for short).

This notion of 𝐺-complete reducibility faithfully generalises the notion of completely reducible representations; see [24, Section 3] for more detail. So far, most studies on 𝐺-complete reducibility consider the case k = k ¯ , for example [17, 26, 18], and not much is known for a non-algebraically closed 𝑘 (in particular, for non-perfect 𝑘) except a few general results and important examples; see [2], [3, Section 5], [30, 31, 33]. We say that a subgroup 𝐻 of 𝐺 is 𝐺-cr (𝐺-ir) if 𝐻 is 𝐺-cr (𝐺-ir) over k ¯ . Now it is natural to ask the following questions.

Question 1.2

Let 𝐻 be a subgroup of 𝐺.

If 𝐻 is 𝐺-cr, then is it 𝐺-cr over 𝑘?
If 𝐻 is 𝐺-cr over 𝑘, then is it 𝐺-cr?

Here is the main result of this paper.

Theorem 1.3

Let 𝑘 be a non-perfect separably closed field of characteristic 2. Let 𝐺 be a simple 𝑘-group of type F 4 . Then there exists a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-cr but not 𝐺-cr over 𝑘, and a connected non-abelian 𝑘-subgroup H ′ of 𝐺 that is 𝐺-cr over 𝑘 but not 𝐺-cr.

Several comments are in order. Firstly, when we consider Question 1.2, we can assume k = k s (a separable closure of 𝑘) by the following result, which is [4, Theorem 1.1].

Proposition 1.4

A subgroup 𝐻 of 𝐺 is 𝐺-cr over 𝑘 if and only if 𝐻 is 𝐺-cr over k s .

In particular, Question 1.2 has an affirmative answer if 𝑘 is perfect. Proposition 1.4 depends on the recently proved and deep centre conjecture of Tits (see Conjecture 5.1) in spherical buildings [24, 27, 21]. The centre conjecture (theorem) has been used to study complete reducibility over 𝑘; see [1, 31]. In this paper, we prove Proposition 5.5 which is related to a rationality problem for the centre conjecture. We assume k = k s throughout.

Secondly, the third author had already found several examples of a subgroup 𝐻 (or H ′ ) satisfying Theorem 1.3 for 𝐺 of type D 4 , G 2 , E 6 , and E 7 and for 𝑘 of characteristic 2 in [29, 30, 32]. We stress that Theorem 1.3 not only extends our collection of such examples, but it is new: we give the first connected such subgroup. In fact, our subgroups 𝐻 and H ′ are just slight modifications of a simple subgroup of type G 2 in F 4 . We are surprised with this pathological example since connected (moreover, simple) subgroups of 𝐺 usually behave nicely.

Thirdly, it is not hard to find examples of such behaviour if we allow 𝐻 (or H ′ ) to be non-𝑘-defined. To find a 𝑘-defined 𝐻, we have used nonseparability of 𝐻 (or a part of H ′ ) in 𝐺. In fact, combining [2, Theorems 1.5 and 9.3] we have that if a 𝑘-subgroup 𝐻 of 𝐺 is separable in 𝐺 and 𝐻 is 𝐺-cr, then it is 𝐺-cr over 𝑘. Recall [29, Definition 1.5].

Definition 1.5

Let 𝐻 and 𝑁 be affine algebraic groups. Suppose that 𝐻 acts on 𝑁 by group automorphisms. The action of 𝐻 on 𝑁 is said to be separable if Lie ⁡ C N ⁢ ( H ) = c Lie ⁡ N ⁢ ( H ) .

Note that the notion of a separable action is a slight generalisation of that of a separable subgroup [5, Definition 1.1]. See [5, 14] for more on separability. In our construction, it is crucial that our 𝐻 (or a part of H ′ ) acts non-separably on the unipotent radical of some proper 𝑘-parabolic subgroup of 𝐺. It is known that if the characteristic 𝑝 of 𝑘 is very good for 𝐺, every subgroup of 𝐺 is separable [5, Theorem 1.2]. This suggests that we need to work with small 𝑝. Such proper non-separable subgroups are hard to find. Only a handful of such examples are known ([5, Section 7], [29, 30, 32]), and all of them are finite subgroups. In our example, 𝐻 (or the non-separable part of H ′ ) is connected (and very close to G 2 ).

Fourthly, our method to construct 𝐻 and H ′ via group-theoretic arguments and geometric invariant theory is almost identical to the constructions in the D 4 , G 2 , E 6 , and E 7 examples mentioned above [29, 30, 32]. Since the same method works for many examples (for finite and connected 𝐻 and H ′ ), we believe that there should be some general phenomenon underlying these constructions (cf. [33, Theorems 1.2, 1.3]).

