Complete reducibility for Lie subalgebras and semisimplification

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Introduction
Let G be a connected reductive linear algebraic group over an arbitrary field k.We revisit the notion of G-complete reducibility over k for a Lie subalgebra h of the Lie algebra g := Lie(G) of G from [5,Def. 5.3].A central theme of the present paper is an extension to the setting of Lie algebras of the construction of a k-semisimplification of a subgroup H of G from [10].For motivation and an overview of the latter and further references, we refer to the introduction of [10].The concept of the semisimplification of a module for a group or algebra is a well-known construction in representation theory.Building on this idea, for H a subgroup of G(k), Serre introduced the concept of a "G-analogue" of semisimplification from representation theory in [40, §3.2.4].As in the subgroup case, one can think of the idea of a k-semisimplification of a subalgebra of g as an analogue of the Jordan-Hölder Theorem.
The definition of a k-semisimplification of a Lie subalgebra of g (Definition 5.4) for arbitrary k is new and generalizes the one for subgroups of G (Definition 5.1).We prove that the k-semisimplification of a subalgebra h of g is unique up to Ad(G(k))-conjugacy (Theorem 5.8).Moreover, for H a closed subgroup of G and H ′ a k-semisimplification of H, we show that Lie(H ′ ) is a k-semisimplification of Lie(H) (Theorem 5.9).
To prove Theorem 5.8 we use the theory of G-complete reducibility for subalgebras of g and some ideas from geometric invariant theory.In Theorem 5.14 we show that when char(k) is large enough, an ideal of a G-completely reducible subalgebra h is G-completely reducible and the process of k-semisimplification behaves well under passing to ideals of h.
We also prove some results of independent interest about G-complete reducibility for subalgebras, including Proposition 4.4.We consider solvable subalgebras in Section 6. Theorem 7.3 gives a necessary condition for a subalgebra to be G-completely reducible when char(k) = p > 0 is sufficiently large, and yields a characterisation of G-completely reducible subalgebras when char(k) = 0. We discuss some related results of Richardson, who pioneered the application of geometric invariant theory to the study of G-complete reducibility.
The notion of G-complete reducibility for subalgebras of g was first defined by McNinch [32] for algebraically closed k and was developed further in [12, §3] by the first, third and fourth authors; the non-algebraically closed case was first studied in [5, §5].The approach via geometric invariant theory stems from work of Richardson [38], who studied subgroups of G and subalgebras of Lie(G) -mainly for algebraically closed k in characteristic 0. Some of our arguments involve extending constructions from op. cit. to positive characteristic.We also use more recent techniques from [7] and [10] including some deep results from the theory of spherical buildings.
The theory of G-complete reducibility for Lie subalgebras closely parallels the theory of G-complete reducibility for subgroups [7], [10].Some arguments are easier in the subalgebra case, as we need not worry about non-connected groups.There are, however, some extra problems for subalgebras, which can be traced back to the failure of certain normalisers to be smooth.It is to avoid these difficulties that we need assumptions on the characteristic of k: see Section 2.3.

Preliminaries
2.1.Basic notation.Let k be a field with char(k) = p ≥ 0. Let k and k s be the algebraic closure and separable closure of k, respectively.Following [15], [13], and [6], we regard an affine variety over a field k as a variety X over the algebraic closure k together with a choice of k-structure.We write X(k) for the set of k-points of X and X(k ′ ) for the set of k ′ -points of X if k ′ /k is a field extension; we often write just X for X(k).By a subvariety of X we mean a closed k-subvariety of X; a k-subvariety is a subvariety that is defined over k.
Throughout, G denotes a connected reductive linear algebraic k-group.Algebraic groups and their subgroups are assumed to be smooth (although we consider certain non-smooth subgroup schemes in Section 2.3).A k-defined affine G-variety X is an affine variety over k equipped with a k-defined morphic action of G. Two important examples of affine G-varieties in this paper are the group G itself, with G acting by inner automorphisms, and the Lie algebra g = Lie(G) of G, with G acting by the adjoint action (that g admits a k-structure is [41, (12.2.3), (4.4.8)]).Recall that these two actions are closely related: if g ∈ G, then we have Inn(g) : G → G, x → gxg −1 , and then Ad(g) := d(Inn(g)) : g → g is the differential (see below).To simplify notation we denote both of these actions by a dot: that is, if g ∈ G, then we let g • x = gxg −1 for each x ∈ G, and g • x = Ad(g)(x) for each x ∈ g.We define ad to be the usual adjoint action of g on g: so ad(x)(y) = [x, y] for x, y ∈ g.If necessary we write Ad G instead of Ad and ad g instead of ad.Given any m ∈ N, we may extend these actions to actions of G on X = G m or g m , the m-fold Cartesian product of G or g, by setting g • (x 1 , . . ., x m ) := (g • x 1 , . . ., g • x m ) for each g ∈ G and (x 1 , . . ., x m ) ∈ X.We refer to this action informally in both cases as the action by simultaneous conjugation.
When char(k) = p > 0, the Lie algebra g is restricted with a p-operation g → g, x → x [p] , [41, (4.4.3)].By a subgroup of G we mean a closed k-subgroup and by a k-subgroup we mean a subgroup that is defined over k.For a subgroup H of G we write H 0 for the identity component of H and Lie(H) for its Lie algebra (which is a subalgebra of g).Subalgebras of g of the form Lie(H) as above are called algebraic.
If H is k-defined, then Lie(H) also admits a k-structure, [41, (12.2.3), (4.4.8)].We write g(k ′ ) for the set of k ′ -points of g if k ′ /k is a field extension, and just g for g(k).By a k-subalgebra of g we mean a Lie subalgebra h over k which is defined over k.
If f : G 1 → G 2 is a homomorphism of algebraic groups then we denote the induced map from Lie(G 1 ) to Lie(G 2 ) by df .

2.2.
Cocharacters and parabolic subgroups.We recall some basic notation and facts concerning cocharacters and parabolic subgroups in connected reductive groups G from [13], [38] and [41].Recall that a cocharacter of G is a homomorphism of algebraic groups λ : G m → G, where G m is the multiplicative group.We define Y k (G) to be the set of kdefined cocharacters of G and Y (G) := Y k (G) to be the set of all cocharacters of G. Given λ ∈ Y (G), we define For ease of reference, we record without proof some basic facts about these subgroups.The unipotent radical of a parabolic subgroup P of G is denoted by R u (P ).The following is [38, 2.3 The following is [13, Lem.2.5 (i)+(iii)].See also [41, 16.1.1].
Lemma 2.3.Let P be a parabolic subgroup of G and L a Levi subgroup of P .
(i) We have P ∼ = L ⋉ R u (P ), and this is a k-isomorphism if P and L are k-defined.
(ii) Let T be a maximal torus of P .Then there is a unique Levi subgroup We denote the canonical projection from P to L by c L ; this is k-defined if P and L are.Given any cocharacter λ of G such that P = P λ and L = L λ then we often write c λ instead of c L .We have c λ (g) = lim a→0 λ(a)gλ(a) −1 for g ∈ P λ ; the kernel of c λ is the unipotent radical R u (P λ ) and the set of fixed points of c λ is L λ .
Let M be a connected reductive subgroup of G and let λ ∈ Y (M).We can perform the above constructions in M and contain subgroups P λ (M) and L λ (M).As above, we write P λ for P λ (G) and L λ for L λ (G).It follows easily from the definitions and results above that All of this transfers over to the Lie algebra g through the adjoint action: since G acts on g by the adjoint action, so does any subgroup of G and, in particular, the image of any cocharacter of G.For λ ∈ Y (G), we define and use throughout p λ := Lie(P λ ) and l λ := Lie(L λ ).We require some standard facts concerning Lie algebras of parabolic and Levi subgroups of G (cf. [38, §2.1]).
2.3.Smoothness of centralisers and normalisers.We are concerned with linear algebraic groups G over k: these are affine group schemes of finite type over k (assumed to be smooth).Note that smoothness is automatic if char(k) = 0, by a result of Cartier [19, II, §6, 1.1].We do need to consider certain non-smooth subgroup schemes of G, as the lack of smoothness has implications for G-complete reducibility.If h is a subalgebra of g then we define N G (h) to be the scheme-theoretic normaliser of h in G, and C G (h) to be the schemetheoretic centraliser of h in G (see [26, I.2.12] or [18, A.1.9]).We define N G (h) and C G (h) to be the setwise normaliser and centraliser of h, respectively.Note that N G (h) and C G (h) are k-defined, but N G (h) and C G (h) need not be k-defined.The group scheme N G (h) (resp.,

