On the Smoothness of Normalisers,

the Subalgebra Structure of Modular Lie Algebras,

and the Cohomology of Small Representations

Sebastian Herpel and David I. Stewart

Received: June 19, 2015 Revised: October 30, 2015

Communicated by Joachim Cuntz

Abstract. We provide results on the smoothness of normalisers in connected reductive algebraic groups G over fields k of positive characteristic p. Specifically we we give bounds on p which guarantee that normalisers of subalgebras of 𝔤 in G are smooth, i.e. so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers.

One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility results for small dimensional modules in the spirit of similar results due to Jantzen, Guralnick, Serre and Bendel–Nakano–Pillen.

2010 Mathematics Subject Classification: Primary 17B45; Secondary 20G07, 20G10. Keywords and Phrases: Smoothness, subgroup structure, complete reducibility, cohomology.

Contents

1 Introduction 2

2 Notation and preliminaries 6

3 Smoothness of normalisers of reductive subgroups 9 4 On exponentiation and normalising, and the proof of The-

orem B(ii) 9

5 Reductive pairs: Proof of Proposition E 11

6 Lifting of normalising tori and the proof of Theorem A 13

7 Non-semisimple subalgebras of classical Lie algebras. Proof of Theorem C 14

8 Complete reducibility and low-degree cohomology for classical Lie algebras: Proof of Theorem D 17

8.1 Cohomology and complete reducibility for small G_{1}-modules . . . . . 18 8.2 Cohomology and complete reducibility for small 𝔤-modules . . . . . . 21

9 Decomposability: the existence of Levi factors 25

10 Proof of Theorem B(i) 27

11 Examples 28

1 Introduction

Let G be an affine group scheme over an algebraically closed field k. We say G is smooth if dim Lie(G) = dim G. A famous theorem of Cartier states that every affine group over a field of characteristic zero is smooth. Therefore, in this situation, the category of smooth group schemes is closed under the scheme-theoretic constructions of taking centres, centralisers, normalisers and transporters. However, Cartier’s theorem fails rather comprehensively in positive characteristic. A classic example of a non-smooth algebraic group is the group scheme μ_{p} whose points are the pth roots of unity; this is not smooth over a field of characteristic p—its Lie algebra is 1-dimensional, but its k-points consist just of the identity element. Furthermore, since μ_{p} is also the schemetheoretic centre of SL_{p}, the centre of this reductive^{1} group is also not smooth over a field of characteristic p. This means that the group-theoretic centre of SL_{p} misses important infinitesimal information about the centre (for instance, the fact that SL_{p} is not adjoint). Nonetheless, centralisers are usually smooth. For example, it is a critical result of Richardson [Ric67, Lem. 6.6], used extensively in the theory of nilpotent orbits, that the centraliser G_{e} = C_{G}(e) of an element e of 𝔤 = Lie(G) is smooth whenever p is a very good prime for G.^{2} (Note that smoothness of the centraliser, or what is the same, the separability of the orbit map G → G · e can be restated as Lie(G_{e}(k)) = 𝔠_{𝔤}(e).) In fact the centralisers of subgroup schemes of a connected reductive group G are usually smooth: work of Bate– Martin–Röhrle–Tange and the first author (cf. Proposition 3.1) gives precise information on the characteristic p of k, depending on the root datum of G, for centralisers of all subgroup schemes of G to be smooth. It suffices, for instance, for p to be very good for G. Furthermore, centralisers of all subgroup schemes of GL_{n} are smooth. The situation for normalisers is much less straightforward, which may explain why results in this direction have been unforthcoming until now. For example, even when G = GL_{n}, for any n ≥ 3 and any p > 0 an arbitrary prime, there are connected smooth subgroups of G with non-smooth normalisers (see Lemma 11.11 below). In light of this situation, perhaps it is surprising that there are any general situations in which normalisers of subgroup schemes are smooth. However, we prove that for sufficiently large p depending on the connected reductive algebraic group G, (a) all normalisers of height one subgroup schemes (in fact the normalisers of all subspaces of the Lie algebra of G); and (b) all normalisers of connected reductive subgroups are indeed smooth. Theorem 3.2 makes (b) precise and the proof is a straightforward reduction to the case of centralisers. Our main result follows.

1We call a smooth algebraic group G reductive provided that R_{u}(G^{0}) = 1.

2Recall that p is good if the following holds: p is not 2 if G contains a factor not of type A, p is not 3 if G contains an exceptional factor and p is not 5 if G contains a factor of type E_{8}. The prime p is very good if it is good and it does not divide n + 1 for any factor of G of type A_{n}.

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Theorem A. There exists a constant c = c(r) such that if p > c and G is any connected reductive group of rank r then all normalisers N_{G}(𝔥) of all subspaces 𝔥 of 𝔤 are smooth. More precisely, let d be the dimension of a minimal faithful representation of G. Then all normalisers of subspaces of 𝔤 are smooth provided that p > 2^{2d}. In particular, if G = GL_{n} we may take p > 2^{2n}.

Remarks 1.1. (a). Clearly, the constant c(r) in the theorem may be defined as 2^{2d′} for d′ the maximal dimension of a minimal faithful module of a connected reductive group of rank r. (b). Note that the maximum is finite since there are only a finite number of isomorphism types of connected reductive groups of a given rank over an algebraically closed field k. Each of these arises by base change from a split reductive group defined over the integers, so one can consider the theorem as a statement that for a fixed group G_{ℤ}, the conclusion holds for each reduction modulo p of G_{ℤ}, whenever p is sufficiently large. It is natural to ask if lower bounds for the constant c in Theorem A exist. In §11, we present a menagerie of examples where smoothness of normalisers fails; in particular, in Example 11.4 we give a p-subalgebra of 𝔤𝔩_{2n+12} with nonsmooth normaliser whenever p|F_{n}, the nth Fibonacci number. Since F_{n} ∼ 1.6^{n} and infinitely many Fibonacci numbers are expected to be prime, we conclude that c(G) should grow exponentially with the rank of G. In other words the bound on p in the theorem is likely to be of the right order. The obstruction to finding linear bounds for c comes from the fact that one cannot, in general, lift the maximal tori of Lie-theoretic normalisers to grouptheoretic normalisers. However, many interesting subalgebras of 𝔤 have normalisers which are generated by nilpotent elements (such as maximal semisimple subalgebras). Adding in this extra, natural hypothesis gives rise to much better bounds. In the following theorem let h = h(G) denote the Coxeter number of (the root system Φ of) G. If Φ is reducible, then h is taken as the maximum over all components.

Theorem B. (i) Let G be a reductive algebraic group and let d be as in Theorem A. Suppose p > d + 1. Then all normalisers N_{G}(𝔥) of p-subalgebras 𝔥 are smooth whenever 𝔫_{𝔤}(𝔥) is generated by nilpotent elements. More precisely, the conclusion holds for normalisers generated by nilpotent elements when G is simple of classical type (that is, the root system of G is of A–D type) and p > h + 1.

(ii) Let p > 2h−2 for the connected reductive group G. Then the normalisers N_{G}(𝔥) of all subspaces 𝔥 of 𝔤 are smooth whenever 𝔫_{𝔤}(𝔥) is generated by nilpotent elements.

Remarks 1.2. (a). The bounds in Theorem B(i) are tight when G is classical of type A, B or C: whenever p ≤ h + 1 the smallest irreducible representation of the first Witt algebra or its adjoint gives rise to a non-smooth normaliser which satisfies the hypotheses. Theorem B(i) is also tight for G_{2}, as it contains a copy of the Witt algebra as a maximal subalgebra when p = 7; more generally, the conclusion of Theorem B(i) fails for all exceptional algebraic groups when p = h + 1 (see [HS16]). (b). Suppose that k is not algebraically closed, and that G is a connected reductive algebraic group defined over k with a closed, k-defined subgroupscheme H. Since smoothness is a geometric property, we have that N_{G}(H) is smooth if and only if N_{G\overline{k}}(H\overline{k}) is smooth. Hence Theorems A and B give sufficient conditions for the smoothness of normalisers over general base fields.

In proving the theorems above we require several auxiliary results which may be of independent interest. The first is necessary in proving Theorem B(i).

Theorem C. Let 𝔤 = Lie(G) for G a simply-connected classical algebraic group over an algebraically closed field k and let p > 2 be a very good prime for G. Then any maximal non-semisimple subalgebra of 𝔤 is parabolic.

Remark 1.3. An announcement of a full classification of the maximal nonsemisimple subalgebras of the Lie algebras of classical groups is given in [Ten87]. We provide a straightforward proof of the stated part in §7 below.

The proof of Theorem B(i) also uses a number of results on cohomology of low-dimensional modules. Such results have something of a history: in [Jan97] Jantzen proved that a module for a connected reductive algebraic group with p ≥ dim V is completely reducible. Building on this, Guralnick tackled the case of finite simple groups in [Gur99]; this time one needs p ≥ dim V + 2 for the same conclusion. In a different direction, Serre proved in [Ser94] that if two semisimple modules V_{1} and V_{2} for an arbitrary group satisfy dim V_{1} +dim V_{2} < p + 2 then their tensor product is semisimple. Extending work of Bendel– Nakano–Pillen, we add analogues of these results for Lie algebras and Frobenius kernels of reductive algebraic groups tackling the ‘crucial case’ of a question of Serre [Ser94, Question 1.2] (though see Footnote 3 below). We summarise our results when G is simple into the following. The extensions to the case G is semisimple or reductive can be found in §8, where also can be found any unexplained terminology.

Theorem D. Suppose G is a simple algebraic group and let G_{r} be its r-th Frobenius kernel with 𝔤 its Lie algebra. Let V be a k-vector space with dim V ≤ p.

(a) Suppose V is a G_{r}-module. Then V is completely reducible unless dim V = p, and either G is of type A_{1} or p = 2 and G is of type C_{n}. In the exceptional cases, V is known explicitly.

(b) Suppose 𝔤 = [𝔤, 𝔤] and V is a 𝔤-module. Then either V is completely reducible or dim V = p, G is of type A_{1} and V is known explicitly.

(c) Let p > h. Then H^{2}(𝔤, L(μ)) = 0, for all μ in the lowest alcove C_{ℤ}, unless G is of type A_{1} and μ = (p − 2); or G is of type A_{2} and μ = (p − 3, 0) or (0, p − 3).

(d) Suppose V and W are semisimple 𝔤-modules with dim V +dim W < p+2. Then V ⊗ W is semisimple and H^{2}(𝔤, V ⊗ W) = 0. ^{3}

We also mention a further tool, used in the proofs of Theorems A and B(i), for which we need a definition due to Richardson: Suppose that (G′, G) is a pair of reductive algebraic groups such that G ⊆ G′ is a closed subgroup. We say that (G′, G) is a reductive pair provided there is a subspace 𝔪 ⊆ Lie(G′) such that Lie(G′) decomposes as a G-module into a direct sum Lie(G′) = Lie(G) + 𝔪. Adapting a result from [Her13] we show

Proposition E. Let (G′, G) be a reductive pair and let H ≤ G be a closed subgroup scheme. Then if N_{G′}(H) is smooth, N_{G}(H) is smooth too.

Acknowledgements

Many thanks to Cornelius Pillen and Brian Parshall who provided a number of helpful hints during the production of this article, and to Alexander Premet for some useful conversations. Also we would like to express our gratitude to the referee for their comments on the paper and encouraging us to elucidate a couple of our arguments along the lines they suggested. The first author acknowledges financial support from ERC Advanced Grant 291512 (awarded to Gunter Malle).

3In [Del13], it is proved that if V and W are semisimple modules for an affine group scheme, satisfying dim V + dim W < p + 2, then V ⊗ W is semisimple. The semisimplicity part of Theorem D(d) can be deduced from this. In fact, in part of Corollary 8.13 we do prove the semisimplicity statement for arbitrary Lie algebras. Our proof is different to Deligne’s, relying just on a theorem of Strade, together with Theorem C.

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2 Notation and preliminaries

Let k be a field of characteristic p ≥ 0 and let G be an algebraic group defined over k. Unless otherwise noted, k will assumed to be algebraically closed. For all aspects to do with the representation theory of a connected reductive algebraic group G we keep notation compatible with [Jan03]. In particular, R is the root system of G, and h is the associated Coxeter number. For a closed subgroup H ≤ G, we consider the scheme-theoretic normaliser N_{G}(H), respectively centraliser C_{G}(H) of H in G. We define N_{G}(H) to be subfunctor of G which takes a k-algebra A and returns the subgroup of elements

N_{G}(H)(A) = {g ∈ G(A) : gH(B)g^{−1} = H(B)}

for all A-algebras B. Similarly, the centraliser is defined via

C_{G}(H)(A) = {g ∈ G(A) : gh = hg for all h ∈ H(B)}.

Since H is closed, N_{G}(H) and C_{G}(H) are closed subgroup schemes of G. By contrast, for any affine algebraic group H over k, we denote by H_{red} the smooth subgroup with k-points H_{red}(k) = H(k). As k is algebraically closed, the existence and uniqueness of such a subgroup is explained for example in [Mil12, Prop. 5.1] and (as we will use in the sequel) we have that N_{G}(H)_{red}(k′) = N_{G(k′)}(H(k′))(k′) (resp. C_{G}(H)_{red}(k′) = C_{G(k′)}(H(k′))(k′)) by [Mil12, §VII.6]. Let 𝔤 be a Lie algebra over k. When the characterstic of k is greater than 0, 𝔤 is often referred to as a modular Lie algebra, and as such our reference for the theory is [SF88]. Recall that a Lie algebra 𝔤 is semisimple if its solvable radical is zero, and that in characteristic p > 0 this is not enough to ensure that it is the direct sum of simple Lie algebras. Sometimes but not all the time, we will have 𝔤 = Lie(G) for G an algebraic group, in which cas we refer to 𝔤 as algebraic; in this case, 𝔤 will carry the structure of a restricted Lie algebra. Bear in mind that Lie(G) may not be semisimple even when G is. Examples of this sort only occur in not-very-good characteristic; for instance, 𝔰𝔩_{2} = Lie(SL_{2}) in characteristic 2 gives a restricted structure on the solvable Lie algebra 𝔰𝔩_{2} with 1-dimensional centre. More generally, all restricted Lie algebras are of the form Lie(H), where H is an infinitesimal group scheme of height one over k. Under this correspondence, the restricted subalgebras of 𝔤 = Lie(G) correspond to height one subgroup schemes of G. If the centre Z(𝔤) = 0, then a Lie algebra 𝔤 has at most one restricted structure. In particular, if two semisimple restricted Lie algebras are isomorphic as Lie algebras, they are isomorphic as restricted Lie algebras. An abelian p-subalgebra 𝔥 of 𝔤 consisting of semsimple elements is called a torus of 𝔤. Cartan subalgebras of algebraic Lie algebras are always toral and in fact the Lie algebras of maximal tori of the associated algebraic group. This follows from [Hum67, Thm. 13.3]. If 𝔤 is a restricted Lie algebra, a representation V is called restricted provided it is given by a morphism of restricted Lie algebras 𝔤 → 𝔤𝔩(V). The following fact follows e.g. from the Kac–Weisfeiler conjecture (see [Pre95, Cor. 3.10]): if G is a simple algebraic group defined in very good characteristic, and if V is an irreducible 𝔤-module with dim V < p, then V is restricted. In particular, it is well-known that V is then obtained by differentiating a simple restricted rational representation of G. When 𝔤 is a Lie algebra, Rad(𝔤) is the solvable radical of 𝔤 and N(𝔤) is the nilradical of 𝔤. If 𝔤 ⊆ 𝔤𝔩(V) there is also the radical of V -nilpotent elements Rad_{V} (𝔤). When 𝔤 is restricted, Rad_{p}(𝔤) is the p-radical of 𝔤, defined to be the biggest p-nilpotent ideal. Further, 𝔤 is p-reductive if the radical Rad_{p}(𝔤) is zero. Recall the following properties from [SF88, §2.1]:

Lemma 2.1. (a) Rad_{p}(𝔤) is contained in the nilradical N(𝔤) and hence in the solvable radical of 𝔤. In particular, semisimple Lie algebras are preductive.