Open Problem 1.6

Suppose that there exists a 𝑘-subgroup 𝐻 of 𝐺 that acts non-separably on the unipotent radical of some proper 𝑘-parabolic subgroup of 𝐺.

If 𝐻 is 𝐺-cr, can one always find some modification H ′ of 𝐻 such that H ′ remains a 𝑘-subgroup, H ′ remains 𝐺-cr, but for which H ′ is not 𝐺-cr over 𝑘?
If 𝐻 is 𝐺-cr over 𝑘, can one always find some modification H ′′ of 𝐻 such that H ′′ remains a 𝑘-subgroup, H ′′ remains 𝐺-cr over 𝑘, but for which H ′′ is not 𝐺-cr?

We have also answered the second part of Question 1.2 using a different method (Weil restriction) and a different language (scheme-theoretic). First, recall [13, Definition 1.1.1]

Definition 1.7

Let 𝑘 be a field. Let 𝐺 be a connected affine algebraic 𝑘-group. If the 𝑘-unipotent radical R u , k ⁢ ( G ) is trivial, 𝐺 is said to be pseudo-reductive.

Weil restriction is a standard tool to construct non-reductive pseudo-reductive groups [13, Example 1.6.1]. Using Weil restriction and a scheme-theoretic argument, we show the following.

Proposition 1.8

Let 𝑘 be a non-perfect field of characteristic 𝑝. Then, for each power p s ( s ∈ N ), there exists a 𝑘-subgroup 𝐻 of G = GL p s that is 𝐺-cr over 𝑘 but not 𝐺-cr.

The subgroup 𝐻 we find in proving Proposition 1.8 is abelian. We can find an abelian example (that is 𝐺-cr over 𝑘 but not 𝐺-cr) without using Weil restriction: take a 𝑘-anisotropic unipotent element of 𝐺 for a finite example, and take its connected centraliser for a connected example; see [31, Remark 5.3]. Remember that a unipotent element is said to be 𝑘-anisotropic if it is not contained in any proper 𝑘-parabolic subgroup of 𝐺. This means the classical construction of Borel–Tits [10, Theorem 2.5] fails for non-perfect 𝑘.

We included Proposition 1.8 in this paper since it shows a relation between rationality problems for complete reducibility and pseudo-reductivity. We think that this topic should be investigated further. For example, the answer to the following open problem is true if 𝑘 is perfect or C G ⁢ ( H ) is pseudo-reductive; see [31, Section 6] for more on this.

Open Problem 1.9

Let 𝑘 be a field. Suppose that a 𝑘-subgroup 𝐻 of 𝐺 is 𝐺-cr over 𝑘. Is C G ⁢ ( H ) then 𝐺-cr over 𝑘?

Here is the structure of the paper. In Section 2, we will set our notion and recall some important results about complete reducibility and related a result from geometric invariant theory (GIT for short). In Section 3, we give a short review of Weil restriction and prove Proposition 1.8. Then, in Section 4, we prove Theorem 1.3. Note that, since our method is almost identical to that in [29, 30, 32], our proof is just a minimum skeleton. Finally, in Section 5, we discuss a rationality problem related to complete reducibility and the centre conjecture.

2 Preliminaries

Throughout, we denote by 𝑘 a separably closed field. Our references for algebraic groups are [8, 9, 13, 15, 25].

Let 𝐻 be a (possibly non-connected) affine algebraic group. We write H ∘ for the identity component of 𝐻. We write [ H , H ] for the derived group of 𝐻. A reductive group 𝐺 is said to be simple as an algebraic group if 𝐺 is connected and all proper normal subgroups of 𝐺 are finite. We write X k ⁢ ( G ) and Y k ⁢ ( G ) for the set of 𝑘-characters and 𝑘-cocharacters of 𝐺 respectively. For k ¯ -characters and k ¯ -cocharacters of 𝐺, we simply say characters and cocharacters of 𝐺 and write X ⁢ ( G ) and Y ⁢ ( G ) respectively.

Fix a maximal 𝑘-torus 𝑇 of 𝐺 (such a 𝑇 exists by [8, Corollary 18.8]). Then 𝑇 is split over 𝑘 since 𝑘 is separably closed. Let Ψ ⁢ ( G , T ) denote the set of roots of 𝐺 with respect to 𝑇. We sometimes write Ψ ⁢ ( G ) for Ψ ⁢ ( G , T ) . Let ζ ∈ Ψ ⁢ ( G ) . We write U ζ for the corresponding root subgroup of 𝐺. We define G ζ := ⟨ U ζ , U - ζ ⟩ . Let ζ , ξ ∈ Ψ ⁢ ( G ) . Let ξ ∨ be the coroot corresponding to 𝜉. Then

