Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

Abstract

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that $p={\operatorname {char}}(k)$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}=\operatorname {Lie}(G)$. Specifically, we show that either $\mathfrak{m}=\operatorname {Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal exotic semidirect product. The conjugacy classes of maximal connected subgroups of $G$ are known thanks to the work of Seitz, Testerman, and Liebeck–Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\mathrm{E}}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\mathrm{E}}_7$.

1. Introduction

Unless otherwise specified, $G$ will denote a simple algebraic group of exceptional type defined over an algebraically closed field $k$ of characteristic $p>0$. We always assume that $p$ is a good prime for $G$, that is, $p>5$ if $G$ is of type ${\mathrm{E}}_8$ and $p>3$ in the other cases. Under this hypothesis, the Lie algebra $\mathfrak{g}=\operatorname {Lie}(G)$ is simple. Being the Lie algebra of an algebraic group, it carries a natural $[p]$th power map $\mathfrak{g}\ni x\mapsto x^{[p]}\in \mathfrak{g}$ equivariant under the adjoint action of $G$. The goal of this paper is to classify the maximal Lie subalgebras $\mathfrak{m}$ of $\mathfrak{g}$ up to conjugacy under the adjoint action of $G$. The maximality of $\mathfrak{m}$ implies that it is a restricted subalgebra of $\mathfrak{g}$, i.e., has the property that $\mathfrak{m}^{[p]}\subseteq \mathfrak{m}$. The main result of Pre17 states that if $\operatorname {rad}(\mathfrak{m})\ne 0$, then $\mathfrak{m}=\operatorname {Lie}(P)$ for some maximal parabolic subgroup of $G$. Therefore, in this paper we are concerned with the case where $\mathfrak{m}$ is a semisimple Lie algebra. In prime characteristic this does not necessarily mean that $\mathfrak{m}$ is a direct sum of its simple ideals.

We write $\mathcal{O}(1;\underline{1})$ for the truncated polynomial ring $k[X]/(X^p)$. The derivation algebra of $\mathcal{O}(1;\underline{1})$, denoted $W(1;\underline{1})$, is known as the Witt algebra. This restricted Lie algebra is a free $\mathcal{O}(1;\underline{1})$-module of rank $1$ generated by the derivative $\partial$ with respect to the image of $X$ in $\mathcal{O}(1;\underline{1})$. A restricted Lie subalgebra $\mathcal{D}$ of $W(1;\underline{1})$ is called transitive if it does not preserve any proper nonzero ideals of $\mathcal{O}(1;\underline{1})$.

Let $\mathfrak{h}$ be any semisimple restricted Lie subalgebra of $\mathfrak{g}$ and let $\operatorname {Soc}(\mathfrak{h})$ denote the socle of the adjoint $\mathfrak{h}$-module $\mathfrak{h}$. This is, of course, the sum of all minimal ideals of $\mathfrak{h}$. As one of the main steps in our classification, we show that if $\operatorname {Soc}(\mathfrak{h})$ is indecomposable and not semisimple, then $\operatorname {Soc}(\mathfrak{h})\cong \mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1})$ and there exists a transitive Lie subalgebra $\mathcal{D}$ of the Witt algebra $W(1;\underline{1})$ such that

$$\begin{equation} \mathfrak{h}\,\cong \, (\mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1}))\rtimes (\operatorname {Id}\otimes \mathcal{D}){\tag{1}} \end{equation}$$

as Lie algebras. We call the semisimple restricted Lie subalgebras of this type exotic semidirect products, esdp’s for short, and we show that under our assumptions on $G$ the Lie algebra $\mathfrak{g}$ contains an esdp if and only if either $G$ is of type $E_7$ and $p\in \{5,7\}$ or $G$ is of type ${\mathrm{E}}_8$ and $p=7$. (Although esdp’s do exist in Lie algebras $\operatorname {Lie}(G)$ of type ${\mathrm{E}}_8$ over fields of characteristic $5$ we ignore them in the present paper as $p=5$ is a bad prime for $G$.) It turned out that for $p>5$ any esdp in type ${\mathrm{E}}_8$ is contained in a proper regular subalgebra of $\mathfrak{g}$ and hence is not maximal in $\mathfrak{g}$. We prove that in type ${\mathrm{E}}_7$ maximal esdp’s of $\mathfrak{g}$ do exist and form a single conjugacy class under the adjoint action of $G$. Furthermore, if $\mathfrak{h}$ as in (1) is maximal in $\mathfrak{g}$, then $\mathcal{D}\cong \mathfrak{sl}_2$ when $p=5$ and $\mathcal{D}=W(1;\underline{1})$ when $p=7$.

As our next step we use the smoothness of centralisers $C_G(V)$, where $V$ is a subspace of $\mathfrak{g}$, to show that if $\operatorname {Soc}(\mathfrak{m})$ is semisimple and contains more than one minimal ideal of $\mathfrak{m}$, then there exists a semisimple (and nonsimple) maximal connected subgroup $M$ of $G$ such that $\mathfrak{m}=\operatorname {Lie}(M)$; see §4.1. We mention that Proposition 4.1 applies to simple algebraic groups of classical types as well.

Having obtained the above results we are left with the case where $\operatorname {Soc}(\mathfrak{m})$ is a simple Lie algebra. In this situation, we prove that if the derivation algebra $\operatorname {Der}(\mathfrak{m})$ is isomorphic to the Lie algebra of a reductive $k$-group, then $\mathfrak{m}=\operatorname {Lie}(M)$ for some maximal connected subgroup of $G$. If $\operatorname {Der}(\mathfrak{m})$ is not of that type, we show that $p-1$ equals the Coxeter number of $G$ and $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, which is unique up to $(\operatorname {Ad}\,G)$-conjugacy by the main result of HS15.

We let $\mathcal{N}(\mathfrak{g})$ denote the nilpotent cone of the restricted Lie algebra $\mathfrak{g}$ and write $\mathcal{O}(D)$ for the adjoint $G$-orbit in $\mathcal{N}(\mathfrak{g})$ with Dynkin label $D$. A Lie subalgebra of $\mathfrak{g}$ is called regular if it contains a maximal toral subalgebra of $\mathfrak{g}$. Our results on semisimple restricted Lie subalgebras $\mathfrak{h}$ of $\mathfrak{g}$ containing a nonsimple minimal ideal do not require the maximality hypothesis.

Theorem 1.1.

Let $\mathfrak{h}$ be a semisimple restricted Lie subalgebra of $\mathfrak{g}$ containing a minimal ideal which is not simple. Then $p\in \{5,7\}$, the group $G$ is of type ${\mathrm{E}}_7$ or ${\mathrm{E}}_8$, and the following hold:

(i)

If $G$ is of type ${\mathrm{E}}_8$, then $\mathfrak{h}$ is contained in a regular subalgebra of type ${\mathrm{E}}_7{\mathrm{A}}_1$ and hence is not maximal in $\mathfrak{g}$.

(ii)

If $G$ is of type ${\mathrm{E}}_7$, then $\operatorname {Soc}(\mathfrak{h})\cong \mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1})$ and $\mathfrak{h}\cong (\mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1}))\rtimes (\operatorname {Id}\otimes \mathcal{D})$ for some transitive restricted Lie subalgebra $\mathcal{D}$ of $W(1;\underline{1})$. In particular, $\mathfrak{h}$ is an esdp.

(iii)

Suppose $G$ is an adjoint group of type ${\mathrm{E}}_7$ and let $N:=N_G(\operatorname {Soc}(\mathfrak{h}))$. Then $N$ is a closed connected subgroup of $G$ acting transitively on the set of all nonzero $\mathfrak{sl}_2$-triples of $\mathfrak{g}$ contained in $\operatorname {Soc}(\mathfrak{h})$. All nilpotent elements of such $\mathfrak{sl}_2$-triples lie in $\mathcal{O}({\mathrm{A}}_3{\mathrm{A}}_2{\mathrm{A}}_1)$ when $p=5$ and in $\mathcal{O}({\mathrm{A}}_2{{\mathrm{A}}_1}^3)$ when $p=7$.

(iv)

Suppose $G$ is as in part (iii) and let $\widetilde{\mathfrak{h}}:=\mathfrak{n}_\mathfrak{g}(\operatorname {Soc}(\mathfrak{h}))$. Then $\widetilde{\mathfrak{h}}=\mathfrak{h}+\operatorname {Lie}(N)$ is a semisimple maximal Lie subalgebra of $\mathfrak{g}$ and $\operatorname {Lie}(N)$ has codimension $1$ in $\widetilde{\mathfrak{h}}$.

(v)

Suppose $\mathfrak{h}$ is a maximal Lie subalgebra of $\mathfrak{g}$. Then $\mathcal{D}\cong \mathfrak{sl}_2$ when $p=5$ and $\mathcal{D}=W(1;\underline{1})$ when $p=7$. Any two maximal esdp’s of $\mathfrak{g}$ are $(\operatorname {Ad}\,G)$-conjugate.

Thus if $G$ is of type ${\mathrm{E}}_7$ and $\mathfrak{h}$ satisfies the conditions of Theorem 1.1, then $\widetilde{\mathfrak{h}}=\mathfrak{n}_\mathfrak{g}(\operatorname {Soc}(\mathfrak{h}))$ is a semisimple maximal Lie subalgebra of $\mathfrak{g}$, unique up to $G$-conjugacy, and its socle coincides with $\operatorname {Soc}(\mathfrak{h})\cong \mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1})$. A more precise description of the group $N$ is given in §3.6.

Let $T$ be a maximal torus $G$ and denote by $\Phi =\Phi (G,T)$ the root system of $G$ with respect to $T$. Let $\Pi$ be a basis of simple roots in $\Phi$ and $\tilde{\alpha }$ the highest root in the positive system $\Phi ^+(\Pi )$. For any $\gamma \in \Phi$ we fix a nonzero element $e_\gamma$ in the root space $\mathfrak{g}_\gamma$. A Zariski closed connected subgroup $M$ of $G$ is said to be a maximal connected subgroup of $G$ if it is maximal among Zariski closed proper connected subgroups of $G$.

Theorem 1.2.

Let $\mathfrak{m}$ be a maximal Lie subalgebra of $\mathfrak{g}$ and suppose that $\mathfrak{m}$ is semisimple and all its minimal ideals are simple Lie algebras. Then one of the following two cases occurs:

(i)

There exists a semisimple maximal connected subgroup $M$ of $G$ such that $\mathfrak{m}=\operatorname {Lie}(M)$.

(ii)

The group $G$ is not of type ${\mathrm{E}}_6$, the Coxeter number of $G$ equals $p-1$, and $\mathfrak{m}$ is $(\operatorname {Ad}\,G)$-conjugate to the Witt subalgebra of $\mathfrak{g}$ generated by the highest root vector $e_{\tilde{\alpha }}$ and the regular nilpotent element $\sum _{\alpha \in \Pi }\,e_{-\alpha }$.

Since all conjugacy classes of maximal connected subgroups of $G$ are known thanks to earlier work of Dynkin Dyn52, Seitz Sei91, Testerman Tes88, and Liebeck–Seitz LS04, Theorems 1.1 and 1.2 give a complete answer to the problem of determining the maximal subalgebras of $\mathfrak{g}$ up to conjugacy.

Let $G$ be a reductive $k$-group and $\mathfrak{g}=\operatorname {Lie}(G)$. Recall that $G$ is said to satisfy the standard hypotheses if $p$ is a good prime for $G$, the derived subgroup $\mathscr{D}G$ is simply connected, and $\mathfrak{g}$ admits a nondegenerate $G$-invariant symmetric bilinear form. Given a Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ we denote by $\operatorname {nil}(\mathfrak{h})$ the largest ideal of $\mathfrak{h}$ consisting of nilpotent elements of $\mathfrak{g}$. We are able to prove the following corollary of our classification, a Lie algebra analogue of the well-known Borel–Tits theorem for algebraic groups.

Corollary 1.3.

If $G$ is a reductive $k$-group satisfying the standard hypotheses, then for any Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ with $\operatorname {nil}(\mathfrak{h})\ne 0$ there exists a parabolic subgroup $P$ of $G$ such that $\mathfrak{h}\subseteq \operatorname {Lie}(P)$ and $\operatorname {nil}(\mathfrak{h})\subseteq \operatorname {Lie}(R_u(P))$.

We stress that Corollary 1.3 breaks down very badly if we drop some of our assumptions on $G$. Indeed, if $G=\operatorname {PGL}(V)$, where $\dim V=p$, then there exists a $2$-dimensional abelian Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}=\mathfrak{pgl}(V)$ with $\mathfrak{h}\subset \mathcal{N}(\mathfrak{g})$ whose inverse image in $\mathfrak{gl}(V)$ acts irreducibly on $V$. This means that $\mathfrak{h}$ cannot be included into a proper parabolic subalgebra of $\mathfrak{g}$. In this example $G=\mathscr{D}G$ is a simple algebraic $k$-group of adjoint type. On the other hand, if $G$ is a simply connected $k$-group of type ${\mathrm{A}}$, ${\mathrm{B}}$, ${\mathrm{C}}$, or ${\mathrm{D}}$ and $p$ is good for $G$, then $\mathfrak{g}$ is isomorphic to one of $\mathfrak{sl}(V)$, $\mathfrak{so}(V)$ or $\mathfrak{sp}(V)$ as a restricted Lie algebra. For such Lie algebras, Corollary 1.3 is a straightforward consequence of the fact that $\operatorname {nil}(\mathfrak{m})$ annihilates a nonzero proper subspace of $V$. This indicates that proving Corollary 1.3 reduces quickly to the case where $G$ is a simple algebraic group of type ${\mathrm{G}}_2$, ${\mathrm{F}}_4$, ${\mathrm{E}}_6$, ${\mathrm{E}}_7$, or ${\mathrm{E}}_8$; see §5.2. Since there are no good substitutes of $V$ for exceptional groups, our proof of Corollary 1.3 relies very heavily on Theorems 1.1 and 1.2.

As an immediate consequence of Corollary 1.3 we obtain the following generalisation of one of the classical results of Lie theory first proved by Morozov in the characteristic zero case; see Bou75, Ch. VIII, §10, Th. 2.

Corollary 1.4.

Suppose $G$ satisfies the standard hypotheses and let $\mathfrak{q}$ be a Lie subalgebra of $\mathfrak{g}=\operatorname {Lie}(G)$ such that $\operatorname {nil}(\mathfrak{q})\ne 0$ and $\mathfrak{q}=\mathfrak{n}_{\mathfrak{g}}(\operatorname {nil}(\mathfrak{q}))$. Then there exists a proper parabolic subgroup $P$ of $G$ such that $\mathfrak{q}=\operatorname {Lie}(P)$ and $\operatorname {nil}(\mathfrak{q})=\operatorname {Lie}(R_u(P))$.

