Sandwich elements and the Richardson property

Let $\mathcal{L}$ be finite dimensional restricted Lie algebra over an algebraically closed field $k$ of characteritic $p>3$. A finite dimensional restricted $\mathcal{L}$-module $V$ is called Richardson if $V$ is faithful and there exists a subspace $R$ of $\mathfrak{gl}(V)$ such that $[\mathcal{L},R]\subseteq R$ and $\mathfrak{gl}(V)=\mathcal{L}\oplus R$, where we identify $\mathcal{L}$ with its image in $\mathfrak{gl}(V)$. In this paper we show that if $\mathcal{L}$ admits an irreducible Richardson module then it is isomorphic (as a restricted Lie algebra) to the Lie algebra of a reductive $k$-group.

According to [Pre90], if MT(L) = s then there exist homogeneous polynomial functions ψ 0 , . . ., ψ s−1 ∈ k[L] such that deg ψ i = p s+e − p i+e for 0 ≤ i ≤ s − 1 and By [Pre90,Pre03b], each ψ i has the property that ψ i (x [p] ) = ψ i (x) p for all x ∈ L and is invariant under the natural action of the restricted automorphism group Aut(L, [p]) on k[L].This implies that the variety N(L) coincides with the zero locus of ψ 0 , . . ., ψ s−1 .Since t ∩ N(L) = {0} for any s-dimensional torus t of L, the Affine Dimension Theorem yields that all irreducible components of the nilpotent cone N(L) have dimension equal to dim L − MT(L).As a consequence, the homogeneous polynomial functions ψ 0 , . . ., ψ s−1 form a regular sequence in k[L] and, in particular, are algebraically independent.In [Pre90], the author conjectured that the variety N(L) is irreducible for any finite dimensional restricted Lie algebra L. This conjecture is verified in many cases, but is still open, in general.
Let t be a maximal toral subalgebra of L. The adjoint action of t on L gives rise to a Cartan decomposition x for all t ∈ t}.
1.2.Motivation for the Richardson property.Given χ ∈ L * we denote by I χ the two-sided ideal of the universal enveloping algebra U (L) generated by all elements x p − x [p] − χ(x) p • 1 with x ∈ L (these elements are central in U (L)).The factor-algebra U χ (L) := U (L)/I χ is called the reduced enveloping algebra associated with χ.It is well-known that for any irreducible L-module V there exists a unique linear function χ = χ V ∈ L * such that (x p − x [p] ) V = χ(x) p • Id V for all x ∈ L. In other words, the action of L on V extends to that of the associative algebra U χ (L).
We denote by I χ the ideal of the symmetric algebra S(L) generated by all elements (x − χ(x)) p with x ∈ L. Since I χ is a Poisson ideal of the Lie-Poisson algebra S(L), the factor-algebra S χ (L) := S(L)/I χ caries a Poisson algebra structure induced by that of S(L).We call S χ (L) the reduced symmetric algebra associted with χ.It is well-known that both U χ (L) and S χ (L) have dimension p dim L over k.The Lie algebra L acts on U χ (L) and S χ (L) by derivations turning both algebras into restricted L-modules.The module structures thus obtained are induced by the adjoint action of L on itself, and we shall use the subscript "ad" to indicate the Lie algebra actions.In general, it is unknown for which restricted Lie algebras the modules U χ (L) ad and S χ (L) ad are isomorphic.Answering this question would be very important from the point of view of the theory of support varieties developed by Friedlander and Parshall [FP86,FP88] in the context of U χ (L)-modules.
, the restricted nullcone of L. The k-subalgebra of U χ (L) generated by a nonzero x ∈ L will be denoted by u χ (x).It follows from the PBW theorem that for x ∈ N p (L) \ {0} the map X → x − χ(x) gives rise to an algebra isomorphism u χ (x) ∼ = k[X](X p ).Given a finite dimensional U χ (L)-module M we write V L (M ) × for the set all nonzero x ∈ N p (L) such that M is not a free u χ (x)-module, and put Let E 1 , . . ., E s be representatives of the equivalence classes of all irreducible U χ (L)-module, and set . This Zariski closed, conical subset of N p (L) contains the rank varieties of all finite dimensional U χ (L)-modules.By an important result of Jantzen [Jan86], we have that V L (0) = N p (L), and it is proved in [Pre99] From the representation-theoretic viewpoint it would be very useful to have a more explicit description of the variety According to [Pre99], there is a very large class of restricted Lie algebras for which the L-modules U χ (L) ad and S χ (L) ad are isomorphic.It includes the restricted Lie algebras L of endomorphisms of a finite dimensional vector space V (over k) admitting direct complements in gl(V ) invariant under the adjoint action of L on gl(V ) as well as the so-called saturated Lie subalgebras of such Lie algebras; see [Pre99,§3] for relevant definitions and more detail.