ζ ∘ ξ ∨ : k ¯ * → k ¯ *

is a 𝑘-homomorphism such that ( ζ ∘ ξ ∨ ) ⁢ ( a ) = a n for some n ∈ Z . Let s ξ denote the reflection corresponding to 𝜉 in the Weyl group of 𝐺. Each s ξ acts on the set of roots Ψ ⁢ ( G ) by s ξ ⋅ ζ = ζ - ⟨ ζ , ξ ∨ ⟩ ⁢ ξ (see [25, Lemma 7.1.8]). By [12, Proposition 6.4.2, Lemma 7.2.1], we can choose 𝑘-homomorphisms ϵ ζ : k ¯ → U ζ so that n ξ ⁢ ϵ ζ ⁢ ( a ) ⁢ n ξ - 1 = ϵ s ξ ⋅ ζ ⁢ ( ± a ) , where n ξ = ϵ ξ ⁢ ( 1 ) ⁢ ϵ - ξ ⁢ ( - 1 ) ⁢ ϵ ξ ⁢ ( 1 ) .

The next result [30, Proposition 1.12] shows complete reducibility behaves nicely under central isogenies.

Definition 2.1

Let G 1 and G 2 be reductive 𝑘-groups. A 𝑘-isogeny f : G 1 → G 2 is central if ker ⁡ d ⁢ f 1 is central in g 1 , where ker ⁡ d ⁢ f 1 is the differential of 𝑓 at the identity of G 1 and g 1 is the Lie algebra of G 1 .

Proposition 2.2

Let G 1 and G 2 be reductive 𝑘-groups. Let H 1 and H 2 be subgroups of G 1 and G 2 respectively. Let f : G 1 → G 2 be a central 𝑘-isogeny.

If H 1 is G 1 -cr over 𝑘, then f ⁢ ( H 1 ) is G 2 -cr over 𝑘.
If H 2 is G 2 -cr over 𝑘, then f - 1 ⁢ ( H 2 ) is G 1 -cr over 𝑘.

The next result [1, Theorem 1.4] is used repeatedly to reduce problems on 𝐺-complete reducibility to those on 𝐿-complete reducibility, where 𝐿 is a Levi subgroup of 𝐺.

Proposition 2.3

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

We recall characterisations of parabolic subgroups, Levi subgroups, and unipotent radicals in terms of cocharacters of 𝐺 (see [25, Proposition 8.4.5]). These characterisations are essential to translate results on complete reducibility into the language of GIT; see [3, 6] for example.

Definition 2.4

Let 𝑋 be an affine 𝑘-variety. Let ϕ : k ¯ * → X be a 𝑘-morphism of affine 𝑘-varieties. We say that lim a → 0 ⁡ ϕ ⁢ ( a ) exists if there exists a 𝑘-morphism ϕ ^ : k ¯ → X (necessarily unique) whose restriction to k ¯ * is 𝜙. If this limit exists, we set lim a → 0 ⁡ ϕ ⁢ ( a ) := ϕ ^ ⁢ ( 0 ) .

Definition 2.5

Let λ ∈ Y ⁢ ( G ) . Define

P λ := { g ∈ G ∣ lim a → 0 ⁡ λ ⁢ ( a ) ⁢ g ⁢ λ ⁢ ( a ) - 1 ⁢ exists } , L λ := { g ∈ G ∣ lim a → 0 ⁡ λ ⁢ ( a ) ⁢ g ⁢ λ ⁢ ( a ) - 1 = g } , R u ⁢ ( P λ ) := { g ∈ G ∣ lim a → 0 ⁡ λ ⁢ ( a ) ⁢ g ⁢ λ ⁢ ( a ) - 1 = 1 } .

Then P λ is a parabolic subgroup of 𝐺, L λ is a Levi subgroup of P λ , and R u ⁢ ( P λ ) is the unipotent radical of P λ . If 𝜆 is 𝑘-defined, P λ , L λ , and R u ⁢ ( P λ ) are 𝑘-defined [23, Section 2.1-2.3]. All 𝑘-defined parabolic subgroups and 𝑘-defined Levi subgroups of 𝐺 arise in this way since 𝑘 is separably closed. It is well known that L λ = C G ⁢ ( λ ⁢ ( k ¯ * ) ) . Note that 𝑘-defined Levi subgroups of a 𝑘-defined parabolic subgroup 𝑃 of 𝐺 are R u ⁢ ( P ) ⁢ ( k ) -conjugate [6, Lemma 2.5 (iii)].

Recall the following geometric characterisation for complete reducibility via GIT [3]. Suppose that a subgroup 𝐻 of 𝐺 is generated by 𝑛-tuple h = ( h 1 , … , h n ) of elements of 𝐺 (or 𝐡 is a generic tuple of 𝐻 in the sense of [2, Definition 9.2]), and 𝐺 acts on G n by simultaneous conjugation.