Indeed, by Corollary 1.3, there is a proper parabolic subgroup $P$ of $G$ such that $\mathfrak{q}\subseteq \operatorname {Lie}(P)$ and $\operatorname {nil}(\mathfrak{q})\subseteq \operatorname {Lie}(R_u(P))$. Let $\mathfrak{u}=\operatorname {Lie}(R_u(P))$ and denote by $\mathfrak{n}$ the normaliser of $\operatorname {nil}(\mathfrak{q})$ in $\mathfrak{u}$. Since $\mathfrak{q}\subseteq \operatorname {Lie}(P)$ normalises both $\mathfrak{u}$ and $\operatorname {nil}(\mathfrak{q})$, it is easy to see that $\mathfrak{n}\subseteq \mathfrak{n}_\mathfrak{g}(\operatorname {nil}(\mathfrak{q}))=\mathfrak{q}$ is an ideal of $\mathfrak{q}$ consisting of nilpotent elements of $\mathfrak{g}$. Suppose for a contradiction that $\operatorname {nil}(\mathfrak{q})\ne \mathfrak{u}$. Then it follows from Engel’s theorem that $\operatorname {nil}(\mathfrak{q})$ annihilates a nonzero vector of the factor space $\mathfrak{u}/\operatorname {nil}(\mathfrak{q})$ forcing $\operatorname {nil}(\mathfrak{q})$ to be a proper ideal of $\mathfrak{n}$. On the other hand, since $\mathfrak{n}$ consists of nilpotent elements of $\mathfrak{g}$ it must be that $\mathfrak{n}\subseteq \operatorname {nil}(\mathfrak{q})$. This contradiction shows that $\operatorname {nil}(\mathfrak{q})=\mathfrak{u}$. But then $\mathfrak{q}=\mathfrak{n}_\mathfrak{g}(\mathfrak{u})$ contains $\operatorname {Lie}(P)$. Since $\mathfrak{q}\subseteq \operatorname {Lie}(P)$, we now deduce that $\mathfrak{q}=\operatorname {Lie}(P)$ and $\operatorname {nil}(\mathfrak{q})=\operatorname {Lie}(R_u(P))$, as wanted.

Corollary 1.4 answers a question posed to one of the authors by Donna Testerman. We finish the introduction by mentioning some interesting open problems related to classifying the maximal subalgebras of Lie algebras of simple algebraic groups.

It is immediate from our classification that if $G$ is an exceptional group and $p$ is good for $G$, then the set of all maximal Lie subalgebras of $\mathfrak{g}$ splits into finitely many orbits under the adjoint action of $G$. We do not know whether the number of $(\operatorname {Ad}\,G)$-orbits of maximal Lie subalgebras of $\mathfrak{g}=\operatorname {Lie}(G)$ is finite in the case where $G$ is a group of type ${\mathrm{B}}$, ${\mathrm{C}}$, ${\mathrm{D}}$ or $G=\mathrm{SL}(V)$ and $p\nmid \dim V$. The problem is closely related with the fact that in dimension $d=p^{n}-s$, where $n\ge 4$ and $s\in \{1,2\}$, there exist infinitely many isomorphism classes of $d$-dimensional simple Lie algebras over $k$ (this was first observed by Kac in the early $1970$s). When $p>3$, it follows from the Block–Wilson–Strade–Premet classification theorem that almost all of them (excepting finitely many isomorphism classes) belong to infinite families of filtered Lie algebras of Cartan type $H$. In order to clarify the situation one would need to describe all irreducible restricted representations $\rho \colon L_p\to \mathfrak{gl}(V)$ of the $p$-envelope $L_p$ of every filtered Hamiltonian algebra $L\cong \operatorname {ad}L$ in $\operatorname {Der}(L)$ and then determine the Lie subalgebras of $\mathfrak{gl}(V)$ containing $\rho (L_p)$.

When $p\,\vert \dim V$ the main result of Pre17 is no longer valid for $\mathfrak{sl}(V)$ which contains maximal subalgebras that are neither semisimple nor parabolic. In fact, every maximal subalgebra of $\mathfrak{sl}(V)$ acting irreducibly on $V$ is neither semisimple nor parabolic as it must contain the scalar endomorphisms of $V$. This leads to intriguing representation-theoretic problems. As a very special example, it is not known at present whether the nonsplit central extension of the Witt algebra $W(1;\underline{1})$ given by the Block–Gelfand–Fuchs cocycle can appear as a maximal subalgebra of $\mathfrak{sl}(V)$ for some vector space $V$ with $p\,\vert \dim V$.

If $\mathcal{H}$ is a simple algebraic $k$-group and $\mathfrak{h}=\operatorname {Lie}({\mathcal{H}})$, then to every linear function $\chi \in \mathfrak{h}^*$ one can attach at least one irreducible $\mathfrak{h}$-module $E$ with $p$-character $\chi$. Let $\rho \colon \,\mathfrak{h}\to \mathfrak{gl}(E)$ denote the corresponding representation of $\mathfrak{h}$. It is known that under mild assumptions on $p$ and $\mathfrak{h}$ the dimension of $E$ is divisible by $p^{d(\chi )}$, where $2d(\chi )$ is the dimension of the coadjoint $\mathcal{H}$-orbit of $\chi$. Another challenging open problem which arises naturally in this setting is to determine all pairs $(\chi , \rho )$ for which $\chi \ne 0$ and $k\,\operatorname {Id}_E+\rho (\mathfrak{h})$ is a maximal Lie subalgebra of $\mathfrak{sl}(E)$. It can be shown by using finiteness of the number of unstable coadjoint $\mathcal{H}$-orbits and earlier results of Block, Kac, and Friedlander–Parshall that the number of such pairs (up to a natural equivalence relation) is finite.

Very little is known about maximal Lie subalgebras of exceptional Lie algebras $\mathfrak{g}$ over fields of bad characteristic. Recent work of Thomas Purslow Pur16 shows that some strange simple Lie algebras which have no analogues in characteristic $p>3$ do appear as maximal subalgebras of $\mathfrak{g}$. It seems that a detailed investigation of the above-mentioned problems could lead to interesting new results in modular representation theory and structure theory of simple Lie algebras.

2. Preliminaries

2.1. Basic properties of restricted Lie algebras

Let $\mathcal{L}$ be a Lie algebra over an algebraically closed field $k$ of characteristic $p>0$. We say that $\mathcal{L}$ is restrictable if $(\operatorname {ad}X)^p$ is an inner derivation for every $X\in \mathcal{L}$. Any restrictable Lie algebra $\mathcal{L}$ can be endowed with a $p$th power map (or a $p$-operation) $X\longmapsto X^{[p]}$ such that

(i)

$(\lambda X)^{[p]}\,=\,\lambda ^p X^{[p]}$ for all $\lambda \in k$ and $X\in \mathcal{L}$;

(ii)

$(\operatorname {ad}(X^{[p]})\,=\,(\operatorname {ad}X)^p$ for all $X\in \mathcal{L}$;

(iii)

$(X+Y)^{[p]}\,=\,X^{[p]}+Y^{[p]}+ \textstyle {\sum }_{i=1}^{p-1}\,s_i(X,Y)$ for all $X,Y\in \mathcal{L}$, where $is_i(X,Y)$ is the coefficient of $t^{i-1}$ in $\operatorname {ad}(tX+Y)^{p-1}(X)$ expressed as a sum of Lie monomials in $X$ and $Y$.

Such a $p$-operation is unique up to a $p$-linear map $\varphi \colon \mathcal{L}\to \mathfrak{z}(\mathcal{L})$, where $\mathfrak{z}(\mathcal{L})$ is the centre of $\mathcal{L}$. Indeed, if $\varphi$ is such a map, then the operation $X\longmapsto X^{[p]}+\varphi (X)$ also satisfies the properties (i), (ii), and (iii). A pair $(\mathcal{L},[p])$, where $\mathcal{L}$ is a restrictable Lie algebra and $[p]$ is a $p$th power map on $\mathcal{L}$, is called a restricted Lie algebra (or a $p$-Lie algebra). If $\mathcal{L}$ is restrictable and centreless, then it admits a unique restricted Lie algebra structure. For any $k$-algebra $\mathcal{A}$ (not necessarily associative or Lie) the Lie algebra $\operatorname {Der}(\mathcal{A})$ of all derivations of $\mathcal{A}$ carries a natural restricted Lie algebra structure which assigns to any $D\in \operatorname {Der}(\mathcal{A})$ the $p$th power of the endomorphism $D$ in $\mathrm{End}(\mathcal{A})$.

Now suppose that $\mathcal{L}$ is a finite-dimensional restricted Lie algebra over $k$. A Lie subalgebra $\mathcal{M}$ of $\mathcal{L}$ is called restricted if $x^{[p]}\in \mathcal{M}$ for all $x\in \mathcal{M}$. For any $x\in \mathcal{L}$ the centraliser $\mathcal{L}_x:=\{y\in \mathcal{L}\,|\,\,[x,y]=0\}$ is a restricted Lie subalgebra of $\mathcal{L}$. Indeed, $[y^{[p]},x]=(\operatorname {ad}y)^p(x)=-(\operatorname {ad}y)^{p-1}([x,y])=0$ for all $y\in \mathcal{L}_x$. If $\mathcal{M}$ is spanned over $k$ by elements $x_1,\ldots , x_r$ such that $x_i^{[p]}\in \mathcal{M}$ for all $1\le i\le r$, then $\mathcal{M}$ is a restricted Lie subalgebra of $\mathcal{L}$. Indeed, it follows from (iii) that $(\sum _{i=1}^r\lambda _ix_i)^{[p]}- \sum _{i=1}^r\lambda _i^px_i^{[p]}\in [\mathcal{M},\mathcal{M}]$ for all $\lambda _i\in k$.

If $\mathcal{L}=\operatorname {Lie}(S)$, where $S$ is a linear algebraic $k$-group, then the Lie algebra $\mathcal{L}$ identifies with the Lie algebra of all left invariant derivations of the coordinate algebra $k[S]$. Since in characteristic $p>0$ the associative $p$th power of any left invariant derivation of $k[S]$ is again a left invariant derivation, $\mathcal{L}$ carries a canonical restricted Lie algebra structure which has the property that $((\operatorname {Ad}\,g)(x))^{[p]}=(\operatorname {Ad}\, g)(x^{[p]})$ for all $g\in S$ and $x\in \mathcal{L}$. Moreover, if $H$ is a closed subgroup of $S$, then it follows from Bor91, Proposition 3.11 that $\operatorname {Lie}(H)$ (regarded with its own canonical $p$th power map) identifies with a restricted Lie subalgebra of $\mathcal{L}$.

An element $x$ of a restricted Lie algebra $\mathcal{L}$ is called semisimple if it lies in the the $k$-span of all $x^{[p]^i}$ with $i\ge 1$. If $x$ is semisimple, then $\operatorname {ad}x$ is a diagonalisable endomorphism of $\mathcal{L}$. A semisimple element $x$ of $\mathcal{L}$ is called toral if $x^{[p]}=x$. If $x$ is toral, then all eigenvalues of $\operatorname {ad}x$ lie in $\mathbb{F}_p$. A restricted Lie subalgebra of $\mathcal{L}$ is called toral if it consists of semisimple elements of $\mathcal{L}$. Since $k$ is algebraically closed, any toral subalgebra $\mathfrak{t}$ of $\mathcal{L}$ is abelian and contains a $k$-basis consisting of toral elements of $\mathcal{L}$. More precisely, the set $\mathfrak{t}^{\mathrm{tor}}$ of all toral elements of $\mathfrak{t}$ is an $\mathbb{F}_p$-form of $\mathfrak{t}$, so that $\mathfrak{t}\cong \mathfrak{t}^{\mathrm{tor}}\otimes _{\mathbb{F}_p}k$ as $k$-vector spaces. From this it is immediate that $\mathfrak{t}$ can be generated under the $p$th power map by a single element $x$, that is, $\mathfrak{t}\,=\,\operatorname {span}\{x^{[p]^i}\,|\,\, i\in \mathbb{Z}_{\ge 0}\}$. The maximal dimension of toral Lie subalgebras of $\mathcal{L}$ is often referred to as the toral rank of $\mathcal{L}$. If $\mathcal{L}=\operatorname {Lie}(S)$ and $H$ is a torus of $S$, then our remark in the previous paragraph implies that $\operatorname {Lie}(H)$ is a toral subalgebra of $\mathcal{L}$. Conversely, it follows from Bor91, Proposition 11.8 and the preceding remark that any toral subalgebra of $\mathcal{L}$ is contained in $\operatorname {Lie}(T)$ for some maximal torus $T$ of $S$. So the toral rank of $\operatorname {Lie}(S)$ coincides with the rank of the group $S$.

Given a finite-dimensional semisimple Lie algebra $L$ over $k$ we denote by $L_p$ the $p$-envelope of $L\cong \operatorname {ad}L$ in $\operatorname {Der}(L)$, that is, the smallest restricted Lie subalgebra of $\operatorname {Der}(L)$ containing $L$. The restricted Lie algebra $L_p$ is semisimple and any finite-dimensional semisimple $p$-envelope of $L$ is isomorphic to $L_p$ in the category of restricted Lie algebras. This is immediate from Str04, Definition 1.1.2 and Corollary 1.1.8. The derived subalgebra of $L_p$ coincides with $[L,L]$ and we have that $L=L_p$ if and only if $L$ is restrictable. If $L$ is simple and nonrestrictable, then the derived subalgebra of the restricted Lie algebra $L_p$ does not admit a restricted Lie algebra structure.

Now suppose that $G$ is a connected reductive algebraic group over $k$ and let $\mathfrak{g}=\operatorname {Lie}(G)$. In contrast with the preceding remark we have the following.

Lemma 2.1.

If $x$ is a semisimple element of $\mathfrak{g}$, then $[\mathfrak{g}_x,\mathfrak{g}_x]$ is a restricted Lie subalgebra of $\mathfrak{g}$.

Proof.

Let $T$ be a maximal torus of $G$ and let $\Phi$ be the root system of $G$ with respect to $T$. By the above discussion we may assume without loss that $x\in \operatorname {Lie}(T)$. Given $\alpha \in \Phi$ we denote by $U_\alpha$ the unipotent root subgroup of $G$ associated with $\alpha$ and write $S_\alpha$ for the subgroup of $G$ generated by $U_\alpha$ and $U_{-\alpha }$. Each $S_\alpha$ is a simple algebraic subgroup of type ${\mathrm{A}}_1$ in $G$. Since $\operatorname {Lie}(U_\alpha )=k e_{\alpha }$ is a restricted Lie subalgebra of $\mathfrak{g}$ it must be that $e_\alpha ^{[p]}=0$. Since $\operatorname {Lie}(S_\alpha )$ is a restricted Lie subalgebra of $\mathfrak{g}$ isomorphic to $\mathfrak{sl}_2(k)$ or $\mathfrak{pgl}_2(k)$, we have that $[e_\alpha ,e_{-\alpha }]^{[p]}\in k[e_\alpha ,e_{-\alpha }]$ (if $k$ has characteristic $2$ and the derived subgroup of $G$ is not simply connected, then it may happen that $[e_\alpha ,e_{-\alpha }]=0$ for some $\alpha \in \Phi$).