1.3.The statement of the main result.A restricted Lie algebra is said to possess the Richardson property if there exists a finite dimensional faithful restricted L-module V with associated representation ρ : Let G be a connected reductive k-group.Recall that p = char(k) is a good prime for G if all coefficients of the positive roots of G expressed via a basis of simple roots are less than p.If p is not good for G then we say that p is bad.The bad primes of connected reductive groups lie in the set {2, 3, 5} and p = 5 is bad for G if and only if G has a component of type E 8 .A good prime p is called very good if G has no components of type A kp−1 with k ≥ 1.We say that G is standard if p is a good prime for G, the derived subgroup of G is simply connected, and the Lie algebra g = Lie(G) admits a non-degenerate (Ad G)-invariant symmetric bilinear form.A rational representation φ : G → GL(V ) is called infinitesimally irreducible if the differential d e φ : g → gl(V ) is an irreducible representation of the Lie algebra g.
The main goal of this note is to confirm the above-mentioned conjecture under the assumption that p > 3.More precisely, we prove the following: Theorem 1.1.Suppose char(k) = p > 3 and let L be a restricted Lie algebra over k possessing the Richardson property with respect to a finite dimensional faithful irreducible restricted representation ρ : L → gl(V ).Then there exists a standard reductive algebraic k-group G with Lie algebra g and an infinitesimally irreducible rational representation φ : G → GL(V ) such that (d e φ)(g) = ρ(L).Moreover, L and g are isomorphic as restricted Lie algebras.By Schur's lemma, if L admits a faithful irreducible representation then the centre of L has dimension ≤ 1.In the final part of this note we show that if p is a very good prime for a standard algebraic group G whose centre has dimension ≤ 1, then the restricted Lie algebra g = Lie(G) possesses the Richardson property with respect to a faithful irreducible representation ρ : g → gl(V ) such that p ∤ dim V .Furthermore, ρ = d e φ, where φ : G → GL(V ) is an infinitesimally irreducible rational representation, and the (Ad G)-invariant trace form (X, Y ) → tr ρ(X) • ρ(Y ) on g is non-degenerate.
In characteristics 2 and 3, we show that if L has the Richardson property with respect to a finite dimensional faithful restricted L-module V , then for any nonzero e ∈ N(L) there exists an element h ∈ L ss such that [h, e] = e.In conjunction with some results of [Pre87b] and [Pre90] mentioned in Subsection 1.1 this implies that all Cartan subalgebras of L are toral and have the same dimension equal to MT(L).In characteristics 3, we show that any e ∈ N(L) lies in the subspace [e, [e, L]].This enables us to deduce that the solvable radical of any subalgebra L(γ 1 , . . ., γ d ) is toral and coincides with the centre of L(γ 1 , . . ., γ d ).We then use Skryabin's classification [Sk98] of simple 3-modular Lie algebras of toral rank 1 to show that for every γ ∈ Γ(L, t) the radical of L(γ) coincides with ker γ = {t ∈ t | γ(t) = 0} and L[γ] = L(γ)/ ker γ is one of sl 2 or psl 3 .It would be interesting to classify the finite dimensional restricted Lie algebras over fields of characteristic 3 having these properties.
1.4.Lie algebras without strong degeneration.The key idea of the proof of Theorem 1.1 is to show that if L has the Richardson property then the factor-algebra L/z(L) does not contain nonzero elements c with (ad c) 2 = 0 and then apply the classification of such Lie algebras obtained in [Pre86,Pre87a].This is the main reason for us to impose the assumption that p > 3.
An element c ∈ L is called a sandwich element if (ad c) 2 = 0.This term, coined by A.I. Kostrikin, has to do the fact that in characteristic p > 2 any c ∈ L with (ad c) 2 = 0 satisfies the sandwich identity It is immediate from this identity that the set C(L) of all sandwich elements of L is closed under taking Lie brackets.In conjunction with the Engel-Jacobson theorem on weakly closed sets this shows that C(L) generates a nilpotent Lie subalgebra of L invariant under the automorphism group of L. We say that L is strongly degenerate if C(L) ̸ = {0}.was first confirmed under various additional assumptions on L; see [Kos67], [Ja71], [Str73] and [Ben77] 1 .In full generality, the conjecture was proved in [Pre87a] for p > 5 and in [Pre86] for p = 5.
In fact, a slightly more general result was proved in [Pre87a,Pre86] Acknowledgement.This work was started during the author's stay at MSRI (Berkeley) in February-April 2018.I would like to thank the Mathematical Sciences Research Institute for its hospitality and creative atmosphere during the programme "Group Representation Theory and Applications".

The case of arbitrary Richardson L-modules
2.1.From now on all L-modules are assumed to be finite dimensional and restricted.We say that a faithful L-module where we identify L with its image in gl(V ).Given a module M for a Lie algebra g and v ∈ M we denote by g v the stabiliser of v in g.We write z(g) for the centre of g and rad g for the radical of g, the largest solvable ideal of g.