Proposition 2.6

A subgroup 𝐻 of 𝐺 is 𝐺-cr if and only if the 𝐺-orbit G ⋅ h is closed in G n .

Combining Proposition 2.6 and a recent result from GIT [6, Theorem 3.3], we have the following proposition.

Proposition 2.7

Let 𝐻 be a subgroup of 𝐺. Let λ ∈ Y ⁢ ( G ) . Suppose that

h ′ := lim a → 0 ⁡ λ ⁢ ( a ) ⋅ h

exists. If 𝐻 is 𝐺-cr, then h ′ is R u ⁢ ( P λ ) -conjugate to 𝐡.

We use a rational version of Proposition 2.7; see [2, Corollary 5.1], [2, Theorem 9.3].

Proposition 2.8

Let 𝐻 be a subgroup of 𝐺. Let λ ∈ Y k ⁢ ( G ) . Suppose that

h ′ := lim a → 0 ⁡ λ ⁢ ( a ) ⋅ h

exists. If 𝐻 is 𝐺-cr over 𝑘, then h ′ is R u ⁢ ( P λ ) ⁢ ( k ) -conjugate to 𝐡.

3 Complete reducibility and pseudo-reductivity

In this section, we use the language of schemes. Recall [13, Section A.5].

Definition 3.1

Let B → B ′ be a finite flat map of noetherian rings, and X ′ a quasi-projective B ′ -scheme. The Weil restriction R B ′ / B ⁢ ( X ′ ) is a 𝐵-scheme of finite type satisfying the universal property

R B ′ / B ⁢ ( X ′ ) ⁢ ( A ) = X ′ ⁢ ( B ′ ⊗ B A )

for all 𝐵-algebras 𝐴.

Now, using a special case of a Weil restriction [13, Example 1.6.1], we have the following proposition.

Proposition 3.2

Let k ′ / k be a finite purely inseparable field extension with k ′ ≠ k and G ′ a non-trivial smooth connected reductive k ′ -group. Then G = R k ′ / k ⁢ ( G ′ ) is a pseudo-reductive and non-reductive 𝑘-group.

Note that G = R k ′ / k ⁢ ( G ′ ) is smooth due to [13, Proposition A.5.11 (1)]. We use the following standard result [24, Example 3.2.2 (a)].

Proposition 3.3

Let 𝐻 be a 𝑘-subgroup of GL ⁢ ( V ) for some finite-dimensional 𝑘-vector space 𝑉. Then 𝐻 is GL ⁢ ( V ) -cr over 𝑘 (resp. GL ⁢ ( V ) -ir over 𝑘) if and only if 𝑉 is a semisimple (resp. irreducible) k ⁢ H -module.

Combining Propositions 3.2 and 3.3 together with the argument in [7, Section 5.2.1] (we reproduce the argument of [7, Section 5.2.1] to make this paper self-contained), we find the following proof.

Proof of Proposition 1.8

Let k ′ / k be a purely inseparable field extension of degree 𝑠 in characteristic 𝑝. Fix a faithful action of G m ⁢ ( k ′ ) on G a ⁢ ( k ′ ) (we regard G m ⁢ ( k ′ ) and G a ⁢ ( k ′ ) as k ′ -groups). Then this action induces an action of H = R k ′ / k ⁢ ( G m ) on the [ k ′ : k ] -dimensional 𝑘-vector group R k ′ / k ⁢ ( G a ) , obtaining an embedding of H = R k ′ / k ⁢ ( G m ) in G = GL p s .

It is easy to see that there are exactly two G m ⁢ ( k ′ ) -orbits on G a ⁢ ( k ′ ) . Thus there are exactly two H ⁢ ( k ) = ( R k ′ / k ⁢ ( G m ) ) ⁢ ( k ) -orbits on G ⁢ ( k ) = ( R k ′ / k ⁢ ( G a ) ) ⁢ ( k ) . This shows that 𝐺 is an irreducible k ⁢ H -module, and then 𝐻 is 𝐺-ir over 𝑘 by Proposition 3.3. Note that 𝐻 is not reductive by Proposition 3.2, so it is not 𝐺-cr by [24, Proposition 4.1]. ∎

4 𝐺-cr vs 𝐺-cr over 𝑘 (Proof of Theorem 1.3)

Let 𝐺 be a simple algebraic group of type F 4 defined over a non-perfect field 𝑘 of characteristic 2. Fix a maximal torus 𝑇 of 𝐺 and a Borel subgroup 𝐵 of 𝐺 containing 𝑇. Let Ψ ⁢ ( G ) = Ψ ⁢ ( G , T ) be the set of roots corresponding to 𝑇, and let Ψ ⁢ ( G ) + = Ψ ⁢ ( G , B , T ) be the set of positive roots of 𝐺 corresponding to 𝑇 and 𝐵. The following Dynkin diagram defines the set of simple roots of 𝐺:

We label Ψ ⁢ ( G ) + as in the following. The corresponding negative roots are labelled accordingly. For example, the roots 1, 2, 3, 17 correspond to 𝛼, 𝛽, 𝛾, 𝛿 respectively.