(b) Rad_{p}(𝔤) is the maximal p-nil (that is, consisting of p-nilpotent elements) ideal of 𝔤.

(c) 𝔤/ Rad_{p}(𝔤) is p-reductive.

In particular, by part (b), if 𝔤 ⊆ 𝔤𝔩(V) is a restricted subalgebra then Rad_{p}(𝔤) = Rad_{V} (𝔤). If 𝔤 ⊆ 𝔤𝔩(V) is a restricted Lie subalgebra and G_{1} is the height one subgroup scheme of GL(V) associated to 𝔤, then 𝔤 is p-reductive if and only if G_{1} is reductive in the sense that is has no connected normal nontrivial unipotent subgroup schemes. For the usual notion of reductivity of smooth algebraic groups only smooth unipotent subgroups are considered. The relation between these two concepts is as follows:

Proposition 2.2 ([Vas05]). Let G be a connected reductive algebraic group. Then G has no non-trivial connected normal unipotent subgroup schemes, except if both p = 2 and G contains a direct factor isomorphic to SO_{2n+1} for some n ≥ 1.

Since there are a number of possible definitions, let us be clear on the following: We define a Borel subalgebra (resp. parabolic subalgebra, resp. Levi subalgebra) of 𝔤 to be Lie(B) (resp. Lie(P), resp. Lie(L)), where B (resp. P, resp. L) is a Borel (resp. parabolic, resp. Levi subgroup of a parabolic) subgroup of G. By P = LQ we will denote a parabolic subgroup of G with unipotent radical Q and Levi factor L. We will usually write 𝔭 = Lie(P) = 𝔩 + 𝔮. A fact that we will use continually during this paper, without proof, is that if H (resp. 𝔥) is a subgroup (resp. subalgebra) of P (resp. 𝔭), such that the projection to the Levi is in a proper parabolic of the Levi, then there is a strictly smaller parabolic P_{1} < P (resp. 𝔭_{1} < 𝔭) such that H ≤ P_{1} (resp. 𝔥 ≤ 𝔭_{1}). See [BT65, Prop. 4.4(c)]. We also use the following fact: If 𝔱 ⊆ 𝔤𝔩_{n} is a torus, then C_{GL_{n}}(𝔱) is a Levi subgroup (this follows e.g. from the construction of a torus T ⊆ GL_{n} in [Die52, Prop. 2] with C_{GL_{n}}(𝔱) = C_{GL_{n}}(T)).

Let V be an 𝔤-module and let λ : V × V → k be a bilinear form on V. We say 𝔤 preserves λ if λ(x(v), w) = −λ(v, x(w)) for all x ∈ 𝔤, v, w ∈ V. We recall definitions of the algebraic simple Lie algebras of classical type: those with root systems of types A–D. Then 𝔬(V) is the set of elements x ∈ 𝔤𝔩(V) preserving the form λ(v, w) = v^{t}w. 𝔰𝔬(V) is the subset of traceless matrices of 𝔬(V). On the other hand when dim V is even, 𝔰𝔭(V) is the set of elements preserving the form λ(v, w) = v^{t}Jw with J = [[0, −I_{n}], [I_{n}, 0]]. If char k ≠ 2 then 𝔰𝔭(V) and 𝔰𝔬(V) are simple (see below). We say 𝔰𝔭(V) is of type C_{n} with 2n = dim V; 𝔰𝔬(V) is of type B_{n} when dim V = 2n + 1, or type D_{n} when dim V = 2n. One fact that we shall use often in the sequel is that that for types B–D, parabolic subalgebras are the stabilisers of totally singular subspaces. (See for example, [Kan79].) Furthermore recall that if G is simple, then 𝔤 is simple at least whenever p is very good. See [Hog82, Cor. 2.7] for a more precise statement. This means in particular that 𝔰𝔩(V) is simple unless p| dim V, in which case the quotient 𝔭𝔰𝔩(V) = 𝔰𝔩(V)/kI is simple; we refer to such algebras as type A_{n} classical Lie algebras, where dim V = n+1. In all cases, we refer to V as the natural module for the algebra in question. We make extensive use of the current state of knowledge of cohomology in this paper, especially in §8. Importantly, recall that the group Ext_{A}^{1}(V, W) (with A either an algebraic group or a Lie algebra) corresponds to the equivalence classes of extensions E of A-modules 0 → W → E → V → 0, and that H^{2}(A, V) measures the equivalence classes of central extensions B of V by A, equivalence classes of exact sequences 0 → V → B → A → 0, where B is either an algebraic group or a Lie algebra. We remind the reader that for restricted Lie algebras, two forms of cohomology are available—the ordinary Lie algebra cohomology, denoted H^{i}(𝔤, V) or the restricted Lie algebra cohomology (where modules respectively morphisms are assumed to be restricted). Since the latter can always be identified with H^{i}(A, V) for A the height one group scheme associated to 𝔤, we shall always use the associated group scheme when we wish to discuss restricted cohomology. Finally, we record the following theorem of Strade which is a central tool in our study of small-dimensional representations. Let char k = p > 0 and let O_{1} = k[X]/X^{p} be the truncated polynomial algebra. Then the first Witt algebra W_{1} is the set of derivations of O_{1}, with basis {X^{r}∂}_{0≤r≤p−1}, where ∂ acts on O_{1} by differentiation of polynomials. For p > 2, W_{1} is simple, and for p > 3, W_{1} is not the Lie algebra of any algebraic group. Since there is a subspace k ≤ O_{1} fixed by W_{1}, we see that W_{1} has a faithful (p−1)-dimensional representation for p > 2.

Theorem 2.3 ([Str73, Main theorem]). Let 𝔤 be a semisimple Lie subalgebra of 𝔤𝔩(V) over an algebraically closed field k of characteristic p > 2 with p > dim V. Then 𝔤 is either a direct sum of algebraic Lie algebras or p = dim V + 1 and 𝔤 is the p-dimensional Witt algebra W_{1}.

3 Smoothness of normalisers of reductive subgroups

Let G be a connected reductive algebraic group and let T be a maximal torus in G with associated roots R, coroots R^{∨}, characters X(T) and cocharacters Y (T). We say that a prime p is pretty good for G provided it is good for R and provided that both X(T)/ℤR and Y (T)/ℤR^{∨} have no p-torsion. We recall the main result of [Her13].

Proposition 3.1. Let G be as above, and let p = char(k). Then p is pretty good for G if and only if all centralisers of closed subgroup schemes in G are smooth.

Theorem 3.2. Let G be a connected reductive algebraic group. Then the normalisers N_{G}(H) of all (smooth) connected reductive subgroups are smooth if p is a pretty good prime for G.

Proof. Let H ≤ G be a closed, connected reductive subgroup of G. We have an exact sequence of group functors

1 → C_{G}(H) → N_{G}(H) ⟶^{int} Aut(H).

Here the first map is the natural inclusion, the second map maps x ∈ G to the automorphism int(x) of H given by conjugation with x, and Aut(H) is the group functor that associates to each k-algebra S the group of automorphisms of the group scheme H_{S}. By [DGd70, XXIV, Cor. 1.7], we have that Aut(H)^{0} = int(H) is smooth, which implies that int(N_{G}(H)) is smooth. By Proposition 3.1, C_{G}(H) is smooth. Thus the outer terms in the exact sequence of affine group schemes

1 → C_{G}(H) → N_{G}(H) → int(N_{G}(H)) → 1

are smooth, which forces N_{G}(H) to be smooth.

Remark 3.3. The implication in the theorem cannot quite be reversed. For example if G is SL_{2}, p = 2 is not pretty good, but a connected reductive subgroup is either trivial, or a torus, whose normaliser is smooth. However, we give examples of non-smooth normalisers of connected reductive subgroups in bad characteristics in Examples 11.6 below.

4 On exponentiation and normalising, and the proof of Theorem B(ii)

Let G be a connected reductive group. We recall the existence of exponential and logarithm maps for p big enough, see [Ser98, Thm. 3] or [Sei00, Prop. 5.2]. We fix a maximal torus T and a Borel subgroup B = T ⋉ U containing T.

Theorem 4.1. Assume that p > h (p ≥ h for G simply connected), where h is the Coxeter number of G. Then there exists a unique isomorphism of varieties log : G^{u} → 𝔤_{nilp}, whose inverse we denote by exp : 𝔤_{nilp} → G^{u}, with the following properties:

(i) log ∘σ = dσ ∘ log for all σ ∈ Aut(G);

(ii) the restriction of log to U is an isomorphism of algebraic groups U → Lie(U), whose tangent map is the identity; here the group law on Lie(U) is given by the Hausdorff formula;

(iii) log(x_{α}(a)) = aX_{α} for every root α and a ∈ k, where X_{α} = dx_{α}(1).

The uniqueness implies that for G = GL(V), p ≥ dim V, exp and log are the usual truncated series. Recall (cf. [Ser98]) that for a G-module V, the number n(V) is defined as n(V) = sup_{λ} n(λ), where λ ranges over all T -weights of V, and where n(λ) = ∑_{α∈R^{+}}⟨λ, α^{∨}⟩. For the adjoint module 𝔤, one obtains n(𝔤) = 2h − 2.

Proposition 4.2. Let ρ : G → GL(V) be a rational representation of G. Suppose that p > h and p > n(V). Let x ∈ 𝔤 be a nilpotent element. Then

ρ(exp_{G} x) = exp_{GL}(dρ(x)).

In particular, if p > 2h − 2, then Ad(exp_{G} x) = exp_{GL}(ad(x)).

Proof. Consider the homomorphism φ : 𝔾_{a} → GL(V) given by φ(t) = ρ(exp_{G}(t.x)). Under our assumptions, it follows from [Ser98, Thm. 5] that φ is a morphism of degree < p, (i.e. the matrix entries of φ are polynomials of degree less than p in t). Moreover, dφ(1) = dρ(x). By [Ser94, §4], this implies that dρ(x)^{p} = 0 and that φ agrees with the homomorphism t ↦ exp_{GL}(t.dρ(x)). The claim follows.

Lemma 4.3. Let X ∈ 𝔤𝔩(V) be a nilpotent element satisfying X^{n} = 0 for some integer n ≤ p. Let l, r ∈ End(𝔤𝔩(V)) be left multiplication with X, respectively right multiplication with −X. Set W = W_{p}(l, r) ∈ End(𝔤𝔩(V)), where W_{p}(x, y) is the the image of \frac{1}{p}((x + y)^{p} − x^{p} − y^{p}) ∈ ℤ[x, y] in k[x, y]. Let 𝔥 be a subset of 𝔤𝔩(V) normalised (resp. centralised) by X. Suppose that 𝔥 ⊆ ker(W). Then exp(X) ∈ GL(V) normalises (resp. centralises) 𝔥. In particular, if p ≥ 2n − 1, then W = 0 and so exp(X) normalises (resp. centralises) every subspace that is normalised (resp. centralised) by X.

Proof. Since the nilpotence degree of X is less than p, the exponential exp(X) = 1 + X + X^{2}/2 + … gives a well-defined element of GL(V). Moreover, for each Y ∈ 𝔥 we have the equality

Ad(exp(X))(Y) = exp(ad(X))(Y) = Y +ad(X)(Y)+ad(X)^{2}(Y)/2+⋯ ∈ 𝔤𝔩(V).

Indeed, we have ad(X) = l + r, and Ad(exp(X)) = exp(l) exp(r). Now by [Ser94, (4.1.7)], exp(l) exp(r) = exp(l + r − W). Since l and r commute with W, we deduce (l + r − W)^{m}(Y) = (l + r)^{m}(Y) for each m ≥ 0. Thus Ad(exp(X))(Y) = exp(l + r)(Y) = exp(ad(X))(Y), as claimed. Hence exp(X) is contained in N_{GL(V)}(𝔥) whenever X ∈ 𝔫_{𝔤𝔩(V)}(𝔥) and exp(X) ∈ C_{GL(V)}(𝔥) whenever X ∈ 𝔠_{𝔤𝔩(V)}(𝔥). Moreover, W_{p}(l, r) = ∑_{i=1}^{p−1} c_{i}l^{i}r^{p−i} for certain non-zero coefficients c_{i} ∈ k. In particular, this expression vanishes for p ≥ 2n − 1.

Corollary 4.4. Let 𝔭 = 𝔮 + 𝔩 ⊆ 𝔤𝔩(V) be a parabolic subalgebra, and suppose that p ≥ dim V. If X ∈ 𝔮 normalises a subset 𝔥 ⊆ 𝔭, then so does exp(X).

Proof. By Lemma 4.3, it suffices to show that 𝔭 ⊆ ker(W). Let 0 = V_{0} ⊆ V_{1} ⊆ ⋯ ⊆ V_{m} = V be a flag with the property

𝔭 = {Y ∈ 𝔤𝔩(V) | Y V_{i} ⊆ V_{i}}
𝔮 = {Y ∈ 𝔤𝔩(V) | Y V_{i} ⊆ V_{i−1}}.

By assumption, we have p ≥ m, and therefore all products X_{1} … X_{p+1} with all X_{i} ∈ 𝔭 and all but one X_{i} ∈ 𝔮 vanish on V. In particular l^{i}r^{p−i}(Y) = 0 for all Y ∈ 𝔭 and hence W (Y) = 0.

Lemma 4.5. Suppose 𝔤 is a subalgebra of 𝔤𝔩(V) generated as a k-Lie algebra by a set of nilpotent elements {X_{i}} of nilpotence degree less than p, and let G = \overline{⟨exp(t.X_{i})⟩} be the closed subgroup of GL(V) generated by exp(t.X_{i}) for each t ∈ k. Then 𝔤 ≤ Lie(G).

Proof. Since Lie(G) contains the element d/dt exp(t.X_{i})|_{t=0} it contains each element X_{i}. Since 𝔤 is generated by the elements X_{i}, we are done.