The following lemma is inspired by a very old observation of Jacobson.
Lemma 2.1.Let V be a Richardson module for L (not necessarily irreducible).
(2) If p > 0 then for any e ∈ N(L) there is an element h ∈ L ss such that [h, e] = e.
Proof.Thanks to our conventions we may assume that e is a nonzero nilpotent element of gl(V ).We first look for x, y ∈ gl(V ) such that e = [e, [e, x]] and e = [y, e].For p > 2 one can argue as in [Jac51, Lemma 2] to observe that e can be included into an sl 2 -triple {e, h, f } ⊂ gl(V ).For p > 0, the Lie algebra gl(V ) admits a Z-grading gl(V ) = i∈Z gl(V ) i such that e ∈ gl(V ) 2 and [e, gl(V ) 0 ] = gl(V ) 2 ; see [Pre03a, Theorem A(ii)].This result is applicable in arbitrary characteristic since gl(V ) admits a non-degenerate (Ad GL(V ))-invariant trace form.
As a consequence, there is y ∈ gl(V ) such that [y, e] = e.Write y = y ′ + y ′′ with y ′ ∈ L and y ′′ ∈ R. As e ∈ L and [L, R] ⊆ R it must be that [y ′′ , e] = 0 and [y ′ , e] = e.As L is a restricted Lie subalgebra of gl(V ) we can replace y ′ by its semisimple part y ′ s ∈ ⟨y ′ ⟩ p to find a semisimple element h ∈ L such that [h, e] = e.This proves (2).
preserves both L and R it must be that [e, [e, f ′ ]] ∈ k × e.This proves (1).□ 2.2.There are examples of restricted Lie algebras admitting infinite families of indecomposable Richardson modules.Indeed, suppose L = sl 2 and p > 3. Let V (m) be the Weyl module for the k-group SL(2) with highest weight m ∈ Z ≥0 .Differentiating the rational action of SL(2) endows V (m) with a natural restricted L-module structure.According to [Pre91] the restricted L-module V (m) is indecomposable if and only if p ∤ (m + 1).It is well-known that the L-module V (m) is simple if and only if m ≤ p − 1.Furthermore, the Steinberg module St = V (p − 1) ∼ = V (p − 1) * is simple and projective over the restricted enveloping algebra U 0 (L).Suppose m = kp + l where k ∈ Z ≥0 and 1 ≤ l ≤ (p − 3)/2.Then [Pre91, Lemma 2.6(i)] shows that where P is a projective U 0 (L)-module isomorphic to a direct sum of k copies of St ⊗ V (kp + 2l + 1).The Weyl module V (kp + 2l + 1) has two composition factors V (2l + 1) and V (p − 3 − 2l), while the composition factors of the projective ) and V (2) ∼ = L as L-modules, the above yields that the L-module gl(V (m)) contains a direct summand R of codimension 3 such that Hom L (L, R) = 0. Identifying the Lie algebra L with its copy in gl(V (m)) we now get If L = sl 2 and p = 3 then L ∼ = V (p − 1) is a projective module over U 0 (L).This means that any non-trivial finite dimensional restricted L-mudule is Richardson for L. Such instances are, of course, extremely rare.2.3.Suppose L admits a Richardson module V (not necessarily irreducible).Our next result provides some insight into the structure of L.
Theorem 2.2.The following are true: (1) If p ≥ 2 the all Cartan subalgebras of L are toral of dimension equal to MT(L).
(2) Suppose p > 2 and let t be any toral subalgebra of L. Then rad c L (t) = z(c L (t)) is a toral subalgebra of L and the factor-algebra c L (t)/z(c L (t)) has no nonzero sandwich elements.
(3) If p > 3 then the factor-algebra L/z(L) is almost classical.
Proof.Let h be a Cartan subalgebra of L. It is well-known (and easy to see) that h = c L (t ′ ) where t ′ = h ∩ L ss , a maximal toral subalgebra of L. Let Γ be the set of roots of L with respect to t ′ and R = R ⊕ γ∈Γ L γ .Then gl(V ) = h ⊕ R and [h, R] ⊆ R, meaning that V is a Richardson module for h.If h contains a nonzero nilpotent element, e say, then Lemma 2.1(2) yields that there is an h ∈ h such that [h, e] = e.But then (ad h) n (e) = e ̸ = 0 for all n ∈ Z >0 .Since h is nilpotent this is impossible.Since h is a restricted subalgebra of L, the semisimple and nilpotent parts of any element of h lie in h.Consequently, h = h ss = t ′ .Now let t be any torus of maximal dimension in L. Then dim t ≥ dim t ′ and dim c L (t) ≤ dim h by the main result of [Pre87b] (see Subsection 1.1 for a related discussion).Since h = t ′ we now deduce that dim h = dim t ′ = MT(L), proving (1).Suppose p > 2 and let t be a toral subalgebra of L (possibly zero).Let L 0 = c L (t) and write Γ(L, t) for the set of roots of L with respect to t (possibly empty).Now set R := R ⊕ γ∈Γ(L,t) L γ .
Then gl(V ) = L 0 ⊕ R and [L 0 , R] ⊆ R, showing that V is a Richardson module for L 0 .Let z be the centre of L 0 , a restricted ideal of L 0 .If z contains a nonzero nilpotent element, e say, then e ∈ [e, [e, L 0 ]] by Lemma 2.1(1).But then 0 ̸ = e ∈ [e, z], a contradiction.Hence z is toral.Suppose L 0 /z contains a nonzero sandwich element.Then there is c ∈ L 0 \z such that [c, [c, L 0 ]] ⊆ z.Since p ≥ 3 it must be that c [p] ∈ z.As z is toral, we may replace c by a suitable element of the form c + z with z ∈ z to assume further that c is nilpotent.By Lemma 2.1(1), we then have c ∈ [c, [c, L 0 ]] ⊆ z, a contradiction.Hence L 0 /z has no nonzero sandwich elements.In particular, L 0 /z has no nonzero abelian ideals.As a consequence, the radical of L 0 /z is trivial.
Let r = rad L 0 .Then z ⊆ r and r/z is a solvable ideal of L 0 /z.The preceding remark shows that r = z, proving (2).Part (3) now follows from (2) and the main results of [Pre87a] and [Pre86] (see our discussion in Subsection 1.4 for more detail).□ In characteristic 3, one can use Skryabin's description of simple Lie lagebras of rank 1 to get more information on the 1-sections of L.
Corollary 2.3.Suppose p = 3 and let α ∈ Γ(L, t) where t is a maximal torus of L. Then rad L(α) = t(α) = ker α and In any event, S is a restricted ideal of L[α].Let S be the inverse image of S in L(α), a restricted ideal of L(α), and let t ′ be a maximal torus of S. Since t ′ contains t(α) it is straightforward to see that dim t ′ = dim t.Applying Theorem 2.2(1) once again yields that t ′ is a self-centralising maximal torus of L(α).As [t ′ , L(α)] ⊆ S, all root subspaces of L(α) with respect to t ′ are contained in S.But then L(α) ⊆ t ′ + S forcing L(α) = S.This completes the proof.□

The case of irreducible Richardson modules
3.1.From now on we assume that p > 3 and our Richardson L-module V is irreducible.In particular, this means that c gl(V ) (L) consists of scalar endomorphisms of V (by Schur's lemma).Therefore, either c gl(V ) (L) = z(L) or L is centreless and c gl(V ) (L) ⊂ R. By Theorem 2.2(3) (and our discussion in Subsection 1.4) there exists a semisimple algebraic k group G of adjoint type such that L/z(L) identifies with a Lie subalgebra of g := Lie(G) containing [g, g].Since G is a group of adjoint type and p > 3, it follows from [BGP09, Lemma 2.7], for example, that the restricted Lie algebra g is isomorphic to Der [g, g].Therefore, we may identify L := L/z(L) with a restricted ideal of g containing [g, g].
Let T be a maximal torus of G and denote by Φ the root system of G with respect to T .Let Π be a set of simple roots in Φ and write Φ + = Φ + (Π) for the set of positive roots with respect to Π.All root subspaces g α with α ∈ Φ are contained in [g, g] and L = t ⊕ α∈Φ g α where t := L ∩ Lie(T ).Since [g α , g −α ] ⊂ t for all α ∈ Φ and p > 3, it follows from Seligman's results that t is a self-centralising maximal torus of L and for every γ ∈ Γ( L, t) there is a unique γ ∈ Φ such that γ = (d e γ)|t; see [Sel67, Ch.II, §3].
Let t be the inverse image of t in L, By Theorem 2.2(1), this is a toral Cartan subalgebra of L.
The above discussion enables us to identify Φ with the set of roots Γ(L, t), i.e. given x ∈ t we write α(x) instead of (d e α)(x) with x = x + z(L).We have L = t ⊕ α∈Φ L α and each root subspace L α = ke α with respect to t is 1-dimensional.We choose root vectors e α in such a way that [[e α , e −α ], e α ] = 2e α for all α ∈ Φ + and embed the torus T of G into Aut(L) by setting t(x) = x and t(e α ) = α(t)e α for all t ∈ T , x ∈ t and α ∈ Φ.

Given
), it has p-character χ when regarded as a module over (L, [p ′ ]).Proof.Suppose for a contradiction that χ does not vanish on s β for some β ∈ Φ + .The above discussion enables us to view V as a U χ (s β )-module.(To ease notation we identify χ with its restriction to s β .)By the choice of β, the stabiliser of χ in s β is a proper Lie subalgebra of s β , hence 1-dimensional.Since the restricted Lie algebra sl 2 has the Richardson property, our discussion in Subsection 1.2 shows that V s β (χ) = N p ((s β ) χ ) is either zero or a single line depending on χ.