Let $\Phi _x=\{\alpha \in \Phi \,|\,\,({\mathrm{d}}_1\alpha )(x)=0\}.$ The Lie algebra $\mathfrak{g}_x$ is spanned over $k$ by $\operatorname {Lie}(T)$ and by all $e_\alpha$ with $\alpha \in \Phi _x$. It follows that the derived subalgebra $[\mathfrak{g}_x,\mathfrak{g}_x]$ is spanned over $k$ by all $e_\alpha$ with ${\mathrm{d}}_1\alpha \ne 0$ and all $[e_\alpha ,e_\beta ]$ with $\alpha ,\beta \in \Phi _x$. Each element $v$ in this spanning set has the property that $v^{[p]}\in kv$. Our earlier remarks in this subsection now show that $[\mathfrak{g}_x,\mathfrak{g}_x]$ is a restricted Lie subalgebra of $\mathfrak{g}$.

■

2.2. Transitive Lie subalgebras of the Witt algebra

We denote by $\mathcal{O}(m;\underline{1})$ the truncated polynomial ring $k[X_1,\ldots ,X_m]/(X_1^p,\ldots , X_m^p)$ in $m$ variables and write $x_i$ for the image of $X_i$ in $\mathcal{O}(m;\underline{1})$. The local $k$-algebra $\mathcal{O}(m;\underline{1})$ inherits a standard degree function from the polynomial ring $k[X_1,\ldots , X_m]$. Given $i\in \mathbb{Z}_{\ge 0}$ we denote by $\mathcal{O}(m;\underline{1})_{(i)}$ the subspace of all truncated polynomials in $\mathcal{O}(m;\underline{1})$ whose initial term has standard degree $\ge i$. Each $\mathcal{O}(m;\underline{1})_{(i)}$ is an ideal of $\mathcal{O}(m;\underline{1})$ and $\mathcal{O}(m;\underline{1})_{(i)}=0$ for $i>m(p-1)$. The maximal ideal of $\mathcal{O}(m;\underline{1})$ is $\mathcal{O}(m;\underline{1})_{(1)}$.

The derivation algebra of $\mathcal{O}(m;\underline{1})$, denoted $W(m;\underline{1})$, is called the $m$th Witt–Jacobson Lie algebra. This restricted Lie algebra is a free $\mathcal{O}(m;\underline{1})$-module of rank $m$ generated by the partial derivatives $\partial _1,\ldots , \partial _m$ with respect to $x_1,\ldots , x_m$. The subspaces $W(m;\underline{1})_{(i)}:=\sum _{i=1}^m\,\mathcal{O}(m;\underline{1})_{(i+1)}\partial _i$ with $-1\le i\le m(p-1)-1$ induce a decreasing Lie algebra filtration

$$\begin{equation*} W(m;\underline{1})=W(m;\underline{1})_{(-1)}\supset W(m;\underline{1})_{(0)}\supset \cdots \supset W(m;\underline{1})_{(m(p-1)-1)}\supset 0 \end{equation*}$$

of $W(m;\underline{1})$ which is called standard. The Lie subalgebra $W(m;\underline{1})_{(0)}$ is often referred to as the standard maximal subalgebra of $W(m;\underline{1})$. This is due to the fact that for $p>3$ it can be characterised as the unique subalgebra of minimal codimension in $W(m;\underline{1})$. Because of that all members of the standard filtration of $W(m;\underline{1})$ are invariant under the action of the automorphism group of $W(m;\underline{1})$.

A restricted Lie subalgebra $\mathcal{D}$ of $W(m;\underline{1})$ is called transitive if it does not preserve any proper nonzero ideals of $\mathcal{O}(m;\underline{1})$. Given a finite-dimensional simple Lie algebra $S$ over $k$ and a restricted transitive Lie subalgebra $\mathcal{D}$ of $W(m;\underline{1})$ we can form a natural semidirect product

$$\begin{equation*} S(m,\mathcal{D})\,:=\,(S\otimes \mathcal{O}(m;\underline{1}))\rtimes (\operatorname {Id}_S\otimes \mathcal{D}). \end{equation*}$$

It is known (and easy to see) that $S(m,\mathcal{D})$ is a semisimple Lie algebra over $k$ and its semisimple $p$-envelope $S(m,\mathcal{D})_p$ is isomorphic to $(S_p\otimes \mathcal{O}(m;\underline{1}))\rtimes (\operatorname {Id}_S\otimes \mathcal{D})$ as restricted Lie algebras, where $S_p$ is the $p$-envelope of $\operatorname {ad}S$ in $\operatorname {Der}(S)$.

The case $m=1$ will play a special role in what follows and we spell out the above in more detail. The Witt algebra $W(1;\underline{1})=\operatorname {Der}(\mathcal{O}(1;\underline{1}))$ has $k$-basis $\{x^i\partial \,|\,\, 0\le i\le p-1\}$, and the Lie bracket in $W(1;\underline{1})$ is given by $[x^i\partial ,x^j\partial ]=(j-i)x^{i+j-1}\partial$ for all $i,j\le p-1$ (here $x^n=0$ for all $n\ge p$). It is well known that $D^p\in kD$ for all $D\in W(1;\underline{1})$; see Pre92, for example. In conjunction with Jacobson’s formula for $[p]$th powers this shows that any Lie subalgebra of $W(1;\underline{1})$ is restricted. It is routine to check that a subalgebra $\mathcal{D}$ of $W(1;\underline{1})$ is transitive if and only if it is not contained in the standard maximal subalgebra $W(1;\underline{1})_{(0)}$.

Note that if $p=2$, then the Witt algebra is solvable, and if $p=3$, then $W(1;\underline{1})\cong \mathfrak{sl}_2(k)$. But things settle for $p>3$ and an old result of Jacobson says that any automorphism of the Witt algebra $W(1;\underline{1})$ is induced by a unique automorphism of $\mathcal{O}(1;\underline{1})$; see Jac43, Theorems 9 and 10. Using this fact it is straightforward to describe the conjugacy classes of transitive Lie subalgebras of the Witt algebra under the action of its automorphism group.

Lemma 2.2.

If $p>3$, then any transitive Lie subalgebra of $W(1;\underline{1})$ is conjugate under the action of $\operatorname {Aut}(W(1;\underline{1}))$ to one of the following:

(1)

$k\partial$;

(2)

$k(1+x)\partial$;

(3)

$k\partial \oplus k(x\partial )$;

(4)

$k\partial \oplus k(x\partial )\oplus k(x^2\partial )\,\cong \, \mathfrak{sl}_2(k)$;

(5)

$W(1;\underline{1})$.

Proof.

Let $\mathcal{G}=\operatorname {Aut}(W(1;\underline{1}))$ and let $\mathcal{D}$ be a transitive Lie subalgebra of $W(1;\underline{1})$. Then there exists $D\in \mathcal{D}$ such that $D\not \in W(1;\underline{1})_{(0)}$. Since $D^p\in kD$ we may assume that either $D^p=0$ or $D^p=D$. If $D^p=0$ (resp., $D^p=D$), then there is $g\in \mathcal{G}$ such that $g(D)=\partial$ (resp., $g(D)=(1+x)\partial$); see Pre92, Lemma 4 and Str04, § 7. This proves the lemma in the case where $\dim \mathcal{D}=1$.

From now on we may assume that $\mathcal{D}$ has dimension $\ge 2$ and intersects nontrivially with the set $\{\partial ,\,(1+x)\partial \}$. We first suppose that $(1+x)\partial \in \mathcal{D}$. The subspace $k(1+x)\partial$ is a self-centralising torus of $W(1;\underline{1})$ and the group $\mathcal{G}$ contains a cyclic subgroup $\Sigma$ of order $p-1$ which transitively permutes the set $\mathbb{F}_p^\times (1+x)\partial$ of all nonzero toral elements of $k(1+x)\partial$; see Pre92, § 1, for example. As $\dim \mathcal{D}\ge 2$ and $\operatorname {ad}(1+x)\partial$ is diagonalisable, it must be that $(1+x)^i\partial \in \mathcal{D}$ for some $i\in \{0,2,\ldots , p-1\}$. Since $\Sigma$ transitively permutes the set of all eigenspaces for $\operatorname {ad}(1+x)\partial$ corresponding to eigenvalues in $\mathbb{F}_p^\times$, there is $\sigma \in \Sigma$ such that $\sigma (\mathcal{D})$ contains both $(1+x)\partial$ and $\partial$. So we may assume without loss that $\partial \in \mathcal{D}$.

Since $\dim \mathcal{D}\ge 2$ there is $f(x)\partial \in \mathcal{D}$ such that $f(x)=a_rx^r+\cdots +a_1x$, where $a_i\in k$ and $a_r\ne 0$ for some $1\le r\le p-1$. Then $(\operatorname {ad}\partial )^{r-1}(f(x)\partial )\in \mathcal{D}$ yielding $x\partial \in \mathcal{D}$. If $\dim \mathcal{D}=2$ we arrive at case (3). If $\dim \mathcal{D}\ge 3$, then $\mathcal{D}\cap W(1;\underline{1})_{(1)}$ contains an eigenvector for $\operatorname {ad}(x\partial )$. Hence $x^r\partial \in \mathcal{D}$ for some $r\in \{2,\ldots , p-1\}$. As $\partial \in \mathcal{D}$, this yields that $x^i\partial \in \mathcal{D}$ for $0\le i\le r$. If $\dim \mathcal{D}=3$, then $r=2$ and we are in case (4). If $\dim \mathcal{D}\ge 4$, then the above shows that $\mathcal{D}$ contains $x^3\partial$. Since the Lie algebra $W(1;\underline{1})$ is generated by $\partial$ and $x^3\partial$, we get $\mathcal{D}=W(1;\underline{1})$ completing the proof.

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2.3. A property of restricted $\mathfrak{sl}_2$-modules

Let $L$ be a Lie algebra over $k$ and let $V$ be a finite-dimensional $L$-module. Given $x\in L$ we set $V_x:=\{v\in V\,|\,\, x. v=0\}.$

In what follows we shall require very detailed information on certain $\mathfrak{sl}_2$-triples $\{e, h, f\}$ of $\mathfrak{g}$ such that $e^{[p]}=f^{[p]}=0$ and $h^{[p]}=h$. In particular, it will be very useful for us to know that $\dim \mathfrak{g}_{h}\le \,\dim \mathfrak{g}_{e}$. Since the $k$-span of $\{e,h,f\}$ is a restricted Lie subalgebra of $\mathfrak{g}$ isomorphic to $\mathfrak{sl}_2(k)$, we may regard $\mathfrak{g}$ as a restricted $\mathfrak{sl}_2(k)$-module.

Lemma 2.3.

Suppose ${\operatorname {char}}(k)>2$ and let $V$ be a finite-dimensional restricted module over the restricted Lie algebra $\mathfrak{s}=\mathfrak{sl}_2(k)$. If $x$ is a nonzero semisimple element of $\mathfrak{s}$, then $\dim V_x\le \dim V_y$ for any $y\in \mathfrak{s}$.

Proof.

Let $\mathcal{S}=\mathrm{SL}_2(k)$ and let $V(m)$ be the Weyl module for $\mathcal{S}$ of highest weight $m\in \mathbb{Z}_{\ge 0}$. The Lie algebra $\mathfrak{s}=\operatorname {Lie}(\mathcal{S})$ acts on $V(m)$ via the differential at identity of the rational representation $\mathcal{S}\to \,\mathrm{GL}(V(m))$. It is well known that any irreducible restricted $\mathfrak{s}$-module is isomorphic to one of the $V(m)$’s with $m\in \{0,1,\ldots , p-1\}$.

In proving this lemma we may assume that $V$ is an indecomposable $\mathfrak{s}$-module. Let $\psi \colon \,\mathfrak{s}\to \,\mathfrak{gl}(V)$ denote the corresponding representation of $\mathfrak{s}$. All such representations are classified in Pre91. To be more precise, it is known that either there is a rational representation $\rho \colon \,\mathcal{S}\to \,\mathrm{GL}(V)$ such that $\psi ={\mathrm{d}}_e\rho$ or $V$ is a maximal $\mathfrak{s}$-submodule of $V(lp+r)$ for some $l\in \mathbb{Z}_{\ge 1}$ and $r\in \{0,1,\ldots , p-2\}$. In the latter case $\dim V=lp$ and $V$ has two composition factors, $V(r)$ and $V(p-2-r)$, both of which appear $l$ times in any composition series of $V$. If $\psi ={\mathrm{d}}_e\rho$, then either $V$ is isomorphic to one of $V(m)$ or $V(m)^*$ with $p\nmid (m+1)$ or $V$ is a projective indecomposable module over the restricted enveloping algebra of $\mathfrak{s}$.

Let $d=\min _{s\in \mathfrak{s}}\,\dim V_s$. Looking at the minors of matrices of the endomorphisms $\psi (s)$ with $s\in \mathfrak{s}$ one observes that the set

$$\begin{equation*} U:=\{s\in \mathfrak{s}\,|\,\,\dim V_s=d\} \end{equation*}$$

is nonempty, Zariski open in $\mathfrak{s}$, and has the property that $k^\times U=U$. If $\psi ={\mathrm{d}}_e\rho$, then $U$ is also $(\operatorname {Ad}\mathcal{S})$-stable. If $\{e,h,f\}$ is a standard basis of $\mathfrak{s}$, then any nonzero semisimple element of $\mathfrak{s}$ is $(\operatorname {Ad}\mathcal{S})$-conjugate to a nonzero multiple of $h$. So if $\psi ={\mathrm{d}}_e\rho$, then $U$ contains all nonzero semisimple elements of $\mathfrak{s}$. This proves the lemma in the present case.

Now suppose $V$ is a maximal submodule of $V(m)$, where $m=lp+r$ and $0\le r\le p-2$. If $r$ is even and $x$ is $(\operatorname {Ad}\,\mathcal{S})$-conjugate to a nonzero multiple of $H$, then $\dim V(r)_x=1$, and if $r$ is odd, then $\dim V(r)_{x}=0$. Since $x^{[p]}=\lambda x$ for some $\lambda \in k^\times$, the endomorphism $\psi (x)$ is diagonalisable. In view of our remarks earlier in the proof this yields that $\dim V_x=l$ for any nonzero semisimple element $x\in \mathfrak{s}$. Consequently, $d=l$ completing the proof.

■

Remark 2.4.

Let $\mathfrak{s}$, $V$, and $\psi$ be as above. It follows from the above-mentioned description of finite-dimensional indecomposable restricted $\mathfrak{s}$-modules that $V$ has a reducible indecomposable direct summand only if $\psi (e)^{p-1}\ne 0$ or $\psi (f)^{p-1}\ne 0$.

2.4. Standard $\mathfrak{sl}_2$-triples

In this subsection we review some results on $\mathfrak{sl}_2$-triples in exceptional Lie algebras over algebraically closed fields of good characteristics. More information on such $\mathfrak{sl}_2$-triples can be found in HS15 and ST16 where the notation is slightly different.

It is well known that the nilpotent cone $\mathcal{N}(\mathfrak{g})$ coincides with the set of all $(\operatorname {Ad}\,G)$-unstable vectors of $\mathfrak{g}$. Therefore, any nonzero $e\in \mathcal{N}(\mathfrak{g})$ admits a cocharacter $\tau \in X_*(G)$ optimal in the sense of the Kempf–Rousseau theory. The adjoint action of the $1$-dimensional torus $\tau (k^\times )$ gives rise to a $\mathbb{Z}$-grading $\mathfrak{g}=\bigoplus _{\,i\in \mathbb{Z}}\,\mathfrak{g}(\tau ,i)$, where the subspace $\mathfrak{g}(\tau ,i)$ consists of all $x\in \mathfrak{g}$ such that $(\operatorname {Ad}\,\tau (\lambda ))(x)=\lambda ^ix$ for all $\lambda \in k^\times$. The optimal parabolic subgroup $P(e)\subset G$ of $e$ is independent of the choice of $\tau$ and $\operatorname {Lie}(P(e))=\bigoplus _{\,i\ge 0}\,\mathfrak{g}(\tau ,i)$. Since the Killing form $\kappa$ of $\mathfrak{g}$ is nondegenerate, we can choose an optimal cocharacter $\tau$ in such a way that $e\in \mathfrak{g}(\tau ,2)$ and $\mathfrak{g}_e\subseteq \operatorname {Lie}(P(e))$; see Pre03, Theorem A. Such optimal cocharacters of $e$ form a single conjugacy class under the adjoint action of the centraliser $G_e\subset P(e)$. Furthermore, it follows from Pre03, Proposition 2.5 that they coincide with the so-called associated cocharacters introduced by Jantzen in Jan04, 5.3. The Lie algebra $\operatorname {Lie}(\tau (k^\times ))$ is a $1$-dimensional torus of $\mathfrak{g}$ spanned by the element $h_\tau :=({\mathrm{d}}_e\tau )(1)$ which has the property that $[h_\tau ,e]=2e$. The centraliser $C_G(\tau )$ of $\tau (k^\times )$ is a Levi subalgebra of $G$ and $\operatorname {Lie}(C_G(\tau ))=\mathfrak{g}(\tau ,0)$.

Put $\mathfrak{g}_e(i)=\mathfrak{g}_e\cap \mathfrak{g}(\tau ,i)$. By Pre03, Theorem A, the group $C_e:=G_e\cap Z_G(\tau )$ is reductive and $\mathfrak{c}_e:=\operatorname {Lie}(C_e)=\mathfrak{g}_e(0)$. Furthermore, $G_e=C_e\cdot R_u(G_e)$. The adjoint $G$-orbit $\mathcal{O}(e)$ of $e$ is uniquely determined by its weighted Dynkin diagram $\Delta =\Delta (e)$ which depicts the weights of $\tau (k^\times )$ on a carefully selected set of simple root vectors of $\mathfrak{g}$. These diagrams are the same as in the characteristic zero case and they can be found in Car93, pp. 401–407 along with the Dynkin labels of the corresponding nilpotent $G$-orbits.

Let $T_e$ be a maximal torus of $C_e$ and let $L=C_G(T_e)$ be a Levi subgroup of $G$. The Lie algebra $\mathfrak{l}'=\operatorname {Lie}(\mathscr{D}L_e)$ is $\tau (k^\times )$-stable and contains $h_\tau$. Moreover, $e$ is distinguished in $\mathfrak{l}'$, that is, $e\in \mathfrak{l}'(\tau ,2)$ and $\dim \mathfrak{l}'(\tau , 0)=\dim \mathfrak{l}'(\tau ,2)$; see Pre03, 2.3–2.7 for detail. Since $\mathfrak{g}_e\subseteq \operatorname {Lie}(P(e))$ and $\dim \mathfrak{l}'(\tau , -2)=\dim \mathfrak{l}'(\tau ,2)$, the map $\operatorname {ad}\,e\colon \, \mathfrak{l}'(\tau ,-2)\to \,\mathfrak{l}'(\tau , 0)$ is bijective. As $h_\tau \in \mathfrak{l}'(\tau ,0)$, there is a unique element $f\in \mathfrak{l}'(\tau ,-2)$ such that $[e,f]=h_\tau$. By construction, $\{e,h_\tau ,f\}$ is an $\mathfrak{sl}_2$-triple in $\mathfrak{g}$. Since $f^{[p]}$ commutes with $e$ for $p>2$, it lies in $\mathfrak{g}_e(\tau ,-2p)$. Since $\tau$ has nonnegative weights on $\mathfrak{g}_e$, this yields $f^{[p]}=0$.

Definition 2.5.

An $\mathfrak{sl}_2$-triple $\{e',h',f'\}$ of $\mathfrak{g}$ is called standard if it is $(\operatorname {Ad}\,G)$-conjugate to one of the $\mathfrak{sl}_2$-triples $\{e,h_\tau , f\}$ described above.

If $\{e,h,f\}$ is a standard $\mathfrak{sl}_2$-triple, then necessarily $e\in \mathcal{N}(\mathfrak{g})$ and $f^{[p]}=0$. However, it may happen in some small characteristics that $e^{[p]}\ne 0$. In particular, if $e^{[p]}\ne 0$, then $e$ and $f$ belong to different nilpotent $G$-orbits. On the other hand, if $e^{[p]}=0$, then there exists a connected subgroup $\mathcal{S}$ of type ${\mathrm{A}}_1$ in $G$ such that $\operatorname {Lie}(\mathcal{S})=ke\oplus kh\oplus kf$; see McN05. In that case $e$ and $f$ are $(\operatorname {Ad}\,G)$-conjugate.

Our earlier remarks in this subsection show that any nilpotent element of $\mathfrak{g}$ can be included into a standard $\mathfrak{sl}_2$-triple. Now suppose $\{e,h,f\}$ is an arbitrary $\mathfrak{sl}_2$-triple in $\mathfrak{g}$ with $e\in \mathcal{N}(\mathfrak{g})$ and $h$ semisimple. Let $\tau$ be an optimal cocharacter for $e$ such that $e\in \mathfrak{g}(\tau ,2)$. All eigenvalues of the toral element $h_\tau$ belong to $\mathbb{F}_p$ and we write $\mathfrak{g}(h_\tau ,\bar{i})$ for the eigenspace of $\operatorname {ad}\,h_\tau$ corresponding to eigenvalue $\bar{i}\in \mathbb{F}_p$. It is straightforward to see that

$$\begin{equation} \mathfrak{g}(h_\tau ,\bar{i})\,=\,\textstyle {\bigoplus }_{\,j\in \mathbb{Z}}\,\mathfrak{g}(\tau ,i+jp). {\tag{2}} \end{equation}$$

Since $e$ and $h-h_\tau$ commute, $h$ is a semisimple element of the restricted Lie algebra $kh_\tau \oplus \mathfrak{g}_e$. Since $\mathfrak{g}_e=\operatorname {Lie}(G_e)$, the latter coincides with the Lie algebra of the normaliser $N_G(k e)=\tau (k^\times )\cdot G_e$. As $\tau (k^\times )\cdot T_e$ is a maximal torus of $N_G(ke)$ contained in $\tau (k^\times )\cdot C_e$, it follows from Bor91, 11.8 that $h$ is conjugate under the adjoint action of $N_G(ke)$ to an element of $kh_\tau \oplus \mathfrak{g}_e(0)$.

So assume from now that $h\in kh_\tau +\mathfrak{g}_e(0)$. Then $h-h_\tau \in \mathfrak{g}_e(0)\cap [e,\mathfrak{g}]$. If $h\ne h_\tau$, then the linear map $(\operatorname {ad}\,h)^2\colon \,\mathfrak{g}(\tau ,-2)\to \,\mathfrak{g}(\tau ,2)$ is not injective. The computations in Pre95 then imply that the orbit $\mathcal{O}(e)$ has Dynkin label ${\mathrm{A}}_{p-1}{\mathrm{A}}_r$ for some $r\ge 0$. In other words, $e$ is a regular nilpotent element of $\operatorname {Lie}(L)$, where $L$ is a Levi subgroup of type ${\mathrm{A}}_{p-1}{\mathrm{A}}_r$ in $G$.

Remark 2.6.

The preceding remark implies that if $\mathfrak{g}$ contains a nonstandard $\mathfrak{sl}_2$-triple $\{e,h,f\}$ with $e\in \mathcal{N}(\mathfrak{g})$ and $h$ semisimple, then $e\in \mathcal{O}({\mathrm{A}}_{p-1}{\mathrm{A}}_r)$ for some $r\ge 0$. As a consequence, $G$ is a group of type ${\mathrm{E}}$ and $p\in \{5,7\}$.

2.5. A remark on exponentiation

Let $K$ be an algebraically closed field of characteristic $0$. In this subsection we assume that $G$ is a simple, simply connected algebraic group over an algebraically closed field $k$ of good characteristic $p>0$ and we write $G_K$ for the simple, simply connected algebraic group over $K$ of the same type as $G$. Both groups are obtained by base change from a Chevalley group scheme $G_\mathbb{Z}$. The Lie algebra $\mathfrak{g}$ of $G$ is obtained by base change from a minimal admissible lattice $\mathfrak{g}_\mathbb{Z}$ in the simple Lie algebra $\mathfrak{g}_K:=\operatorname {Lie}(G_K)$. For any $p$-power $q\in \mathbb{Z}$ the field $k$ contains a unique copy of the finite field $\mathbb{F}_q$ and the finite Lie algebra $\mathfrak{g}_{\mathbb{F}_q}:=\mathfrak{g}_\mathbb{Z}\otimes _\mathbb{Z}\mathbb{F}_q$ is an $\mathbb{F}_q$-form of $\mathfrak{g}$ closed under taking $[p]$th powers in $\mathfrak{g}$. The restricted nullcone $\mathcal{N}_p(\mathfrak{g})$ consists of all $x\in \mathfrak{g}$ with $x^{[p]}=0$. This is a Zariski closed, conical subset of the nilpotent variety $\mathcal{N}(\mathfrak{g})$ and it arises naturally when one studies exponentiation. Indeed, if $U$ is a one-parameter unipotent subgroup of $G$, then $\operatorname {Lie}(U)$ is a $1$-dimensional $[p]$-nilpotent restricted subalgebra of $\mathfrak{g}$ and hence lies in $\mathcal{N}_p(\mathfrak{g})$.

It is well known that each nilpotent $(\operatorname {Ad}\,G)$-orbit $\mathcal{O}$ has a representative $e\in \mathfrak{g}_{\mathbb{F}_p}$ such that $e=\tilde{e}\otimes _\mathbb{Z}1$ for some nilpotent element $\tilde{e}$ of $\mathfrak{g}_K$ contained in $\mathfrak{g}_\mathbb{Z}$. By Pre03, 2.6, one can choose $\tilde{e}$ in such a way that the unstable vectors $e\in \mathfrak{g}$ and $\tilde{e}\in \mathfrak{g}_K$ admit optimal cocharacters obtained by base-changing a cocharacter $\tau \in X_*(G_\mathbb{Z})$. Moreover, $\tau (\mathbb{G}_{m,\mathbb{Z}})$ is contained in a split maximal torus $T_\mathbb{Z}$ of $G_\mathbb{Z}$ and $\dim _k\mathcal{O}=\dim _K\,(\operatorname {Ad}\,G_K)\,\tilde{e}$. Various properties of the cocharacters $\tau$ have already been discussed in §2.4 and we are going to use the notation introduced there for both $e$ and $\tilde{e}$. In particular, we write $\mathfrak{g}_{\tilde{e},K}$ for the centraliser of $\tilde{e}$ in $\mathfrak{g}_K$ and $\mathfrak{g}_{\tilde{e},K}(i)$ for the intersection of $\mathfrak{g}_{\tilde{e},K}$ with the $i$-weight space $\mathfrak{g}_{K}(\tau , i)$ of $\operatorname {Ad}\,\tau (K^\times )$. More generally, given a commutative ring $A$ with $1$, we set $\mathfrak{g}_A:=\mathfrak{g}_\mathbb{Z}\otimes _\mathbb{Z}A$. If $A\subseteq k$, we put $\mathfrak{g}_{e,A}:=\,\mathfrak{g}_{e}\cap \mathfrak{g}_{A}$ and $\mathfrak{g}_{e,A}(i):=\, \mathfrak{g}(\tau ,i)\cap \mathfrak{g}_{e,A}$. If $A\subseteq K$, we define $\mathfrak{g}_{\tilde{e},A}$ and $\mathfrak{g}_{\tilde{e},A}(i)$ in a similar fashion.

Our next result shows that by modifying the exponentiation techniques of Tes95, §1 one can construct one-parameter unipotent subgroups of $G_e$ whose Lie algebras are spanned by prescribed elements of $\mathcal{N}_p(\mathfrak{g})$ contained in the nilradical of $\mathfrak{g}_{e}$. These subgroups respect filtrations associated with optimal cocharacters of $e$.

Proposition 2.7.

Let $e$ be a nonzero nilpotent element of $\mathfrak{g}$ and let $\tau$ be an optimal cocharacter for $e$ such that $e\in \mathfrak{g}(\tau ,2)$. If $d$ is a positive integer, then for any nonzero $x\in \bigoplus _{i\ge d}\,\mathfrak{g}_{e}(i)$ with $x^{[p]}=0$ there exists a collection of endomorphisms $\{X^{(i)}\,|\,\,0\le i\le p^2-1\}$ of $\mathfrak{g}$ and a one-parameter unipotent subgroup $\mathcal{U}_x=\{x(t)\,|\,\,t\in k\}$ of $G_e$ with Lie algebra $kx$ such that $X^{(i)}=\frac{1}{i!}(\operatorname {ad}x)^i$ for $0\le i\le p-1$ and

$$\begin{equation*} \big (\operatorname {Ad}\,x(t)\big )(v)=\sum _{i=0}^{p^2-1}\,t^iX^{(i)}(v)\qquad \quad \big (\forall \,v\in \mathfrak{g}\big ). \end{equation*}$$

Furthermore, each endomorphism $X^{(i)}$ can be expressed as a sum of weight vectors of weight $\ge d i$ with respect to the natural action of the torus $\tau (k^\times )$ on $\mathrm{End}(\mathfrak{g})$.

Proof.

First assume the root system of $G$ is classical. This case is more elementary. The group $G$ admits a rational representation $\rho \colon \,G\to \mathrm{GL}(V)$ defined over $\mathbb{F}_p$ whose kernel is central and whose image is either $\mathrm{SL}(V)$ or the stabiliser in $\mathrm{SL}(V)$ of a nondegenerate bilinear form $\Psi$ on $V$. Let $\bar{X}=({\mathrm{d}}_e\rho )(x)$ and $\bar{X}^{(i)}=\frac{1}{i!}\bar{X}^i$, where $0\le i\le p-1$. Since $x^{[p]}=0$ and ${\mathrm{d}}_e\rho \colon \,\mathfrak{g}\to \mathfrak{gl}(V)$ is a faithful restricted representation of $\mathfrak{g}$, the exponentials $\bar{x}(t):=\exp (t\bar{X})\,=\,\sum _{i=0}^{p-1}\,t^i\bar{X}^i$ with $t\in k$ form a one-parameter unipotent subgroup of $\mathrm{SL}(V)$. Moreover, each $\bar{X}^j$ with $0\le j\le p-1$ is a sum of weight vectors of weight $\ge d\cdot j$ with respect to the conjugation action of $\rho (\tau (k^\times ))$ on $\mathrm{End}(V)$. If $\rho (G)$ fixes $\Psi$, then $\bar{X}$ is skew-adjoint with respect to $\Psi$ and hence $\big \{\bar{x}(t)\,|\,\,t\in k\big \}$ is a one-parameter unipotent subgroup of $\rho (G)$. Also,

$$\begin{align} \big (\operatorname {Ad}\,\bar{x}(t)\big )(Y) &=\exp (t\bar{X})\cdot Y\cdot \exp (-t\bar{X}){\tag{3}}\\ &= \sum _{r=0}^{2p-2}\,t^r\big (\sum _{i=0}^r\,(-1) ^{i}\bar{X}^{(r-i)}\cdot Y\cdot \bar{X}^{(i)}\big ) \end{align}$$

for all $Y\in \mathfrak{sl}(V)$. It should be mentioned here that $2p-2\le p^2-1$ for any prime number $p$.

Restricting $\operatorname {Ad}\,\bar{x}(t)$ to $({\mathrm{d}}_e\rho )(\mathfrak{g})$ and identifying the latter with $\mathfrak{g}$ gives rise to a one-parameter unipotent subgroup of $\operatorname {Aut}(\mathfrak{g})^\circ$ which we call $U_x$. As $x=\bar{X}$ commutes with $e$, it follows from (3) that $U_x$ fixes $e$. Taking the identity component of the inverse image of $U_x$ under a central isogeny $G\twoheadrightarrow \operatorname {Aut}(\mathfrak{g})^\circ$ we then obtain a one-parameter unipotent subgroup of $G_e$ that satisfies all our requirements.

Now suppose $G$ is exceptional. By the remarks immediately preceding the proposition, we may assume that $e=\tilde{e}\otimes _\mathbb{Z}1$ and $\tau \in X_*(G_\mathbb{Z})$. Let $\mathcal{X}$ be the set of all tuples $\big (x,X^{(p)},\ldots , X^{(p^2-1)}\big )$ in $\big (\mathcal{N}_p(\mathfrak{g})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)\big )\times \mathrm{End}(\mathfrak{g})^{p^2-p}$ such that each $X^{(i)}\in \bigoplus _{j\ge d i}\big (\mathrm{End}(\mathfrak{g})\big )(\tau ,j)$ annihilates $e$ and the set

$$\begin{equation*} \Big \{\sum _{i=0}^{p-1}\frac{t^i}{i!}(\operatorname {ad}x)^i+\sum _{i=p}^{p^2-1}t^iX^{(i)}\,|\,\,t\in k\Big \} \end{equation*}$$

forms a one-parameter subgroup of $\operatorname {Aut}(\mathfrak{g})$. If we choose $k$-bases of $\mathfrak{g}$ and $\mathrm{End}(\mathfrak{g})$ contained in $\mathfrak{g}_{\mathbb{F}_p}$ and $\mathrm{End}(\mathfrak{g}_{\mathbb{F}_p})$, respectively, then the above conditions can be rewritten in the form of polynomial equations with coefficients in $\mathbb{F}_p$ on the coordinates of $x$ and the $X^{(i)}$’s. In other words, $\mathcal{X}$ is a Zariski closed subset of $\big (\mathcal{N}_p(\mathfrak{g})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)\big )\times \mathrm{End}(\mathfrak{g})^{p^2-p}$ defined over $\mathbb{F}_p$. The projection $\pi \colon \,\mathcal{X}\longrightarrow \mathcal{N}_p(\mathfrak{g})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)$ sending $\big (x,X^{(p)},\ldots , X^{(p^2-1)}\big )$ to $x$ is a morphism of affine varieties defined over $\mathbb{F}_p$. Evidently, we wish to show this map is surjective.

Let $k_0$ denote the algebraic closure of $\mathbb{F}_p$ in $k$. We claim that

$$\begin{equation} \pi \colon \mathcal{X}(k_0)\longrightarrow \, \mathcal{N}_p(\mathfrak{g}_{k_0})\cap \textstyle {\bigoplus }_{i\ge d}\,\mathfrak{g}_{e,k_0}(i)\ \ \text{ is surjective.}\tag{*} \end{equation}$$

Given the claim, since $k_0$ is an algebraically closed subfield of $k$, it follows from general results of algebraic geometry that the morphism $\pi \colon \mathcal{X}(k)\!\longrightarrow \! \mathcal{N}_p(\mathfrak{g})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)$ is surjective as well; see GW01, Exercise 10.6, for example. This means that a one-parameter unipotent group $U_x$ with the required properties exists for any $x\in \mathcal{N}_p(\mathfrak{g})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)$. Since $G$ is simply connected and $p$ is good for $G$, there exists a central isogeny $\iota \colon \, G\twoheadrightarrow \operatorname {Aut}(\mathfrak{g})^\circ$. So we can take for $\mathcal{U}_x$ the identity component of $\iota ^{-1}(U_x)$.

The claim will follow if we can show that $\pi _{\mathbb{F}_q}:\mathcal{X}(\mathbb{F}_q)\longrightarrow \mathcal{N}_p(\mathfrak{g}_{\mathbb{F}_q})\cap \bigoplus _{i\ge d}\,\mathfrak{g}_{e,\mathbb{F}_q}(i)$ is surjective, since $k_0$ is the union of its finite subfields. Thus we assume that $x\in \mathfrak{g}_{\mathbb{F}_q}$ for some $q=p^n$. As usual, we denote by $\mathbb{Q}_p$ and $\mathbb{Z}_p$ the field of $p$-adic numbers and the ring of $p$-adic integers, respectively. Let $K$ be an algebraic closure of $\mathbb{Q}_p$ and let $L/\mathbb{Q}_p$ be an unramified Galois extension of degree $n$ (we can take for $L$ the field $\mathbb{Q}_p(\zeta _n)$ where $\zeta _n$ is a primitive $(p^n-1)$st root of unity in $K$). Let $A$ be the ring of integers of $L$. Since the extension $L/\mathbb{Q}_p$ is unramified, the field $A/pA$ has degree $n$ over its subfield $\mathbb{Z}_p/p\mathbb{Z}_p\cong \mathbb{F}_p$. Therefore, $A/pA\cong \mathbb{F}_q$.

By construction, $A$ is a local ring with maximal ideal $pA$ and any bad prime for $G$ is invertible in $A$. Since the torus $T_\mathbb{Z}$ is split over $\mathbb{Z}$ and $\tau (\mathbb{G}_m)\subset T_\mathbb{Z}$ we have that $\mathfrak{g}_\mathbb{Z}\,=\,\bigoplus _{i\in \mathbb{Z}}\,\mathfrak{g}_\mathbb{Z}(\tau ,i)$. Since $\tau (K^\times )$ is optimal for $\tilde{e}$, arguing as in Spa84, p. 285 one observes that each $A$-module $\big [\tilde{e},\mathfrak{g}_A(\tau , i)\big ]$ is a direct summand of $\mathfrak{g}_A(\tau , i+2)$ and $[\tilde{e},\mathfrak{g}_A(\tau , i)]\,=\,\mathfrak{g}_A(i+2)$ for all $i\ge 0$. From this it is follows that each $\mathfrak{g}_{\tilde{e},A}(i)$ is a direct summand of the free $A$-module $\mathfrak{g}_A(\tau , i)$. Since $A/pA\cong \mathbb{F}_q$ the natural map $\mathfrak{g}_A\twoheadrightarrow \mathfrak{g}_A\otimes _A\mathbb{F}_q\cong \mathfrak{g}_{\mathbb{F}_q}$ sends $\tilde{e}$ onto $e$ and $\mathfrak{g}_{\tilde{e},A}$ onto $\mathfrak{g}_{e,\mathbb{F}_q}$. It follows that there exists an element $\tilde{x}\in \bigoplus _{i\ge d}\,\mathfrak{g}_{\tilde{e},A}(i)$ such that $x=\tilde{x}\otimes _A 1$.

Recall that $G$ is an exceptional group. Since $p$ is a good prime for $G$, the tables in Ste17 show that $(\operatorname {ad}\tilde{x})^{p^2-1}=0$ unless either $G$ is of type ${\mathrm{E}}_8$, $p=7$, and $e$ is regular or $G$ is of type ${\mathrm{E}}_7$, $p=5$, and $e\in \mathcal{N}(\mathfrak{g})$ is regular or subregular. In view of LT11, pp. 122, 185 this implies that if $(\operatorname {ad}\tilde{x})^{p^2-1}\ne 0$, then $\mathfrak{g}_e(i)=0$ for $i$ odd and $\mathfrak{g}_e(2)=ke$. Since in all these cases $e^{[p]}\ne 0$ and $x\in \bigoplus _{i\ge d}\,\mathfrak{g}_e(i)$ lies in $\mathcal{N}_p(\mathfrak{g})$ by our assumption, we must have $d\ge 4$. From this it is immediate that $(\operatorname {ad}\tilde{x})^{p^2-1}=0$ in all cases.

We now put $\tilde{X}^{(i)}:=\frac{1}{i!}(\operatorname {ad}\tilde{x})^i$ for $0\le i\le p^2-1$. The set $U_{\tilde{x}}$ of all linear operators

$$\begin{equation*} \exp (t\,\operatorname {ad}\tilde{x})\,=\,\sum _{i=1}^{p^2-1} \,t^i\tilde{X}^{(i)} \end{equation*}$$

with $t\in K$ forms a one-parameter unipotent subgroup of $\operatorname {Aut}(\mathfrak{g}_{K})^\circ$. Since $(\operatorname {ad}x)^p=0$ and $\operatorname {ad}x=\operatorname {ad}(\tilde{x}\otimes _A 1)=(\operatorname {ad}\tilde{x})\otimes _A 1$ as endomorphisms of $\mathfrak{g}_{\mathbb{F}_p}$ we have that $(\operatorname {ad}\tilde{x})^p( \mathfrak{g}_{A}) \subseteq p\cdot \mathfrak{g}_A$. But then $\frac{1}{p}(\operatorname {ad}\tilde{x})^p(\mathfrak{g}_A)\subseteq \mathfrak{g}_A$. Since $(p-1)!$ is invertible in $A$, it follows that $\tilde{X}^{(p)}$ preserves $\mathfrak{g}_A$. Any positive integer $n\le p^2-1$ can be uniquely presented as $n=ap+b$ with $0\le a,b\le p-1$. Since $n!\,=\,p^a\cdot r_n$ for some $r_n\in \mathbb{Z}$ coprime to $p$ and $(\operatorname {ad}\tilde{x})^n\,=\,\big ((\operatorname {ad}\tilde{x})^p\big )^a\cdot (\operatorname {ad}\tilde{x})^b$, we have that

$$\begin{equation*} \tilde{X}^{(n)}\,=\,\frac{1}{r_n}\big (\tilde{X}^{(p)}\big )^a\cdot (\tilde{X}^{(1)})^b, \quad \ r_n\in A^\times . \end{equation*}$$

This implies that each endomorphism $\tilde{X}^{(i)}$ preserves $\mathfrak{g}_A$. As a consequence, the closed subgroup $U_{\tilde{x}}$ of $\operatorname {Aut}(\mathfrak{g}_{K})^\circ$ is defined over $A$. We now set $X^{(i)}:=\tilde{X}^{(i)}\otimes _A 1$. In view of our earlier remarks it is straightforward to check that the collection of endomorphisms $\{X^{(i)}\,|\,\,0\le i\le p^2-1\}$ of $\mathfrak{g}$ possesses all required properties. Since the unipotent group $U_{\tilde{x}}$ is defined over $A$, the set $U_x:=\big \{\sum _{i=0}^{p^2-1}\,t^iX^{(i)}\,|\,\,t\in k\big \}$, obtained by base-changing $U_{\tilde{x}}$, forms a one-parameter unipotent subgroup of $\operatorname {Aut}(\mathfrak{g})^\circ$.

The claim is proved.

■

Remark 2.8.

It seems plausible that one can always find a one-parameter subgroup $\mathcal{U}_x$ in $G_e$ satisfying the conditions of Proposition 2.7 and such that $X^{(i)}=0$ for $i>2p-2$. By McN05, for any nonzero $x\in \mathcal{N}_p(\mathfrak{g})$ the optimal parabolic $P(x)$ contains a nice one-parameter subgroup $U_x$ with $\operatorname {Lie}(U_x)=kx$. In fact, $U_x$ lies in a connected subgroup of type ${\mathrm{A}}_1$ whose Lie algebra contains $x$. By Sei00, the number of nonzero $X^{(i)}$’s with $i>0$ associated with $U_x$ is always bounded by $2p-2$. Moreover, it follows from Sob15, Lemma 4.2 and Corollary 4.3(i) that if $[x,e]=0$ for some nonzero $e\in \mathcal{N}(\mathfrak{g})$, then $U_x\subseteq G_e$. However, it is not clear from the constructions in loc. cit. that the endomorphisms $X^{(i)}$ with $i\ge p$ coming from the distribution algebra of $U_x$ have the desired weight properties with respect to an optimal cocharacter for $e$.

3. Lie subalgebras with nonsemisimple socles and exotic semidirect products

3.1. The general setup

In this section we always assume that $G$ is an exceptional algebraic group of rank $\ell$ defined over an algebraically closed field $k$ of characteristic $p>0$. We let $\mathfrak{h}$ be a semisimple restricted Lie subalgebra of $\mathfrak{g}=\operatorname {Lie}(G)$ whose socle is not semisimple. Since $\mathfrak{h}$ is restricted, it follows from Block’s theorem Str04, Corollary 3.3.5 that $\mathfrak{h}$ contains a minimal ideal $I$ such that $I\cong S\otimes \mathcal{O}(m;\underline{1})$ for some simple Lie algebra $S$ and $m\ge 1$, and $\mathfrak{h}/\mathfrak{c}_\mathfrak{h}(I)$ is sandwiched between $S\otimes \mathcal{O}(m;\underline{1})$ and $(\operatorname {Der}(S)\otimes \mathcal{O}(m;\underline{1}))\rtimes (\operatorname {Id}_S\otimes W(m;\underline{1}))$. Moreover, the canonical projection $\pi \colon \,\mathfrak{h}\to W(m;\underline{1})$ maps $\mathfrak{h}$ onto a transitive Lie subalgebra of $W(m;\underline{1})$. Recall the latter means that $\pi (\mathfrak{h})\subseteq W(m;\underline{1})$ does not preserve any nonzero proper ideals of $\mathcal{O}(m;\underline{1})$.

We identify $I$ with $S\otimes \mathcal{O}(m;\underline{1})$ and let $\mathcal{I}$ and $\mathcal{S}$ denote the $p$-envelopes of $I$ and $S\otimes 1$ in $\mathfrak{g}$, respectively. Since $\mathfrak{h}$ is restricted, $\mathcal{I}$ is an ideal of $\mathfrak{h}$. Since $\mathfrak{h}$ is semisimple and $\mathfrak{z}(\mathcal{I})$ is an abelian ideal of $\mathfrak{h}$, it must be that $\mathfrak{z}(\mathcal{I})=0$. Since $\mathfrak{z}(\mathcal{S})\subseteq \mathfrak{z}(\mathcal{I})$ due to the nature of Lie multiplication in $S\otimes \mathcal{O}(m;\underline{1})$, we must have $\mathfrak{z}(\mathcal{S})=0$. Our discussion in Section 2 then shows that $\mathcal{S}\cong S_p$ as restricted Lie algebras.