Proof of Theorem B(ii). Let 𝔥 be a subspace of 𝔤 and let 𝔫 = 𝔫_{𝔤}(𝔥) be the Lie-theoretic normaliser of 𝔥 in 𝔤. Let {x_{1}, …, x_{r}} be a set of nilpotent elements generating 𝔫. To show that N_{G}(𝔥) is smooth, it suffices to show that each x_{i} belongs to the Lie algebra of N_{G}(𝔥)_{red}. But for a nilpotent generator x_{i}, we may consider the smooth closed subgroup M_{i} = \overline{⟨exp(t.x_{i}) | t ∈ k⟩} of G. By Proposition 4.2, M_{i} ⊆ N_{G}(𝔥)_{red} and hence x_{i} ∈ Lie(M_{i}) ⊆ Lie(N_{G}(𝔥)_{red}), as required.

5 Reductive pairs: Proof of Proposition E

The following definition is due to Richardson [Ric67].

Definition 5.1. Suppose that (G′, G) is a pair of reductive algebraic groups such that G ⊆ G′ is a closed subgroup. Let 𝔤′ = Lie(G), 𝔤 = Lie(G). We say that (G′, G) is a reductive pair provided there is a subspace 𝔪 ⊆ 𝔤′ such that 𝔤′ decomposes as a G-module into a direct sum 𝔤′ = 𝔤 ⊕ 𝔪.

With p sufficiently large, reductive pairs are easy to find.

Lemma 5.2 ([BHMR11, Thm. 3.1]). Suppose p > 2 dim V − 2 and G is a connected reductive subgroup of GL(V). Then (GL(V), G) is a reductive pair.

We need a compatibility result for normalisers of subgroup schemes of height one.

Lemma 5.3. Let H ⊆ G be a closed subgroup scheme of height one, with 𝔥 = Lie(H). Then N_{G}(H) = N_{G}(𝔥) (scheme-theoretic normalisers).

Proof. We have a commutative diagram

Hom(H, H) ⟶ Hom_{p−Lie}(𝔥, 𝔥)
↓  ↓
Hom(H, G) ⟶ Hom_{p−Lie}(𝔥, 𝔤),

where the horizontal arrows are given by differentiation and are bijective (cf. [DG70, II, §7, Thm. 3.5]). Now if x ∈ N_{G}(𝔥), the map Ad(x)_{𝔥} in the bottom right corner may be lifted via the top right corner to a map in Hom(H, H). The commutativity of the diagram shows that conjugation by x stabilises H, and hence x ∈ N_{G}(H). This works for points x with values in any k-algebra, and hence proves the containment of subgroup schemes N_{G}(𝔥) ⊆ N_{G}(H). The reverse inclusion is clear.

We show that the smoothness of normalisers descends along reductive pairs. Let us restate and then prove Propostion E.

Proposition 5.4. Let (G′, G) be a reductive pair and let H ⊆ G be a closed subgroup scheme. If N_{G′}(H) is smooth, then so is N_{G}(H). In particular, if 𝔥 ⊆ 𝔤 is a restricted subalgebra and if N_{G′}(𝔥) is smooth, then so is N_{G}(𝔥).

Proof. The last assertion follows from Lemma 5.3. Let H ⊆ G be a closed subgroup scheme. We follow the proof of [Her13, Lem. 3.6]. Let 𝔤′ = 𝔤 ⊕ 𝔪 be a decomposition of G-modules. By [DG70, II, §5, Lem. 5.7], we have

dim Lie(N_{G′}(H)) = dim 𝔥 + dim(𝔤′/𝔥)^{H} = dim 𝔥 + dim(𝔤/𝔥)^{H} + dim 𝔪^{H}

= dim Lie(N_{G}(H)) + dim 𝔪^{H} ≥ dim N_{G}(H) + dim 𝔪^{H}.

On the left hand side, as N_{G′}(H) is smooth by assumption, we have dim N_{G′}(H) = dim Lie(N_{G′}(H)). Thus to show that N_{G}(H) is smooth, it suffices to show that dim N_{G′}(H) − dim N_{G}(H) ≤ dim 𝔪^{H}. Now as in [Her13, Lem. 3.6], one shows that there is a monomorphism of quotient schemes N_{G′}(H)/N_{G}(H) ↪ (G′/G)^{H}, and that the tangent space on the right hand side identifies as Tē(G′/G)^{H} ≅ 𝔪^{H}. The claim follows.

6 Lifting of normalising tori and the proof of Theorem A

In this section we let G = GL(V) and 𝔥 be a subspace of 𝔤. We would like to lift a normaliser 𝔫_{𝔤}(𝔥) to a subgroup N normalising 𝔥 such that Lie(N) = 𝔫_{𝔤}(𝔥). It turns out that the hardest part of this is to find a lift of a maximal torus normalising 𝔥. This is the content of the next lemma.

Lemma 6.1. Let G = GL_{n} with p > 2^{2n} and let 𝔥 ⊆ 𝔤 be any subspace of 𝔤 = Lie(G). Suppose that 𝔠 ⊆ 𝔤 is a torus normalising 𝔥. Then 𝔠 = Lie(C) for a torus C ⊆ N_{G}(𝔥).

Proof. Let T be a diagonal maximal torus of GL_{n} and 𝔱 = Lie(T). Since 𝔠 consists of semisimple elements, we may assume 𝔠 ⊆ 𝔱. Since 𝔠 is restricted, it has a basis defined over 𝔽_{p} of elements Z_{1}, …, Z_{s} with Z_{i} = diag(z_{i1}, …, z_{in}) and each z_{ij} ∈ 𝔽_{p}. By [Die52, Prop. 2] we may assume that 𝔠 is a maximal torus of 𝔫_{𝔤}(𝔥), which we do from now on. Since k is algebraically closed, we may take a decomposition of 𝔥 into weight spaces for 𝔠. We have 𝔥 = 𝔥_{0} ⊕ ⨁_{α} 𝔥_{α} where 𝔥_{0} is some set of elements commuting with 𝔠, α is a non-trivial linear functional 𝔠 → k and each 𝔥_{α} is a subspace of 𝔤𝔩_{n} with [c, X] = α(c)X for c ∈ 𝔠 and X ∈ 𝔥_{α}. Let {X_{i}} be a basis for 𝔥 with each X_{i} ∈ 𝔥_{0} or 𝔥_{α} for some α as above. Then 𝔠 = ⋂_{i} 𝔫_{𝔱}(⟨X_{i}⟩). Suppose c = diag(c_{1}, …, c_{n}). The condition c ∈ 𝔫_{𝔱}(⟨X_{i}⟩) puts a set of conditions on the c_{i}. If only one entry of the matrix X_{i} is non-zero or X_{i} is diagonal, then 𝔱 normalises X_{i}, hence the set of conditions is empty. Otherwise, if (X_{i})_{j,k} and (X_{i})_{l,m} are non-zero, then c normalising ⟨X_{i}⟩ implies c_{j} − c_{k} = c_{l} − c_{m}. Letting c = (c_{1}, …, c_{n}) this condition can be rewritten as a linear equation rc = 0, where r is an appropriate row vector whose entries are all 0, except for up to four, where the non-zero entries take the values, up to signs or permutations, (1, −1), (2, −2), (1, −2, 1) or (1, −1, −1, 1) according to the values of j, k, l and m. The collection of these, say m relations, across i and all pairs of non-zero entries in X_{i} gives an m × n integral matrix R so that c ∈ 𝔠 if and only if it satisfies the equation Rc = 0 modulo p. Similarly, if χ(t) = diag(t^{a_{1}}, …, t^{a_{n}}) is a cocharacter with image in T, then one checks that χ(t) normalises 𝔥 if the integral equation Ra = 0 where a = (a_{1}, …, a_{n}). If the nullity of R is the same modulo p as it is over the integers then for any c ∈ 𝔫_{𝔱}(𝔥), there exists a cocharacter χ of N_{T} (𝔥) with d/dt|_{t=1}(χ(t)) = c and we are done. But if the nullity of R modulo p differs from the nullity of R over the integers, then we must have that p|d_{i} for d_{i} one of the non-zero elementary divisors of R. Now by the theory of Smith Normal Form, if r ∈ ℕ is taken maximal so that there exists a non-vanishing r × r minor, then the elementary divisors of R are all at most the greatest common divisor of all non-zero r × r minors. Let M be such an r × r minor. We are going to argue by induction on r that | det(M)| ≤ 2^{2r}. Since r ≤ n, the hypothesis will then show that p is not a prime factor of det(M), as required. We must have r ≤ n. If there is a row of M containing only elements of modulus 2, then at most 2 of these are non-zero and 2 is a prime factor of det M; Laplace’s formula implies that the remaining matrix has determinant at most 2 det M′ where M′ is a certain r − 1 × r − 1 minor of M, so that we are done by induction. If there are no entries of modulus 2, then each row contains at most 4 entries of modulus 1 and Laplace’s formula then implies that det M ≤ 4 det M′ where M′ is a certain r −1×r −1 minor of M of the required form, so that we are done again by induction. Otherwise there is at least one row with non-zero entries (1, −2) or (1, −2, 1). By Laplace’s formula and induction, it is now easy to see that | det M| ≤ 2^{2n−2} + 2.2^{2n−2} + 2^{2n−2} = 2^{2n} and we are done.

We are now in a position to prove Theorem A.

Proof of Theorem A. First consider the case G = GL_{n}. Let 𝔥 be a subspace of 𝔤 and let 𝔫 = 𝔫_{𝔤}(𝔥) be the Lie-theoretic normaliser of 𝔥 in 𝔤. As before, by definition, 𝔫 is a restricted subalgebra of 𝔤. Hence, applying the Jordan decomposition for restricted Lie algebras, we see that 𝔫 is generated by its nilpotent and semisimple elements. Let {x_{1}, …, x_{r}, y_{1}, …, y_{s}} be such a generating set with x_{1}, …, x_{r} nilpotent and y_{1}, …, y_{s} semisimple. To show that N_{G}(𝔥) is smooth, it suffices to show that all the elements x_{i} and y_{j} belong to the Lie algebra of N_{G}(𝔥)_{red}. For a nilpotent generator x_{i}, of nilpotence degree at most n < p, consider the smooth closed subgroup M_{i} = \overline{⟨exp(t.x_{i}) | t ∈ k⟩} of G. Since p > 2h − 2, we may apply Proposition 4.2, to obtain M_{i} ⊆ N_{G}(𝔥)_{red} and hence x_{i} ∈ Lie(M_{i}) ⊆ Lie(N_{G}(𝔥)_{red}), as required. It remains to consider the semisimple generators y_{i}. Let 𝔱_{i} := ⟨y_{i}⟩_{p} ≤ 𝔫 be the torus generated by the p-powers of y_{i}. By hypothesis, p > 2^{2n} and so we may apply Lemma 6.1 to find a torus T_{i} ≤ N_{G}(𝔥) such that Lie(T_{i}) = 𝔱_{i}. In particular y_{i} ∈ Lie(N_{G}(𝔥)_{red}). This finishes the proof in the case G = GL(V). If G is a reductive algebraic group suppose G → GL(V) ≅ GL_{d} is a minimal faithful module for G. Now since p > 2^{2 dim V}, we have that normalisers of all subspaces of GL(V) are smooth. But now, by Lemma 5.2, (GL(V), G) is a reductive pair, so that invoking Proposition 5.4 we obtain that N_{G}(𝔥) is smooth. This completes the proof.

7 Non-semisimple subalgebras of classical Lie algebras. Proof of Theorem C

Suppose char k > 2 for this section. This section provides proofs for some of the claims made in [Ten87]. Here we tackle the proof of Theorem C.

Proposition 7.1 (see [SF88, §5.8, Exercise 1]). Let 𝔤 ≤ 𝔤𝔩(V) be a Lie algebra acting irreducibly on an 𝔤-module V such that 𝔤 preserves a non-zero bilinear form. Then 𝔤 is semisimple.

Proof. Assume otherwise. Then Rad(𝔤) ≠ 0 and we can find an abelian ideal 0 ≠ J ◁ 𝔤. Take x ∈ J. As [x^{p}, y] = ad(x)^{p}y ∈ J^{(1)} = 0, x^{p} centralises 𝔤 and we have that v ↦ x^{p}v is a 𝔤-homomorphism V → V. Since k is algebraically closed and V is irreducible, Schur’s lemma implies that x^{p}v = α(x)v for some map α : J → k. Since λ ≠ 0 there are v, w with λ(v, w) = 1. Now α(x) = λ(x^{p}v, w) = −λ(v, x^{p}w) = −α(x) so α(x) = 0. Thus x^{p}v = 0 for all x ∈ J. Hence J acts nilpotently on V and so Engel’s theorem gives an element 0 ≠ v ∈ V annihilated by J. Since V is irreducible, it follows that JV = J(𝔤v) ≤ 𝔤Jv = 0. Thus J = 0 and 𝔤 is semisimple.

Since any subalgebra of a classical simple Lie algebra of type B, C or D preserves the associated (non-degenerate) form we get

Corollary 7.2. If 𝔥 is a non-semisimple subalgebra of a classical simple Lie algebra 𝔤 of type B, C or D then 𝔥 acts reducibly on the natural module V for 𝔤.

Remark 7.3. If 𝔤 = 𝔤_{2} (resp. 𝔣_{4}, 𝔢_{7}, 𝔢_{8}) then a subalgebra acting irreducibly on the self-dual modules V_{7} (resp. V_{26}, or V_{25} if p = 3, V_{56}, V_{248} = 𝔢_{8}) is semisimple. Here V_{n} refers to the usual irreducible module of dimension n. A subalgebra is maximal rank if it is proper and contains a Cartan subalgebra (CSA) of 𝔤. (Note that CSAs of simple algebraic Lie algebras are tori.) Call a subalgebra 𝔥 of 𝔤 an R-subalgebra if 𝔥 is contained in a maximal rank subalgebra of 𝔤. For the following, notice that if p| dim V then 𝔰𝔩(V) is not simple, though provided 𝔰𝔩(V) ≠ 𝔰𝔩_{2} in characteristic 2, the central quotient 𝔭𝔰𝔩(V) is simple. Now, a subalgebra 𝔥 of 𝔭𝔰𝔩(V) is an R-subalgebra of 𝔭𝔰𝔩(V) if and only if its preimage π^{−1}𝔥 under π : 𝔰𝔩(V) → 𝔭𝔰𝔩(V) is an R-subalgebra. We say 𝔥 acts reducibly on V if π^{−1}𝔥 does.

Proposition 7.4. Let 𝔤 be a simple algebraic Lie algebra of classical type and let 𝔥 ≤ 𝔤 act reducibly on the natural module V for 𝔤. Then 𝔥 is an R-subalgebra unless 𝔤 = 𝔰𝔬(V) with dim V = 2n with 𝔥 ≤ 𝔰𝔬(W) × 𝔰𝔬(W′) stabilising a decomposition of V into two odd-dimensional, non-degenerate subspaces W and W′ of V.