It follows from Lemma 3.1 that e
[p] ±α = 0 and h

[p]
α = h α for all α ∈ Φ + .This must hold in the case where L is centreless as well, since L then admits a unique [p]-structure.Now one can check directly the action of the maximal torus T of G on L described in Subsection 3.1 respects the [p]-structure of L, i.e. has the property that t(x [p] ) = t(x) [p] for all t ∈ T and x ∈ L (one has to keep in mind here that t is restricted and T acts trivially on t).As a consequence, T embeds into the automorphism group of the restricted enveloping algebra U 0 (L).
Lemma 3.2.The module V admits a rational T -action compatible with the action of T on L.
Proof.By [GG82, Prop.3.5] and [Jan00, Corollary 2], the simple U 0 (L)-module V is gradable, i.e. decomposes into a direct sum of T -weight spaces V = λ∈X(T ) V µ in such a way that each weight space V µ is t-stable and t(e α .v)= (µ + α)(t)v for all t ∈ T , α ∈ Φ and v ∈ V µ .□ Since G is a group of adjoint type the torus T acts faithfully on [g, g].As L embeds into gl(V ), this implies that T acts faithfully on V .We may thus identify T with a connected subgroup of GL(V ).Since T ⊂ N GL(V ) (L) the Lie algebra Lie(T ) normalises L.
Recall from Subsection 3.1 that L = L/z(L) is sandwiched between g and [g, g].
Proposition 3.3.We have the equality L = g.

Proof.
Let n(L) denote the normaliser of L in gl(V ).Given x ∈ n(L) we write Lemma 3.4.The derived subalgebra of g has codimension ≤ 1 in g.

Proof.
In view of Proposition 3.3, in order to prove the lemma it suffices to show that L has codimension ≤ 1 in L. Since L is a direct summand of gl(V ), a self-dual L-module, the coadjoint L-module L * must be a direct summand of gl(V ) as well.Since the trivial L-submodule (L * ) L of L * identifies canonically with the dual space (L/[L, L]) * , we have that If follows that either g is perfect or [g, g] has codimension 1 in g. □ Lemma 3.4 shows that the group G cannot have more that one component of type A kp−1 , and its proof implies that [L, L] has codimension ≤ 1 in L.
where G 1 has no components of type A kp−1 and G 2 ∼ = PGL kp .Setting g i := Lie(G i ) for i = 1, 2 we get two commuting restricted ideals of g such that g = g 1 ⊕ g 2 .Furthermore, [g 1 , g 1 ] = g 1 is centreless and is a completely reducible ad g-module.Since pgl kp is a direct summand of g, this is impossible.By contradiction, the result follows.□ , where the G i 's are as in the proof of Lemma 3.5, we let G = G 1 × G 2 be the reductive k-group such that G 1 is a semisimple, simply connected cover of G 1 and G 2 = GL kp .
If G has no components of type A kp−1 and L is centreless, we set G := G 1 .Finally, if G has no components of type A kp−1 and dim z(L) = 1 , we set G := T 0 × G 1 where T 0 is a 1-dimensional central torus of G.In all cases, the derived subgroup of G is semisimple and simply connected.Since p > 3, the Lie algebra g 1 ∼ = Lie( G 1 ) is a direct sum of simple ideals.By [Pre97, Lemma 2.2], the Lie algebra g := Lie( G) admits a non-degenerate symmetric (Ad G)-invariant bilinear form, say b : g × g → k, and it follows from [Gar09, Theorem A] that b can be chosen to be a trace form (associated with a rational representation of G) provided that p is a good prime for G 1 .
Proposition 3.6.In all cases, the restricted Lie algebra L is isomorphic to g. .
Since the above discussion implies that [g, g] admits a unique non-trivial central extension (up to equivalence), it follows from Lemma 3.5 that [L, L] ∼ = g 1 ⊕ sl kp as ordinary Lie algebras.Since Lemma 3.1 entails that each subspace [L α , L −α ] is spanned by a toral element of L, this is, in fact, an isomorphism of restricted Lie algebras.
Let {α 1 , . . .α kp−1 } ⊆ Π be the simple roots of G 2 with respect to T numbered as in [Bou68, Planche I].Since g 2 ∼ = pgl kp there exists h ∈ Lie(T ) such that [h, g 1 ] = 0, α 1 (h) = 1, and α i (h) = 0 for i > 0 Since h ̸ ∈ [g, g] we have that g = kh ⊕ [g, g].Since z(L) is a torus, it is immediate from Proposition 3.3 that there exists a toral element ĥ ∈ L which maps onto h under the canonical homomorphism L ↠ L = g.Since L = k ĥ ⊕ [L, L] and [L, L] ∼ = g 1 ⊕ sl kp , applying Lemma 3.1 once again we deduce that L ∼ = g 1 ⊕ gl kp ∼ = Lie( G) as restricted Lie algebras.This completes the proof.□ 3.6.Proposition 3.6 enables us to identify the restricted Lie algebras L and g and regard V as an irreducible restricted g-module.We may also assume that T is a maximal torus of the reductive group G. Let n + and n − denote the k-spans of all e α and all e −α with α ∈ Φ + , respectively.By Lemma 3.2, there is a rational action of T on V compatible with that of g.Since V is irreducible, the fixed-point space has dimension 1 and is spanned by a highest weight vector v λ for T , where λ ∈ X(T ).Also, V = U 0 (n − ) .v λ .Since the derived subgroup of G is simply connected, [Jan00, Theorem 2] shows that λ ∈ X(T ) can be chosen to be dominant and p-restricted, that is 0 ≤ ⟨λ, α ∨ ⟩ ≤ p − 1 for all α ∈ Π.We denote by L(λ) the irreducible rational G-module with highest weight λ and write ρ for the corresponding representation of G in GL(L(λ)).By the general theory of linear algebraic groups, the differential d e ρ : g → gl(L(λ)) is a restricted representation of g.
Proof.Since λ ∈ X(T ) is a p-restricted dominant weight, [Jan03, Part II, Prop.9.24(b)] yields that the g-module L(λ) is irreducible.On the other hand, it is well-known that the irreducible restricted g-modules are determined up to isomorphism by their highest weights; see [Jan03, Part II, Prop.3.10], for example.This shows that the restricted g-modules V and L(λ) are isomorphic.□ 3.7.By our discussion in Subsection 3.5, the derived subgroup of G is simply connected and the Lie algebra Lie( G) admits a non-degenerate (Ad G)-invariant symmetric bilinear form.In view of Lemma 3.7, in order to finish the proof of Theorem 1.1 it remains show that p is a good prime for G.As p > 3 by our general assumption, we just need to rule out the case where p = 5 and G has components of type E 8 .
Lemma This completes the proof of Lemma 3.8, and Theorem 1.1 follows.
3.8.In this subsection we assume that G is a simple, simply connected algebraic k-group and p is a very good prime for G.All standard results on representations of reductive groups used in what follows can be found in [Jan03].Given a dominant weight λ ∈ X(T ) we denote by V (λ) the Weyl module of highest weight λ.The quotient L(λ) := V (λ)/rad V (λ) is a simple rational G-module of highest weight λ and dim V (λ) can be computed by using the Weyl dimension formula.The Lie algebra g = Lie(G) acts on L(λ) via the differential at e ∈ G of the irreducible representation ρ λ : G → GL(L(λ)) and d e ρ λ : g → gl(L(λ)) is irreducible if and only if λ is p-restricted (as defined in Subsection 3.6).As before, we adopt Bourbaki's numbering of simple roots in Π and let θ be the highest root of Φ with respect to Π. Since ρ is already engaged we write δ for the half-sum of the roots in Φ + .
Given a finite dimensional representation ρ : g → gl(M ) we denote by tr M (h θ ) 2 the trace of the endomorphism ρ(h θ ) 2 .It is not hard to see for any two finite dimensional restricted g-modules M and N one has (one should keep in mind that M and N decompose into a direct sum of weight spaces with restpect to t and h θ ∈ [g, g] has zero trace on both M and N ).
Our goal in this subsection is to find a p-restricted dominant λ ∈ X(T ) such that p ∤ dim L(λ) and tr L(λ) (h θ ) 2 ̸ = 0.In the characteristic zero case similar quantities are often computed by using the notion of Dynkin index; see [Dyn57, Ch.I, §2].Therefore, it will be convenient for us to regard G as the group of k-points of a simply connected Chevalley group scheme G Z with the same root datum as G.We shall also assume that T is obtained by base-changing a maximal split torus of T Z of G Z , so that X(T ) = X(T Z ).Then h θ = H θ ⊗ Z 1 for the semisimple root vector H θ = d e (θ ∨ ) ∈ Lie(T Z ).
Let g Z be the Lie algebra of G Z and denote by V Z (λ) the Weyl module for G Z with highest weight λ ∈ X(T ).Then g = g Z ⊗ Z k and V (λ) = V Z ⊗ Z k.We denote by X(λ) the set of all T Z -weights of V Z (λ) and write n µ for the multiplicity of µ ∈ It Here V Q (λ) = V Z (λ) ⊗ Z Q and g Q = g Z ⊗ Z Q. Dynkin's original proof of the integrality of d(λ) involved some case-by-case considerations, but a shorter argument was later found in [Oni94, Ch.I, §3.10].We refer to [Pan09, §1] for more detail on the history of this formula.