Since $S\otimes 1$ is a simple Lie subalgebra, it follows from HS15, Theorem 1.3 that either $S\cong \operatorname {Lie}(\mathcal{H})$ for some simple algebraic $k$-group $\mathcal{H}$, or $S\cong \mathfrak{psl}_{rp}$ for $r\ge 1$, or $S\cong W(1;\underline{1})$. In any event, $S$ is a restricted Lie algebra, so that $S=S_p$. Since $\mathfrak{psl}_{rp}=\mathfrak{sl}_{rp}/\mathfrak{z}(\mathfrak{sl}_{rp})$ contains toral subalgebras of dimension $rp-2$ and $\operatorname {rk}(G)\le 8$, the case $S\cong \mathfrak{psl}_{rp}$ may occur only if $r=1$.

We let $\mathfrak{t}$ be a toral subalgebra of maximal dimension in the restricted Lie algebra $S$ and denote by $\mathfrak{t}_{\mathrm{reg}}$ the set of all $t\in \mathfrak{t}$ with the property that $\mathfrak{t}=\operatorname {span}\{t^{[p]^i}\,|\,i\ge 1\}$. As $\mathfrak{t}^{\mathrm{tor}}$ is an $\mathbb{F}_p$-form of $\mathfrak{t}$, it is a routine exercise to check that $\mathfrak{t}_{\mathrm{reg}}$ is nonempty and Zariski open in $\mathfrak{t}$.

The above discussion then shows that $\mathfrak{t}\otimes 1$ is a toral subalgebra of $\mathfrak{g}$. We pick $z\in \mathfrak{t}_{\mathrm{reg}}$. Since $\operatorname {ad}\,(z\otimes 1)$ is semisimple and $[z\otimes 1,\mathfrak{h}]\subseteq I$ we also have that $\mathfrak{h}=\mathfrak{h}_{z\otimes 1}+I$, where $\mathfrak{h}_{z\otimes 1}$ denotes the centraliser $\mathfrak{c}_\mathfrak{h}(z\otimes 1)$ of $z\otimes 1$ in $\mathfrak{h}$. The latter yields

$$\begin{equation} \pi (\mathfrak{h})=\pi (\mathfrak{h}_{z\otimes 1}){\tag{4}} \end{equation}$$

showing that $\pi (\mathfrak{h}_{z\otimes 1})$ is a transitive Lie subalgebra of $W(m;\underline{1})$. Since $\mathfrak{c}_\mathfrak{h}(I)\subseteq \mathfrak{h}_{z\otimes 1}$, the factor algebra $\mathfrak{h}_{z\otimes 1}/\mathfrak{c}_\mathfrak{h}(I)$ identifies with a Lie subalgebra of $(\operatorname {Der}(S)_z\otimes \mathcal{O}(m;\underline{1}))\rtimes (\operatorname {Id}_S\otimes \pi (\mathfrak{h}_{z\otimes 1}))$.

Since $z\otimes 1$ is a semisimple element of $\mathfrak{g}$, Lemma 2.1 shows that $[\mathfrak{g}_{z\otimes 1},\mathfrak{g}_{z\otimes 1}]$ is a restricted Lie subalgebra of $\mathfrak{g}$. On the other hand, our characterisation of $S$ yields that $\mathfrak{t}$ is a self-centralising torus of $S$. Hence $I_{z\otimes 1}=\mathfrak{t}\otimes \mathcal{O}(m;\underline{1})$. Let $\mathcal{O}(m;\underline{1})_{(1)}$ denote the maximal ideal of the local ring $\mathcal{O}(m;\underline{1})$. Since $\mathfrak{t}\otimes \mathcal{O}(m;\underline{1})_{(1)}$ is stable under the action of $\operatorname {Der}(S)_z\otimes \mathcal{O}(m;\underline{1})$ and $\pi (\mathfrak{h}_{z\otimes 1})$ is a transitive Lie subalgebra of $W(m;\underline{1})$, there exist $u\in \mathfrak{h}_{z\otimes 1}$ and $v\in k z\otimes \mathcal{O}(m;\underline{1})_{(1)}$ such that

$$\begin{equation*} [u,v]\,\equiv z\otimes 1 \mod \mathfrak{t}\otimes \mathcal{O}(m;\underline{1})_{(1)}. \end{equation*}$$

Since $\mathfrak{h}\cong \mathfrak{h}_p$ is semisimple, the uniqueness of the restricted Lie algebra structure on $\mathfrak{h}$ gives $\big (S\otimes \mathcal{O}(m;\underline{1})_{(1)}\big )^{[p]}=0$. As $\mathfrak{t}\otimes \mathcal{O}(m;\underline{1})$ is abelian and $[\mathfrak{g}_{z\otimes 1},\mathfrak{g}_{z\otimes 1}]$ is restricted, we deduce that $[u,v]^{[p]}=z^{[p]}\otimes 1$ lies in $[\mathfrak{g}_{z\otimes 1},\mathfrak{g}_{z\otimes 1}]$. Since $z\in \mathfrak{t}_{\mathrm{reg}}$ and $\mathfrak{z}(\mathfrak{g}_{z\otimes 1})$ is restricted, we obtain that

$$\begin{equation} \mathfrak{t}\otimes 1\subseteq \mathfrak{z}(\mathfrak{g}_{z\otimes 1})\cap [\mathfrak{g}_{z\otimes 1},\mathfrak{g}_{z\otimes 1}].{\tag{5}} \end{equation}$$

3.2. Describing the socle of ${\mathfrak{h}}$

In this subsection we are going to give a more precise description of the socle of $\mathfrak{h}$. A Lie algebra $L$ is said to be decomposable if it can be presented as a direct sum of two commuting nonzero ideals of $L$.

Lemma 3.1.

Let $\mathfrak{h}$, $I$, $z$, and $m$ be as in §3.1. Then $G$ is of type ${\mathrm{E}}$ and the following hold:

(i)

$\mathfrak{g}_{z\otimes 1}$ is a Levi subalgebra of type ${\mathrm{A}}_{p-1}{\mathrm{A}}_{\ell -p}$ in $\mathfrak{g}$ and $\mathfrak{z}(\mathfrak{g}_{z\otimes 1})=k(z\otimes 1)$.

(ii)

$S$ is either $\mathfrak{sl}_2$ or $W(1;\underline{1})$ and $m=1$.

(iii)

$\mathfrak{h}$ has no minimal ideals isomorphic to $\mathfrak{psl}_{rp}$.

(iv)

$I$ is the unique nonsimple minimal ideal of $\mathfrak{h}$ and $\operatorname {Soc}(\mathfrak{h})=I\oplus \mathfrak{c}_\mathfrak{h}(I)$. Any minimal ideal of $\mathfrak{c}_\mathfrak{h}(I)=\operatorname {Soc}(\mathfrak{c}_\mathfrak{h}(I))$ is isomorphic to the Lie algebra of a simple algebraic $k$-group.

(v)

If $\mathfrak{c}_\mathfrak{h}(I)\ne 0$, then $\mathfrak{h}$ is decomposable. More precisely, $\mathfrak{h}\cong (\mathfrak{h}/\mathfrak{c}_\mathfrak{h}(I))\oplus \mathfrak{c}_\mathfrak{h}(I)$ as Lie algebras and $\mathfrak{h}/\mathfrak{c}_\mathfrak{h}(I)\cong (S\otimes \mathcal{O}(1;\underline{1}))\rtimes (\operatorname {Id}_S\otimes \mathcal{D})$ for some transitive Lie subalgebra $\mathcal{D}$ of $W(1;\underline{1})$.

Proof.

Replacing $z\otimes 1$ by an $(\operatorname {Ad}G)$-conjugate we may assume that $\mathfrak{g}_{z\otimes 1}=\operatorname {Lie}(G_{z\otimes 1})$ is a standard Levi subalgebra of $\mathfrak{g}$. Let $a=\dim \big ([\mathfrak{g}_{z\otimes 1},\mathfrak{g}_{z\otimes 1}]\cap \mathfrak{z}(\mathfrak{g}_{z\otimes 1})\big )$. By PSt16, 2.1, the restricted Lie algebra $\mathfrak{g}_{z\otimes 1}$ decomposes into a direct sum of its ideals each of which either has form $\mathfrak{gl}_{rp}$ for some $r\ge 0$ or is isomorphic to $\operatorname {Lie}(\mathcal{H})$ for some simple algebraic $k$-group $\mathcal{H}$. Since $\dim \mathfrak{z}(\mathfrak{gl}_{rp})=\dim \mathfrak{z}(\mathfrak{sl}_{rp})=1$ for all $r\ge 1$ and each $\operatorname {Lie}(\mathcal{H})$ is simple, $a$ coincides with the number of irreducible components of type ${\mathrm{A}}_{rp-1}$ with $r\ge 1$ of the standard Levi subgroup $G_{z\otimes 1}$.

In view of 5 we have that $a\ge \dim \mathfrak{t}$. In particular, $a\ge 1$. Since $p$ is a good prime for $G$, examining the subdiagrams of the Dynkin diagram of $G$ yields that $G$ is of type ${\mathrm{E}}$ and $r=1$. Since $\mathfrak{t}$ is a torus of maximal dimension in $S$ and $1=d\ge \dim \mathfrak{t}$, we now deduce that $S$ is either $\mathfrak{sl}_2$ or $W(1;\underline{1})$. Since $z\otimes 1$ lies in the Lie algebra of the ${\mathrm{A}}_{p-1}$-component of $G_{z\otimes 1}$, it also follows that $G_{z\otimes 1}$ has type ${\mathrm{A}}_{p-1}A_{\ell -p}$ proving (i).

Suppose $I'$ is a simple ideal of $\mathfrak{h}$ isomorphic to $\mathfrak{psl}_{rp}$ for some $r\ge 1$. Then $[I,I']\subseteq I\cap I'=0$ showing that $I'\subseteq \mathfrak{g}_{z\otimes 1}$. The restriction of the Killing form $\kappa$ of $\mathfrak{g}$ to the Levi subalgebra $\mathfrak{g}_{z\otimes 1}$ is nondegenerate. Since $\dim I'\ge p^2-2>(\dim \mathfrak{g}_{z\otimes 1})/2$ by part (i), it follows that the Lie algebra $\mathfrak{psl}_{rp}$ admits a faithful representation with a nonzero trace form. This, however, contradicts Bl62, proving (iii). In view of BGP09, Lemma 2.7 and our earlier remarks this entails that any minimal ideal $I'=S'\otimes \mathcal{O}(m';\underline{1})$ of $\mathfrak{h}$ has the property that $\operatorname {Der}(S')=\operatorname {ad}S'$. Applying Str04, Corollary 3.6.6 then yields that $\mathfrak{h}$ is sandwiched between $\operatorname {Soc}(\mathfrak{h})=\bigoplus _{i=1}^d(S_i\otimes \mathcal{O}(m_i;\underline{1}))$ and $\bigoplus _{i=1}^d\,(S_i\otimes \mathcal{O}(m_i;\underline{1}))\rtimes (\operatorname {Id}_{S_i}\otimes W(m_i;\underline{1}))$. For $1\le i\le d$ we denote by $\pi _i$ the projection from $\mathfrak{h}$ to $W(m_i;\underline{1})$. Each $\pi _i(\mathfrak{h})$ acts transitively on $\mathcal{O}(m_i;\underline{1})$.

To keep the notation introduced earlier we assume that $I_1=I$ so that $m_1=m$, $S_1=S$, and $\pi _1=\pi$. For $2\le i\le d$ let $I_i$ be the minimal ideal $S_i\otimes \mathcal{O}(m_i;\underline{1})$. Since $[I,I_i]=0$ and $\pi _i(\mathfrak{h})=\pi _i(\mathfrak{h}_{z\otimes 1})$ by (4), each $I_i$ is a minimal ideal of $\mathfrak{h}_{z\otimes 1}$. Since $\mathfrak{g}_{z\otimes 1}\cong \mathfrak{gl}_p\times \mathfrak{sl}_{l-p}$ acts faithfully on a vector space of dimension $\ell +1=p+(\ell -p+1)$ and $I_i=[I_i,I_i]$, the Lie algebra $\mathfrak{h}_{z\otimes 1}$ affords an irreducible restricted representation $\rho _i$ of dimension $\le \ell +1$ such that $\rho (I_i)\ne 0$. We let $V_i$ be the corresponding $\mathfrak{h}_{z\otimes 1}$-module.

Suppose $m_i\ge 1$ for some $i\in \{2,\ldots ,m\}$. Then it follows from Block’s theorem on differentiably simple modules that there is a faithful $S_i$-module $V_i'$ such that $V_i\cong V_{i}'\otimes \mathcal{O}(m_i;\underline{1})$ as vector spaces and $\rho _i(\mathfrak{h}_{z\otimes 1})$ embeds into the Lie subalgebra

$$\begin{equation*} (\mathfrak{gl}(V_{i}')\otimes \mathcal{O}(m_i;\underline{1})) \rtimes (\operatorname {Id}_{\mathfrak{gl}(V_{i}')}\otimes W(m_i;\underline{1})) \end{equation*}$$

of $\mathfrak{gl}(V_i)$; see Str04, Theorem 3.3.10 and Corollary 3.3.11. Since the $S_i$-module $V_{i}'$ is faithful, it is of dimension at least $2$ and so it must be that $\dim V_i= p^{m_i}\dim V_{i}'\ge 2p^{m_i}$. But then $\ell +1\ge \dim V\ge 2p$ which is impossible as $p$ is a good prime for $G$. We thus deduce that $m_i=0$ for all $i\ge 2$. It follows that $\mathfrak{h}$ contains a restricted ideal $\mathfrak{h}'$ isomorphic to $(S\otimes W(m;\underline{1}))\rtimes (\operatorname {Id}_S\otimes \mathcal{D})$ for some transitive Lie subalgebra $\mathcal{D}$ of $W(m;\underline{1})$ and such that $\mathfrak{h}=\mathfrak{h}'\oplus \mathfrak{c}_\mathfrak{h}(\mathfrak{h}')$. Moreover, any minimal ideal of $\mathfrak{c}_\mathfrak{h}(\mathfrak{h}')=\mathfrak{c}_\mathfrak{h}(I)$ (if any) is isomorphic to the Lie algebra of a simple algebraic $k$-group. This proves (iv).

In order to finish the proof of the lemma it remains to show that $m=1$. Suppose $m>1$. If $m\ge 3$, then $\dim I\ge 3p^3>\dim \mathfrak{g}$ which is false. Hence $m=2$ and $\dim I\ge 3p^2>(\dim \mathfrak{g})/2$. It follows that the restriction of the Killing form $\kappa$ to $I$ is nonzero. Write $\bar{\kappa }$ for the restriction of $\kappa$ to $\mathfrak{h}'$. Since $I$ is a minimal ideal of $\mathfrak{h}'$ and $\bar{\kappa }(I,I)\ne 0$ it must be that $I\cap \operatorname {Rad}\,\bar{\kappa }=0$.