Proof. Let V be the natural module for 𝔤 and let W ≤ V be a minimal 𝔥submodule, so that 𝔥 ≤ Stab_{𝔤}(W). If 𝔤 is of type A then Stab_{𝔤}(W) is Lie(P) for a (maximal) parabolic P of SL(V). Hence 𝔥 is an R-subalgebra of 𝔤. If 𝔤 is of type B, C or D, then consider U = W ∩ W^{⊥}; this is the subspace of W whose elements v satisfy λ(v, w) = 0 for every w ∈ W. Since M preserves λ, this is a submodule of W, hence we have either U = 0 or U = W by minimality of W. If the latter, W is totally singular. Thus Stab_{𝔤}W is Lie(P) for a parabolic subgroup P of the associated algebraic group. On the other hand, U = 0 implies that W is non-degenerate. Then V = W ⊕ W^{⊥} is a direct sum of 𝔥-modules and we see that Stab_{𝔤}W is isomorphic to (i) 𝔰𝔭_{2r} × 𝔰𝔭_{2s} in case L is of type C, dim W = 2s and 2r + 2s = dim V

(ii) 𝔰𝔬_{r} × 𝔰𝔬_{s} in case L is of type B or D, dim W = s and r + s = dim V.

Note that by [Bou05, VII, §2, No. 1, Prop. 2] the dimensions of the CSA of a direct product is the sum of the dimensions of the CSAs of the factors. In case (i), the subalgebra described has the (r + s)-dimensional CSA arising from the two factors. In case (ii), if dim V = 2n + 1 is odd then one of r and s is odd. If r is odd then 𝔰𝔬_{r} has a CSA of dimension (r − 1)/2, and 𝔰𝔬_{s} has a CSA of dimension s/2, so that the two together give a CSA of dimension s/2 + (r − 1)/2 = n. (Similarly if s is odd.) Otherwise dim V = 2n is even. If dim W is even then Stab_{𝔤}W contains a CSA of dimension r/2 + s/2 = n. If dim W is odd then we are in the exceptional case described in the proposition.

Remark 7.5. In the exceptional case, note that 𝔰𝔬_{2r+1} × 𝔰𝔬_{2s+1} contains a CSA of dimension r+s, whereas 𝔰𝔬_{2n+2} contains a CSA of dimension n+1 = r+s+1.

Corollary 7.6. Let 𝔤 be of type B, C or D. If 𝔥 is a maximal non-semisimple subalgebra of 𝔤, then 𝔥 is Lie(P) for P a maximal parabolic of G. In particular, if 𝔥 is any non-semisimple subalgebra of 𝔤, it is an R-subalgebra.

Proof. Assume otherwise. Then 𝔥 fixes no singular subspace on V. Suppose 𝔥 preserves a decomposition V = V_{1} ⊥ V_{2} ⊥ ⋯ ⊥ V_{n} on V with n as large as possible, with the V_{i} all non-degenerate. Then 𝔥 ≤ 𝔤_{1} = 𝔰𝔬(V_{1}) × ⋯ × 𝔰𝔬(V_{n}) or 𝔥 ≤ 𝔤_{1} = 𝔰𝔭(V_{1}) × ⋯ × 𝔰𝔭(V_{n}). Since 𝔥 is non-semisimple, the projection 𝔥_{1} of 𝔥 in 𝔰𝔬(V_{1}) or 𝔰𝔭(V_{1}), say, is non-semisimple. Then Proposition 7.1 shows that 𝔥 acts reducibly on V_{1}. Since 𝔥 stabilises no singular subspace, the proof of Proposition 7.4 shows that 𝔥 stabilises a decomposition of V_{1} into two nondegenerate subspaces, a contradiction of the maximality of n.

Let 𝔥 be a restricted Lie algebra, I ≤ 𝔥 an abelian ideal and V an 𝔥-module. Let λ ∈ I^{∗}. Recall from [SF88, §5.7] that 𝔥^{λ} = {x ∈ 𝔥|λ([x, y]) = 0 for all y ∈ I} and V^{λ} = {v ∈ V |x.v = λ(x)v for all x ∈ I}.

Proposition 7.7. Let 𝔥 be a non-semisimple subalgebra of 𝔰𝔩(V) with V irreducible for 𝔥. Then p| dim V.

Proof. Let 𝔥 be as described and let I be a nonzero abelian ideal of 𝔥. If 𝔥_{p} denotes the closure of 𝔥 under the p-mapping, then by [SF88, 2.1.3(2),(4)], I_{p} is an abelian p-ideal of 𝔥_{p}. Thus Rad 𝔥_{p} ≠ 0 and 𝔥_{p} is non-semisimple. Hence we may assume from the outset that 𝔥 = 𝔥_{p} is restricted with nonzero abelian ideal I. Since 𝔥 acts irreducibly on V, by [SF88, Corollary 5.7.6(2)] there exist S ∈ 𝔥^{∗}, λ ∈ I^{∗} such that

V ≅ Ind_{𝔥^{λ}}^{𝔥}(V^{λ}, S).

If λ is identically 0 on I then V^{λ} is an 𝔥-submodule. We cannot have V^{λ} = 0 (or else V = 0) so V^{λ} = V and I acts trivially on V, a contradiction since I ≤ 𝔰𝔩(V).

Hence λ(x) ≠ 0 for some x ∈ I. Suppose V^{λ} = V. Then as x ∈ 𝔰𝔩(V), we have tr_{V} (x) = dim V · λ(x) = 0 and thus p| dim V and we are done. If dim V^{λ} < dim V, then by [SF88, Prop. 5.6.2] we have dim V = p^{dim L/L^{λ}} · dim V^{λ}. Thus again p| dim V, proving the theorem.

Corollary 7.8. If p ∤ dim V then any non-semisimple subalgebra 𝔥 of 𝔰𝔩(V) acts reducibly on V. Hence it is contained in Lie(P) for P a maximal parabolic of SL(V). In particular 𝔥 is an R-subalgebra.

Putting together Corollaries 7.6 and 7.8, this completes the proof of Theorem C. As a first application, the following lemma uses Theorem C to show that preductive implies strongly p-reductive. Recall that a restricted Lie algebra is strongly p-reductive if it is the direct sum of a central torus and a semisimple ideal.

Lemma 7.9. Let 𝔥 ⊆ 𝔤𝔩_{n} be a subalgebra and let p > n. If 𝔥 is p-reductive, it is strongly p-reductive.

Proof. Take 𝔭 = 𝔩 + 𝔮 a minimal parabolic subalgebra with 𝔥 ≤ 𝔭. Set 𝔥_{l} to be the image of 𝔥 under the projection π : 𝔭 → 𝔩. Since p > n, we have 𝔩 ≅ 𝔤𝔩(W_{1}) × ⋯ × 𝔤𝔩(W_{s}) ≅ 𝔰𝔩(W_{1}) × … 𝔰𝔩(W_{s}) × 𝔷, where 𝔷 is a torus. Let 𝔰_{i} be the projection of 𝔥_{𝔩} to 𝔰𝔩(W_{i}), and let 𝔷′ be the projection of 𝔥_{𝔩} to 𝔷. If the projection of Rad(𝔥_{𝔩}) to 𝔰𝔩(W_{i}) is non-trivial, then 𝔰_{i} is not semisimple. By Theorem C, W_{i} is not irreducible for 𝔰_{i}. Thus 𝔭 is not minimal subject to containing 𝔥, a contradiction, proving that all the 𝔰_{i} are semisimple. Moreover, 𝔷′ = Z(𝔥_{𝔩}), as the projection of 𝔷 to each 𝔰𝔩(W_{i}) must vanish. This forces 𝔥_{𝔩} ⊆ 𝔰_{1} × ⋯ × 𝔰_{s} × Z(𝔥_{𝔩}) to be strongly p-reductive. As 𝔥 is p-reductive, we have that π is injective on 𝔥, and hence 𝔥 ≅ 𝔥_{l} is strongly p-reductive.

8 Complete reducibility and low-degree cohomology for classical Lie algebras: Proof of Theorem D

Let G be a connected reductive algebraic group with root system R and let G_{r} ◁ G be the rth Frobenius kernel for any r ≥ 1. It is well-known that the representation theory of G_{1} and 𝔤 are very closely related. In this section we recall results on the cohomology of small G_{r}-modules and use a number of results of Bendel, Nakano and Pillen to prove that small G_{r}-modules are completely reducible with essentially one class of exceptions. We do this by examining Ext_{G_{r}}^{1}(L(λ), L(μ)) for two simple modules L(λ) and L(μ) of bounded dimension or weight. While we are at it, we also get information about H^{2}(G_{1}, L(λ)). In a further subsection, we then go on to use this to prove the analogous statements for 𝔤-modules. One crucial difference we notice is with central extensions: H^{2}(𝔤, k) tends to be zero, whereas H^{2}(G_{1}, k) is almost always not; c.f. Corollary 8.2 and Theorem 8.9. All the notation in this section is as in [Jan03, List of Notations, p. 569]: In particular, for a fixed maximal torus T ≤ G, we denote by R the corresponding root system, by R^{+} a choice of positive roots with corresponding simple roots S ⊆ R^{+}, by X(T)_{+} ⊆ X(T) the dominant weights inside the character lattice, by L(λ) the simple G-module of highest weight λ ∈ X(T)_{+}, by H^{0}(λ) the module induced from λ with socle L(λ), by C_{ℤ} (resp. C̄_{ℤ}) the dominant weights inside the lowest alcove (respectively, in the closure of the lowest alcove). If G is simply connected, we write ω_{i} ∈ X(T)_{+} for the fundamental dominant weight corresponding to α_{i} ∈ S = {α_{1}, …, α_{l}}. Let us recall some results from [McN02] which show the interplay between the conditions that, relative to p, (i) modules are of small dimension; (ii) their high weights are small; and (iii) the Coxeter number is small.

Proposition 8.1 ([McN02, Prop. 5.1]). Let G be simple and simply connected, let L be a simple non-trivial restricted G-module with highest weight λ ∈ X(T)_{+} and suppose that dim L ≤ p. Then

(i) We have λ ∈ C̄_{ℤ}.

(ii) We have λ ∈ C_{ℤ} if and only if dim L < p.

(iii) We have p ≥ h. If moreover dim L < p then p > h.

(iv) If R is not of type A and dim L = p then p > h. If p = h and dim L = p then R = A_{p−1} and λ = ω_{i} with i ∈ {1, p − 1}.

8.1 Cohomology and complete reducibility for small G_{1}-modules

We need values of H^{i}(G_{1}, H^{0}(μ)) for μ ∈ C̄_{ℤ} and i = 1 or 2. Thus H^{0}(μ) = L(μ).

Proposition 8.2. Let G be simple and simply connected and suppose L = L(μ) with μ ∈ C̄_{ℤ} and p ≥ 3. Then:

(i) we have H^{1}(G_{1}, L) = 0 unless G is of type A_{1}, L = L(p − 2) and in that case H^{1}(G_{1}, L)^{[−1]} ≅ L(1); (ii) suppose p > h. Then we have H^{2}(G_{1}, L) = 0 unless: L = k and H^{2}(G_{1}, k)^{[−1]} ≅ 𝔤^{∗}; or G = SL_{3}, with H^{2}(G_{1}, L(p − 3, 0))^{[−1]} ≅ L(0, 1) and H^{2}(G_{1}, L(0, p − 3))^{[−1]} ≅ L(1, 0).

Proof. Part (i) is immediate from [BNP02, Corollary 5.4 B(i)]. The A_{1} result is well known. Part (ii) requires some argument. If H^{2}(G_{1}, H^{0}(μ)) ≠ 0 then since p > h we may assume μ ∈ C_{ℤ}. Now, the values of H^{2}(G_{1}, H^{0}(μ))^{[−1]} are known from [BNP07, Theorem 6.2]. It suffices to find those that are non-zero for which μ ∈ C_{ℤ} \ {0}. All of these have the form μ = w.0 + pλ for l(w) = 2 and λ ∈ X(T)_{+}. Now, if l(w) = 2, we have −w.0 = α + β for two distinct roots α, β ∈ R^{+} (cf. [BNP07, p. 166]). To have w.0 + pλ in the lowest alcove, one needs ⟨w.0 + pλ + ρ, α_{0}^{∨}⟩ < p. Now ⟨pλ, α_{0}^{∨}⟩ ≥ p so ⟨w.0 + ρ, α_{0}^{∨}⟩ < 0. Thus m := ⟨α + β, α_{0}^{∨}⟩ > h − 1. Now one simply considers the various cases. If G is simply-laced, then the biggest value of ⟨α, α_{0}^{∨}⟩ is 2, when α = α_{0} and 1 otherwise, thus m > h − 1 implies h ≤ 3. Thus we get G = SL_{3}, and this case is calculated in [Ste12, Prop. 2.5]. If G = G_{2} we have m at most 5, giving h at most 5, a contradiction. If G is type B, C or F, then m is at most 4, so G = Sp_{4}, p ≥ 5 and this is calculated in [Ibr12, Prop. 4.1]. One checks that all μ such that H^{2}(G_{1}, L(μ)) ≠ 0 have μ ∉ C_{ℤ}.

Remark 8.3. All the values of H^{2}(G_{r}, H^{0}(λ))^{[−1]} are known for all λ by [BNP07, Theorem 6.2] (p ≥ 3) and [Wri11] (p = 2). For example, H^{2}(G_{1}, k)^{[−1]} ≅ 𝔤^{∗} also when G is of type A_{1} and p = 2. Even for λ = 0 there are quite a few exceptional cases when p = 2: see [Wri11, C.1.4]. There are also two exceptional cases for p = 3, for A_{2} and G_{2}, see [BNP07, Theorem 6.2].

One can go further in the case of 1-cohomology to include extensions between simple modules:

Lemma 8.4 ([BNP02, Corollary 5.4 B(i)]). Let G be a simple, simply connected algebraic group not of type A_{1}. If p > 2 then Ext_{G_{r}}^{1}(L(λ), L(μ)) = 0 for all λ, μ ∈ C̄_{ℤ}.

We will use the above result to show that small G_{r}-modules are completely reducible, but we must first slightly soup it up before we use it.

Lemma 8.5. Let G be a simple, simply connected algebraic group not of type A_{1} and p > 2.

(i) We have Ext_{G_{r}}^{1}(L(λ)^{[s]}, L(μ)^{[t]}) = 0 for all λ, μ ∈ C̄_{ℤ} and s, t ≥ 0. (ii) For λ, μ ∈ X_{r}(T), let λ = λ_{0} + pλ_{1} + ⋯ + p^{r−1}λ_{r−1} and μ = μ_{0} + pμ_{1} + ⋯ + p^{r−1}μ_{r−1} be their p-adic expansions. Suppose we have λ_{i}, μ_{i} ∈ C̄_{ℤ} for each i. Then Ext_{G_{r}}^{1}(L(λ), L(μ)) = 0.