Let λ = 13 ∨ = 2 ⁢ α ∨ + 4 ⁢ β ∨ + 3 ⁢ γ ∨ + 2 ⁢ δ ∨ . Then

P λ = ⟨ T , U ζ ∣ ζ ∈ Ψ ⁢ ( G ) + ∪ { - 1 , … , - 9 } ⟩ , L λ = ⟨ T , U ζ ∣ ζ ∈ { ± 1 , … , ± 9 } ⟩ , R u ⁢ ( P λ ) = ⟨ U ζ ∣ ζ ∈ Ψ ⁢ ( G ) + \ { 1 , … , 9 } ⟩ .

Note that L λ is of type B 3 . In this section, we use the commutator relations [15, Lemma 32.5, Propositions 33.3 and 33.4] repeatedly. Let

M = ⟨ ϵ 2 ⁢ ( x 1 ) , ϵ - 2 ⁢ ( x 2 ) , ϵ 1 ⁢ ( x 3 ) ⁢ ϵ 3 ⁢ ( x 3 ) , ϵ - 1 ⁢ ( x 4 ) ⁢ ϵ - 3 ⁢ ( x 4 ) ∣ x i ∈ k ¯ ⟩ .

Using the commutator formulae for root elements of 𝐺 (we used those found in Magma [11]), it is straightforward to check that the above generators for 𝑀 satisfy commutator relations so that 𝑀 is a simple algebraic group of type G 2 with the given generators as root elements; cf. [12, pp. 72–77]. Let a ∈ k \ k 2 and v ⁢ ( a ) = ϵ 20 ⁢ ( a ) ⁢ ϵ 21 ⁢ ( a ) . In the rest of the paper, the dot action always represents simultaneous conjugation. Define

H := v ( a ) ⋅ M = ⟨ ϵ 2 ( x 1 ) , ϵ - 2 ( x 2 ) , ϵ 1 ⁢ ( x 3 ) ⁢ ϵ 3 ⁢ ( x 3 ) ⁢ ϵ 14 ⁢ ( a ⁢ x 3 ) , ϵ - 1 ⁢ ( x 4 ) ⁢ ϵ - 3 ⁢ ( x 4 ) ⁢ ϵ 12 ⁢ ( a ⁢ x 4 ) ⁢ ∣ x i ∈ k ¯ ⟩ .

The first main result in this section is the following.

Proposition 4.1

𝐻 is connected, 𝑘-defined, and 𝐺-cr, but not 𝐺-cr over 𝑘.

Proof

It is clear that 𝐻 is generated by connected subgroups, each of which is defined over 𝑘, so 𝐻 is connected and 𝑘-defined by [8, AG. 11]. Also, 𝐻 is 𝐺-cr since 𝑀 is L λ -ir by [19, Table 10, ID 3]. We show that 𝐻 is not 𝐺-cr over 𝑘. Suppose the contrary. Choose b ∈ k with b 3 = 1 and b ≠ 1 . Let 𝐡 be a generic tuple of 𝐻 containing β ∨ ⁢ ( b ) and ϵ 1 ⁢ ( 1 ) ⁢ ϵ 3 ⁢ ( 1 ) ⁢ ϵ 14 ⁢ ( a ) . Then h ′ := lim a → 0 ⁡ λ ⁢ ( a ) ⋅ h exists since H < P λ . Then, by Proposition 2.8, h ′ must be R u ⁢ ( P λ ) ⁢ ( k ) -conjugate to 𝐡. Let v = ( β ∨ ⁢ ( b ) , ϵ 1 ⁢ ( 1 ) ⁢ ϵ 3 ⁢ ( 1 ) ⁢ ϵ 14 ⁢ ( a ) ) . Then

v ′ := lim a → 0 ⁡ λ ⁢ ( a ) ⋅ v = ( β ∨ ⁢ ( b ) , ϵ 1 ⁢ ( 1 ) ⁢ ϵ 3 ⁢ ( 1 ) ) .

Thus there exists an element u ∈ R u ⁢ ( P λ ) ⁢ ( k ) with v = u ⋅ v ′ , which implies that 𝑢 commutes with β ∨ ⁢ ( b ) . By [25, Proposition 8.2.1], we can set u = ∏ i = 10 24 ϵ i ⁢ ( x i ) for some x i ∈ k . Then x i = 0 for i ∈ { 11 , 12 , 14 , 15 , 18 , 19 , 22 , 23 } . The equation v = u ⋅ v ′ also implies

ϵ 1 ⁢ ( 1 ) ⁢ ϵ 3 ⁢ ( 1 ) ⁢ ϵ 14 ⁢ ( a ) = ( ϵ 10 ( x 10 ) ϵ 13 ( x 13 ) ϵ 16 ( x 16 ) ϵ 17 ( x 17 ) ϵ 20 ( x 20 ) ϵ 21 ( x 21 ) ϵ 24 ( x 24 ) ) ⋅ ( ϵ 1 ( 1 ) ϵ 3 ( 1 ) ) = ϵ 1 ⁢ ( 1 ) ⁢ ϵ 3 ⁢ ( 1 ) ⁢ ϵ 11 ⁢ ( x 10 ) ⁢ ϵ 14 ⁢ ( x 21 2 ) ⁢ ϵ 18 ⁢ ( x 17 ) ⁢ ϵ 22 ⁢ ( x 20 + x 21 ) .

This means a = x 21 2 , which is a contradiction. ∎

Remark 4.2

From the calculation above, we see that the curve

C ⁢ ( x ) := { ϵ 20 ⁢ ( x ) ⁢ ϵ 21 ⁢ ( x ) }

is not contained C R u ⁢ ( P λ ) ⁢ ( M ) , but the corresponding element in Lie ⁡ ( R u ⁢ ( P λ ) ) , that is, e 20 + e 21 is in c Lie ⁡ ( R u ⁢ ( P λ ) ) ⁢ ( M ) . Then the argument in the proof of [29, Proposition 3.3] shows that 𝑀 (hence 𝐻) acts non-separably on R u ⁢ ( P λ ) .

We move on to the second main result in this section. We use the same 𝑘, 𝑎, 𝑏, 𝐺, 𝑀, and 𝜆 as above. Let v ⁢ ( a ) = ϵ - 20 ⁢ ( a ) ⁢ ϵ - 21 ⁢ ( a ) . Define

H ′ := ⟨ v ⁢ ( a ) ⋅ M , ϵ 18 ⁢ ( x ) ∣ x ∈ k ¯ ⟩ .

Proposition 4.3

H ′ is connected, 𝑘-defined, and 𝐺-cr over 𝑘, but not 𝐺-cr.

Proof

We have

v ( a ) ⋅ M = ⟨ ϵ 2 ( x 1 ) , ϵ - 2 ( x 2 ) , ϵ 1 ⁢ ( x 3 ) ⁢ ϵ 3 ⁢ ( x 3 ) ⁢ ϵ - 12 ⁢ ( a ⁢ x 3 ) , ϵ - 1 ⁢ ( x 4 ) ⁢ ϵ - 3 ⁢ ( x 4 ) ⁢ ϵ - 14 ⁢ ( a ⁢ x 4 ) ⁢ ∣ x i ∈ k ¯ ⟩ .

Since H ′ is generated by 𝑘-defined connected subgroups of 𝐺, it is connected and 𝑘-defined by [8, AG. 11]. In the following, we use the non-separability of 𝑀 in 𝐺 again. We show that H ′ is not 𝐺-cr. From [19, Table 38, X = G 2 , ID 3, p = 2 ], C G ⁢ ( M ) ∘ is of type A 1 . By a direct computation using commutator relations, we have G 13 ≤ C G ⁢ ( M ) ∘ . Then C G ⁢ ( M ) ∘ = G 13 . Now an easy calculation gives C G ⁢ ( H ′ ) ∘ = v ⁢ ( a ) ⋅ U 13 . (The point is that U - 13 is not centralised by U 18 .) Thus C G ⁢ ( H ′ ) ∘ is unipotent. Then, by the classical result of Borel and Tits [10, Proposition 3.1], C G ⁢ ( H ′ ) ∘ is not 𝐺-cr. Since C G ⁢ ( H ′ ) ∘ is a normal subgroup of C G ⁢ ( H ′ ) , by [6, Example 5.20], C G ⁢ ( H ′ ) is not 𝐺-cr. Then H ′ is not 𝐺-cr by [3, Corollary 3.17]. Now we show that H ′ is 𝐺-cr over 𝑘. Note that

v ⁢ ( a ) - 1 ⋅ H ′ = ⟨ M , ϵ 18 ⁢ ( x ) ⁢ ϵ - 5 ⁢ ( a ⁢ x ) ∣ x ∈ k ¯ ⟩ .