and in this case
There is a criterion for smoothness in terms of the Lie algebra.Define  (a) Let Γ be a subgroup of G(k) consisting of unipotent elements.We call Γ plongeable if there is a parabolic subgroup P of G such that Γ ⊆ R u (P )(k) (see [45,Sec. 1]).(b) Let h be a k-subalgebra of g consisting of nilpotent elements.We call h plongeable if there is a parabolic subgroup P of G such that h ⊆ Lie(R u (P )).
Remark 2.6.Let M be a connected reductive k-subgroup of G and let Γ be a subgroup of (v) It is immediate that if condition (a) of Definition 2.7 holds for G regarded as a k-group then it also holds for G regarded as a k-group.
Let h be the abelian subalgebra of g spanned by x 1 := 0 1 0 0 and x 2 := 0 0 1 0 .It is easy to check that h is not contained in Lie(B) for any Borel subgroup B of G. Similar examples are produced in [29] for any G such that p is a torsion prime for G. Now let m be the subalgebra of g spanned by x 1 .Then C G (m)(k) is (the image of) the group of upper unitriangular matrices and N G (m)(k) is (the image of) the group of upper triangular matrices, but c g (m) = h and n g (m) = g.Hence neither C G (m) nor N G (m) is smooth.A similar calculation shows that neither C G (h) nor N G (h) is smooth.Suppose k = k.Owing to [21,Thm. 1.1], centralisers of subgroups of G and centralisers of subalgebras of g are smooth if and only if char(k) is 0 or a "pretty good prime" p for G, ([21,Def. 2.11]).The latter implies that p is not a torsion prime for the root system of G.It then follows from [29, Thm.2.2 and Rems.(a)] that condition (c) of Definition 2.7 holds.Hence condition (a) implies condition (c) in this instance (see also [22,Thm. 9.2] and [36,Cor. 1.3]).We now extend this result to arbitrary fields (Proposition 2.13), using results from [45] on plongeable unipotent subgroups.First we show that condition (a) is inherited by Levi subgroups and certain quotients of G.
The proof for centralisers of k-subalgebras is similar.
Remark 2.11.Let P be a k-parabolic subgroup of G and let L a Levi k-subgroup of P .Let Γ be a unipotent subgroup of P (k) and suppose there is Lemma 2.12.Suppose the centraliser of every k-subgroup of G and every k-subalgebra of g is smooth.Then every unipotent subgroup of G(k) is plongeable.
Proof.Let Γ be a subgroup of G(k) consisting of unipotent elements.Without loss Γ = 1.We use induction on dim(G).The result is trivial if dim(G) = 0. Let N be the algebraic subgroup of G generated by Γ.There is no harm in taking Γ to be N(k).Let Z be the last nontrivial term of the descending central series of N (this makes sense as N is unipotent).Then Z is nontrivial and k-defined (see [33,Ch. 6f.]).For any u ∈ Z(k s ), C G (u) is smooth by hypothesis, so u is k s -plongeable by [45,Prop. 2].Hence by [45,Cor. 3] there is a k sdefined parabolic subgroup P of G such that Z ⊆ R u (P ).We can take P to be the subgroup P ks (Z(k s )) in the sense of [45, Sec.1]; then P is k-defined by [45, Cor.1] and N(k s ) ⊆ P (k s ) by [45,Cor. 2].Note that P = G since N = 1.
Let L be a Levi k-subgroup of P .By Lemma 2.10(a) and our induction hypothesis, c L (Γ) is plongeable in L. Hence Γ is plongeable in G by Remark 2.11.Proposition 2.13.Suppose that centralisers of k-subgroups of G and k-subalgebras of g are smooth.Then for any k-subalgebra h of g consisting of nilpotent elements, there is a parabolic k-subgroup P of G such that h ⊆ Lie(R u (P )).
Proof.We can assume without loss that G is connected and h = 0. We use induction on dim(G).The result holds trivially if dim(G) = 0. Let h be a k-subalgebra of g consisting of nilpotent elements.Then h is a nilpotent Lie algebra by Engel's Theorem [23, 3.2 Since Z centralises M, Z centralises m; in particular, Z centralises z.Hence Z ⊆ M, and we deduce that Z = Z(M).It follows that Z is commutative.Since Z is smooth, z belongs to Lie(Z), so dim(Z) > 0. Fix a maximal k-torus S of Z. Since M ⊆ C G (S) and h ⊆ m, we can assume without loss by Remark 2.6 and Lemma 2.10(a) that G = C G (S).Let ψ : G → G/S be the canonical projection.Then ψ(Z) is unipotent, so by Lemma 2.10(b) and Lemma 2.12 there is a parabolic k Let L be a Levi subgroup of P .Then dim(L) < dim(P ), so c Lie(L) (h) is plongeable in L by our induction hypothesis.It follows from Remark 2.11 that h is plongeable in G, so we are done.
We finish by showing that if p is sufficiently large then p is fabulous for G.
Proposition 2.14.For r ≥ 0, there is a prime p 0 = p 0 (r) with the following property: for any prime p ≥ p 0 , any field k of characteristic p and any connected reductive k-group G of rank at most r, p is fabulous for G.
Proof.Since there are only finitely many possibilities for G up to isomorphism (as a k-group), we can ensure that p is very good for G by taking p sufficiently large.Then p is pretty good for G, and then -as observed above -condition (a) of Definition 2.7 holds, which implies that condition (c) holds by Proposition 2.13.Condition (b) holds for large enough p by [22,Thm.A].The result follows.
Remark 2.15.We do not have an analogous result to Proposition 2.14 if we also require normalisers of subgroups to be smooth: see [22,Lem. 11.11].