Let $x_1$, $x_2$ be generators of the local ring $\mathcal{O}(2;\underline{1})$ contained in $\mathcal{O}(2;\underline{1})_{(1)}$ and put $x^\mathbf{p-1}:=x_1^{p-1}x_2^{p-1}$. Pick any nonzero element $e\in S$. Then $e\otimes x^\mathbf{p-1}\in I$ and the above yields that $\kappa (e\otimes x^\mathbf{p-1},y)\ne 0$ for some $y\in \mathfrak{h}'$. Let $V$ be any composition factor of the $(\operatorname {ad}\mathfrak{h}')$-module $\mathfrak{g}$ and write $\psi$ for the corresponding (restricted) representation of $\mathfrak{h}'$. If $\psi (I)=0$, then, of course, $\psi (e\otimes x^\mathbf{p-1})\circ \psi (y)=0$. If $\psi (I)\ne 0$, then it follows from Block’s theorem on differentiably simple modules that there is a $k$-vector space $V_0$ and a linear isomorphism $V\cong V_0\otimes \mathcal{O}(2;\underline{1})$ such that $\psi (\mathfrak{h}')$ embeds into $(\mathfrak{gl}(V_0)\otimes \mathcal{O}(2;\underline{1})) \rtimes (\operatorname {Id}_{\mathfrak{gl}(V_0)}\otimes W(2;\underline{1}))$ and $\psi \big (S\otimes \mathcal{O}(2;\underline{1})_{(1)}\big )$ embeds into $\mathfrak{gl}(V_0)\otimes \mathcal{O}(2;\underline{1})_{(1)}$; see Str04, 3.3.

Given a Lie algebra $L$ we write $L^d$ for the $d$th member of the lower central series of $L$. As $S$ is simple, $e\in S^{\,2p-2}$. From this it is immediate that $e\otimes x^\mathbf{p-1}\in S\otimes \mathcal{O}(2;\underline{1}\big )_{(2p-2)}$. But then $\psi (e\otimes x^\mathbf{p-1})=E\otimes x^\mathbf{p-1}$ for some $E\in \mathfrak{gl}(V_0)$. As $\psi (y)\in (\mathfrak{gl}(V_0)\otimes \mathcal{O}(2;\underline{1})) \rtimes (\operatorname {Id}_{\mathfrak{gl}(V_0)}\otimes W(2;\underline{1}))$, this implies that $\psi (e\otimes x^\mathbf{p-1})\circ \psi (y)$ is a square-zero endomorphism. It follows that $\operatorname {ad}(e\otimes x^\mathbf{p-1})\circ \operatorname {ad}y$ acts nilpotently on any composition factor of the $(\operatorname {ad}\mathfrak{h}')$-module $\mathfrak{g}$. But then $\kappa (e\otimes x^\mathbf{p-1},y)=\mathrm{tr}(\operatorname {ad}(e\otimes x^\mathbf{p-1})\circ \operatorname {ad}y)=0$ contrary to our choice of $y$. Therefore, $m=1$ and our proof is complete.

■

Lemma 3.1(i) shows that $z$ is a scalar multiple of a nonzero toral element of $S$. Since $S$ is either $\mathfrak{sl}_2$ or $W(1;\underline{1})$ by Lemma 3.1(ii) we may assume without loss that there is an $\mathfrak{sl}_2$-triple $\{e,h,f\}\subset S$ such that $h=z$. Indeed, this is clear when $S=\mathfrak{sl}_2$ and when $S=W(1;\underline{1})$ it follows from the fact that any toral element of $S$ is conjugate under the action of $\operatorname {Aut}(S)$ to either $(1+x)\partial$ or to a multiple of $x\partial$; see Pre92, for example.

3.3. Determining the conjugacy class of $S\otimes 1$

Recall from §3.2 that $S$ is either $\mathfrak{sl}_2$ or $W(1;\underline{1})$ and $S\otimes 1$ is a restricted subalgebra of $\mathfrak{g}$. It follows that $(e\otimes 1)^{[p]}=(f\otimes 1)^{[p]}=0$ and $(h\otimes 1)^{[p]}=h\otimes 1$.

Proposition 3.2.

Let $\mathfrak{h}$ and $S$ be as in §3.2 and suppose further that $\mathfrak{h}$ is indecomposable. Then the following hold:

(i)

$G$ is of type ${\mathrm{E}}_7$ or ${\mathrm{E}}_8$ and $p\in \{5,7\}$.

(ii)

If $p=5$, then $G$ is of type ${\mathrm{E}}_7$ and $e\otimes 1\in \mathcal{O}({\mathrm{A}}_3{\mathrm{A}}_2{\mathrm{A}}_1)$.

(iii)

If $p=7$, then $G$ is of type ${\mathrm{E}}_7$ or ${\mathrm{E}}_8$ and $e\otimes 1\in \mathcal{O}({\mathrm{A}}_2{{\mathrm{A}}_1}^3)$.

(iv)

The $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is standard and $S \cong \mathfrak{sl}_2$.

Proof.

Since $p$ is a good prime for $G$, it follows from Lemma 3.1(i) and our choice of $h\in S$ that $\dim \mathfrak{g}_{h\otimes 1}=28$ if $G$ is of type ${\mathrm{E}}_6$, $\dim \mathfrak{g}_{h\otimes 1}=33$ if $G$ is of type ${\mathrm{E}}_7$ and $p=5$, $\dim \mathfrak{g}_{h\otimes 1}=49$ if $G$ is of type ${\mathrm{E}}_7$ and $p=7$, and $\dim \mathfrak{g}_{h\otimes 1}=52$ if $G$ is of type ${\mathrm{E}}_8$. On the other hand, $\dim G_{e\otimes 1}=\dim \mathfrak{g}_{e\otimes 1}\ge \dim \mathfrak{g}_{h\otimes 1}$ by Lemma 2.3. Looking through the tables in Car93, pp. 402–407 it is now straightforward to see that if $e\otimes 1\in \mathcal{O}({\mathrm{A}}_{p-1}{\mathrm{A}}_r)$ for some $r\ge 0$, then $G$ is of type ${\mathrm{E}}_7$, $p=5$, and $r=0$. If $e\otimes 1$ is not of that type, then the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ must be standard by Remark 2.6.

Let $\Pi =\{\alpha _1,\ldots ,\alpha _\ell \}$ be a basis of simple roots in the root system $\Phi =\Phi (G,T)$ and let $\tilde{\alpha }= \sum _{i=1}^\ell n_i\alpha _i$ be the highest root of the positive system $\Phi ^+(\Pi )$. In what follows we are going to use Bourbaki’s numbering of simple roots in $\Pi$; see Bou68, Planches I–IX. Given a nilpotent $G$-orbit $\mathcal{O}\subset \mathfrak{g}$ we write $\Delta =(a_1,\ldots , a_\ell )$ for the weighted Dynkin diagram of $\mathcal{O}$. We know that $a_i\in \{0,1,2\}$ and there is a nice representative $e'\in \mathcal{O}$ which admits an optimal cocharacter $\tau \in X_*(T)$ such that $e'\in \mathfrak{g}(\tau ,2)$ and $(\alpha _i\circ \tau )(\lambda )=\lambda ^{a_i}$ for all $\lambda \in k^\times$ and $1\le i\le \ell$; see Pre03, 2.6. We put $\Delta '=(a_1',\ldots ,a_\ell '):=\frac{1}{2}\Delta$ if all $a_i$ are even, and $\Delta '=\Delta$ if $a_j$ is odd for some $j\le \ell$.

We first suppose that the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is standard. If $e\otimes 1\in \mathcal{O}(\Delta )$ and $\Delta '=(a'_1,\ldots , a'_\ell )$, then looking through the tables inCar93, pp. 402–407 once again one observes that in the majority of cases specified at the beginning of the proof the inequality

$$\begin{equation} \sum _{i=1}^\ell n_ia_i'<p{\tag{6}} \end{equation}$$

holds. If this happens, then $\sum _{i=1}^\ell m_ia_i'<p$ for all positive roots $\beta =\sum _{i=1}^\ell m_i\alpha _i\in \Phi ^+(\Pi )$ and (2) yields that $\mathfrak{g}(h_\tau ,\bar{0})\cong \mathfrak{g}(\tau ,0)$ as Lie algebras. In type ${\mathrm{E}}_6$ the inequality (6) holds in all cases of interest and there is no $\tau$ as above for which $\mathfrak{g}(\tau ,0)$ has type ${\mathrm{A}}_{p-1}{\mathrm{A}}_{\ell -p}$. In view of Remark 2.6 this rules out the case where $G$ is of type ${\mathrm{E}}_6$ thereby proving (i).

Suppose $G$ is of type ${\mathrm{E}}_7$ and $p=5$. We first consider the case where $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is a standard $\mathfrak{sl}_2$-triple. If (6) holds for $\Delta '$, then $\mathfrak{g}_{h\otimes 1}\cong \mathfrak{g}(\tau ,0)$ must have type ${\mathrm{A}}_4{\mathrm{A}}_2$. Since $\dim G_{e\otimes 1}\ge 33$, examining the Dynkin diagrams in Car93, p. 403 one observes that $e\otimes 1\in \mathcal{O}({\mathrm{A}}_3{\mathrm{A}}_2{\mathrm{A}}_1)$. If (6) does not hold for $\Delta '$, then Car93, p. 403 reveals that $\mathcal{O}$ must have one of the following labels:

$$\begin{equation*} {\mathrm{A}}_3,\, {{\mathrm{A}}_2}^2{\mathrm{A}}_1,\, ({\mathrm{A}}_3{\mathrm{A}}_1)',\, {\mathrm{A}}_3{{\mathrm{A}}_1}^2,\,{\mathrm{D}}_4,\,{\mathrm{D}}_4({\mathrm{a}}_1){\mathrm{A}}_1,\,{\mathrm{A}}_3{\mathrm{A}}_2. \end{equation*}$$

Since a root system of type ${\mathrm{A}}_4{\mathrm{A}}_2$ does not contain subsystems of type ${\mathrm{D}}_4$, ${\mathrm{A}}_3{{\mathrm{A}}_1}^2$, ${\mathrm{A}}_2{{\mathrm{A}}_1}^3$, and ${\mathrm{A}}_5$, looking at the simple roots in $\Pi$ corresponding to those $i$ for which $a_i=0$ one can see that only the case where $e\otimes 1\in \mathcal{O}\big ({\mathrm{D}}_4({\mathrm{a}}_1){\mathrm{A}}_1\big )$ is possible. Since $p=5$, one can check directly that in this case the root system of $\mathfrak{g}(h_\tau ,\bar{0})$ with respect to $T$ has basis consisting of $\alpha _1$, $\alpha _4$, $\alpha _5$, $\alpha _6$, and $\beta ={\scriptstyle {1\atop }\!{2\atop }\!{3\atop 2}\!{2\atop }\! {1\atop }\!{1\atop }}$. As a consequence, it has type ${\mathrm{A}}_4{\mathrm{A}}_1$, a contradiction.

Now suppose that $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is a nonstandard $\mathfrak{sl}_2$-triple. Then $e\otimes 1\in \mathcal{O}({\mathrm{A}}_4)$ by our remarks at the beginning of the proof. By LT11, p. 104, the reductive subgroup $C_{e\otimes 1}$ of $G_e$ has form $T_1\cdot \mathscr{D}C_{e\otimes 1}$, where $\mathscr{D}C_{e\otimes 1}$ is of type ${\mathrm{A}}_2$ and $T_1$ is a $1$-dimensional central torus in $C_{e\otimes 1}$. It is straightforward to see that $\mathfrak{c}':=\operatorname {Lie}(\mathscr{D}C_{e\otimes 1})$ is generated by the simple root vectors $e_{\pm \alpha _6}$ and $e_{\pm \alpha _7}$. Also, $T_0=\varpi _5^\vee (k^\times )$ where, as usual, $\varpi _i^\vee \in X_*(T)$ stands for the fundamental coweight of $\alpha _i\in \Pi$. It follows that the restriction of the Killing form of $\mathfrak{g}$ to $\mathfrak{c}'$ is nondegenerate. Since $[e\otimes 1,\mathfrak{g}(\tau ,-2)]$ is orthogonal to $\mathfrak{g}_e(0)=\operatorname {Lie}(C_{e\otimes 1})$ with respect to $\kappa$ and $(h\otimes 1)-h_\tau \in \mathfrak{g}_{e\otimes 1}(0)\cap [e\otimes 1,\mathfrak{g}(\tau ,-2)]$ by our discussion in §2.4, it must be that $(h\otimes 1)-h_\tau \in \operatorname {Lie}(\varpi _5^\vee (k^\times ))$.

For $1\le i\le \ell$ set $t_i:=({\mathrm{d}}_e\varpi _i^\vee )(1)$ and write $s_i$ for the simple reflection in the Weyl group $W=N_G(T)/T$ corresponding to the simple root $\alpha _i\in \Pi$. The toral elements $t_1,t_2,\ldots , t_7$ span $\operatorname {Lie}(T)$. Since $h\otimes 1$ is a toral element of $\operatorname {Lie}(T)$ and $\operatorname {Lie}(\varpi _5^\vee (k^\times ))=k t_5$ we have that $\frac{1}{2}(h\otimes 1)= \frac{1}{2}h_\tau +r t_5$ for some $r\in \mathbb{F}_5$. It is immediate from LT11, p. 104 that $\frac{1}{2}h_\tau =t_1+t_2+t_3+t_4-3t_5$. Since $p=5$, direct computation shows that $s_2s_4s_3s_1(\textstyle {\frac{1}{2}}h_\tau )\,=\, \textstyle {\frac{1}{2}}h_\tau +3t_5.$ As $s_2s_4s_3s_1$ fixes $t_5$, this yields that all elements in the set $\frac{1}{2}h_\tau +\mathbb{F}_5\cdot t_5$ are conjugate under the action of $W$. But then $h\otimes 1$ and $h_\tau$ have isomorphic centralisers. Since it is immediate from Car93, p. 403 that the centraliser of $h_\tau$ has type ${\mathrm{D}}_4{\mathrm{A}}_1$, this contradicts Lemma 3.1(i). We thus conclude that when $p=5$ the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is standard and statement (ii) holds.

Next suppose that $G$ is of type ${\mathrm{E}}_7$ and $p=7$. Since in this case $\dim \mathfrak{g}_{e\otimes 1} \ge 49$ by Lemma 2.3, analysing the weighted Dynkin diagrams in Car93, p. 403 shows that $e\otimes 1\not \in \mathcal{O}({\mathrm{A}}_6)$ and the inequality (6) holds for $\Delta '(e\otimes 1)$. Moreover, there is a unique weighted Dynkin diagram $\Delta '$ for which the centraliser of $h_\tau$ has type ${\mathrm{A}}_6$ and the corresponding nilpotent orbit has Dynkin label ${\mathrm{A}}_2{{\mathrm{A}}_1}^3$. This proves (iii) and shows that the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is standard in type ${\mathrm{E}}_7$.

Now suppose $G$ is of type ${\mathrm{E}}_8$. In this case, $p=7$ and $\mathfrak{g}_{h\otimes 1}$ has type ${\mathrm{A}}_6{\mathrm{A}}_1$. Since $\dim \mathfrak{g}_{e\otimes 1}\ge 52$ by Lemma 2.3, it follows from Car93, p. 406 that the orbit $\mathcal{O}(e\otimes 1)$ is not of type ${\mathrm{A}}_6{\mathrm{A}}_r$ for $r\ge 0$. So the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ must be standard. Analysing the weighted Dynkin diagrams in Car93, pp. 405, 406 shows that either (6) holds for $\Delta '(e\otimes 1)$ or the orbit $\mathcal{O}(e\otimes 1)$ has one of the following labels:

$$\begin{equation*} {\mathrm{A}}_4{\mathrm{A}}_1,\, {\mathrm{D}}_5({\mathrm{a}}_1),\, {\mathrm{A}}_5,\,{{\mathrm{A}}_3}^2,\, {\mathrm{D}}_5({\mathrm{a}}_1){\mathrm{A}}_1,\, {\mathrm{A}}_4{\mathrm{A}}_2{\mathrm{A}}_1,\,{\mathrm{D}}_4{\mathrm{A}}_1,\, {\mathrm{A}}_4{{\mathrm{A}}_1}^2. \end{equation*}$$

The first three labels cannot occur since in each of them the root system of $\mathfrak{g}(\tau ,0)$ contains a subsystem of type ${\mathrm{D}}_4$. The fourth label cannot occur either since a root system of type ${\mathrm{A}}_6{\mathrm{A}}_1$ does not contain subsystems of type ${{\mathrm{A}}_3}^2$. The label ${\mathrm{D}}_5({\mathrm{a}}_1){\mathrm{A}}_1$ cannot occur because $e_{\scriptstyle {1\atop }\!{3\atop }\!{5\atop 2}\!{4\atop }\!{3\atop }\!{2\atop }\!{1\atop }}$ commutes with $h_\tau$ and hence the root system of $\mathfrak{g}(h_\tau ,\bar{0})$ contains a subsystem of type ${\mathrm{A}}_4{\mathrm{A}}_3$, forcing $\mathfrak{g}(h_\tau ,\bar{0})\not \cong \mathfrak{g}_{h\otimes 1}$. The label ${\mathrm{A}}_4{\mathrm{A}}_2{\mathrm{A}}_1$ cannot occur since $e_{\scriptstyle {2\atop }\!{4\atop }\!{5\atop 2}\!{4\atop }\!{3\atop }\!{2\atop }\!{1\atop }}$ commutes with $h_\tau$. This implies that the root system of $\mathfrak{g}(h_\tau ,\bar{0})$ contains a subsystem of type ${\mathrm{A}}_4{\mathrm{A}}_2{\mathrm{A}}_1$ which is not the case for the root system of $\mathfrak{g}_{h\otimes 1}$. The label ${\mathrm{A}}_4{{\mathrm{A}}_1}^2$ cannot occur for the same reason: $e_{\scriptstyle {2\atop }\!{4\atop }\!{6\atop 3}\!{4\atop }\!{3\atop }\!{2\atop }\!{1\atop }}$ commutes with $h_\tau$ and hence the root system of $\mathfrak{g}(h_\tau ,\bar{0})$ contains a subsystem of type ${\mathrm{A}}_4{\mathrm{A}}_2{\mathrm{A}}_1$.

Finally, suppose $e\otimes 1\in \mathcal{O}({\mathrm{D}}_4{\mathrm{A}}_1)$. By Car93, p. 405, $h_\tau$ is conjugate to $h'_\tau :=t_2+t_7+2t_8$ under the action of the Weyl group $W=N_G(T)/T$. Using Bou68, Planche VII (and the fact that $p=7$) it is straightforward to see that the roots $\alpha _1,\alpha _3,\alpha _4,\alpha _5,\alpha _6$, and ${\scriptstyle {1\atop }\!{3\atop }\!{5\atop 3}\!{4\atop }\!{3\atop }\!{2\atop }\!{1\atop }}$ form a basis of simple roots of the root system of $\mathfrak{g}(h'_\tau ,\bar{0})$ with respect to $T$. So the present case cannot occur and hence (6) holds for $\Delta '(e\otimes 1)$. Looking once again at the weighted Dynkin diagrams in Car93, p. 405 for which (6) holds one finds out that $\mathcal{O}({\mathrm{A}}_2{{\mathrm{A}}_1}^3)$ is the only nilpotent orbit in $\mathfrak{g}$ for which $\mathfrak{g}(\tau ,0)=\mathfrak{g}(h_\tau ,\bar{0})$ has type ${\mathrm{A}}_6{\mathrm{A}}_1$. This proves (iii).

It remains to show that $S\cong \mathfrak{sl}_2$. So suppose the contrary. Then $S$ may be identified with $\operatorname {Der}(\mathcal{O}(1;\underline{1}))$ by Lemma 3.1(ii). The $k$-algebra $\mathcal{O}(1;\underline{1})$ is generated by an element $x$ with $x^p=0$ and $S$ is spanned by the derivations $x^i\partial$ with $0\le i\le p-1$. We may choose $\{e,h,f\}\subset S$ so that $f=\partial$, $h=2x\partial$, and $e=x^2\partial$. Let $u=x^{p-1}\partial$. Then $[e,u]=0$ and $[h,u]=2(p-2)u=-\bar{4}u$. Since we have already proved that the $\mathfrak{sl}_2$-triple $\{e\otimes 1,\mathfrak{h}\otimes 1,f\otimes 1\}\subset \mathfrak{g}$ is standard we may assume further that $h\otimes 1=h_\tau$. Since $e\otimes 1\in \mathcal{O}({\mathrm{A}}_3{\mathrm{A}}_2{\mathrm{A}}_1)$ if $p=5$ and $e\otimes 1\in \mathcal{O}({\mathrm{A}}_2{{\mathrm{A}}_1}^3)$ if $p=7$, it follows from LT11, pp. 97, 105 that

$$\begin{equation*} \mathfrak{g}_{e\otimes 1}\cap \mathfrak{g}(h_\tau , -\bar{4})\,=\,\mathfrak{g}_{e\otimes 1}(p-4) \,=\,\{0\} \end{equation*}$$

if $G$ is of type ${\mathrm{E}}_7$. Since $u\otimes 1\in \mathfrak{g}_{e\otimes 1}(-\bar{4})$, we conclude that $G$ is a group of type ${\mathrm{E}}_8$. In this case we may assume that $e\otimes 1=\,\sum _{\,i\ne 4,6,8}\,e_{\alpha _i}$. By LT11, p. 127, $u\otimes 1\in \mathfrak{g}_{e\otimes 1}(3)$ and the $C_{e\otimes 1}$-module $\mathfrak{g}_{e\otimes 1}(3)$ is generated by $e_{\scriptstyle {2\atop }\!{4\atop }\!{5\atop 3}\!{4\atop }\!{3\atop }\!{2\atop }\!{1\atop }}$ as in the characteristic zero case. From this it follows that $(\operatorname {ad}\,(f\otimes 1))^4(x)=0$ for all $x\in \mathfrak{g}_{e\otimes 1}(3)$. Since this contradicts the fact that $(\operatorname {ad}\,\partial )^4(x^6\partial )\ne 0$, we now deduce that the case where $S\cong W(1;\underline{1})$ is impossible. Then $S\cong \mathfrak{sl}_2$ by Lemma 3.1(ii) and our proof is complete.

■

Corollary 3.3.

If $G$ is of type ${\mathrm{E}}_8$ and $\mathfrak{h}$ is as in Proposition 3.2, then there exists an involution $\sigma \in G$ such that $G^\sigma$ is of type ${\mathrm{E}}_7{\mathrm{A}}_1$ and $\mathfrak{h}\subset \mathfrak{g}^\sigma$. In particular, $\mathfrak{h}$ is not maximal in $\mathfrak{g}$.

Proof.

Since $G$ is of type ${\mathrm{E}}_8$, we have $p=7$. By Proposition 3.2, the $\mathfrak{sl}_2$-triple $\{e\otimes 1,h\otimes 1,f\otimes 1\}$ is standard and $e\otimes 1\in \mathcal{O}({\mathrm{A}}_1{{\mathrm{A}}_1}^3)$. So we may assume that $e\otimes 1=\,\sum _{\,i\ne 4,6,8}\,e_{\alpha _i}$ and $h\otimes 1=h_\tau$, where the optimal cocharacter $\tau \in X_*(T)$ is as in LT11, p. 127. Let $\sigma =\tau (-1)$. Using loc. cit. it is straightforward to see that a root element $e_\beta \in \mathfrak{g}^\sigma$ if and only if $\beta =\sum _{\,i=1}^8\,m_i\alpha _i$ and $m_8$ is even. From this it is immediate that $G^\sigma$ is a group of type ${\mathrm{E}}_7{\mathrm{A}}_1$. Clearly $e\otimes \mathcal{O}(1;\underline{1})\subseteq \mathfrak{g}(h_\tau ,\bar{2})$ and $f\otimes \mathcal{O}(1;\underline{1})\subseteq \mathfrak{g}(h_\tau ,-\bar{2})$. As $\big [e\otimes \mathcal{O}(1;\underline{1}),f\otimes \mathcal{O}(1;\underline{1})\big ]=h\otimes \mathcal{O}(1;\underline{1})$ and $\mathfrak{g}(h_\tau ,\pm \bar{2})=\mathfrak{g}(\tau ,\pm 2)$ by Car93, p. 405, we have that $S\otimes \mathcal{O}(1;\underline{1})\subset \mathfrak{g}^\sigma$. As

$$\begin{equation*} \mathfrak{h}=(S\otimes \mathcal{O}(1;\underline{1}))\rtimes (\operatorname {Id}\otimes \mathcal{D})\subseteq (S\otimes \mathcal{O}(1;\underline{1}))+\mathfrak{c}_{e\otimes 1} \end{equation*}$$

and $\mathfrak{c}_{e\otimes 1}=\operatorname {Lie}(C_{e\otimes 1})\subset \mathfrak{g}(\tau ,0)\subset \mathfrak{g}^\sigma$ we obtain $\mathfrak{h}\subset \mathfrak{g}^\sigma$, as claimed.

■

3.4. The existence of ${\mathfrak{h}}$

The goal of this subsection is to give explicit examples of exotic semidirect products $\mathfrak{h}\cong (\mathfrak{sl}_2\otimes \mathcal{O}(1;\underline{1}))\rtimes (\operatorname {Id}\otimes \mathcal{D})$. In view of our discussion in §3.3 we shall assume that $G$ is of type ${\mathrm{E}}_7$ and $p\in \{5,7\}$. The notation introduced in the previous subsections will be used without further notice. We write $U^+$ and $U^-$ for the maximal unipotent subgroups of $G$ generated by the root subgroups $U_\gamma$ with $\gamma \in \Phi ^+(\Pi )$ and $\gamma \in -\Phi ^+(\Pi )$, respectively. Combining Proposition 3.2 with Car93, p. 403 one observes that $(\operatorname {ad}(e\otimes 1))^{p-1}=(\operatorname {ad}(f\otimes 1))^{p-1}=0$. In view of Remark 2.4 this implies that $\mathfrak{g}$ is a completely reducible $\operatorname {ad}(S\otimes 1)$-module. As a consequence, $(\operatorname {ad}(f\otimes 1))^{3}$ annihilates $\mathfrak{g}_{\,e\otimes 1}(2)$.

We first suppose that $p=5$. In view of Proposition 3.2 we may assume that $e\otimes 1=\sum _{\,i\ne 4}\,e_{\alpha _i}$ and $h\otimes 1=h_\tau$, where $\tau \colon \,k^\times \to T$ is as in LT11, p. 104. The group $C_e$ is connected of type ${\mathrm{A}}_1$ and contains $T_0:=\varpi ^\vee _4(k^\times )$ as a maximal torus. The group $C_{e\otimes 1}^+:=C_{e\otimes 1}\cap U^+$ is normalised by $T_0$ and $B_{e\otimes 1}^+:=T_0\cdot C_{e\otimes 1}^+$ is a Borel subgroup of $C_{e\otimes 1}$. In the notation of LT11 the maximal unipotent subgroup $C_{e\otimes 1}\cap U^-$ of $C_{e\otimes 1}$ consists of all $x_{-\beta _1}(t)$ with $t\in k$. (Here $-\beta _1$ is not a root of $G$ with respect to $T$.) By the theory of rational $\mathrm{SL}(2)$-modules, there exist nilpotent endomorphisms $X^{(i)}$ of $\mathfrak{g}$ such that

$$\begin{equation*} \big (\operatorname {Ad}\,x_{-\beta _1}(t)\big )(v)\,=\,\sum _{i\ge 0}\,t^iX^{(i)}(v)\qquad \ \big (\forall \,v\in \mathfrak{g}\big ) \end{equation*}$$

(these endomorphisms may be different from those in Proposition 2.7). Each $X^{(i)}(v)$ is a weight vector of weight $2i$ for the action of $T_0$ on $\mathrm{End}(\mathfrak{g})$. Furthermore, there exists a generator $X$ of $\operatorname {Lie}(C_{e\otimes 1}\cap U^-)$ such that $X^{(i)}=\frac{1}{i!}(\operatorname {ad}X)^i$ for $0\le i\le p-1$.

As mentioned in LT11, p. 105, in characteristic $5$ the $C_{e\otimes 1}$-submodule generated by $e_{\tilde{\alpha }}\in \mathfrak{g}_{\,e\otimes 1}(2)$ is contained in the $10$-dimensional indecomposable summand of $\mathfrak{g}_{e\otimes 1}(2)$ isomorphic to the tilting module $T_{{\mathrm{A}}_1}(8)$. The other indecomposable summand of the $15$-dimensional $C_{e\otimes 1}$-module $\mathfrak{g}_{\,e\otimes 1}(2)$ is isomorphic to the Steinberg module $V(4)$. Since $T_{{\mathrm{A}}_1}(8)$ is a projective module over the restricted enveloping algebra of $\mathfrak{sl}_2$ by Do93, §2, the vector space $\mathfrak{g}_{X}\cap \mathfrak{g}_{\,e\otimes 1}(2)$ is $3$-dimensional with basis consisting of $T_0$-weight vectors of weight $-8$, $-4$, and $0$. It follows that the fixed point space $(\mathfrak{g}_X\cap \mathfrak{g}_{\,e\otimes 1}(2))^{T_0}$ is spanned by $e\otimes 1$. On the other hand, since the tilting module $T_{{\mathrm{A}}_1}(8)$ admits a Weyl filtration, the $C_{e\otimes 1}$-submodule of $\mathfrak{g}_{\,e\otimes 1}(2)$ generated by the highest weight vector $e_{\tilde{\alpha }}$ of weight $8$ for $B_{e\otimes 1}^+$ is isomorphic to the Weyl module $V(8)$. Since $X^{[5]}=0$ we have that $(\operatorname {ad}\,X)^4(e_{\tilde{\alpha }})\in (\mathfrak{g}_X\cap \mathfrak{g}_{\,e\otimes 1}(2))^{T_0}$, implying $k(e\otimes 1)\,=\, k(\operatorname {ad}\,X)^4(e_{\tilde{\alpha }})$.

Let $A$ denote the $k$-span of all