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Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group of type E7
Abstract
Let G be a connected reductive algebraic group defined over an algebraically closed field k. The aim of this paper is to present a method to find triples (G, M, H) with the following three properties. Property 1: G is simple and k has characteristic 2. Property 2: H and M are closed reductive subgroups of G such that H < M < G, and (G, M) is a reductive pair. Property 3: H is G-completely reducible, but not M-completely reducible. We exhibit our method by presenting a new example of such a triple in G = E7 . Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction.
Keywords: algebraic groups, separable subgroups, complete reducibility
Introduction
Let G be a connected reductive algebraic group defined over an algebraically closed field k of characteristic p. In [15, Sec. 3], J.P. Serre defined that a closed subgroup H of G is G-completely reducible (G-cr for short) if whenever H is contained in a parabolic subgroup P of G, H is contained in a Levi subgroup L of P . This is a faithful generalization of the notion of semisimplicity in representation theory since if G = GLn(k), a subgroup H of G is G-cr if and only if H acts complete reducibly on kn [15, Ex. 3.2.2(a)]. It is known that if a closed subgroup H of G is G-cr, then H is reductive [15, Prop. 4.1]. Moreover, if p = 0, the converse holds [15, Prop. 4.2]. Therefore the notion of G-complete reducibility is not interesting if p = 0. In this paper, we assume that p > 0.
Completely reducible subgroups of connected reductive algebraic groups have been much studied [9], [10], [15]. Recently, studies of complete reducibility via Geometric Invariant Theory (GIT for short) have been fruitful [1], [2], [3]. In this paper, we see another application of GIT to complete reducibility (Proposition 3.6).
Here is the main problem we consider. Let H and M be closed reductive subgroups of G such that H ≤ M ≤ G. It is natural to ask whether H being M-cr implies that H is G-cr and vice versa. It is not difficult to find a counterexample for the forward direction. For example, take H = M = P GL2(k) and G = SL3(k) where p = 2 and H sits inside G via the adjoint representation. Another such example is [1, Ex. 3.45]. However, it is hard to get a counterexample for the reverse direction, and it necessarily involves a small p. In [3, Sec. 7], Bate et al. presented the only known counterexample for the reverse direction where p = 2, H =∼ S3, M =∼ A1A1, and G = G2, which we call “the G2 example”. The aim of this paper is to prove the following.
Theorem 1.1. Let G be a simple algebraic group of type E7 defined over k of characteristic p = 2. Then there exists a connected reductive subgroup M of type A7 of G and a reductive subgroup H =∼ D14 (the dihedral group of order 14) of M such that (G, M) is a reductive pair and H is G-cr but not M-cr.
Our work is motivated by [3]. We recall a few relevant definitions and results here. We denote the Lie algebra of G by Lie G = g. From now on, by a subgroup of G, we always mean a closed subgroup of G.
Definition 1.2. Let H be a subgroup of G acting on G by inner automorphisms. Let H act on g by the corresponding adjoint action. Then H is called separable if Lie CG(H) = cg(H).
Recall that we always have Lie CG(H ) ⊆ cg(H). In [3], Bate et al. investigated the relation-ship between G-complete reducibility and separability, and showed the following [3, Thm. 1.2, Thm. 1.4].
Proposition 1.3. Suppose that p is very good for G. Then any subgroup of G is separable in G.
Proposition 1.4. Suppose that (G, M) is a reductive pair. Let H be a subgroup of M such that H is a separable subgroup of G. If H is G-cr, then it is also M-cr.
Recall that a pair of reductive groups G and M is called a reductive pair if Lie M is an M-module direct summand of g. This is automatically satisfied if p = 0. Propositions 1.3 and 1.4 imply that the subgroup H in Theorem 1.1 must be non-separable, which is possible for small p only.
Now, we introduce the key notion of separable action, which is a slight generalization of the notion of a separable subgroup.
Definition 1.5. Let H and N be subgroups of G where H acts on N by group automorphisms. The action of H is called separable in N if Lie CN (H) = cLie N (H). Note that the condition means that the fixed points of H acting on N, taken with their natural scheme structure, are smooth.