Cocharacter-closed orbits and G-complete reducibility
We introduce the notion of complete reducibility for subalgebras of g and explain the link with geometric invariant theory (GIT).At the end of the section, we extend to arbitrary k the main results from [32].As in [10], our main tool from GIT is the notion of cocharacterclosure, introduced in [13] and [6].Definition 3.1.Let X be a k-defined affine G-variety and let x ∈ X (we do not require x to be a k-point).We say that the orbit The following is [6, Thm.Next recall the notion of G-complete reducibility for subgroups of G.
For more on G-complete reducibility for subgroups of G, see [39], [40], [7]; our main focus in this paper is the analogue for Lie algebras.We first recall the definition of G-complete reducibility for Lie algebras and then also the link between this concept and GIT using generating tuples, due to Richardson.Most of what follows was originally written down in [5, §5].Definition 3.4.A subalgebra h of g is G-completely reducible over k (G-cr over k) if for any parabolic k-subgroup P of G such that h ⊆ Lie(P ), there is a Levi k-subgroup L of P such that h ⊆ Lie(L) (see [5,Def. 5.3]).We say that h is G-irreducible over k (G-ir over k) if h is not contained in Lie(P ) for any proper parabolic k-subgroup of G at all.We say that h is G-indecomposable over k (G-ind over k) if h is not contained in Lie(L) for any proper Levi k-subgroup L of G.
As in the subgroup case, we say that h is G-completely reducible (resp., G-irreducible, G-indecomposable) if it is G-completely reducible over k (resp., G-irreducible over k, Gindecomposable over k).
Example 3.5.Let k, G, h and m be as in Example 2.9.Then h is G-ir but the ideal m of h is not G-cr.
Example 3.6.If p is fabulous for G and 0 = h is a subalgebra of g consisting of nilpotent elements then h is not G-cr: for h ⊆ Lie(B) for some Borel subgroup B of G, but clearly h ⊆ Lie(T ) for any maximal torus T of B.
Remark 3.7.The notion of G-complete reducibility for subgroups and subalgebras generalizes the concept of semisimplicity from representation theory in the following sense.If H is a subgroup of GL(V ) or SL(V ), or h is a subalgebra of gl(V ) or sl(V ), then H (resp. h) is GL(V )-cr or SL(V )-cr if and only if V is a semisimple H-module (resp.h-module).See [39], [40, (3.2.2)] for the case of subgroups; the case of subalgebras is almost identical.Remark 3.8.Let H be a subgroup of G and h a subalgebra of g.If k ′ /k is an algebraic field extension then we may also regard G as a k ′ -group, and it therefore makes sense to ask whether H and h are G-cr over k ′ as well as whether they are G-cr over k.Remark 3.9.Note that Definitions 3.3 and 3.4 make sense even if H and h are not k-defined.We also note that since p g•λ = g • p λ and l g•λ = g • l λ for any λ ∈ Y (G) and any g ∈ G (see, e.g., [7, §6]), it follows that h is G-cr over k if and only if every Ad(G(k))-conjugate of h is.More generally, one can show that if h is G-cr over k (resp., G-ir over k, resp., G-ind over k) then so is dφ(h), for any k-defined automorphism φ of G. Similar observations hold for subgroups.
We now recall the link to GIT.For this, we need the following definition.Definition 3.10.Let h be a Lie algebra.We call x = (x 1 , . . ., x m ) ∈ h m , for some m ∈ N, a generating tuple for h if x 1 , . . ., x m is a generating set for h as a Lie algebra.
The next theorem shows the relevance of the previous definition to the study of complete reducibility.It is [5,Thm. 5.4]; see also [32, Thm.1(i)] for the case k = k.Theorem 3.11.Let h be a subalgebra of g = Lie(G).Let x ∈ g m be a generating tuple for h, and let G act on g m by simultaneous conjugation.Then h is G-completely reducible over k if and only if the G(k)-orbit of x is cocharacter-closed in g m over k.
The next result is [5,Thm. 5.5].Note that if k is perfect, then part (i) implies the equivalence of G-complete reducibility over k and G-complete reducibility (over k), because the extension k/k is separable for perfect k.Proposition 3.12.Let h be a subalgebra of g.
(i) Suppose h is k-defined, and let k ′ /k be a separable algebraic field extension.Then h is G-completely reducible over k ′ if and only if h is G-completely reducible over k. (ii) Let S be a k-defined torus of C G (h) and set L = C G (S). Then h is G-completely reducible over k if and only if h is L-completely reducible over k.
The final ingredient we need for our first main result is the connection between complete reducibility of subgroups of G and the notion of complete reducibility in the spherical building of G. Recall that the spherical building ∆ k = ∆ k (G) of G over k can be identified as a simplicial complex with the poset of k-parabolic subgroups of G, with inclusion reversed [43]; the parabolic subgroup G corresponds to the empty simplex.For each k-parabolic subgroup P of G, we let σ P denote the corresponding simplex in ∆ k .We note that k-points of G induce simplicial automorphisms of ∆ k : for g ∈ G(k) and a k-parabolic subgroup P of G, we can define g • σ P = σ gP g −1 .Recall that the simplicial complex ∆ k has a geometric realisation, which we denote by ∆ k [1, A.1.1].We have the following definitions.(iii) A convex subcomplex Σ of ∆ k is said to be ∆ k -completely reducible (∆ k -cr) if for every σ ∈ Σ, there exists τ ∈ Σ with τ opposite to σ.
The following argument forms part of the proof of [32,Lem. 4]; it provides the key link between G-complete reducibility and the building ∆ k .We give the idea of the proof for completeness.
Proposition 3.14.Suppose h is a subalgebra of g.Then ∆ h k := {σ P | h ⊆ Lie(P )} is a convex subcomplex of ∆ k , and it is ∆ k -completely reducible if and only if h is G-completely reducible over k.
Sketch Proof.For the first assertion, the key observation is that for the intersection of any two connected subgroups P and Q of G that are normalized by T , we have Lie(P ∩ Q) = Lie(P ) ∩ Lie(Q) [15,Cor. 13.21].Now the result follows from Serre's criterion [40,Prop. 3.1] for recognising a convex subcomplex of ∆ k : a collection Σ of simplices is a convex subcomplex if whenever P, Q, R are k-parabolic subgroups of G with σ P , σ Q ∈ Σ and To see that h is G-cr over k if and only if ∆ h k is ∆ k -cr, the key again is the smoothness of the intersection of two k-parabolic subgroups.This implies that for two k-parabolic subgroups P and Q of G, we can detect whether or not they are opposite (and hence correspond to opposite simplices of ∆ k ) on the level of their Lie algebras.Remark 3.15.An analogous result holds for subgroups, with a very similar proof [40].
Since each subalgebra h of g gives rise to a subcomplex ∆ h k of ∆ k , we can use the socalled Centre Conjecture of Tits, which in fact is a theorem for subcomplexes, see [34] (G of classical type or type G 2 ), [28] (G of type F 4 or E 6 ) and [37] (G of type E 7 or E 8 ).We give a version which is suitable for our purposes.Theorem 3.16 (Tits' Centre Theorem).Let ∆ k be a thick spherical building, and Σ a convex subcomplex of ∆ k .Then (at least) one of the following holds: (i) Σ is ∆ k -completely reducible; (ii) there exists a nonempty simplex σ ∈ Σ which is fixed by all simplicial automorphisms of ∆ k that stabilize Σ.
For a definition of thick see [1, 4.2].In our case of algebraic groups, ∆ k is always thick, see the comment after [46,Thm. 3.2.6].The typical use of this theorem (as we see in the next proof below) is in guaranteeing the existence of a simplex as in part (ii) when Σ is not ∆ k -cr; such a simplex is often referred to as a centre of Σ.We can now prove the main result of this section.Over k, this is [32, Thm.1(ii)]; see also [13,Thm. 5.27,Ex. 5.29].However, the proofs given there, which use the technology of optimal destabilising subgroups, do not go through over arbitrary k.Instead, we use the Centre Theorem.Theorem 3.17.Suppose H is a k-defined subgroup of G.If H 0 is G-completely reducible over k, then so is Lie(H).
Proof.Since Lie(H) = Lie(H 0 ), and the hypothesis is that H 0 is G-cr, it is no loss to assume that H is connected.Let h = Lie(H).Using Proposition 3.12(i) (and its analogue for subgroups [8, Thm.1.1]), it is enough to prove the result over the separable closure k s .These reductions imply that we may assume that for some λ ∈ Y k (C G (H)), Proposition 3.12(ii) (and its analogue for subgroups [5,Thm. 1.4]) implies that H and h are G-cr if and only if they are L-cr, so we may replace G with L. Then H is G-cr over k but is not contained in any proper Levi k-subgroup of G, so H is G-ir over k.
We now proceed with the proof of the theorem by contradiction.Suppose that h is not G-cr over k.The subgroup H(k) acts on the building ∆ k by simplicial automorphisms, and the subcomplex ∆ h k is stabilized by the action of H(k) since H stabilizes its own Lie algebra.We are assuming that h is not G-cr over k, so this subcomplex is not ∆ k -cr, by Proposition 3.14, and hence there is a nonempty H(k)-fixed simplex σ in ∆ h k by Theorem 3.16.This simplex has the form σ P for some proper k-parabolic subgroup P of G, and the fact that σ P is H(k)-fixed translates into the fact that P is normalized by H(k).Since parabolic subgroups are self-normalizing, this shows that H(k) ⊆ P , which in turn gives H ⊆ P because H(k) is dense in H.This contradicts the conclusion of the first paragraph, that H is G-ir over k, so we are done.Remarks 3.18.(i).We observe that the converse of Theorem 3.17 is false, already in the algebraically closed case.See the counterexample due to the third author in [32, §1]; see also Example 5.13 below.
(ii).If k is perfect, then Theorem 3.17 also holds with the hypothesis that H (instead of H 0 ) is G-cr.For, as has already been observed, if k is perfect then H is G-cr over k if and only if H is G-cr, and the same for Lie(H).Now over k, if H is G-cr then H 0 is G-cr too [7, Thm.3.10], and the hypotheses of the theorem hold.