Proof. (i) Clearly we may assume s, t < r. When r = 1 the result follows from Lemma 8.4. So assume r > 1. Without loss of generality (dualising if necessary) we may assume s ≤ t. Suppose s > 0 and consider the following subsequence of the five-term exact sequence of the LHS spectral sequence applied to G_{s} ◁G_{r} (see [Jan03, I.6.10]):

0 → Ext_{G_{r−s}}^{1}(L(λ), L(μ)^{[t−s]}) → Ext_{G_{r}}^{1}(L(λ)^{[s]}, L(μ)^{[t]})
→ Hom_{G_{r−s}}(L(λ), Ext_{G_{s}}^{1}(k, k)^{[−s]} ⊗ L(μ)^{[t−s]}) → 0.

Since Ext_{G_{s}}^{1}(k, k) = 0, we have

Ext_{G_{r−s}}^{1}(L(λ), L(μ)^{[t−s]}) ≅ Ext_{G_{r}}^{1}(L(λ)^{[s]}, L(μ)^{[t]}),

and the left-hand side vanishes by induction, so we may assume s = 0. There is another exact sequence

0 → Ext_{G_{r−1}}^{1}(k, Hom_{G_{1}}(L(λ), L(μ)^{[t]})^{[−1]}) → Ext_{G_{r}}^{1}(L(λ), L(μ)^{[t]})
→ Hom_{G_{r−1}}(k, Ext_{G_{1}}^{1}(L(λ), L(μ)^{[t]})^{[−1]}) = 0,

where the last term vanishes by induction. If t = 0 then as λ ≠ μ, the first term of the sequence vanishes and we are done. So we may assume t > 0. Now we can rewrite the first term as Ext_{G_{r−1}}^{1}(k, Hom_{G_{1}}(L(λ), k)^{[−1]} ⊗ L(μ)^{[t−1]}). If this expression is non-trivial we have λ = 0 and Ext_{G_{r−1}}^{1}(k, L(μ)^{[t−1]}) vanishes by induction, which completes the proof. (ii) Suppose i is the first time either λ_{i−1} or μ_{i−1} is non-zero. Without loss of generality, λ_{i−1} ≠ 0. Write λ = λ^{i} + p^{i}λ′ and take a similar expression for μ. Then there is an exact sequence

0 → Ext_{G_{r−i}}^{1}(L(λ′), Hom_{G_{i}}(L(λ^{i}), L(μ^{i}))^{[−i]} ⊗ L(μ′)) → Ext_{G_{r}}^{1}(L(λ), L(μ))
→ Hom_{G_{r−i}}(L(λ′), Ext_{G_{i}}^{1}(L(λ^{i}), L(μ^{i}))^{[−i]} ⊗ L(μ′)).

We have L(λ^{i}) = L(λ_{i−1})^{[i−1]} and L(μ^{i}) = L(μ_{i−1})^{[i−1]}. Hence the right-hand term vanishes by part (i). The left-hand term is non-zero only if λ^{i} = μ^{i} and then we get Ext_{G_{r−i}}^{1}(L(λ′), L(μ′)) ≅ Ext_{G_{r}}^{1}(L(λ), L(μ)). Thus the result follows by induction on r.

We put these results together to arrive at an analogue of Jantzen’s well-known result [Jan97] that G-modules for which dim V ≤ p are completely reducible.

Proposition 8.6. Let G be a simple, simply connected algebraic group and let dim V ≤ p be a G_{r}-module. Then exactly one of the following holds:

(i) V is a semisimple G_{r}-module; (ii) G is of type A_{1}, p > 2, r = 1, dim V = p and V is uniserial, with composition factors L(p − 2 − s) and L(s) with 0 ≤ s ≤ p − 2; (iii) G is of type C_{n} with n ≥ 1, p = 2 and V is uniserial with two trivial composition factors.

Proof. Assume V has only trivial composition factors. We have Ext_{G_{r}}^{1}(k, k) ≠ 0 if and only if p = 2 and G is of type C_{n}, in which case Ext_{G_{r}}^{1}(k, k)^{[−r]} ≅ L(ω_{1}); [Jan03, II.12.2]. This is case (iii). Otherwise, p > 2 and Ext_{G_{r}}^{1}(L(λ), L(λ)) = 0 for all λ ∈ X_{r}(T) by [Jan03, II.12.9]. Assume G is not of type A_{1}. By assumption, V has a non-trivial composition factor with dim V ≤ p. Then p > 2 and the hypotheses of Lemma 8.4 hold. Since dim V ≤ p, by Proposition 8.1 any G_{r}-composition factor L(λ) of V has a p-adic expansion λ = λ_{0} + ⋯ + p^{r−1}λ_{r} with each λ_{i} ∈ C̄_{ℤ}. If there were a non-split extension 0 → L(λ) → V → V/L(λ) → 0 then there would be a non-split extension of L(λ) by L(μ) for L(μ) a composition factor of V, also of the same form. But by Lemma 8.5(ii) we have Ext_{G_{r}}^{1}(L(λ), L(μ)) = 0, hence this is impossible and L(λ) splits off as a direct summand. Induction on the direct complement completes the proof in this case. If G is of type A_{1} then the G_{r}-extensions of simple modules are well known. If r > 1 with λ, μ ∈ X_{r}(T) then dim Ext_{G_{r}}^{1}(L(λ), L(μ)) = dim Ext_{G}^{1}(L(λ), L(μ)) and this must vanish whenever dim L(λ)+dim L(μ) ≤ p. If r = 1, then the only pairs of G_{1}-linked weights are s and p − 2 − s with Ext_{G_{1}}^{1}(L(s), L(p − 2 − s)) ≅ L(1)^{[1]} as G-modules. Here we have dim L(s) + dim L(p − s − 2) = p giving case (ii).

The following two corollaries are immediate, in the first case, the passage from G being simple to being reductive is trivial (consider the cover of G by the product of the radical and the simply connected cover of the derived group).

Corollary 8.7. Let G be a connected reductive algebraic group and let V be a G_{r}-module with p > dim V. Then V is semisimple.

Corollary 8.8. Let G be connected reductive and G_{r} ≤ GL(V) with dim V ≤ p. Then either G_{r} is completely reducible on V or dim V = p, G is of type A_{1}, r = 1 and G_{r} is in a maximal parabolic of GL(V) acting indecomposably on V as described in case (ii) of Proposition 8.6. Moreover, if 𝔤 is a p-reductive subalgebra of GL(V) with dim V < p then 𝔤 acts semisimply on V.

Proof. If G is not simple, it can be written as HK with H and K non-trivial mutually centralising connected reductive subgroups with maximal tori S and T say. The Frobenius kernels H_{1}, K_{1} ≤ G_{1} ≤ G_{r} are also mutually centralising, so that H_{1} is in the centraliser of T_{1}. Now the centraliser of T_{1} is a proper Levi subgroup of GL(V), hence restriction of V to H_{r} has at least one trivial direct factor, with direct complement W say, dim W < p. Thus by Corollary 8.7, W is completely reducible for H_{r} and by symmetry, for K_{r}. Thus W is completely reducible for K_{r}H_{r} = G_{r}. Otherwise, G is simple and Proposition 8.6 gives the result (note that case (iii) does not occur due to dimension restrictions). For the last part, Lemma 7.9 implies that 𝔤 is the direct sum of a semisimple ideal and a torus, and we may hence assume that 𝔤 is a semisimple restricted subalgebra of 𝔤𝔩(V). If 𝔤 is not irreducible on V, then by Theorem 2.3 there exists a semisimple group G with 𝔤 = Lie(G). Now the result follows from the case G_{1} above.

8.2 Cohomology and complete reducibility for small 𝔤-modules

We now transfer our results to the ordinary Lie algebra cohomology for 𝔤. Recall the exact sequence [Jan03, I.9.19(1)]:

0 → H^{1}(G_{1}, L) → H^{1}(𝔤, L) → Hom^{s}(𝔤, L^{𝔤})
→ H^{2}(G_{1}, L) → H^{2}(𝔤, L) → Hom^{s}(𝔤, H^{1}(𝔤, L))  (1)

The following theorem is the major result of this section.

Theorem 8.9. Let 𝔤 = Lie(G) be semisimple. Then:

(a) If p > h with μ ∈ C̄_{ℤ} then either H^{2}(𝔤, L(μ)) = 0, or one of the following holds: (i) 𝔤 contains a factor 𝔰𝔩_{3} and L(μ) contains a tensor factor of L(p − 3, 0) or L(0, p − 3) for this 𝔰𝔩_{3}; (ii) 𝔤 contains a factor 𝔰𝔩_{2} and L(μ) has a tensor factor L(p − 2) for this 𝔰𝔩_{2}.

(b) If p > 2 is very good for G then H^{2}(𝔤, k) = 0.

(c) If p > 2 is very good for G and λ, μ ∈ C̄_{ℤ} we have Ext_{𝔤}^{1}(L(λ), L(μ)) = 0, or G contains a factor of type A_{1}, L(λ) and L(μ) are simple modules for that factor, λ = s < p − 1, μ = p − 2 − s and we have Ext_{𝔤}^{1}(L(λ), L(μ))^{[−1]} ≅ L(1).

Proof. We may assume that G is simply connected, since the condition on p implies that 𝔤 = 𝔤_{1} × 𝔤_{2} ⋯ × 𝔤_{s}. Now one can reduce to the case that G is simple using a Künneth formula. To begin with, any simple module L(λ) for 𝔤 = 𝔤_{1} × 𝔤_{2} × ⋯ × 𝔤_{s} is a tensor product of simple modules L(λ_{1}) ⊗ ⋯ ⊗ L(λ_{s}) for the factors. Then by the Künneth formula dim Ext_{𝔤}^{1}(L(λ), L(μ)) ≠ 0 implies that λ_{i} = μ_{i} for all i ≠ j, some 1 ≤ j ≤ s and Ext_{𝔤}^{1}(L(λ), L(μ)) ≅ Ext_{𝔤_{j}}^{1}(L(λ_{j}), L(μ_{j})). This means we may assume G to be simple in (c). For H^{2}(𝔤, L(λ)) to be non-zero one must have all λ_{i} = 0 for all i ≠ j, k some 1 ≤ j < k ≤ s and then

H^{2}(𝔤, L(λ)) = H^{2}(𝔤_{j}, L(λ_{j})) ⊗ H^{0}(𝔤_{k}, L(λ_{k})) ⊕ H^{1}(𝔤_{j}, L(λ_{j})) ⊗ H^{1}(𝔤_{k}, L(λ_{k}))
⊕ H^{0}(𝔤_{j}, L(λ_{j})) ⊗ H^{2}(𝔤_{k}, L(λ_{k})).

Now first suppose that both λ_{j} and λ_{k} are non-trivial. Then only the second direct summand in H^{2}(𝔤, L(λ)) survives, and by (1) it coincides with the tensor product of the 1-cohomology groups of the corresponding Frobenius kernels. By Proposition 8.2, non-vanishing would force λ_{j} = p − 2 = λ_{k} and 𝔤_{j} = 𝔤_{k} = 𝔰𝔩_{2} giving one exceptional case. Next we treat the case λ_{k} = 0 and λ_{j} non-trivial. Again by (1) and Proposition 8.2, we obtain H^{2}(𝔤, L(λ)) = H^{2}(𝔤_{j}, L(λ_{j})), and we are in the case where G is simple and L(λ) non-trivial. In case 𝔤 = 𝔰𝔩_{2}, the result follows from [Dzh92]. So suppose 𝔤 ≠ 𝔰𝔩_{2}. Setting L = L(μ) in (1) we see that if μ ≠ 0 we have H^{1}(𝔤, L) ≅ H^{1}(G_{1}, L) and the right-hand side is zero by Lemma 8.4. Thus we also have H^{2}(𝔤, L) ≅ H^{2}(G_{1}, L) and the latter is zero by Proposition 8.2 unless 𝔤 = 𝔰𝔩_{3} and the exception is as in the statement of the Theorem, since we have excluded the A_{1} case. Finally, the case λ_{j} = λ_{k} = 0 reduces by the above to the case G simple, L = k and the claim that H^{2}(𝔤, k) = 0. Here we have H^{1}(𝔤, k) ≅ (𝔤/[𝔤, 𝔤])^{∗} and this is zero since p is very good and 𝔤 is semisimple. We also have H^{2}(G_{1}, k)^{[−1]} ≅ 𝔤^{∗}. The injective map Hom^{s}(𝔤, L^{𝔤}) → H^{2}(G_{1}, L) is hence an isomorphism, which forces H^{2}(𝔤, k) = 0 in the sequence (1). This also proves (b). Now we prove the statement (c) under the assumption that G is simple. We have an isomorphism Ext_{𝔤}^{1}(L(λ), L(μ)) ≅ H^{1}(𝔤, L(μ)⊗L(λ)^{∗}). Let M = L(μ)⊗ L(λ)^{∗}. If λ ≠ μ, then applying the exact sequence (1) to M yields H^{1}(𝔤, M) ≅ H^{1}(G_{1}, M) and the latter is zero by Lemma 8.4 if G is not of type A_{1} and well-known if G is of type A_{1}. Hence we may assume λ = μ. The assignation of L to the sequence (1) is functorial, thus, associated to the G-map k → M ≅ Hom_{k}(L, L), there is a commutative diagram

0 ⟶ H^{1}(𝔤, k) = 0 ⟶ Hom^{s}(𝔤, k^{𝔤}) ≅ (𝔤^{∗})^{[1]} ⟶^{≅} H^{2}(G_{1}, k) ↓  ≅↓  ^{θ}↓ 0 ⟶ H^{1}(𝔤, M) ⟶ Hom^{s}(𝔤, M^{𝔤}) ≅ (𝔤^{∗})^{[1]} ⟶^{ζ} H^{2}(G_{1}, M),

where the natural isomorphism k^{𝔤} → M^{𝔤} induces the middle isomorphism and the top right isomorphism has been discussed already. We want to show that ζ is injective, since then it would follow that H^{1}(𝔤, M) = 0. To do this it suffices to show that θ is an injection (𝔤^{∗})^{[1]} → H^{2}(G_{1}, M) and for this, it suffices to show that the simple G-module (𝔤^{∗})^{[1]} does not appear as a submodule of H^{1}(G_{1}, M/k). Now since λ ∈ C̄_{ℤ} we have L(λ) ≅ H^{0}(λ) and so by [Jan03, II.4.21], M has a good filtration. The socle of any module H^{0}(μ) with μ ∈ X^{+} is simple. Thus the submodule k ≤ M constitutes a section of this good filtration, with M/k also having a good filtration. The G-modules H^{1}(G_{1}, H^{0}(μ)) have been well-studied by Jantzen [Jan91] and others. In order to have (𝔤^{∗})^{[1]} a composition factor of H^{1}(G_{1}, H^{0}(μ)), we would need 𝔤 ≅ 𝔤^{∗} ≅ H^{0}(ω_{α}) where μ = pω_{α} − α and α is a simple root with ω the corresponding fundamental dominant weight; [BNP04, Theorem 3.1(A,B)]. Now for type A_{n}, with p |̸n + 1, we have 𝔤 = L(2ω_{1}) if n = 1 and 𝔤 = L(ω_{1} + ω_{n}) else; and for type B_{2}, we have 𝔤 = L(2ω_{2}), ruling these cases out. For the remaining types, we have

\begin{matrix}Type & B_{n},C_{n} & D_{n} & E_{6} & E_{7} & E_{8} & F_{4} & G_{2} \\ 𝔤 ≅ L(ω_{α}) for ω_{α} = & ω_{2} & ω_{2} & ω_{2} & ω_{1} & ω_{8} & ω_{1} & ω_{2} \\ ⟨pω_{α} − α, α_{0}^{∨}⟩ & 2p & 2p & 2p − 1 & 2p − 1 & 2p − 1 & 2p & 3p \end{matrix}

On the other hand, since λ ∈ C̄_{ℤ} it satisfies ⟨λ + ρ, α_{0}^{∨}⟩ ≤ p, i.e. ⟨λ, α_{0}^{∨}⟩ ≤ p−h+1. Hence any high weight μ of M = L⊗L^{∗} satisfies ⟨μ, α_{0}^{∨}⟩ ≤ 2p−2h+2. Looking at the above table, it is easily seen that this is a contradiction. Thus (𝔤^{∗})^{[1]} is not a composition factor of H^{1}(G_{1}, M/k) and the result follows.