Here is a brief sketch of our method. Note that in our construction, p needs to be 2.
1. Pick a parabolic subgroup P of G with a Levi subgroup L of P . Find a subgroup K of L such that K acts non-separably on the unipotent radical Ru(P ) of P . In our case, K is generated by elements corresponding to certain reflections in the Weyl group of G.
2. Conjugate K by a suitable element v of Ru(P ), and set H = vKv−1. Then choose a connected reductive subgroup M of G such that H is not M-cr. Use a recent result from GIT (Proposition 2.4) to show that H is not M-cr. Note that K is M-cr in our case.
3. Prove that H is G-cr.
Remark 1.6. It can be shown using [17, Thm. 13.4.2] that K in Step 1 is a non-separable subgroup of G.
First of all, for Step 1, p cannot be very good for G by Proposition 1.3 and 1.4. It is known that 2 and 3 are bad for E7 . We explain the reason why we choose p = 2, not p = 3 (Remark 2.9). Remember that the non-separable action on Ru(P ) was the key ingredient for the G2 example to work. Since K is isomorphic to a subgroup of the Weyl group of G, we are able to turn a problem of non-separability into a purely combinatorial problem involving the root system of G (Section 3.1). Regarding Step 2, we explain the reason of our choice of v and M explicitly (Remarks 3.4, 3.5). Our use of Proposition 2.4 gives an improved way for checking G-complete reducibility (Remark 3.7). Finally, Step 3 is easy.
In the G2 and E7 examples, the G-cr and non-M-cr subgroups H are finite. The following is the only known example of a triple (G, M, H) with positive dimensional H such that H is G-cr but not M-cr. It is obtained by modifying [1, Ex. 3.45].
Example 1.7. Let p = 2, m ≥ 4 be even, and (G, M) = (GL2m(k ), Sp2m(k)). Let H be a copy of Spm (k) diagonally embedded in Spm(k) × Spm(k). Then H is not M-cr by the argument in [1, Ex. 3.45]. But H is G-cr since H is GLm(k ) × GLm(k)-cr by [1, Lem. 2.12]. Also note that any subgroup of GL(k) is separable in GL(k) (cf. [1, Ex. 3.28]), so (G, M) is not a reductive pair by Proposition 1.4.
In view of this, it is natural to ask:
Open Problem 1.8. Is there a triple H < M < G of connected reductive algebraic groups such that (G, M) is a reductive pair, H is non-separable in G, and H is G-cr but not M-cr?
Beyond its intrinsic interest, our E7 example has some important consequences and ap-plications. For example, in Section 4, we consider a rationality problem concerning complete reducibility. We need a definition first to explain our result there.
Definition 1.9. Let k0 be a subfield of an algebraically closed field k. Let H be a k0 -defined closed subgroup of a k0-defined reductive algebraic group G. Then H is called G-cr over k0 if whenever H is contained in a k0-defined parabolic subgroup P of G, it is contained in some k0-defined Levi subgroup of P .
Note that if k0 is algebraically closed then G-cr over k0 means G-cr in the usual sense. Here is the main result of Section 4.
Theorem 1.10. Let k0 be a nonperfect field of charecteristic p = 2, and let G be a k0-defined split simple algebraic group of type E7. Then there exists a k0-defined subgroup H of G such that H is G-cr over k, but not G-cr over k0.
As another application of the E7 example, we consider a problem concerning conjugacy classes. Given n ∈ N, we let G act on Gn by simultaneous conjugation:
g • (g1, g2, . . . , gn) = (gg1g−1, gg2g−1, . . . , ggng−1).
In [16], Slodowy proved the following fundamental result applying Richardson’s tangent space argument, [12, Sec. 3], [13, Lem. 3.1].
Proposition 1.11. Let M be a reductive subgroup of a reductive algebraic group G defined over k. Let n ∈ N, let (m1, . . . , mn) ∈ Mn and let H be the subgroup of M generated by m1, . . . , mn. Suppose that (G, M) is a reductive pair and that H is separable in G. Then the intersection G • (m1, . . . , mn) ∩ Mn is a finite union of M-conjugacy classes.