Maps induced by inclusion
In this section we assume k is algebraically closed.We need some material on quotients in GIT.Recall that if G acts on an affine variety X then we can form the quotient variety X/ /G.The coordinate algebra k[X/ /G] is by definition the subring of invariants k[X] G of the coordinate ring k[G], and the inclusion of k[X] G in k[G] induces a morphism π X (or π X,G ) from X to X/ /G.If x ∈ X, there is a unique closed G-orbit C(x) contained in the closure of the orbit G • x.By the Hilbert-Mumford Theorem, there exists λ ∈ Y (G) such that lim a→0 λ(a) • x belongs to C(x).Given x 1 , x 2 ∈ X, we have π if and only if x 1 and x 2 lie in the same G-orbit.Hence points of X/ /G correspond to closed G-orbits in X.For background on quotient varieties and GIT, see [35,Ch. 3] or [3].Here is a particular instance of this set-up.The group G acts on G m by simultaneous conjugation, so we can form the quotient G m / /G.Likewise we can form the quotient g m / /G.There is a direct connection with G-complete reducibility arising from Theorem 3.11 as follows: points in g m / /G correspond to closed G-orbits G•(x 1 , . . ., x m ), and (x 1 , . . ., x m ) ∈ g m yields a closed orbit if and only if the subalgebra h generated by the x i is G-cr.Analogous statements hold for G m / /G.Now let H be a reductive subgroup of G and let h = Lie(H).The inclusion ι of H m in G m induces a map Ψ : H m / /H → G m / /G.The third author proved that Ψ is a finite morphism of varieties [31,Thm. 1.1]; this has various consequences for the theory of G-completely reducible subgroups (see, e.g., [7,Cor. 3.8]).It follows that the image of Ψ is closed.In particular, π G m (ι(H m )) is a closed subset of G m / /G.Now consider the analogous situation for Lie algebras.The differential dι of ι maps h m to g m .Now dι gives rise to a map ψ : h m / /H → g m / /G mapping π h m ,H (x 1 , . . ., x m ) to π g m ,G (dι(x 1 , . . ., x m )), but we see in Example 4.2 below that this map need not be finite.
First we need a preliminary result.This shows that the fibre ψ −1 (π g 2 ,G (0, 0)) is infinite, so ψ cannot be finite.
If we put a restriction on p then the situation improves.Recall that if V is a rational G-module then v ∈ V is said to be unstable if 0 belongs to the closure of G • v, and semistable otherwise (cf.[38,Sec. 1.4]).
Lemma 4.3.Let R and S be commutative rings graded by N 0 with R 0 = S 0 = k.Set R + = i>0 R i and S + = i>0 S i .Let φ : R → S be a graded k-algebra homomorphism; we regard S as an R-module via φ.Let I be Sφ(R + ) (regarded as an ideal of S).Suppose S is finitely generated as a k-algebra and S + is the radical of I. Then S is a finitely generated R-module.
Proof.Choose homogeneous elements f 1 , . . ., f r ∈ S + such that the f i generate S as a kalgebra.There exist n 1 , . . ., n r ∈ N such that f n i i ∈ I for each i.The elements of the form , where the a i are non-negative integers, span S over k.If a i ≥ n i for some i then Hence S/I is spanned over k by the images of the elements of the form f a 1 1 • • • f ar r with 0 ≤ a i < n i .This shows that S/I is finitely generated as an R-module.This shows that S/I is finitely generated as an R-module.It follows from a corollary [30,Cor. 3] to the graded Nakayama lemma that S is a finite R-module; note that Lemma 1 of op.cit.still holds even when the module M is not finitely generated (see [20,Ex. 4.6]).This completes the proof.Proposition 4.4.Suppose p is fabulous for G. Then ψ is a finite map.
Proof.Suppose (x 1 , . . ., x m ) ∈ h m is unstable for the G-action.Then the subalgebra m generated by the x i consists of nilpotent elements, by Proposition 4.1.The hypothesis on p implies that m ⊆ Lie(U) for some maximal unipotent subgroup U of H. Hence (x 1 , . . ., x m ) is unstable for the H-action by Proposition 4.1.It follows from Proposition 4.1 again that (4.5) ] is a polynomial algebra, so it has a natural grading by N 0 .Since G acts linearly on g m , the ring of invariants