This shows that v ⁢ ( a ) - 1 ⋅ H ′ ≤ P λ . Thus H ′ ≤ v ⁢ ( a ) ⋅ P λ . Using the argument in [32, Claim 3.6] word-for-word, we have that P λ is not 𝑘-defined. (The same argument works since v ⁢ ( a ) is not a 𝑘-point and v ⁢ ( a ) ∉ P λ .) In the following, we show that v ⁢ ( a ) ⋅ P λ is the unique proper parabolic subgroup of 𝐺 containing H ′ , which implies that H ′ is 𝐺-ir over 𝑘.

Let P μ be a proper parabolic subgroup containing v ⁢ ( a ) - 1 ⋅ H ′ . Then we have M ≤ P μ . Since 𝑀 is 𝐺-cr, there exists a Levi subgroup 𝐿 of P μ containing 𝑀. Since Levi subgroups of P μ are R u ⁢ ( P μ ) -conjugate, we may assume L = L μ . Note that L μ = C G ⁢ ( μ ⁢ ( k ¯ * ) ) , so μ ⁢ ( k ¯ * ) must centralise 𝑀. Since C G ⁢ ( M ) ∘ = G 13 , we have μ = g ⋅ 13 ∨ = g ⋅ λ for some g ∈ G 13 . Using the Bruhat decomposition, 𝑔 is of one of the following forms:

g = λ ⁢ ( t ) ⁢ ϵ 13 ⁢ ( x 1 ) or g = ϵ 13 ⁢ ( x 1 ) ⁢ n 13 ⁢ λ ⁢ ( t ) ⁢ ϵ 13 ⁢ ( x 2 )

for some x 1 , x 2 ∈ k ¯ and t ∈ k ¯ * . We rule out the second case. Suppose 𝑔 is of the second form. We have

ϵ 18 ⁢ ( 1 ) ⁢ ϵ - 5 ⁢ ( a ) ≤ v ⁢ ( a ) ⋅ H ′ ≤ P μ = P g ⋅ λ = g ⋅ P λ .

So it is enough to show that g - 1 ⋅ ϵ 18 ⁢ ( 1 ) ⁢ ϵ - 5 ⁢ ( a ) ∉ P λ . Since U 13 and λ ⁢ ( k ¯ * ) are contained in P λ , we can assume g = n 13 . A direct computation shows that n 13 - 1 ⋅ ϵ 18 ⁢ ( 1 ) ⁢ ϵ - 5 ⁢ ( a ) = ϵ - 23 ⁢ ( 1 ) ⁢ ϵ - 5 ⁢ ( a ) ∉ P λ . Thus 𝑔 is of the first form, but this implies P μ = P λ . We are done. ∎

Remark 4.4

One might try to get another example of 𝐻 that is 𝐺-cr but not 𝐺-cr over 𝑘 (or a subgroup H ′ that is 𝐺-cr over 𝑘 but not 𝐺-cr) by applying the special graph automorphism 𝜎 of F 4 (in the sense of [12, Proposition 12.3.3]) on 𝐻 in Proposition 4.1 or on H ′ in Proposition 4.3. Remember that

σ ⁢ ( ϵ ζ ⁢ ( x ) ) = ϵ σ ⁢ ( ζ ) ⁢ ( x f ⁢ ( ζ ) ) ,

where f ⁢ ( ζ ) = 2 if 𝜁 is short and f ⁢ ( ζ ) = 1 if 𝜁 is long (we abuse the notation 𝜎 for an automorphism of Ψ ⁢ ( G ) ). This method fails in both cases since σ ⁢ ( v ⁢ ( a ) ) becomes a 𝑘-point.

Remark 4.5

The last remark gives a first counterexample to [30, Open Problem 3.14] that asked: does the second part of Proposition 2.2 hold without assuming 𝑓 central? We supply some details. We set f = σ . Then 𝜎 is a (non-central) 𝑘-isogeny. We use 𝐻, 𝑀, and L λ in the proof of Proposition 4.1. We have shown that 𝐻 is not 𝐺-cr over 𝑘. It is easy to see that σ ⁢ ( M ) is of type G 2 and is contained in σ ⁢ ( L λ ) of type C 3 . By [19, Table 9, ID 4], σ ⁢ ( M ) is σ ⁢ ( L λ ) -ir, thus σ ⁢ ( L λ ) -ir over 𝑘. Then σ ⁢ ( M ) is 𝐺-cr over 𝑘, and then σ ⁢ ( H ) is 𝐺-cr over 𝑘 since σ ⁢ ( H ) is G ⁢ ( k ) -conjugate to σ ⁢ ( M ) . Since σ - 1 ⁢ ( σ ⁢ ( H ) ) = H , we are done. For an easy counterexample to the first part of Proposition 2.2 without the centrality assumption (using Frobenius map), see [30, Example 3.12].