Proposition 1.11 has many consequences. See [1], [16], and [18, Sec. 3] for example. In [3, Ex. 7.15], Bate et al. found a counterexample for G = G2 showing that Proposition 1.11 fails without the separability hypothesis. In Section 5, we present a new counterexample to Proposition 1.11 without the separability hypothesis. Here is the main result of Section 5.
Theorem 1.12. Let G be a simple algebraic group of type E7 defined over an algebraically closed k of characteristic p = 2. Let M be the connected reductive subsystem subgroup of type A7. Then there exists n ∈ N and a tuple m ∈ Mn such that G • m ∩ Mn is an infinite union of M-conjugacy classes. Note that (G, M) is a reductive pair in this case.
Now, we give an outline of the paper. In Section 2, we fix our notation which follows [4], [8], and [17]. Also, we recall some preliminary results, in particular, Proposition 2.4 from GIT. After that, in Section 3, we prove our main result, Theorem 1.1. Then in Section 4, we consider a rationality problem, and prove Theorem 1.10. Finally, in Section 5, we discuss a problem concerning conjugacy classes, and prove Theorem 1.12.
Preliminaries
Notation
Throughout the paper, we denote by k an algebraically closed field of positive characteristic p. We denote the multiplicative group of k by k∗. We use a capital roman letter, G, H, K, etc., to represent an algebraic group, and the corresponding lowercase gothic letter, g, h, k, etc., to represent its Lie algebra. We sometimes use another notation for Lie algebras: Lie G, Lie H, and Lie K are the Lie algebras of G, H, and K respectively.
We denote the identity component of G by G◦. We write [G, G] for the derived group of G. The unipotent radical of G is denoted by Ru(G). An algebraic group G is reductive if Ru(G) = {1}. In particular, G is simple as an algebraic group if G is connected and all proper normal subgroups of G are finite.
In this paper, when a subgroup H of G acts on G, H always acts on G by inner auto-morphisms. The adjoint representation of G is denoted by Adg or just Ad if no confusion arises. We write CG(H ) and cg(H) for the global and the infinitesimal centralizers of H in G and g respectively. We write X(G) and Y (G) for the set of characters and cocharacters of G respectively.
2.2 Complete reducibility and GIT
Let G be a connected reductive algebraic group. We recall Richardson’s formalism [14, Sec. 2.1–2.3] for the characterization of a parabolic subgroup P of G, a Levi subgroup L of P , and the unipotent radical Ru(P ) of P in terms of a cocharacter of G and state a result from GIT (Proposition 2.4).
Definition 2.1. Let X be an affine variety. Let φ : k∗ → X be a morphism of algebraic ˆ X (necessarily varieties. We say that lim φ(a) exists if there exists a morphism φ : k → a→0 unique) whose restriction to k∗ is φ. If this limit exists, we set lim φ(a) = φˆ(0).
Definition 2.2. Let be a cocharacter of . Define λ:={ a→0 | a→0 } ∈ λ G P g G lim λ(a)gλ(a)−1 exists , L λ:={ g ∈ G lim λ(a)gλ(a)−1 = g } , R u (P λ ) := { g ∈ G | lim λ(a)gλ(a)−1 = 1 . | a 0 a 0 }
Note that Pλ is a parabolic subgroup of G, Lλ is a Levi subgroup of Pλ, and Ru(Pλ) is a unipotent radical of Pλ [14, Sec. 2.1-2.3]. By [17, Prop. 8.4.5], any parabolic subgroup P of G, any Levi subgroup L of P , and any unipotent radical Ru(P ) of P can be expressed in this form. It is well known that Lλ = CG(λ(k∗)).
Let M be a reductive subgroup of G . Then, there is a natural inclusion Y (M ) ⊆ Y (G) of cocharacter groups. Let λ ∈ Y (M). We write Pλ(G) or just Pλ for the parabolic subgroup of G corresponding to λ, and Pλ(M) for the parabolic subgroup of M corresponding to λ. It is obvious that Pλ(M) = Pλ(G) ∩ M and Ru(Pλ(M)) = Ru(Pλ(G)) ∩ M.
Definition 2.3. Let λ ∈ Y (G). Define a map cλ : Pλ → Lλ by cλ(g) := lim0 λ(a)gλ(a)−1. 10 a→
Note that the map cλ is the usual canonical projection from Pλ to Lλ =∼ Pλ/Ru(Pλ). Now, we state a result from GIT (see [1, Lem. 2.17, Thm. 3.1], [2, Thm. 3.3]).
Proposition 2.4. Let H be a subgroup of G. Let λ be a cocharacter of G with H ⊆ Pλ. If H is G-cr, there exists v ∈ Ru(Pλ) such that cλ(h) = vhv−1 for every h ∈ H.
Root subgroups and root subspaces
Let G be a connected reductive algebraic group. Fix a maximal torus T of G . Let Ψ(G, T ) denote the set of roots of G with respect to T . We sometimes write Ψ(G) for Ψ(G, T ). Fix a Borel subgroup B containing T . Then Ψ(B, T ) = Ψ+(G) is the set of positive roots of G defined by B. Let Σ(G, B) = Σ denote the set of simple roots of G defined by B. Let ζ ∈ Ψ(G).
We write Uζ for the corresponding root subgroup of G and uζ for the Lie algebra of Uζ . We define Gζ := hUζ , U−ζ i.
Let H be a subgroup of G normalized by some maximal torus T of G. Consider the adjoint representation of T on h. The root spaces of h with respect to T are also root spaces of g with respect to T , and the set of roots of H relative to T , Ψ(H, T ) = Ψ(H) = {ζ ∈ Ψ(G) | gζ ⊆ h}, is a subset of Ψ(G).
Let ζ, ξ ∈ Ψ(G). Let ξ∨ be the coroot corresponding to ξ. Then ζ ◦ ξ∨ : k∗ → k∗ is a homomorphism such that (ζ ◦ ξ∨)(a) = an for some n ∈ Z. We define hζ, ξ∨i := n. Let sξ denote the reflection corresponding to ξ in the Weyl group of G. Each sξ acts on the set of roots Ψ(G) by the following formula [17, Lem. 7.1.8]: sξ • ζ = ζ − hζ, ξ∨iξ. By [5, Prop. 6.4.2, Lem. 7.2.1], we can choose homomorphisms ζ : k → Uζ so that nξ ζ (a)nξ−1 = sξ•ζ (±a), where nξ = ξ(1) −ξ(−1) ξ(1). (2.1)
We define eζ := ζ0(0). Then we have Ad(nξ)eζ = ±esξ•ζ . (2.2)
Now, we list four lemmas which we need in our calculations. The first one is [17, Prop. 8.2.1].
Lemma 2.5. Let P be a parabolic subgroup of G. Any element u in Ru(P ) can be expressed uniquely as i(ai), for some ai ∈ k, where the product is taken with respect to a fixed ordering of Ψ (Ru(P )).
The next two lemmas [8, Lem. 32.5 and Lem. 33.3] are used to calculate CRu(P )(K).
Lemma 2.6. Let ξ, ζ ∈ Ψ(G). If no positive integral linear combination of ξ and ζ is a root of G, then
ξ(a) ζ (b) = ζ (b) ξ(a).
Lemma 2.7. Let Ψ be the root system of type A2 spanned by roots ξ and ζ. Then ξ(a) ζ (b) = ζ (b) ξ(a) ξ+ζ (±ab).
The last result is used to calculate cLie (Ru(P ))(K). ξ where ξ ∈ Ψ(G) (the group W is isomorphic to the Weyl group of G). Let K be a subgroup of W . Let {Oi | i = 1 • • • m} be the set of orbits of the action of K on Ψ (Ru(P )). Then,Lemma2.8.Supposethatp=2.LetWbeasubgroupofGgeneratedbyall the n cLie(Ru(P ))(K) = ai ∈ eζ ai p n e = e . Then an easy calculation gives the desired
Proof. When = 2, (2.2) yields Ad( ξ) ζ nξ •ζ
result.
Remark 2.9. Lemma 2.8 holds in p = 2 but fails in p = 3.
Introduction
Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group of type E7
Non-separability and complete reducibility: En examples with an application to a question of K¨ulshammer
Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields I
Complete reducibility of subgroups of reductive algebraic groups