k-semisimplification
First, we recall the notion of k-semisimplification for subgroups of G and the main theorem from [10].
Definition 5.1.Let H be a subgroup of G.We say that a subgroup H ′ of G is a ksemisimplification of H (for G) if there exist a parabolic k-subgroup P of G and a Levi k-subgroup L of P such that H ⊆ P , H ′ = c L (H) and H ′ is G-completely reducible (or equivalently L-completely reducible, by [10,Prop. 3.6]) over k, where c L : P → L is the canonical projection.We say the pair (P, L) yields H ′ .Remarks 5.2.(i).Let H be a subgroup of G.If H is G-cr over k then clearly H is a k-semisimplification of itself, yielded by the pair (G, G).
(ii).Given any subgroup H of G, [10,Rem. 4.3] guarantees the existence of a k-semisimplification of H.
Here is the main result [10,Thm. 4.5] from [10], which was proved in the special case k = k in [13, Prop.5.14(i)], cf.[40,Prop. 3.3(b)].The uniqueness asserted in Theorem 5.3 is akin to the theorem of Jordan-Hölder.We now come to the analogue of Definition 5.1 for subalgebras of g.Definition 5.4.Let h be a Lie subalgebra of g.We say that a Lie subalgebra h ′ of g is a k-semisimplification of h (for G) if there exist a parabolic k-subgroup P of G and a Levi k-subgroup L of P such that h ⊆ Lie(P ), h ′ = c Lie(L) (h) and h ′ is G-completely reducible (or equivalently, by Proposition 3.12(ii), L-completely reducible) over k.We say the pair (P, L) yields h ′ .Remarks 5.5.(i).Let h be a subalgebra of g.If h is already G-cr over k then clearly h is a k-semisimplification of itself, yielded by the pair (G, G).
(ii).Suppose (P, L) yields a k-semisimplification h ′ of h.Let L 1 be another Levi ksubgroup of P .Then L 1 = uLu −1 for some u ∈ R u (P )(k) by Lemma 2.3(iii), so consequently c Lie(L 1 ) (h) = u • c Lie(L) (h).Hence (P, L 1 ) also yields a k-semisimplification of h.Because of this, when the choice of L doesn't matter we simply say that P yields a k-semisimplification of h.
(iii).It is straightforward to check that if φ is an automorphism of G (as a k-group), h is a subalgebra of g and (P, L) yields a k-semisimplification h ′ of h then dφ(h ′ ) is a ksemisimplification of dφ(h), yielded by (φ(P ), φ(L)).
The following is immediate from Lemma 2.1.
As in the group case (Remark 5.2(ii)) we always have the existence of a k-semisimplification of an arbitrary subalgebra of g, due to the rational Hilbert-Mumford Theorem 3.2, as the following remark shows.
Remark 5.7.Suppose h is a subalgebra of g.Let h = (h 1 , . . ., h m ) ∈ h m be a generating tuple for h.Then c λ (h) = (c λ (h 1 ), . . ., c λ (h m )) is a generating tuple for c λ (h), for any λ ∈ Y k (G), and hence c λ (h) is a k-semisimplification of h if and only if G(k) • c λ (h) is cocharacter-closed over k, by Theorem 3.11.It follows from Theorem 3.2 that h admits at least one k-semisimplification: Here is the analogue of the main result Theorem 4.5 from [10] in the Lie algebra setting, which can be viewed as a kind of Jordan-Hölder theorem.Since the adjoint action is k-linear, the proof is easier to the one in [10], where a descending chain argument is needed.
Theorem 5.8.Let h be a subalgebra of g.Then any two k-semisimplifications of h are Ad(G(k))-conjugate.
Proof.Let h 1 , h 2 be k-semisimplifications of h.By Lemma 5.6, there exist λ 1 , λ 2 ∈ Y k (G) such that (P λ 1 , L λ 1 ) realizes h 1 and (P λ 2 , L λ 2 ) realizes h 2 .Let x ∈ h m be a generating tuple for h.Then c λ i (x) is a generating tuple for h i for i = 1, 2, and each orbit G(k) • c λ i (x) is cocharacter-closed over k.It follows from the uniqueness result in the rational Hilbert-Mumford Theorem 3.2 that the two orbits G(k) • c λ 1 (x) and G(k) • c λ 2 (x) have to be equal.Thus there exists g ∈ G(k) such that g • c λ 1 (x) = c λ 2 (x).This means that the spanning set c λ 1 (x) of h 1 is Ad(G(k))-conjugate to the one of h 2 .Since the adjoint action is k-linear, h 1 and h 2 are also Ad(G(k))-conjugate.
Next we study the connection between the notions of k-semisimplifications for subgroups and subalgebras.It turns out that they are compatible in a natural fashion.Theorem 5.9.Let H be a subgroup of G and let H ′ be a k-semisimplification of H. Then Lie(H ′ ) is a k-semisimplification of Lie(H).
Proof.By Lemma 2.1, there is a λ ∈ Y k (G) such that (P λ , L λ ) yields H ′ = c λ (H).It follows from Theorem 3.17 that Lie(H ′ ) is G-cr over k.Since the differential of conjugation is the adjoint action, we have dc L λ = c l λ .Using Lemma 2.4, it follows that Lie(H ′ ) = c l λ (Lie(H)), and so the pair (P λ , L λ ) also yields Lie(H ′ ).Next we revisit the example of Remark 3.18(i) in the context of Theorems 5.8 and 5.9.
Example 5.10.Suppose char(k) = p > 0. Let H be a non-trivial connected semisimple group and let ̺ i : H → SL(V i ) be representations of H for i = 1, 2 with ̺ 1 semisimple and ̺ 2 not semisimple.Let ̺ : not semisimple as a J-module, J is not G-cr (cf.Remark 3.7).However, Lie(J) = Im d̺ 1 ⊕ 0 ⊆ g is G-cr, showing that the converse of Theorem 3.17 fails.Now let J ′ be a k-semisimplification of J.It follows from Theorem 5.9 that Lie(J ′ ) is a k-semisimplification of Lie(J) and thus, by Remark 5.5(a) and Theorem 5.8, that Lie(J ′ ) and Lie(J) are Ad(G(k))-conjugate.
The following is the analogue of [10,Def. 4.6].
Definition 5.11.Let h be a subalgebra of g.We define D k (h) to be the set of Ad(G(k))conjugates of any k-semisimplification of h in g.This is well-defined by Theorem 5.8.
In the following two examples we show that not every element of D k (h) need be a k-semisimplification of h and that there is no direct relation between the notions of ksemisimplification and k-semisimplification of a subalgebra of g.Example 5.12.Let h be a subalgebra of g.As noted in Remark 5.5(a), if h is G-cr over k then h is a k-semisimplification of itself, yielded by the pair (G, G).If h is G-ir, then h is not contained in Lie(P ) for any proper parabolic subgroup P of G, so h is the only k-semisimplification of itself.
Example 5.13.There are many examples in the literature of the following: a reductive group G over an imperfect field k, and a subgroup H of G such that H is G-cr over k but not G-cr, or H is G-cr but not G-cr over k; see [2, Thm.1.3], for example.Not all of these instances give rise to similar ones on the level of Lie algebras, even when the subgroup H is connected, because of problems like that in Example 5.10.However, one of the most basic families of examples does work, as we now describe.
Let k be an imperfect field of characteristic p, and let a ∈ k \ k p .Let t be a p th root of a in k; then the extension k ′ = k(t) is purely inseparable over k of degree p.Given a k ′ -group H ′ , we may form the Weil restriction R k ′ /k (H ′ ), which is a k-group, see [18, A.5] for example.(It is easiest to describe Weil restriction functorially: if we view H ′ as a functor from k ′ -algebras to groups, then R group with a (p − 1)-dimensional unipotent radical, but no connected normal unipotent k-subgroup (it is a basic example of a so-called pseudo-reductive group, see [18,Ex. 1.1.3]).The natural action of H ′ on k ′ by multiplication Weil restricts to give a p-dimensional representation of H over k which is irreducible: in terms of coordinates, this arises by writing down a k-basis for k ′ and interpreting the action through that basis, see [14, §5.2].However, after base changing to k, this module becomes indecomposable and not irreducible [14,Rem. 4.7(i)].This means that H is GL p -cr over k but not GL p -cr.This example is due to McNinch (see [7,Ex. 5.11]).
If we turn our attention to the Lie algebras, we may identify the Lie algebra Lie(H ′ ) with the additive group over k ′ , and we may view the multiplicative group H ′ as an open subset.Doing this is compatible with the action of H ′ and Lie(H ′ ) on k ′ by multiplication.Therefore, after Weil restricting, we may identify the Lie algebra of H with a p-dimensional abelian Lie algebra over k containing H as an open subset, with the same compatibility between the corresponding representations (see [18, A.7.6] for more details on the Lie algebra of a Weil restriction).Therefore, in this case, we also have that Lie(H) is GL p -cr over k but not GL p -cr.Note that C GLp (Lie(H)) is smooth.
The following theorem shows that under a mild restriction on k, the process of k-semisimplification behaves well under passing to ideals.
Theorem 5.14.Suppose p is fabulous for G. Let h be a subalgebra of g and let m be an ideal of h.Then: (a) If h is G-completely reducible over k then so is m.(b) Every parabolic subgroup P of G which yields a k-semisimplification of h also yields one for m.In particular, there exist k-semisimplifications h ′ of h and m ′ of m such that m ′ is an ideal in h ′ .
Proof.For (a), by the same reductions as at the start of the proof of Theorem 3.17, we may reduce to the case that k = k s and h is G-ir over k.We again proceed from this point using a contradiction argument invoking Tits' Centre Theorem 3.16.So suppose m is not G-cr over k.Then the subcomplex ∆ m k of the building ∆ k is not ∆ k -cr, by Proposition 3.14, and hence there is a proper k-parabolic subgroup P of G such that σ P ∈ ∆ where the equality in the middle follows from smoothness of N G (m) again.This gives the required contradiction, as we had reduced to the case that h is G-ir over k.
For (b), pick any λ ∈ Y k (G) such that (P λ , L λ ) yields a k-semisimplification h ′ := c λ (h) of h.Then c λ (h) is G-cr over k and, as c λ is a Lie algebra homomorphism, c λ (m) is an ideal in c λ (h).Now c λ (h) and c λ (m) satisfy the hypotheses of the theorem, so c λ (m) is G-cr over k by (a).Hence (P λ , L λ ) yields a semisimplification m ′ := c λ (m) of m as well, and m ′ is an ideal in h ′ .Remark 5.15.Both parts of Theorem 5.14 are false without the assumption on p: see Example 3.5.