Proposition 3.9.If p is a very good prime for G, then there exists a p-restricted dominant weight λ ∈ X(T ) such that p ∤ dim L(λ) and tr L(λ) (h θ ) 2 ̸ = 0.
Proof.If G is SL n or Sp 2n with p ∤ n, then the natural G-module L(ϖ 1 ) satisfies all our requirements since p ∤ dim L(ϖ 1 ) and tr L(ϖ 1 ) (h θ ) 2 = 2.If G = Sp 2n and p | n we can take L(ϖ 2 ) instead.Indeed, [PSu83, Cor.2] shows that in this case rad V (ϖ 2 ) ∼ = L(0), implying that dim L(ϖ 2 ) = 2n 2 − 2. Since d(ϖ 2 ) = 2n − 2 by [LaSo97, p. 504], we have that tr L(ϖ 2 ) (h θ ) 2 = tr V (ϖ 2 ) (h θ ) 2 = 2d(ϖ 2 ) mod p ∈ F × p .If G is of type B n or D n , then [LaSo97, p. 504] shows that we can take for λ the minuscule weight ϖ n as both d(ϖ n ) and dim L(ϖ n ) = dim V (ϖ n ) are powers of 2 (and p > 2).Now suppose G is an exceptional group.Since p is a good prime for G, the Killing form of g is non-degenerate.This is well-known and follows, for example, from the fact that d(θ) = 2h ∨ , where h ∨ is the dual Coxeter number of Φ; see [Gar09, §1] for more detail.If p ∤ dim g, we can take for L(λ) the adjoint G-module g ∼ = L(θ).Indeed, our assumptions on p and G imply that g is a simple Lie algebra and p ∤ h ∨ .
From now on we may assume that G is exceptional and p | dim g.If G is of type E 6 we can take for λ the minuscule weight ϖ 1 .In this case V (ϖ 1 ) ∼ = L(ϖ 1 ) has dimension 27 and d(ϖ 1 ) = 6 by [LaSo97,p. 504].If G is of type E 7 and p = 7 we can take for λ the fundamental weight ϖ 6 .Then d(ϖ 6 ) = 648 by [LaSo97, p. 504], whilst a quick look at [Lüb01, 6.52] reveals that rad V (ϖ 6 ) ∼ = L(0) has dimension 1 and dim L(ϖ 6 ) = 1538.Since both 648 and 1538 are invertible Since d e ρ is faithful, we see that V is a Richardson module for g.Furthermore, the complement R is invariant under the conjugation action of ρ(G) on gl(V ).
Remark 3.12.Suppose G is as in Corollary 3.10 (with s ≥ 1) and consider G = DG × GL kp where k ≥ 1.It is well-known (and straightforward to see) that there exist u, v ∈ N p (gl kp ) such that [u, v] = 1 kp .Let a denote the 3-dimensional restricted Lie subalgebra of gl kp spanned by u, v and 1 kp .Representation theory of nilpotent Lie algebras shows that all faithful irreducible a-modules have the same dimension equal to p. Consequently, any faithful irreducible gl kp -module has dimension divisible by p.Using (3.1) one observes that in contrast with Corollary 3.10 the trace forms associated with the irreducible faithful representations of Lie( G) are always degenerate.

3
. Therefore, if m(m + 1)(m + 2) is divisible by 9, then the trace form of gl(V (m)) vanishes on (d e ρ m )(g).This example shows that there exist indecomposable Richardson modules V for a restricted Lie algebra L such that the trace form of gl(V ) vanishes on L ⊂ gl(V ).
L has the Richardson property then for any χ ∈ L * the L-modules S χ (L) ad and U χ (L) ad are isomorphic.In [Pre99, 3.6], the author speculated that if L possesses the Richardson property with respect to an irreducible faithful restricted L-module, then there exists a reductive algebraic group G over k such that L ∼ = Lie(G) as restricted Lie algebras.This conjecture was recently mentioned in the survey article [BF15] by Benkart-Feldvoss as one of the interesting open problems in the theory of modular Lie algebras; see Problem 7(c) in loc.cit.
Sandwich elements play an important role in the study of Engel Lie algebras and in the classification theory of finite dimensional simple Lie algebras over algebraically closed fields of characteristic p > 3; see[Kos90],[KZ90],[Pre94],[PS97],[Str17],[Zel91].In his talks at the ICM's inStockholm  (1962)  and Nice (1971), Kostrikin conjectured that a finite dimensional simple Lie algebra L over an algebraically closed field of characteristic p > 3 is either strongly degenerate or classical in the sense of Seligman, that is, has the form L = [Lie(G), Lie(G)] for some simple algebraic k-group G of adjoint type.Kostrikin's conjecture attracted a lot of attention in 1960's and 1970's and . A finite dimensional Lie algebra L over an algebraically closed field of characteristic p > 3 is called almost classical if there exists a semisimple algebraic k-group G of adjoint type such that L is isomorphic to a subalgebra of Lie(G) containing [Lie(G), Lie(G)].Note that under our assumptions on p the Lie algebra Lie(G) identifies with the derivation algebra of the restricted Lie algebra [Lie(G), Lie(G)].The latter decomposes into a direct sum of classical simple Lie algebras which may include components isomorphic to psl rp (k) with r > 0. The main result of [Pre86, Pre87a] states that for p > 3 a finite dimensional Lie algebra L over k is almost classical if and only if C(L) = {0}.

Proof.
By construction, z( g) lies in [ g, g] and coincides with the radical of the restriction of b to [ g, g].Therefore, b gives rise to a non-degenerate g-invariant symmetric bilinear form on [g, g] ∼ = [ g, g]/z( g); we call it b.Since Der [g, g] ∼ = g and g ∼ = g/z( g) as Lie algebras, the bilinear form b is invariant under all derivations of [g, g].It follows that for any 2-cocycle φ on [g, g] with values in k there is an element h ∈ g such that φ(x, y) = b([h, x], y) for all x, y ∈ [g, g].Furthermore, the central extension of [g, g] associated with φ is trivial if and only if h ∈ [g, g] α ∈ Φ + we put h α := [e α , e −α ].Obviously, {e α , h α , e −α } is an sl 2 -triple in L and we write s α for the k-span of e ±α and h α .It is straightforward to see that L admits a natural restricted Lie algebra structure x → x [p ′ ] with respect to which all h α are toral and all e α have the property that e ′ ]) may be very different if dim z(L) = 1.In order to proceed further we have to investigate this problem.
[p ′ ] α = 0. Unfortunately, the restricted Lie algebras (L, [p]) and (L, [p Suppose that z(L) is spanned by z acts faithfully [g, g] (and hence on L) the restriction of the first projection n(L) → L to Lie(T ) is injective.Since g = Lie(T ) + α∈Φ Lα this implies that dim g = dim L.
As L ⊆ g, the claim follows.□ 3.4.Suppose H is a simple algebraic k-group of adjoint type and p > 3. Then Lie(H) is a perfect Lie algebra whenever H has type other than A kp−1 , and [Lie(H), Lie(H)] has codimension 1 in Lie(H) if H ∼ = PGL kp .Indeed, in the latter case Lie(H) ∼ = pgl kp and [Lie(H), Lie(H)] ∼ = psl kp .Since our algebraic group G is isomorphic to a a direct product of simple algebraic groups of adjoint type, dim(g/[g, g]) equals the number of simple components of G having types A kp−1 with k ∈ Z >0 .
3.8.If G has a component of type E 8 then p > 5.Proof.Let H be a simple component of type E 8 in G.It is immediate from our description of G that it contains a connected normal subgroup H ′ such that G ∼ = H × H ′ .Let h = Lie(H) and h ′ = Lie(H ′ ).The ideals h and h ′ of g commute and g = h ⊕ h ′ .To ease notation we identify g with L. Since h is a direct summand of g, the g-module V is Richardson for h.Let R be a subspace ofgl(V ) such that [h, R] ⊆ R and gl(V ) = h ⊕ R.It is well-known that the nilpotent variety N(h) is irreducible and contains a unique open (Ad H)orbit O reg .Pick e ∈ O reg and regard it as a nilpotent element of gl(V ).Since the closed subgroup ρ(H) of GL(V ) normalises h = (d e ρ)(h), the tangent space T e (Ad ρ(H) .e) is contained in h.On the other hand, since all adjoint GL(V )-orbits are smooth,T e (Ad ρ(H) .e)⊆ h ∩ T e ((Ad GL(V ) .e) = h ∩ [gl(V ), e] = h ∩ [h, e] ⊕ [ R, e] = [h, e],for [h, e] ⊆ h and h ∩ [ R, e] ⊆ h ∩ R = {0}.Therefore, dim (Ad H) .e ≤ dim [h, e] = dim h − dim h e , forcing dim C H (e) ≥ dim h e .As Lie(C H (e)) ⊆ h e , we deduce the adjoint H-orbit of e ∈ O reg is smooth.Thanks to a well-known result of Springer, this entails that p is a good prime for H; see [Spr66, Theorem 5.9].□ is well-known that d(λ) is an integer; see [LaSo97, 2.3], for example.For all fundamental dominant weights ϖ i with 1 ≤ i ≤ rk(G Z ), the integers d(ϖ i ) are computed in [Dyn57, Table 5], and the three misprints in type E 8 are corrected in [LaSo97, p. 504].
Let ( • , • ) be the scalar product on the X(T Z ) ⊗ Z R invariant under the action of the Weyl group W (Φ) and such that (θ, θ) = 2. Dynkin proved in [Dyn57] that (3.2)