Remarks 8.10. (i) When λ ≠ μ in the proof of the above proposition, one also sees that there is an isomorphism Ext_{G_{1}}^{2}(L(λ), L(μ)) ≅ Ext_{𝔤}^{2}(L(λ), L(μ)) but we do not use this fact in the sequel. (ii) The conclusion of the theorem is incorrect if G is reductive but not semisimple. For example, if G is a torus, then 𝔤 is an abelian Lie algebra, and H^{1}(𝔤, k) is non-trivial. For instance the two-dimensional non-abelian Lie algebra is a non-direct extension of k by k. One also has H^{2}(k × k, k) ≠ 0 by the Künneth formula: for example the Heisenberg Lie algebra is a non-split extension of k by k × k. (iii) When p = 3 and G = SL_{3}, then H^{2}(G_{1}, k)^{[−1]} ≅ 𝔤^{∗} ⊕ L(ω_{1}) ⊕ L(ω_{2}), by [BNP07, Theorem 6.2]. Thus the same argument shows that H^{2}(𝔤, k) ≅ L(ω_{1}) ⊕ L(ω_{2}). It follows from the Künneth formula that if G is a direct product of n copies of SL_{3} then H^{2}(𝔤, k) ≅ [L(ω_{1}) ⊕ L(ω_{2})]^{⊕n}.

(iv) In part (a) of the theorem, one can be more specific. If 𝔤 = 𝔰𝔩_{2} then [Dzh92] shows that H^{2}(𝔤, L(p − 2)) is isomorphic to L(1)^{[1]} as a G-module. If 𝔤 = \underbrace{𝔰𝔩_{2} × ⋯ × 𝔰𝔩_{2}}_{n times} ×𝔥 then one can show moreover that H^{2}(𝔤, L(μ)) is non-zero

only if

L(μ) ≅ L(μ_{1}) ⊗ ⋯ ⊗ L(μ_{n}) ⊗ L(μ_{n+1})

with each μ_{i} ∈ {0, p−2} and μ_{n+1} = 0. Let r be the number of times μ_{i} = p−2. Then, the Künneth formula shows that

dim H^{2}(𝔤, L(μ)) = \begin{cases}0 if r = 0; \\ 2 if r = 1; \\ 4 if r = 2; \\ 0 otherwise. \end{cases}

We use the theorem above to get analogues of Corollary 8.8 for Lie algebra representations.

Proposition 8.11. Let G be a simple algebraic group with 𝔤 = [𝔤, 𝔤] and let dim V ≤ p be a 𝔤-module. Then exactly one of the following holds:

(i) V is a semisimple 𝔤-module; (ii) G is of type A_{1}, dim V = p and V is uniserial, with composition factors L(p − 2 − s) and L(s).

Proof. The proof is similar to Proposition 8.6. Since dim V ≤ p, any composition factor of V is a restricted simple 𝔤-module, or V is simple. Since Ext_{𝔤}^{1}(k, k) = H^{1}(𝔤, k) ≅ (𝔤/[𝔤, 𝔤])^{∗} = 0, if V consists only of trivial composition factors then V is semisimple. Thus we may assume that 𝔤 contains a non-trivial composition factor L. Then either dim L = p and V is simple, or p > h by Lemma 8.1(iii). By the condition on V, any two distinct composition factors, L(λ) and L(μ) satisfy λ, μ ∈ C_{ℤ} by Lemma 8.1(ii). If G is not of type A_{1}, then Ext_{𝔤}^{1}(L(λ), L(μ)) = 0 by Theorem 8.9 and the exceptional case, where G = A_{1}, is well known.

As before there is a corollary:

Corollary 8.12. Let G be a semisimple algebraic group and let V be a 𝔤module with p > dim V. Assume that 𝔤 = [𝔤, 𝔤]. Then V is semisimple.

The next corollary uses a famous result of Serre on the semisimplicity of tensor products to extend our results a little further. This result will be crucial for showing the splitting of certain non-semisimple Lie algebras.

Corollary 8.13. Let 𝔤 be a Lie algebra and V, W two semisimple 𝔤-modules with dim V + dim W < p + 2. Then V ⊗ W is semisimple. Furthermore, let 𝔤 = Lie(G) for G a semisimple algebraic group with p > 2 and p very good. Then H^{2}(𝔤, V ⊗ W) = 0 unless 𝔤 contains a factor 𝔰𝔩_{2} and V ⊗ W contains a composition factor of the 𝔰𝔩_{2}-module L(p−2). Also H^{1}(𝔤, V ⊗W) = 0, unless one of V and W is isomorphic to k and we are in one of the exceptional case of Theorem 8.9.

Proof. For the first statement, we begin with some reductions as in [Ser94]. If W = 0 or k there is nothing to prove. If W is at least 2-dimensional, then either p = 2 and V is trivial (so that the result holds), or both dim V and dim W < p. We may assume that both V and W are simple. Further, we may replace 𝔤 by the restricted algebra generated by its image in 𝔤𝔩(V ⊕ W). As V ⊕ W is a semisimple module, we may thus assume 𝔤 is p-reductive. Now 𝔤 ⊆ 𝔤𝔩(V) × 𝔤𝔩(W) = 𝔰𝔩(V) × 𝔰𝔩(W) × 𝔷, where 𝔷 is a torus, and where the projections of 𝔤 onto the first two factors are irreducible, hence semisimple by Theorem B. We thus may assume 𝔤 ⊆ 𝔰𝔩(V) × 𝔰𝔩(W) is a semisimple restricted subalgebra. By Theorem 2.3, either (i) 𝔤 has a factor W_{1}, the first Witt algebra and V is the (p − 1)-dimensional irreducible module for W_{1}; or (ii) 𝔤 is Lie(G) for a direct product of simple algebraic groups, and V and W are (the differentials of) prestricted modules for G. In case (i), as p > 2, we would have W ≅ k⊕k for W_{1} and the result holds. So we may assume that (ii) holds. Now [Ser94, Prop. 7] implies that V ⊗ W is the direct sum of simple modules with restricted high weights λ satisfying λ ∈ C_{ℤ}. Since each of these composition factors is simple also for 𝔤, V ⊗ W is semisimple with those same composition factors. For the remaining statements, let 𝔥 be the image of 𝔤 in 𝔤𝔩(V ⊕ W), so that 𝔤 = 𝔥 ⊕ 𝔰 with 𝔰 acting trivially. Let h be the coxeter number of 𝔥. Now if W = k, say, then since p is very good for 𝔤 we can have p = dim V by Proposition 8.1 only for p > h, so otherwise dim V < p. And if dim W > 1 then dim V < p also. Now dim V < p also implies by Proposition 8.1 that p > h. Also a summand L(λ) of V ⊗ W has λ ∈ C_{ℤ}. Now Theorem 8.9 implies that H^{1}(𝔤, V ⊗ W) = H^{2}(𝔤, V ⊗ W) = 0, unless we are in the exceptional cases described. However, if 𝔤 = 𝔰𝔩_{3} then the module L(p − 3, 0) or its dual has dimension (p − 1)(p − 2)(p − 3)/2 > ((p + 1)/2)^{2} hence it cannot appear as a composition factor of V ⊗ W.

Remark 8.14. If 𝔤 = W_{1} the conclusion of the second part is false, since H^{1}(𝔤, V) ≠ 0 when V is the irreducible (p − 1)-dimensional module for 𝔤.

Proof of Theorem D:. We must just give references for the statements made. For (a), see Proposition 8.6; for (b), see Proposition 8.11; for (c), see Theorem 8.9; for (d), see Corollary 8.13. This completes the proof of Theorem D.

9 Decomposability: the existence of Levi factors

Let 𝔥 be a restricted subalgebra of 𝔤𝔩(V) with p > dim V. In this section we show, in Theorem 9.2, a strong version of the Borel–Tits Theorem in this context.

Let G be connected reductive. Recall, say from [ABS90] that if 𝔭 = 𝔩 + 𝔮 is a parabolic subalgebra of 𝔤 = Lie G then 𝔮 has a central filtration such that successive quotients have the structure of 𝔩-modules. We record a specific case:

Lemma 9.1. In case G = GL_{n}, a parabolic subalgebra 𝔭 = 𝔩+𝔮 has the property that 𝔩 is a direct product 𝔤𝔩(V_{1})×𝔤𝔩(V_{2})×⋯×𝔤𝔩(V_{r}) and 𝔮 has a central filtration with successive factors being modules of the form V_{i} ⊗V_{j}^{∗}, each factor occurring exactly once.

Theorem 9.2. Let 𝔥 be a restricted Lie subalgebra of 𝔤𝔩(V) with dim V < p, and let 𝔯 = Rad_{p}(𝔥) (= Rad_{V} (𝔥)). Then there is a parabolic subalgebra 𝔭 = 𝔩 + 𝔮, with 𝔯 ≤ 𝔮 and containing a complement 𝔰 to 𝔯 in 𝔥, with 𝔰 ≤ 𝔩 and 𝔥 = 𝔰 + 𝔯 as a semidirect product. Furthermore, 𝔰 acts completely reducibly on V and is the direct sum of a torus and a semisimple ideal.

Proof. As in the proof of Lemma 7.9 we take a minimal parabolic subgroup P = LQ so that its Lie algebra 𝔭 = 𝔩 + 𝔮 contains 𝔥 and so that the projection 𝔥_{𝔩} := π(𝔥) of 𝔥 to the Levi subalgebra 𝔩 is strongly p-reductive and we may write 𝔥_{𝔩} = 𝔥_{s} ⊕ 𝔷 where 𝔥_{s} is semisimple and 𝔷 = Z(𝔥_{𝔩}). We also have 𝔯 ≤ 𝔮, since 𝔥_{𝔩} is p-reductive. Now by Theorem 2.3, either 𝔥_{s} = W_{1}, 𝔥 = 𝔥_{𝔩}, 𝔭 = 𝔩 = 𝔤𝔩(V) and we are done; or 𝔥_{𝔩} is isomorphic to a direct product of classical Lie algebras 𝔰_{i} and 𝔷. We first lift 𝔷 to 𝔥. Let π′ : 𝔥 → 𝔷 be the composition of π with the projection onto 𝔷. By [SF88, Lemma 2.4.4(2)], there is a torus 𝔷′ ≤ Z(𝔩) + 𝔮 so that 𝔥 = 𝔷′ + ker(π′). Now since 𝔷′ is a torus, it is linearly reductive, we may replace 𝔥 by a conjugate by Q so that 𝔷′ ⊆ Z(𝔩). Let us rewrite 𝔷 = 𝔷′ and identify 𝔷 with its image in 𝔩 under π. Next we construct a complement to 𝔯 in 𝔥. Let π′′ : 𝔥 → 𝔥_{s} be the composition of π with the projection onto 𝔥_{s} and let 𝔥′ ⊆ 𝔥 be a vector space complement to ker(π′′). Then 𝔯 + 𝔥′ ≤ 𝔥 is a subalgebra, and we have an exact sequence

0 → 𝔯 → 𝔯 + 𝔥′ →^{π′′} 𝔥_{s} → 0.

We show this sequence is split. By Lemma 9.1, the nilpotent radical 𝔮 of 𝔩 has a filtration 𝔮 = 𝔮_{1} ⊇ 𝔮_{2} ⊇ ⋯ ⊇ 𝔮_{m} = 0 with each 𝔮_{i}/𝔮_{i+1} having the structure of an 𝔩-module M_{i} ⊗ N_{i} with M_{i} and N_{i} irreducible modules for the projections of 𝔥_{𝔩} to distinct factors of the Levi. Since dim M_{i} + dim N_{i} < p, we have by Corollary 8.13 that M_{i} ⊗ N_{i} is a direct sum of irreducible modules for 𝔥_{s} with H^{2}(𝔥_{s}, M_{i} ⊗ N_{i}) = 0. By intersecting with 𝔯, we get a filtration 𝔯 = 𝔯_{1} ⊇ 𝔯_{2} ⊇ ⋯ ⊇ 𝔯_{m} = 0 by 𝔥_{s}-modules so that each 𝔯_{i}/𝔯_{i+1} is a submodule of M_{i} ⊗ N_{i}, hence also a semisimple module with H^{2}(𝔥_{s}, 𝔯_{i}/𝔯_{i+1}) = 0. By an obvious induction on the length m of the filtration {𝔯_{i}} we now see that the sequence

0 → 𝔯 → 𝔯 + 𝔥′ → 𝔥_{s} → 0

is split. Thus we may set 𝔥′_{s} a complement to 𝔯 in 𝔥′ + 𝔯.

We would like to set 𝔰 = 𝔥′_{s} + 𝔷, however this vector space may not be a subalgebra of 𝔤. Write 𝔮 = 𝔠_{𝔮}(𝔷) + [𝔮, 𝔷]. (This can be done, for instance by [SF88, Lemma 2.4.4(1)].) Any element h of 𝔥′_{s} can be written as h_{1} +q_{1} +q_{2} for h_{1} ∈ 𝔩, q_{1} in 𝔠_{𝔮}(𝔷) and q_{2} ∈ [𝔮, 𝔷]. As 𝔥 is stable under ad 𝔷, with 𝔷 centralising h_{1} and q_{1}, we conclude that q_{2} ∈ 𝔥. Thus we have the element h′ = h_{1} +q_{1} ∈ 𝔥. Thus we may form the subspace 𝔥′′_{s} ≤ 𝔥 with 𝔥′′_{s} ≤ 𝔩 + 𝔠_{𝔮}(𝔷). Using that 𝔥′_{s} ≤ 𝔥 is a subalgebra, that 𝔠_{𝔮}(𝔷) is 𝔩 = 𝔠_{𝔤𝔩(V)}(𝔷)-invariant and that [𝔮, 𝔷] is an ideal in 𝔮, one checks that 𝔥′′_{s} is indeed a subalgebra,^{4} with 𝔥′′_{s} also a complement to 𝔯 in 𝔥′_{𝔰} + 𝔯. Now we have guaranteed that 𝔰 = 𝔥′′_{s} + 𝔷 is a subalgebra of 𝔥, a complement to 𝔯 in 𝔥. Now, by Corollary 8.12, 𝔥′′_{s} acts completely reducibly. Also, since 𝔷 is a torus, 𝔷 is linearly reductive on restricted representations, hence also acts completely reducibly. Thus 𝔰 is completely reducible on V. In particular, we may replace 𝔩 with a Levi subalgebra of 𝔭 that contains 𝔰, which finishes the proof.

10 Proof of Theorem B(i)

Proof. We first prove the statement in the case that G = GL(V), so we assume p > dim V + 1. By assumption, 𝔥 is a restricted subalgebra of 𝔤. Let 𝔫 = 𝔫_{𝔤}(𝔥). By Theorem 9.2 we may decompose both 𝔫 and 𝔥. Let 𝔫 = 𝔫_{𝔩} + 𝔫_{𝔮} ≤ 𝔭 = 𝔩 + 𝔮 with 𝔫_{𝔩} ≤ 𝔩 and 𝔫_{𝔮} ≤ 𝔮, with 𝔫_{𝔩} = 𝔫_{s} + 𝔷, 𝔷 a torus and 𝔫_{s} is by Theorem 2.3 isomorphic to a direct product of classical Lie algebras acting completely reducibly on V; also set 𝔥_{𝔮} = 𝔥 ∩ 𝔮 and 𝔥_{𝔩} = π(𝔥) the projection to 𝔩. Since 𝔫 is generated by nilpotent elements we have 𝔷 = 0 and 𝔥_{𝔩} = 𝔥_{s}. Since the complement to 𝔥_{𝔮} in 𝔥 obtained by Theorem 9.2 is completely reducible on V and hence conjugate to a subalgebra of 𝔩, we may assume that 𝔥 = 𝔥_{𝔮} + 𝔥_{𝔩} is this splitting. Furthermore, 𝔥_{𝔩} ≤ 𝔫_{𝔩} is an ideal of a direct product of simple subalgebras, hence is a direct product of some subset of those simples. Since V has dimension less than p, V |_{𝔫_{𝔩}} is a restricted module for 𝔫_{𝔩}. Hence there is a connected algebraic group N_{𝔩} with Lie N_{𝔩} ≅ 𝔫_{𝔩}, N_{𝔩} ≤ GL(V) and V |_{Lie(N_{𝔩})} ≅ V |_{𝔫_{𝔩}}. Hence, replacing N_{𝔩} by a conjugate if necessary, we have Lie(N_{𝔩}) = 𝔫_{𝔩}. Moreover if L is a Levi subgroup of GL(V) chosen so that Lie(L) = 𝔩 then we may produce N_{𝔩} ≤ L. Clearly N_{𝔩} normalises any direct factor of 𝔫_{𝔩}, in particular, 𝔥_{𝔩}. Now, since the 𝔩-composition factors of 𝔮 are all of the form W_{1} ⊗ W_{2} for dim W_{1} + dim W_{2} < p and W_{1}, W_{2} irreducible for 𝔫_{𝔰}, [Ser94, Prop. 7] implies that 𝔮 is a restricted semisimple module for N_{𝔩} and 𝔫_{𝔩}. Since 𝔫_{𝔩} normalises 𝔥_{𝔮} = 𝔥∩𝔮, this space also appears as an N_{𝔩}-submodule in 𝔮, hence N_{𝔩} normalises 𝔥_{𝔮}.

[h_{1} + q_{1} + q_{2}, h′_{1} + q′_{1} + q′_{2}] = \underbrace{[h_{1}, h_{2}]}_{∈𝔥_{𝔩}} + \underbrace{[h_{1}, q′_{1}] + [q_{1}, h′_{1}] + [q_{1}, q′_{1}]}_{∈𝔠_{𝔮}(𝔷)} +x,

4The calculation is as follows: if h_{1} + q_{1} + q_{2} and h′_{1} + q′_{1} + q′_{2} are two elements of 𝔥′_{s} then where x ∈ [𝔮, 𝔷] by the Jacobi identity. But projecting to 𝔥′′_{s} one simply deletes q_{2}, q′_{2} and x to get the analagous calculation.

Documenta Mathematica 21 (2016) 1–37

It remains to construct a unipotent algebraic group N_{𝔮} such that Lie N_{𝔮} = 𝔫_{𝔮} with N_{𝔮} normalising 𝔥. For this we use Corollary 4.4. Let N_{𝔮} = \overline{⟨exp x : x ∈ 𝔫_{𝔮}⟩}. Then N_{𝔮} is a closed subgroup, which by Corollary 4.4 consists of elements normalising 𝔥. By Lemma 4.5, 𝔫_{𝔮} ≤ Lie(N_{𝔮}). Let N be the smooth algebraic group given by N = ⟨N_{𝔩}, N_{𝔮}⟩. We have shown that N normalises 𝔥 and that 𝔫 ⊆ Lie N. Since also Lie N ⊆ 𝔫 we are done for the case G = GL(V). To prove the remaining part, we appeal to Proposition E again. Let G be a simple algebraic group with minimal dimensional representation V. Then since p > dim V, (GL(V), G) is a reductive pair. Indeed, the assumption on p guarantees that the trace form associated to V is non-zero, see [Gar09, Fact 4.4]. This implies the reductive pair property (cf. the proof of [Gar09, Prop. 8.1]). The theorem now follows by invoking Proposition E.

11 Examples

In this section we mainly collect, in a number of statements, examples which demonstrate the tightness of some of our bounds. First let us just point out that there are some rather general situations in which smooth normalisers can be found.

Example 11.1 ([MT09, Theorem B]). Suppose G is a quasi-split reductive group over a field k of very good characteristic. Then the normaliser N = N_{G}(C) of the centraliser C = C_{G}(e) of a regular nilpotent element e of 𝔤 = Lie(G) is smooth.

Example 11.2 ([HS16, Proof of Lem. 3.1]). Suppose G is reductive over an algebraically closed field k of very good characteristic and e is a nilpotent element of 𝔤 = Lie(G), then the normaliser N_{G}(⟨e⟩) of the 1-space ⟨e⟩ of 𝔤 is smooth.

We will first give the promised example discussed after the statement of Theorem A. For this, we will need a lemma.

Lemma 11.3. Let B = T U be a Borel subgroup of a reductive algebraic group G containing a maximal torus T with unipotent radical U. Suppose N_{B}(𝔥) is smooth and s ∈ 𝔱 = Lie(T) an element normalising a subspace 𝔥 of 𝔲 = Lie(U). Then ⟨s⟩ = Lie(χ(𝔾_{m})) for a cocharacter χ : 𝔾_{m} → N_{B}(𝔥), such that χ(𝔾_{m}) is conjugate by an element of C_{U}(s) to a cocharacter with image in T.

Proof. Since N_{B}(𝔥) is smooth, we may, by [Hum67, Thm. 13.3], write any maximal torus 𝔰 of 𝔫_{𝔟}(𝔥) as Lie(S) for S a maximal torus of N_{B}(𝔥). By [Die52, Prop. 2], for any semisimple element s ∈ 𝔰 we may write ⟨s⟩ = Lie(S_{1}) for S_{1} ⊆ S. Defining an appropriate isomorphism 𝔾_{m} → S_{1}, we may even write s = \frac{d}{dt} |_{t=1}χ(t) for χ a cocharacter of N_{B}(𝔥). As the maximal tori of B are conjugate by elements of U, we have that S is conjugate to its projection to T, say via u ∈ U; in particular, uχ(t)u^{−1} ∈ T. Since projection to T is B-equivariant, we have on differentiating, that \frac{d}{dt} |_{t=1}(uχ(t)u^{−1}) = s, so that usu^{−1} = s, i.e. that u ∈ C_{U}(s).

Example 11.4. Let n ≥ 4. This example depends on three fixed parameters λ_{1}, λ_{2}, λ_{3} together with variables {a_{i}}_{1≤i≤n}, {b_{i}}_{1≤i≤n−1}, c, d, and e, each taking values in k = 𝔽̄_{p}. Let us define the following matrices:

A := ( \begin{matrix}0 & a_{1} & a_{2} & ∗ & ∗ & ∗ & … & ∗ & ∗ & ∗ \\ & 0 & a_{1} & β_{2} & ∗ & ∗ & … & ∗ & ∗ & ∗ \\ & & 0 & a_{2} & β_{3} & ∗ & … & ∗ & ∗ & ∗ \\ & & & 0 & a_{3} & β_{4} & ... & ⋮ & ⋮ & ⋮ \\ & & & & 0 & a_{4} & ... & ∗ & ∗ & ∗ \\ & & & & & 0 & ... & β_{n−2} & c & e + λ_{1}β_{n−2} \\ & & & & & & ... & a_{n−2} & β_{n−1} & (1 + λ_{1})a_{n−2} + d \\ & & & & & & & 0 & a_{n−1} & b_{n−1} \\ & & & & & & & & 0 & a_{n} \\ & & & & & & & & & 0 \end{matrix} ) ,

with β_{i} = a_{i+1} + b_{i−1} for i = 2, …, n − 1,

B := ( \begin{matrix}0 & a_{1} & b_{1} & ∗ & … & ∗ \\ & 0 & a_{2} & b_{2} & ... & ⋮ \\ & & 0 & a_{3} & ... & ∗ \\ & & & 0 & ... & b_{n−1} \\ & & & & ... & a_{n} \\ & & & & & 0 \end{matrix} ) ,

C := ( \begin{matrix}0 & a_{n−3} & a_{n−1} + b_{n−3} & c & e + λ_{2}(a_{n−1} + b_{n−3}) \\ & 0 & a_{n−2} & a_{n} + b_{n−2} & (1 + λ_{2})a_{n−2} + d \\ & & 0 & a_{n−1} & b_{n−1} \\ & & & 0 & a_{n} \\ & & & & 0 \end{matrix} ) ,

D := ( \begin{matrix}0 & a_{n−2} & a_{n} + b_{n−2} & d + λ_{3}a_{n−2} \\ & 0 & a_{n−1} & b_{n−1} \\ & & 0 & a_{n} \\ & & & 0 \end{matrix} )

Then the reader may check that for each choice of λ_{1}, λ_{2} and λ_{3}, the following set defines a subalgebra 𝔥 of the strictly upper triangular matrices:

\{( \begin{matrix}A & ∗ & ∗ & ∗ \\ 0 & B & ∗ & ∗ \\ 0 & 0 & C & ∗ \\ 0 & 0 & 0 & D \end{matrix} ) : a_{i} ∈ k, b_{j} ∈ k, c, d, e ∈ k\} .

Let F_{i} denote the ith Fibonacci number, so that F_{0} = F_{1} = 1 and F_{2} = 2 and suppose that r is chosen so that F_{r+1} = p is the prime characteristic of k, and let us suppose that N_{G}(𝔥) is smooth. Since every entry of the superdiagonal is non-zero for some element in 𝔥, it is easy to check that N_{G}(𝔥) ⊆ B. Thus N_{G}(𝔥) = N_{B}(𝔥) and we may employ Lemma 11.3. Suppose s = diag(s_{1}, …, s_{2n+12}) is an arbitrary element of the diagonal torus 𝔱 = Lie(T). Then one can calculate the dimension of 𝔫_{𝔱}(𝔥) by enumerating the linear conditions amongst the t_{i} necessary to normalise 𝔥. For example, setting all indeterminates in a general matrix of 𝔥 to be zero, except for a_{1} = 1 gives a matrix M, which spans a 1-space ⟨M⟩ of 𝔥. One can see by inspection that s will normalise 𝔥 only if it normalises ⟨M⟩. However, calculating [s, M], we see that to normalise ⟨M⟩ implies the following condition must hold:

s_{1} − s_{2} = s_{2} − s_{3} = s_{2n+4} − s_{2n+5}.

Repeating over other 1-spaces leads to a collection of relations which can be expressed by a system of linear equations Rs = 0 for some matrix R and the vector s = (s_{1}, …, s_{2n+12}). The kernel of R modulo p then determines the dimension of 𝔫_{𝔱}(𝔥). To determine the dimension of N_{T} (𝔥), one searches for cocharacters χ(t) = diag(t^{k_{1}}, t^{k_{2}}, …, t^{k_{2n+12}}) which normalise 𝔥 by conjugation. This leads to an identical set of relations on the entries of the vector k = (k_{1}, …, k_{2n+12}), so that the equation Rk = 0 must be solved over the integers. Then the dimension of N_{T} (𝔥) is the nullity of R. The nullities of R over ℤ and over ℤ/p are identical if and only if s can be lifted to a diagonal cocharacter χ(t) so that d/dt|_{t=1}χ(t) = s. By explicit calculation of R in our particular case, one sees its elementary divisors are 0^{4}, 1^{2n+7}, F_{r+1}. Thus since p = F_{r+1} the nullity of R modulo p is bigger than over ℤ. Thus there is a toral element s, which cannot be lifted to a diagonal cocharacter. In our case, 𝔥 has an obvious centraliser whose elements are:

diag(\underbrace{s_{1}, …, s_{1}}_{r+2}, \underbrace{s_{2}, …, s_{2}}_{r+1}, \underbrace{s_{3}, …, s_{3}}_{5}, \underbrace{s_{4}, …, s_{4}}_{4})

which accounts also for the four-dimensional kernel over the integers. One also checks that the subalgebra 𝔥 is normalised by the toral element

s :=diag(1, 2, 3, 5, 8, …, F_{r}, F_{r+1}, F_{r+2})

⊕ diag(F_{2} + 4 = 6, F_{3} + 4 = 7, …, F_{r} + 4, F_{r+1} + 4, F_{r+2} + 4 = F_{r} + 4)

⊕ diag(F_{r−2} + 1, F_{r−1} + 1, F_{r} + 1, F_{r+1} + 1, F_{r+2} + 1)
⊕ diag(F_{r−1} + 2, F_{r} + 2, F_{r+1} + 2, F_{r+2} + 2),

where for the direct sum A ⊕ B of two square matrices A and B we mean the block diagonal matrix having A and B on the diagonal. Note the congruence amongst the entries in t, F_{r} = F_{r+2} mod p. Thus on each line, the last and pen-penultimate entries are the same modulo p. Furthermore, since this element does not centralise 𝔥, it can have no lift to a diagonal cocharacter. By assumption, N_{G}(𝔥) = N_{B}(𝔥) is smooth. Thus ⟨s⟩ lifts to the image of a cocharacter χ′ which, by Lemma 11.3 is conjugate by C_{U}(s) to a diagonal cocharacer χ. Since by inspection, only five entries of s are the same, s is a regular toral element of 𝔫_{𝔤}(𝔥) and one checks

C_{U}(s) = ⟨1 + te_{r,r+2}, 1 + te_{2r+1,2r+3}, 1 + te_{2r+6,2r+8}, 1 + te_{2r+10,2r+12} : t ∈ k⟩.

The action of the second listed element in turn normalises 𝔥 and the first, third and fourth simply change the values of λ_{1}, λ_{2}, λ_{3}. Thus if g ∈ C_{U}(s) then one computes a new relation matrix R′ computing the normaliser 𝔫_{𝔱}(𝔥^{g}) which, by virtue of being independent of the values of λ_{i}, is identical to R. In particular, ⟨s⟩ still normalises 𝔥 but there is still no lift to a diagonal cocharacter. This contradicts the conclusion of Lemma 11.3, hence N_{G}(𝔥) is not smooth.

The next example will show the necessity of the bound in Theorem 3.2. We first collect some miscellaneous auxiliary results in the following lemma. Recall that a subgroup H of a connected reductive group G is called G-irreducible if it is in no proper parabolic subgroup of G.

Lemma 11.5. Suppose G is a connected reductive algebraic group and H is a (possibly disconnected) closed reductive subgroup of G.

(i) We have N_{G}(H)_{red}^{∘} = H^{∘}C_{G}(H)_{red}^{∘}. (ii) If H is G-irreducible, then C_{G}(H)_{red}^{∘} = Rad(G), where Rad(G) = Z(G)_{red}^{∘}. (iii) Suppose H ≤ M ≤ G is an intermediate reductive subgroup with Rad(G) ≤ Rad(M) and that H is G-irreducible. Write Z(M)^{∘} = Rad(M) × μ_{M} for an infinitesimal subgroup scheme μ_{M}. Then either μ_{M} ≤ Z(H) or N_{G}(H) is non-smooth.

Proof. (i) and (ii) follow from [Mar03, Lemmas 6.2 and 6.8]. For (iii), clearly μ_{M} ≤ Z(M) ≤ N_{G}(H). If N_{G}(H) is smooth, then by parts (i) and (ii) we have μ_{M} ≤ Z(M)^{∘} ≤ H^{∘}C_{G}(H)_{red}^{∘} = H^{∘} Rad(G). This forces μ_{M} ≤ H^{∘}.

Examples 11.6. Lemma 11.5 can be used to produce reductive subgroups H of G with non-smooth normalisers in bad characteristic. We use [Her13, Example 4.1], in which the first author constructs examples of non-smooth centralisers for each reductive group over a field of characteristic p for which p is not a very good prime for G. All the subgroups constructed in loc. cit. are maximal rank reductive subgroups M such that C_{G}(M) = Z(M) is non-smooth, hence μ_{M} ≠ 1 in Lemma 11.5(iii) above. In many cases, we may take a further connected, reductive G-irreducible subgroup H of M such that p is pretty good for H. Thus its centre is in fact smooth, and being finite, cannot contain μ_{M}. Thus by Lemma 11.5(iii) the normaliser N_{G}(H) is non-smooth. Let us list some triples (G, p, M, H) which work for this process. By V_{n} we denote a natural module of dimension n for the classical group M; by M̃_{1} we mean a subgroup of type M_{1} corresponding to short roots.

\begin{matrix}G & p & M & H \\ G_{2} & 3 & A_{2} & A_{1} ↪ M; V_{3}|_{H} = L(2) \\ F_{4} & 2 & A_{1}^{4} & A_{1} ↪ M; x ↦ (x, x^{2}, x^{4}, x^{16}) \\ F_{4} & 3 & A_{2}Ã_{2} & (A_{1}, A_{1}) ↪ M; (V_{3}, V_{3})|_{H} = (L(2), L(2)) \\ E_{8} & 5 & A_{4}^{2} & A_{1}^{2} ↪ M; (V_{5}, V_{5})|_{H} = (L(4), L(4)) \\ SL_{p} & p > 2 & SL_{p} & A_{1} ↪ M; V_{p}|_{H} = L(p − 1). \end{matrix}

Remark 11.7. A complete list of conjugacy classes of simple G-irreducible subgroups of exceptional groups has been compiled by A. Thomas, see [Tho15] for the cases of rank at least 2 and [Tho16] for the rank 1 case. For the G_{2} example one may consult [Ste10, Theorem 1, Corollary 3].

The next example shows the promised tightness of Theorem B(i) as stated in Remark 1.2(a).

Lemma 11.8. Let G = GL(V) with dim V ≥ p − 1 ≥ 3 and take any subspace W ≤ V with dim W = p − 1. Then if W_{1} ≤ 𝔤𝔩(W) is the first Witt algebra in its p − 1-dimensional representation we have N_{G}(W_{1}) is not smooth.

Proof. Since W_{1} is irreducible on W, the normaliser 𝔫_{𝔤𝔩(V)}(W_{1}) = 𝔫_{𝔰𝔩(W)}(W_{1})⊕ 𝔷⊕𝔤𝔩(U) for V = W ⊕U and 𝔷 the centre of 𝔤𝔩(W). Moreover as W_{1} is irreducible on W, so is 𝔫 = 𝔫_{𝔰𝔩(W)}(W_{1}). By Theorem C, 𝔫 is semisimple, hence, as W_{1} is simple, it must be a direct factor of 𝔫, say 𝔫 = W_{1} ⊕ 𝔥. But now the action of ad 𝔥 on W is a W_{1}-module homomorphism, hence is a scalar by Schur’s lemma. Thus 𝔥 ≤ 𝔷(𝔰𝔩(W)) = 0. It follows that 𝔫 = W_{1}. Now N_{G}(W_{1}) sends W to another W_{1}-invariant subspace of the same dimension, hence N_{G}(W_{1}) ≤ GL(W) × GL(U). Since W_{1} is self-normalising, if N_{G}(W_{1}) were smooth we would have Lie N_{G}(W_{1}) = 𝔫_{𝔤}(W_{1}) = W_{1} ⊕ 𝔤𝔩(U). This shows that W_{1} is algebraic, a contradiction.

We now justify the remark after Theorem B that the bound in Theorem B(i) is tight for G = Sp_{2n}.

Lemma 11.9. The p-dimensional Witt algebra W_{1} is a maximal subalgebra of 𝔰𝔭_{p−1}. Furthermore, its normaliser in any Sp_{p−1}-Levi of Sp_{2n} with 2n ≥ p − 1 is non-smooth.

Proof. Since W_{1} stabilises the element

X ∧X^{p−1}+ \frac{1}{2}X^{2}∧X^{p−2}+ \frac{1}{3}X^{3}∧X^{p−3}+⋯+ \frac{2}{p − 1}X^{(p−1)/2}∧X^{(p+1)/2} ∈ ⋀^{2}V

we find that W_{1} is contained in 𝔰𝔭_{p−1}, acting irreducibly on the p − 1dimensional module. Exponentiating a set of nilpotent generators of the Witt algebra as in the proof of Theorem B(ii) gives an irreducible subgroup W ≤ Sp_{p−1}. We claim that we must have equality. From this claim it follows that W_{1} is in no proper classical algebraic subalgebra of 𝔰𝔭_{p−1}, hence, by Theorem 2.3, is maximal. To prove the claim, suppose W is a proper subgroup of G = Sp_{p−1}. Since W is irreducible on the p − 1-dimensional module, W is it no parabolic of G. Thus it is in a connected reductive maximal subgroup M. We must have that M is simple, or else W_{1} would be in a parabolic of G. Now since the lowest dimensional non-trivial representation of W_{1} is p − 1, it follows that M can have no lower-dimensional non-trivial representation. Since p > 2, Sp_{p−1} has no simple maximal rank subgroup. All classical groups of rank lower than \frac{p−1}{2} have natural modules of smaller dimension than p − 1, so M is of exceptional type. The lowest dimensional representations of the exceptional types are 6 (p = 2), 7, 25 (p = 3), 26, 27, 56 and 248. The only time one of these is p − 1 is when p = 57 and M = E_{7}. But if p = 57 then p > 2h − 2 for E_{7}, then by Theorem B(ii) all maximal semisimple subalgebras are algebraic and so W_{1} is not a subalgebra of E_{7}. This proves the claim, hence gives the first part of the lemma. For the second, with 2n > p − 1, we have W_{1} ≤ 𝔰𝔭_{p−1} ⊕ 𝔰𝔭_{2n−p+1} with W_{1} sitting in the first factor. Then its normaliser is evidently W_{1} ⊕ 𝔰𝔭_{2n−p+1}, however this is not algebraic for p > 3, hence the normaliser N_{G}(W_{1}) cannot be smooth. Thus we have shown that normalisers of all subalgebras of 𝔰𝔭_{2n} are smooth only if p > h + 1.

If n ≥ p there is a more straightforward example of a (non-restricted) subalgebra of 𝔤𝔩_{n} whose normaliser in GL_{n} is not smooth.

Example 11.10. Let 𝔤 = 𝔤𝔩_{n}, take J_{p} a Jordan block of size p and take the abelian one-dimensional Lie algebra 𝔥 = k(I_{p} + J_{p}) where I_{p} is an identity block of size p. Then one can show with elementary matrix calculations that the normaliser of N_{G}(𝔥) is non-smooth.

The next example shows that even the normalisers of smooth groups are not smooth, even in GL(V), and even when p is arbitrarily large.

Lemma 11.11. Let G = GL(V) with dim V ≥ 3 and let W be a 3-dimensional subspace. Let U ≤ GL(W) be defined as the smooth subgroup whose k-points are

U(k) = \{[ \begin{matrix}1 & 0 & t \\ 0 & 1 & t^{p} \\ 0 & 0 & 1 \end{matrix} ] : t ∈ k\} .

Write V = W ⊕W′ for some complement W′ to W and set H = U ⊕GL(W′) ≤ GL(V). Then N_{G}(H) is non-smooth.

Proof. From the reductivity of GL(W′) it follows that N_{G}(H) = N_{GL(W)}(U) ⊕ GL(W′) so it suffices to show that N_{GL(W)}(U) is non-smooth. This is a routine calculation. For example, if x is an element of a k-algebra A, with x^{p} = 0 then one checks that the matrix

[ \begin{matrix}1 & x & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] ∈ N_{GL(W)}(U)(A).

Now, the normaliser of U of course normalises Lie(U). Since

Lie(U) = k [ \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ] ,

the normaliser of Lie(U) is the product of the centraliser of a certain (nilpotent) element and the image of a cocharacter associated with that element. In particular, the normaliser of Lie(U) is contained in the upper triangular Borel subgroup. Write V for the unipotent radical of that Borel subgroup, so V is 3-dimensional; a typical element has the form

[ \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ] .

In fact, the condition a^{p} = 0 defines the scheme-theoretic normaliser in V of U, and the condition a = 0 defines the corresponding smooth subgoup of V whose k-points form the group-theoretic normaliser of U(k) in V (k). The lemma follows.

Now we show that normalisers of height two or more subgroup schemes are not smooth.

Example 11.12. Let G be any connected reductive algebraic group over an algebraically closed field k of characteristic p > 2 and set F : G → G to be the Frobenius endomorphism. Let B = T U be a Borel subgroup of G with T an F -stable maximal torus, and let U the non-trivial unipotent radical. Let T_{r} be the kernel in T of F^{r} and U_{1} the kernel in U of F. Finally set H = T_{r} ⋉ U_{1}. Then N_{G}(H) = T ⋉ U_{1}, hence is not smooth.

The next example shows that if p = dim V, then the normaliser of a smooth connected solvable non-diagonalisable algebraic subgroup of GL(V) can even be irreducible on V, thus a fortiori it is not smooth. This also gives an example for when p = 2 and dim V = 2 that the normalisers in SL(V) and GL(V) of subalgebras of the respective Lie algebras are not smooth.

Example 11.13. By [Ten87, Lemma 3] the Lie algebra W_{1} + O_{1} formed as the semidirect product of W_{1} and O_{1}, where O_{1} acts on itself by multiplication, is a maximal subalgebra of 𝔰𝔩_{p} = 𝔰𝔩(k[X]/X^{p}). We imitate the embedding of O_{1} in 𝔤𝔩_{p} by a solvable subgroup of GL_{p}. Define the height ht(α) of a root α to be the sum of the coefficients of the simple roots. Let U be the subgroup ⟨∏_{α∈R^{−};ht(α)=i} x_{α}⟩_{1≤i≤p−1}. By construction U is connected and unipotent and one can show that dim U = p − 1 and that Lie H = O_{1}, where H is the smooth solvable subgroup Z(GL_{p})U. Now it can be shown that there is a subgroup scheme W corresponding to W_{1} in GL_{p} which normalises H and for which W ⋉ H is irreducible. It immediately follows that N_{G}(H) cannot be smooth.

Finally we show that if p ≤ 2n − 1 the normalisers in GL_{n} and SL_{n} of subspaces of their Lie algebras are not all smooth, even when these normalisers are generated by nilpotent elements, showing that the bound in Theorem B(ii) cannot be improved for general subspaces.

Lemma 11.14. If p < 2n − 1, normalisers of subspaces of 𝔤𝔩_{n} (or 𝔰𝔩_{n}) are not necessarily smooth.

Proof. Let p = 2n − 3 and let 𝔥 = 𝔰𝔩_{2} = Lie H with H = SL_{2} over a field k of characteristic p. Then the action of H on the simple module L((p + 1)/2) gives an (irreducible) embedding H → GL_{n}. Restricting the adjoint representation of 𝔤𝔩_{n} on itself to H gives a module

L((p + 1)/2) ⊗ L((p + 1)/2)^{∗} ≅ T (p + 1) ⊕ M,

where M is a direct sum of irreducibles for H (and 𝔥) and T (p + 1) is a tilting module, uniserial with successive composition factors L(p−3)|L(p+1)|L(p−3). Now for the algebraic group H = SL_{2} we have L(p + 1) ≅ L(1) ⊗ L(1)^{[1]} by Steinberg’s tensor product formula. Restricting to 𝔥, L(p + 1) is isomorphic to L(1) ⊕ L(1). Now it is easy to show the restriction map Ext_{G}^{1}(L(p + 1), L(p − 3)) → Ext_{𝔤}^{1}(L(1), L(p−3))⊕Ext_{𝔤}^{1}(L(1), L(p−3)) is injective. Hence T (p+1)|_{𝔤} contains a submodule M isomorphic to L(1)/L(p − 3). Now, the Lie theoretic normaliser of M contains 𝔥 but the scheme-theoretic stabiliser does not contain H. It follows that the normaliser of this subspace is not smooth. Indeed, as 𝔥 acts irreducibly on the n-dimensional natural representation for 𝔤𝔩_{n}, it is in no parabolic of 𝔤𝔩_{n} (or 𝔰𝔩_{n}). However, the set of k-points N_{H}(M)(k) = N_{GL_{n}}(M)(k) ∩ H is in a parabolic of H, hence in a parabolic of GL_{n}.

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Sebastian Herpel Ruhr-Universität Bochum Bochum, Germany David Stewart University of Cambridge Cambridge, UK dis20@cam.ac.uk 38

Documenta Mathematica 21 (2016)