G-toral and solvable subalgebras
We assume in this section that k is algebraically closed.We study G-complete reducibility properties of solvable and G-toral subalgebras h of g (see Definition 6.4 for the latter).Definition 6.1.We call a Lie subalgebra h of g Jordan-closed if for every x ∈ h, the semisimple part x s and nilpotent part x n of x both belong to h. 1 We define the Jordan closure h J of h to be the smallest Jordan-closed Lie subalgebra of g that contains h.Remarks 6.2.(a).It is clear that h J is well-defined.Here is an explicit construction.We define an increasing chain of subalgebras h i of g as follows.Set h 0 = h.Given h i , let h i+1 be the subalgebra generated by the elements of the form x s and x n for x ∈ h i .For dimension reasons, the chain becomes stationary and we have h n = h J for n sufficiently large.
(b).If h is algebraic then h is Jordan-closed.In particular, if P is a parabolic subgroup of G and L is a Levi subgroup of P then Lie(P ) and Lie(L) are Jordan-closed.It follows easily that for any subalgebra m of g, m is G-cr (resp., G-ir, resp., G-ind) if and only if m J is.Also, if char(k) = 0, h is a subalgebra of g and h is semisimple then h is algebraic (see [38,Lem. 3.2]), so h is Jordan-closed.Remark 6.3.Let h be a subalgebra of g and let f : G → M be a homomorphism of connected reductive groups.We have df (x) s = df (x s ) and df (x) n = df (x n ) for any x ∈ h, so df (h) is Jordan-closed if h is.The converse obviously holds if f is an embedding, for then df is injective.Definition 6.4.A subalgebra h of g is G-toral if h ⊆ Lie(S) for some torus S of G. Remark 6.5.Recall that a Lie algebra h is said to be toral if every element of h is ad hsemisimple; in this case, h is abelian [23,Lem. 8.1].Clearly, if h is G-toral then h is toral, but the converse is false: e.g., take h to be a nonzero abelian subalgebra consisting of nilpotent elements.
The following is the counterpart of [25,Lem. 11.24] for subalgebras of g.Lemma 6.6.Let h be a G-toral subalgebra of g.Then h is G-completely reducible.
Proof.Choose a torus S such that h ⊆ Lie(S).By Proposition 3.12(ii) it suffices to prove that h is C G (S)-cr.But this is clear because S is central in C G (S), so Lie(S) is contained in Lie(P ) and Lie(L) for every parabolic subgroup P and every Levi subgroup L of C G (S). Remark 6.7.Let m be a G-toral subalgebra of g: say, m ⊆ Lie(S), where S is a torus of G. Then N G (m) is reductive.To see this, note that N G (m) contains T , where T is a maximal torus of G containing S, so N G (m) 0 is generated by T and by certain root groups U α with respect to T .But if α is a root, g ∈ U α and x ∈ Lie(S) then g • x − x ∈ Lie(U α ), so U α normalises m if and only if U α centralises m if and only if dα annihilates m.We deduce that U α ⊆ N G (m) if and only if U −α ⊆ N G (m), and reductivity of N G (m) follows.This argument also shows that Suppose further that p is fabulous for G. Then centralisers and normalisers are smooth, so Proof.(a).Since c λ (h) is solvable and G-cr, c λ (h) consists of semisimple elements by Lemma 6.12.Hence c λ kills every nilpotent element of h, so n ⊆ Lie(R u (P λ )).But Lie(R u (P λ )) consists of nilpotent elements, so n = h ∩ Lie(R u (P λ )).This is an ideal of h because h ⊆ p λ .(b).Let s be a maximal G-toral subalgebra of h.We claim that c λ (s) = c λ (h).Suppose not.Now s is abelian and consists of ad-semisimple elements, so s acts completely reducibly on h.Let h 0 = c h (s) be the trivial weight space for the action of s on h; then, since the s-weight spaces of h corresponding to non-zero weights consist of nilpotent elements and c λ annihilates all such, we have c λ (h 0 ) = c λ (h).Note that s ⊆ h 0 as s is abelian.
By hypothesis, there exists x ∈ h 0 such that c λ (x) ∈ c λ (s).As p is fabulous for G, c g (s) = Lie(C G (s)) is Jordan-closed, by Remarks 6.2(b), and h is Jordan-closed by assumption, so h 0 = c h (s) = c g (s) ∩ h is Jordan-closed.Hence x s ∈ h 0 and c λ (x s ) = c λ (x) ∈ c λ (s); in particular, x s ∈ s.Since x s is semisimple and commutes with s, the subalgebra s ⊕ k • x s of h is toral by Lemma 6.10.But this contradicts the choice of s.We deduce that c λ (s) = c λ (h), which implies that h = s ⊕ n.