5 Tits’ centre conjecture

In [27], Tits conjectured the following.

Conjecture 5.1

Let 𝑋 be a spherical building. Let 𝑌 be a convex contractible simplicial subcomplex of 𝑋. If 𝐻 is a subgroup of the automorphism group of 𝑋 stabilising 𝑌, then there exists a simplex of 𝑌 fixed by 𝐻.

This so-called centre conjecture of Tits was proved by case-by-case analyses by Tits, Mühlherr, Leeb, and Ramos-Cuevas [16, 20, 22]. Recently, a uniform proof was given in [21]. In relation to the theory of complete reducibility, Serre [24] showed the following.

Proposition 5.2

Let 𝐺 be a reductive 𝑘-group. Let Δ ⁢ ( G ) be the building of 𝐺. If 𝐻 is not 𝐺-cr over 𝑘, then the convex fixed point subcomplex Δ ⁢ ( G ) H is contractible.

We identify the set of proper 𝑘-parabolic subgroups of 𝐺 with Δ ⁢ ( G ) in the usual sense of Tits [28]. Note that, for a subgroup 𝐻 of 𝐺, N G ⁢ ( k ) ⁢ ( H ) induces a group of automorphisms of Δ ⁢ ( G ) stabilising Δ ⁢ ( G ) H . Thus, combining the centre conjecture with Proposition 5.2, we obtain the following.

Proposition 5.3

If a subgroup 𝐻 of 𝐺 is not 𝐺-cr over 𝑘, then there exists a proper 𝑘-parabolic subgroup of 𝐺 containing 𝐻 and N G ⁢ ( k ) ⁢ ( H ) .

Proposition 5.3 was an essential tool in proving various theoretical results on complete reducibility over non-perfect 𝑘 in [30, 31]. We have asked the following in [31, Remark 6.5].

Question 5.4

If H < G is not 𝐺-cr over 𝑘, then does there exist a proper 𝑘-parabolic subgroup of 𝐺 containing H ⁢ C G ⁢ ( H ) ?

The answer is affirmative if C G ⁢ ( H ) is 𝑘-defined (or 𝑘 is perfect). In this case, the set of 𝑘-points is dense in C G ⁢ ( H ) (since we assume k = k s ), and the result follows from Proposition 5.3. The main result in this section is to present the first counterexample to Question 5.4 when 𝑘 is non-perfect and 𝐻 is connected. (A counterexample with discrete 𝐻 was given in [32, Theorem 4.5].)

Proposition 5.5

Let 𝑘 be non-perfect of characteristic 2. Let 𝐺 be simple of type F 4 . Then there exists a non-abelian connected 𝑘-subgroup 𝐻 of 𝐺 such that 𝐻 is not 𝐺-cr over 𝑘 but H ⁢ C G ⁢ ( H ) is not contained in any proper 𝑘-parabolic subgroup of 𝐺.

Proof

We use the same 𝐻, 𝑀, v ⁢ ( a ) , and 𝜆 as in the proof of Proposition 4.1. We have shown that 𝐻 is not 𝐺-cr over 𝑘. We had G 13 ≤ C G ⁢ ( M ) , so

⟨ M , G 13 ⟩ ≤ M ⁢ C G ⁢ ( M ) .

By an argument similar to the proof of Proposition 4.3, we can show that the unique proper parabolic subgroup of 𝐺 containing ⟨ M , U 13 ⟩ is P λ (since we have n 13 ⋅ 13 = - 13 ). It is clear that P λ does not contain U - 13 . So there is no proper parabolic subgroup of 𝐺 containing M ⁢ C G ⁢ ( M ) . Thus there is no proper parabolic subgroup of 𝐺 containing v ⁢ ( a ) ⋅ ( M ⁢ C G ⁢ ( M ) ) = H ⁢ C G ⁢ ( H ) . ∎

Remark 5.6

Proposition 5.5 (and its proof) shows that it is hard to control C G ⁢ ( H ) when C G ⁢ ( H ) is not 𝑘-defined even if 𝐻 is connected. This makes Open Problem 1.9 difficult even for connected 𝐻.

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 19K14516

Funding statement: While undertaking the work for this article, the second and third authors were supported by Alexander von Humboldt Fellowships. The third author also acknowledges the financial support of JSPS Grant-in-Aid for Early-Career Scientists (19K14516).

Acknowledgements

The authors would like to thank Gerhard Röhrle and Michael Bate for helpful comments.

Communicated by: Timothy C. Burness

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Received: 2020-11-27

Revised: 2021-07-28

Published Online: 2021-11-30

Published in Print: 2022-05-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2020-0191