Characteristic 0
We keep our assumption that k is algebraically closed.If H is a G-cr subgroup of G then H is reductive, and the converse also holds if char(k) = 0 [40,Prop. 4.2].This gives an intrinsic characterisation of G-cr subgroups in characteristic 0: a subgroup H is G-cr if and only if it is reductive.(Note that a reductive subgroup of G need not be G-cr in positive characteristic: e.g., see [7,Ex. 3.45].)The situation for Lie algebras is not so clear-cut.
Example 7.1.Consider the 1-dimensional Lie algebra k.We may embed k in sl 2 as the subalgebra of traceless diagonal matrices, or as the subalgebra of strictly upper triangular matrices.It is clear that in the first case, the image of k is SL 2 -cr, while in the second it is not.
This example shows that even in characteristic 0, we cannot determine whether a subalgebra h of g is G-cr by looking only at the intrinsic properties of h.We can trace the problem to the lack of a Jordan decomposition for h: there is no intrinsic notion of semisimple and nilpotent elements.We do have notions of ad-semisimple and ad-nilpotent elements, but they don't detect properties of z(h).
There is an intrinsic characterisation for when a restricted subalgebra h of g is G-cr when the characteristic is positive and sufficiently large, using the notion of "p-reductive" subalgebras: see [22,Cor. 10.3].In loc.cit.one makes use of the extra structure arising from the p-power map on h.Below we give a counterpart to [22,Cor. 10.3] by providing a characterisation of G-cr subalgebras of g in characteristic 0 (Theorem 7.3), and we give an explicit description of a semisimplification of an arbitrary subalgebra in characteristic 0. Different methods from those of op.cit.are required, as there is no notion of restricted structure here.
The content of Theorem 7.3 in characteristic 0 follows quickly from work of Richardsonsee Remark 7.8 below.We give an independent proof, however, as some of our results hold under the weaker hypothesis that p is fabulous for G.
The next result is a Lie algebra counterpart of [7,Thm. 3.46]; the analogues to the separability conditions in loc.cit.hold here because of the assumption on p. Proposition 7.2.Suppose p is fabulous for G. Let h be a subalgebra of g such that h acts semisimply on g.Then h is G-completely reducible.
Proof.Suppose for a contradiction that h is not G-cr.Choose x = (x 1 , . . ., x m ) ∈ h m for some m ∈ N such that the x i span h.The orbit G • x is not closed (Theorem 3.11), so there exists λ ∈ Y (G) such that x ′ := lim a→0 λ(a) • x exists and G • x ′ is closed.This implies that dim(G • x) > dim(G • x ′ ).Consequently, letting h ′ be the subalgebra generated by the components of x ′ , we have dim(C G (h ′ )) > dim(C G (h)), so dim(c g (h ′ )) = dim(Lie(C G (h ′ ))) > dim(Lie(C G (h))) = dim(c g (h)), where the equalities hold because p is fabulous for G. Now since ad g (h) is GL(g)-cr and ad g (x) is a generating tuple of ad g (h), GL(g) • ad g (x) is closed, by Theorem 3.11.Hence ad g (x ′ ) = ad g (lim a→0 λ(a) • x) = lim a→0 λ(a) • ad g (x) is GL(g)-conjugate to ad g (x).But c g (h ′ ) and c g (h) are precisely the subsets of g annihilated by ad g (h ′ ) and ad g (h), respectively, so c g (h ′ ) and c g (h) are GL(g)-conjugate, which implies they have the same dimension.This gives a contradiction.We conclude that h must be G-cr after all, so we are done.
Before we proceed to Theorem 7.3, we need to recall some standard Lie algebra theory.If h is a Lie algebra then we define rad(h) to be the solvable radical of h: that is, the unique largest solvable ideal of h.We say that h is semisimple if rad(h) = 0, [23, §3.1].If char(k) = 0 then any finite-dimensional representation of a semisimple Lie algebra is completely reducible Remark 7.7.The equivalence of (ii) and (iii) in Theorem 7.3 can fail in positive characteristic.For example, let p, G and h be as in Example 2.9.We observed earlier that h is G-ir, but it is easy to check that every element of h is nilpotent.
Conversely, let k be algebraically closed of characteristic 3, let M = SL 2 , let V be the natural module for M, and consider the action of M on the third symmetric power W := S 3 Vthis gives an embedding of M inside G := GL 4 and gives rise to a faithful representation of the Lie algebra m.We claim that W is not semisimple as an m-module, so m is not G-cr even though is has trivial radical.The four-dimensional module W has a basis x 3 , x 2 y, xy 2 , y 3 , where x and y can be identified with the standard basis vectors for V .The basis vectors x 3 and y 3 are killed by the action of m, so span a two-dimensional submodule with a trivial maction.The quotient by this submodule is a copy of the natural module V , which is simple, but there is no submodule of W isomorphic as an m-module to V , as direct calculation with the standard generators for m ∼ = sl 2 will easily verify.
Corollary 7.6 can also fail in positive characteristic.For, let char(k) = 2 and let G = PGL 2 (k) × k * × k * and let h be the subalgebra of g spanned by the elements of the form 0 a 0 0 , a, 0 and 0 0 b 0 , 0, b for a, b ∈ k.Clearly h is G-cr but is not Jordanclosed.
Remark 7.8.Richardson's seminal paper [38] laid the foundations for the study via GIT of Gcomplete reducibility for subgroups and subalgebras.We explain how to obtain Theorem 7.3 in the characteristic 0 case from Richardson's results.Suppose char(k) = 0.If h is a subalgebra of g then there is a unique smallest subalgebra h alg of g such that h ⊆ h alg and h alg is algebraic, and there is a unique connected subgroup A(h) of G such that h alg = Lie(A(h)).
In fact, the map K → Lie(K) gives an inclusion-preserving bijection from the set of connected subgroups of G to the set of algebraic subalgebras of g [24, 13.1 Thm.].Now let m ∈ N, let x 1 , . . ., x m ∈ g, let h be the subalgebra of g generated by the x i and let M = A(h).Richardson showed that the following are equivalent (see [38,Lem. 3.5 and Thm. 3.6]): (a) G • (x 1 , . . ., x m ) is closed; (b) M is reductive; (c) M acts semisimply on g.These conditions are equivalent to h being G-cr, by Theorem 3.11.Note that if M is reductive then rad(h alg ) = rad(Lie(M)) = Lie(Z(M) 0 ) is G-toral, and it is not hard to see that the converse holds.Moreover, it also straightforward to show that rad(h alg ) = rad(h) alg , and that rad(h) alg is G-toral if and only if rad(h) is G-toral.The characteristic 0 case of Theorem 7.3 now follows.
Example 7.9.Assume char(k) = 0. We now give an explicit description of the k-semisimplification of a subalgebra h of g.Write h = k ⊕ m, where k is semisimple and m = rad(h).Then k is Jordan-closed (see the proof of Corollary 7.6), and it follows easily that h J = k⊕m J .Now m J is solvable by Proposition 6.13.Write m J = s ⊕ n as in Corollary 6.14.We claim that k ⊕ s is a semisimplification of both h and h J .
To see this, choose a parabolic subgroup P of G such that P yields a semisimplification of h J .Since k ⊕ s is G-cr by Theorem 7.3, we can choose λ ∈ Y (G) such that P λ = P and λ centralises k ⊕ s.Now m J is an ideal of h J , so (P λ , L λ ) also yields a semisimplification of m J by Theorem 5.

Lemma 2 . 1 .
a→0 λ(a)gλ(a) −1 exists} and L λ = C G (Im(λ)) ⊆ P λ (for the definition of a limit, see [41, §3.2.13]).We have P λ = L λ = G if and only if Im(λ) is contained in the centre of G. Any parabolic subgroup (resp., any Levi subgroup of a parabolic subgroup of G) is of the form P λ (resp., L λ ) [41, Prop.8.4.5].In fact, this also holds for k-defined parabolic and Levi subgroups of G, by the following, which is [41, Lem.15.1.2(ii)]:Every pair (P, L) consisting of a parabolic k-subgroup P of G and a Levi k-subgroup L of P is of the form (P, L) = (P λ , L λ ) for some λ ∈ Y k (G), and vice versa.

Lemma 2 . 4 .
Let x ∈ g.Then with the notation from above we have (i) x ∈ p λ if and only if lim a→0 λ(a) • x exists; (ii) x ∈ l λ if and only if lim a→0 λ(a) • x = x if and only if the image of λ centralizes x; (iii) x ∈ Lie(R u (P λ )) if and only if lim a→0 λ(a) • x exists and equals 0.The map c λ := c l λ : p λ → l λ given by x → lim a→0 λ(a) • x, where the action is given by adjoint action, coincides with the usual projection of p λ onto l λ .The kernel of c λ is Lie(R u (P λ )), and the set of fixed points of c λ is l λ .Let m ∈ N and recall that G acts on g m by simultaneous conjugation.Given λ ∈ Y (G), we have a map p m λ → l m λ given by x → lim a→0 λ(a) • x; we abuse notation slightly and also call this map c λ .For any x ∈ p m λ , there exists a Levi k-subgroup L of P λ with x ∈ l m if and only

Lemma 2 . 10 .
Let G be connected.Suppose the centraliser of every k-subgroup of G and every k-subalgebra of g is smooth.(a) Let S be a k-torus of G and let L = C G (S). Then the centraliser in L of every k-subgroup of L and every k-subalgebra of Lie(L) is smooth.(b) Let S be a k-subgroup of Z(G) 0 .Then the centraliser in L of every k-subgroup of G/S and every k-subalgebra of Lie(G/S) is smooth.Proof.(a) Let H be a k-subgroup of L. Set M = C G (H) and m = Lie(M).Then S ⊆ M and m = c g (H) by assumption.We have C L (H) = C M (S) and c l (H) = c m (S).But centralisers of tori are smooth, so Lie 1.3].Theorem 3.2 (Rational Hilbert-Mumford Theorem).Let G, X, x be as above.Then there is a unique G(k)-orbit O such that (a) O is cocharacter-closed over k, and (b) there exists λ ∈ Y k (G) such that lim a→0 λ(a) • x belongs to O.

Definition 3 .
13. ([39, 2.1.4,2.1.5,2.2.1]) (i) Two simplices σ, τ in ∆ k are called opposite if when we write σ = σ P and τ = σ Q for k-parabolic subgroups P and Q of G, then P and Q are opposite in G; that is, P ∩ Q is a common Levi subgroup of P and Q. (ii) A subcomplex Σ of ∆ k is called convex if the corresponding subset Σ ⊆ ∆ k of the geometric realisation is convex.

Proposition 4 . 1 .
Let (x 1 , . . ., x m ) ∈ g m and let m be the subalgebra spanned by the x i .Then the following are equivalent:(a)π g m ,G (x 1 , . . ., x m ) = π g m ,G (0, . . ., 0).(b) For every nonconstant homogeneous f ∈ k[g m ] G , we have f (x 1 , . . ., x m ) = 0. (c) There exists λ ∈ Y (G) such that lim a→0 λ(a) • (x 1 , . . ., x m ) = (0, . . ., 0).(d) There exists a maximal unipotent subgroup U of G such that m ⊆ Lie(U).Proof.The equivalence of (a), (b) and (c) follows from the results given at the start of the section.The equivalence of (c) with (d) follows from Lemma 2.4(iii), since a maximal unipotent subgroup of G is the unipotent radical of a Borel subgroup.Example 4.2.Let p = 2 and let G = SL 3 (k).Let H = PGL 2 (k).The adjoint representation of SL 2 (k) on its Lie algebra gives rise to an embedding i of H in G; with a suitable choice of basis, this takes the form i pair (x 1 , x 2 (a)) spans the subalgebra h from Example 2.9, which is H-ir by Example 3.5.It follows from Theorem 3.11 that H • (x 1 , x 2 (a)) is closed for each a.It is easily seen that the (x 1 , x 2 (a)) are pairwise non-conjugate under the H-action.Hence the points π h 2 ,H (x 1 , x 2 (a)) are pairwise distinct in h 2 / /H.On the other hand, di(h) ⊆ Lie(B), where B is the Borel subgroup of upper triangular matrices in G, so π g 2 ,G (x 1 , x 2 (a)) = π g 2 ,G (0, 0) for all a, by Proposition 4.1.

Theorem 5 . 3 .
Let H be a subgroup of G. Then any two k-semisimplifications of H are G(k)-conjugate.
m k is fixed by all simplicial automorphisms of ∆ k stabilizing ∆ m k , by Theorem 3.16.Since N G (m)(k) clearly stabilizes ∆ m k , we can conclude that N G (m)(k) ⊆ P .Now, because p is fabulous, N G (m) is smooth, and because k = k s this means that the k-points of N G (m) are dense by [15, AG.13.3 Cor.].Therefore, we may conclude that N G (m) ⊆ P .Finally, this gives m ⊆ h ⊆ n g (m) = Lie(N G (m)) ⊆ Lie(P ), 14(b).Hence c λ kills n by Corollary 6.14(a).It follows that c λ (hJ ) = k ⊕ s.If a ∈ k * then λ(a) • m ⊆ p λ and c λ (λ(a) • m) = c λ (m) since λ centralises s.Hence c λ (m) = c λ (m ′ ), where m ′ is the subspace of p λ spanned by the subspaces λ(a) • m for a ∈ k * .
for all y ∈ h}.(H) = {x ∈ g | h•x = x for all h ∈ H}, and C G (H) is smooth if and only if C G (H) = C G (H) if and only if c g (H) = Lie(C G (H)).We recall a useful fact [15, III.9.2 Cor.]: if S is a torus in a linear algebraic group M then C M (S) is smooth.
of Lie(M) consisting of nilpotent elements and h is plongeable in M then h is plongeable in G.Remarks 2.8.(i) Suppose k = k and let h be a subalgebra of g.Then h is plongeable if and only if h ⊆ Lie(B) for some Borel subgroup B of G, and this is the case if and only if h ⊆ Lie(U) for some maximal unipotent subgroup U of G.
and the comorphism ψ * : R → S is graded because the inclusion of h m in g m is linear.It follows from Eqn. (4.5) that S + is the radical of Sψ * (R + ), so S is a finite R-module by Lemma 4.3.Hence ψ is finite.
This implies that Z is a torus, as proper Levi subgroup L of G. Then h is L-cr by Proposition 3.12(ii) and dim(L) < dim(G), so h is L-toral by our induction hypothesis.Hence h is G-toral, as required.We can now give a classification result for maximal solvable subalgebras of g when p is fabulous for G: compare [22, Thm.D(b)].Proposition 6.13.Suppose p is fabulous for G. Let h be a solvable subalgebra of g.Then h ⊆ h J ⊆ Lie(B) for some Borel subgroup B of G.In particular, a maximal solvable subalgebra of g is the Lie algebra of some Borel subgroup.Proof.Since Lie(B) is Jordan-closed for any Borel subgroup B of G (Remarks 6.2(b)), it suffices to prove that h ⊆ Lie(B) for some Borel subgroup B. Let λ ∈ Y (G) such that (P λ , L λ ) yields a k-semisimplification of h.Then s := c λ (h) ⊆ l λ is solvable and G-cr and h ⊆ s + Lie(R u (P λ )).Now s is L λ -cr by Proposition 3.12(ii), so s is L λ -toral by Lemma 6.12: say, s ⊆ Lie(S) for some torus S of L λ .It follows that h ⊆ Lie(S) + Lie(R u (P λ )) = Lie(SR u (P λ )).But SR u (P λ ) is contained in a Borel subgroup of G, so the first assertion follows.The second assertion is immediate.
Corollary 6.14.Suppose p is fabulous for G. Let h be a Jordan-closed solvable subalgebra of g and let n be the set of nilpotent elements of h.Let λ ∈ Y (G) such that (P λ , L λ ) yields a semisimplification of h.Then: