A new family of irreducible subgroups of the orthogonal algebraic groups

Let $n\geq 3,$ and let $Y$ be a simply connected, simple algebraic group of type $D_{n+1}$ over an algebraically closed field $K.$ Also let $X$ be the subgroup of type $B_n$ of $Y,$ embedded in the usual way. In this paper, we correct an error in a proof of a theorem of Seitz, resulting in the discovery of a new family of triples $(X,Y,V),$ where $V$ denotes a finite-dimensional, irreducible, rational $KY$-module, on which $X$ acts irreducibly. We go on to investigate the impact of the existence of the new examples on the classification of the maximal closed connected subgroups of the classical algebraic groups.


Introduction
Let K be an algebraically closed field of characteristic p ≥ 0. In the 1950's, Dynkin determined the maximal closed connected subgroups of the simple classical type linear algebraic groups defined over K, assuming char(K) = 0 (see [11,12]); in 1987, Seitz [23] established an analogous classification in the case where char(K) > 0. The main step in both of these classifications is the determination of all triples (X, Y, V ), where Y is a simple linear algebraic group defined over K, X is a proper closed connected subgroup of Y , and V is a non-trivial irreducible, finite-dimensional (p-restricted if char(K) = p > 0) KY -module on which X acts irreducibly. The determination of these so-called "irreducible triples" is covered in the work of Dynkin [11,12] (in case char(K) = 0), Seitz [23] (in case char(K) > 0 and Y is a classical group), and Testerman [26] (in case char(K) > 0 and Y is of exceptional type). The existence of an irreducible triple of the form (X, Y, V ) as above, arising from a rational representation ρ : Y → GL(V ), indicates that ρ(X) is not maximal in the smallest classical group Isom(V ) containing both ρ(X) and ρ(Y ), while the large majority of tensor-indecomposable irreducible representations of a simple algebraic group give rise to maximal subgroups of the smallest classical group containing the image. weights for D 4 and we label Dynkin diagrams as in [3]). The restriction of the highest weight to a maximal torus of B 3 shows the existence of a B 3 -composition factor of highest weight ω 1 + ω 2 + ω 3 , (ω i , i = 1, 2, 3, a set of fundamental weights for B 3 ) and consulting [19], one sees that these modules are both of dimension 384, and hence X acts irreducibly on V .
The absence of this example from [23, Table 1] is the result of an error in the proof of [23, 8.7]. Here, we correct the error in this proof and, in so doing, establish the existence of a whole new family of modules V for the group Y = D n+1 on which X = B n acts irreducibly. For a fixed n and a fixed p, there are finitely many modules V , but for each n there exist infinitely many primes p for which there is a new example. The precise description of the family is given in Theorem 1.2 below.
The goal of this paper is two-fold: first we concentrate on the embedding X = B n ⊂ Y = D n+1 and determine all p-restricted irreducible representations of Y whose restriction to X is irreducible, thereby correcting [23, 8.7]; see Theorem 1.2 below. The second goal of the paper is to show that the existence of the new examples has no further influence on the main results of [23] and [26]. Indeed, the proofs of the main theorems in these two articles depend on an inductive hypothesis, concerning the list of examples for smaller rank groups. The new family of examples for the pair (B n , D n+1 ) alters the inductive hypothesis and therefore requires one to take these new examples into consideration when working through all other possible embeddings. This is precisely what has been carried out in the proofs of Proposition 1.5 and Theorem 1.7.  Table 1] for pairs (X, Y ) different from (B n , D n+1 ), while Theorem 1.2 covers the embedding B n ⊂ D n+1 . This assertion is dependent upon our Hypothesis 1.4 below, where we state explicitly which results from [23] are assumed for the proofs of our main results.

Statement of results
Let Y = Spin 2n+2 (K) (n ≥ 2) be a simply connected, simple algebraic group of type D n+1 over K. Also let X be the subgroup of type B n , embedded in Y in the usual way, as the stabilizer of a non-singular 1-dimensional subspace of the natural module for Y. Fix T Y a maximal torus of Y and B Y ⊂ Y a Borel subgroup containing T Y . Denote by {λ 1 , . . . , λ n+1 } the corresponding set of fundamental weights for T Y , ordered as in [3], where the natural (2n + 2)-dimensional KY -module has highest weight λ 1 . Let σ be a graph automorphism of Y stabilizing T Y , and with X = Y σ , the group of σ-fixed points. Our first main result is the following; the proof is given in Section 3. Theorem 1.2. Let Y = Spin 2n+2 (K) be a simply connected, simple algebraic group of type D n+1 over K, n ≥ 2, and let X be the subgroup of type B n , as above. Consider a non-trivial, irreducible KY -module V having p-restricted highest weight λ = n+1 r=1 a r λ r . Then X acts irreducibly on V if and only if λ or σ(λ) is equal to n r=1 a r λ r , with a n = 0, and for all 1 ≤ i < j ≤ n such that a i a j = 0 and a r = 0 for all i < r < j, we have p | (a i + a j + j − i).
So we see that the new examples are a generalization of those found by Seitz, where one congruence condition is replaced by a set of congruence conditions. (Note that there are new examples only if n ≥ 3.) It is perhaps informative to point out precisely what error occurs in the proof of [23, 8.7], where the embedding B n ⊂ D n+1 is considered. In the proof, Seitz defines a certain vector in the irreducible KY -module of highest weight λ and shows that this vector is annihilated by all simple root vectors in the Lie algebra of X, which then implies that X does not act irreducibly. However if λ satisfies the congruence conditions, the vector is in fact the zero vector in V and so does not give rise to a second composition factor as claimed. It is also natural to ask how one might discover the given set of congruence conditions, and here we must give credit to the work of Ford in [13], where he studied irreducible triples of the form (H, G, V ), G a simple classical type algebraic group over K, H a disconnected closed subgroup of G with H • simple, and V an irreducible KG-module on which H acts irreducibly. He discovered a family of irreducible triples for the embedding D n .2 ⊂ B n , where the highest weight of the irreducible KB n -module satisfies similar congruence conditions. His methods were later applied by Cavallin in [9] when studying irreducible KB n -modules having precisely two D n -composition factors.
The second goal of this article is to show that the existence of the new examples for the pair (X, Y ) = (B n , D n+1 ), described by Theorem 1.2, has no further influence on the main theorems in [23,26]. To explain the issue which must be addressed and our approach to the problem, we must describe to some extent the strategy of the proof of [23,Theorem 1]. First note that the assumption that X acts irreducibly on some KY -module implies that X is semisimple. One of the main techniques used to determine the triples (X, Y, V ) as above involves arguing inductively, working with a suitable embedding P X = Q X L X ⊂ P Y = Q Y L Y of parabolic subgroups, where Q X = R u (P X ) ⊂ R u (P Y ) = Q Y . Indeed, [23, 2.1] implies that if X acts irreducibly on V, then the derived subgroup L ′ X acts irreducibly on the commutator quotient V /[V, Q Y ], an irreducible KL ′ Y -module. Moreover, the highest weight of V /[V, Q Y ] as KL ′ Y -module is the restriction of the highest weight of V to an appropriate maximal torus of L ′ Y . (This is a variation of a result of [24].) Thus, Seitz and Testerman proceed by induction on the rank of X; Seitz treats the case X of type A 1 by ad hoc methods, exploiting the fact that all weights of an irreducible KA 1 -module are of multiplicity one. Now Theorem 1.2 above introduces a new family of examples of irreducible triples. As a consequence, one needs to reinvestigate all embeddings X ⊂ Y where the pair B m ⊂ D m+1 , m ≥ 3, may arise when considering the projection of a Levi factor L ′ X of X into a simple component of a Levi factor L ′ Y of Y , under the additional hypothesis that X acts irreducibly on a KY -module whose highest weight has restriction to the D m+1 -component of L ′ Y among the new examples described by Theorem 1.2. This is precisely what we consider in Proposition 1.5 below. In order to state the result, we introduce the following terminology. Definition 1.3. We will say a p-restricted dominant weight λ for Y = D m+1 , m ≥ 3, satisfies the congruence conditions if (iii) there exists i < j ≤ m − 1 with a i a j = 0, and (iv) for all 1 ≤ i < j ≤ m such that a i a j = 0 and a r = 0 for all i < r < j, we have p | (a i + a j + j − i).
Note that the above congruence conditions are precisely those satisfied by the highest weights in Theorem 1.2 but not appearing in [23,  For the proofs of Proposition 1.5 and Theorem 1.7, we require the following inductive hypotheses.
Hypothesis 1.4. Assume char(K) = p > 0. Let G be a simple algebraic group defined over K and H a semisimple, proper, closed subgroup of G, where the pair (H, G) is one of the following: Let V be a p-restricted irreducible KG-module, with corresponding representation ρ : G → GL(V ). Then H acts irreducibly on V if and only if the triple (ρ(H), ρ(G), V ) appears in [23, Table 1], where the highest weight of V is given up to graph automorphisms of G.

The classical case
Let Y be of classical type. The next result ensures that, under the assumption of Hypothesis 1.4 for all embeddings H ⊂ G with rank(G) < rank(Y ), the only new examples of irreducible triples (X, Y, V ) are those listed in Theorem 1.2. Proposition 1.5. Let Y be a simply connected, simple algebraic group of type D n+1 , n ≥ 4, X a semisimple, proper, closed subgroup of Y acting irreducibly on a p-restricted irreducible KY -module V of highest weight λ, V not the natural module for Y . Assume Hypothesis 1.4 for all embeddings H ⊂ G with rank(G) < rank(Y ). Moreover, if X is simple, assume the following conditions are satisfied.
(i) P X is a maximal proper parabolic subgroup of X, with Levi factor L X of type B m−1 , m ≥ 4.
(ii) P Y is a parabolic subgroup of Y with P X ⊂ P Y and R u (P X ) ⊂ R u (P Y ).
(iii) For a Levi factor L Y of P Y , writing L ′ Y = L 1 L 2 · · · L t , a commuting product of simple groups, L t is of type D m , and L ′ X projects non-trivially into L t .
(v) The highest weight of the KL t -module M t satisfies the congruence conditions. Then one of the following holds.
(a) X ⊂ B k × B n−k for some 0 < k < n, and λ = λ n or λ n+1 , or (b) X = B n and the embedding of X in Y is the usual embedding of B n in D n+1 , that is, B n is the stablizer of a nonsingular 1-space on the natural (2n + 2)-dimensional KYmodule.
Moreover, in case (a) above, the subgroup B k × B n−k acts irreducibly on the KY -modules of highest weights λ n and λ n+1 . Remark 1.6. Now to go on to determine the irreducible triples satisfying (a), we rely on Hypothesis 1.4 (i), and for those satisfying (b), we apply Theorem 1.2.

The exceptional case
We now turn to the consideration of the case where Y is a simply connected, simple algebraic group of exceptional type over K and X is a proper closed, connected subgroup of Y acting irreducibly on some p-restricted irreducible KY -module. As usual, X is then semisimple, and once again, we must consider the possibility of a parabolic embedding P X ⊂ P Y , with Levi factor L X of P X , of type B m , Levi factor L Y of P Y , having a simple factor of type D m+1 , with the action on the commutator quotient arising from a weight which satisfies the congruence conditions. In particular, Y is of type E n , for n = 6, 7 or 8. Theorem 1.7. Let Y be a simply connected simple algebraic group of type E n , 6 ≤ n ≤ 8, defined over K and let X be a semisimple, proper, closed, connected subgroup of Y, having a proper parabolic subgroup with Levi factor of type B m , for some m ≥ 3. Assume Hypothesis 1.4 for all embeddings H ⊂ G with rank(G) < rank(Y ). Let V be a non-trivial irreducible KY -module with p-restricted highest weight λ. Then X acts irreducibly on V if and only if Y = E 6 , X = F 4 , and one of the following holds.
Note that the existence of the examples arising in Theorem 1.7 had already been established by Testerman [26,Main Theorem]. The proof of the only if direction requires us to treat, eventually ruling out, several new potential configurations that arise from Theorem 1.2 in the inductive process, as explained in Section 5.

About the proofs
We conclude this section with a brief discussion of the methods and further remarks on our inductive assumption (Hypothesis 1.4). In order to prove Theorem 1.2, we first show that it is enough to work with the Lie algebras of Y and X. Indeed, as λ is p-restricted, the irreducible KY -module V is generated by a maximal vector v + for B Y as a module for the universal enveloping algebra U Y of Lie(Y ). Therefore in order to show that V | X is irreducible, it suffices to show that U Y v + = U X v + , where U X is the universal enveloping algebra of Lie(X). We rely on the fact that any irreducible module for Y is self-dual as a KX-module (see 3.1 below), and apply the techniques developed by Ford in [13], further investigated by Cavallin in [9], to establish this generation result.
For the proof of Proposition 1.5, we carry out an analysis used by Seitz in [23, Section 8], but applied specifically to the group Y = D m+1 . He first shows that a proper closed connected subgroup X acts irreducibly on a non-trivial irreducible KY -module only if either X acts irreducibly and tensor indecomposably on the natural module for Y , or the triple (X, Y, V ) is known. This part of our proof is not at all original, but we include it for completeness. At this point, however, our proof proceeds along different lines; we compare the commutator series for two different parabolic embeddings and obtain conditions on the highest weight which are compatible with the given congruence conditions only if the pair (X, Y ) is (B m , D m+1 ), which is handled by Theorem 1.2.
For the proof of Theorem 1.7, we proceed differently than in [26]; we use the classification of the maximal closed positive-dimensional subgroups of the exceptional type algebraic groups, given in [18], which was not available when [26] was written. Hence, we first consider the case where X is maximal, find only the two examples of the theorem and conclude using the main result of [26] for the group Y = F 4 .
In addition to Hypothesis 1.4, we rely upon two further results in [23], namely [23,Theorem 4.1] and [23, 6.1]. The first result classifies the irreducible triples (X, Y, V ) when rank(X) = rank(Y ), the second covers the case where rank(X) = 1. The proofs of these results are completely independent of the results in [23,Section 8]. Finally, we will use the results of [23, Section 2] concerning parabolic embeddings and commutator series in irreducible modules for semisimple groups.

Preliminaries
In this section, we introduce the notation that shall be used in the remainder of the paper, and recall some basic properties of rational modules for simple linear algebraic groups. We rely on the standard reference [16] for a treatment of this general theory.

Notation
Let K be an algebraically closed field of characteristic p ≥ 0, and let G be a simply connected, simple linear algebraic group over K. (All algebraic groups considered here will be linear algebraic groups, even if we omit to say so explicitly.) Also fix a Borel subgroup B = U T of G, where T is a maximal torus of G and U denotes the unipotent radical of B. Let rank(G) = ℓ and let Π = {α 1 , . . . , α ℓ } be the corresponding base of the root system Φ = Φ + ⊔ Φ − of G, where Φ + and Φ − denote the sets of positive and negative roots, respectively. Throughout we use the ordering of simple roots as in [3]. Let W be the Weyl group of G, and for α ∈ Φ, denote by s α the corresponding reflection. In addition, let X(T ) = Hom(T, K * ) denote the character group of T and write (−, −) for the usual W -invariant inner product on the space X(T ) R = X(T ) ⊗ R. Also let λ 1 , . . . , λ ℓ be the fundamental dominant weights for T corresponding to our choice of base Π, that is, for λ, α ∈ X(T ), α = 0. Set X + (T ) = {λ ∈ X(T ) : λ, α ≥ 0 for all α ∈ Π} and call a character λ ∈ X + (T ) a dominant T -weight (or simply dominant weight, if the choice of torus is clear in the context). Finally, we say that µ ∈ X(T ) is under λ ∈ X(T ) (and we write µ λ) if λ − µ = ℓ r=1 c r α r for some c 1 , . . . , c ℓ ∈ Z ≥0 . We also write µ ≺ λ to indicate that µ λ and µ = λ.

Rational modules
In this section, we recall some elementary facts on weights and multiplicities, as well as basic properties of Weyl and irreducible modules for G. Let V be a finite-dimensional, rational KG-module.
where for µ ∈ X(T ), we have V µ = {v ∈ V : tv = µ(t)v for all t ∈ T }. A weight µ ∈ X(T ) is called a weight of V if V µ = 0, in which case V µ is said to be its corresponding weight space. Also, we denote by m V (µ) the multiplicity of µ in V, and let Λ(V ) = {µ ∈ X(T ) : V µ = 0} denote the set of weights of V and write Λ + (V ) = Λ(V ) ∩ X + (T ) for the set of dominant weights of V. It is well-known that each weight of V is W -conjugate to a unique dominant weight in Λ + (V ). Also, if λ ∈ X + (T ), then wλ λ for every w ∈ W , and all weights in a W -orbit have the same multiplicity.
A non-zero vector v + ∈ V is called a maximal vector of weight λ ∈ Λ(V ) for the pair (B, T ) if v + ∈ V λ and Bv + ⊆ v + K . Now for λ ∈ X + (T ) a dominant weight, we write V G (λ) for the Weyl module having highest weight λ, and denote by L G (λ) the unique irreducible quotient of V G (λ). In other words, where rad(λ) is the unique maximal submodule of V G (λ), called the radical of V G (λ). We write Λ(λ) for Λ(V G (λ)) and Λ + (λ) for Λ + (V G (λ)). Also, we denote by H 0 (λ) the induced KG-module having highest weight λ. Recall that H 0 (λ) has a unique simple submodule, isomorphic to L G (λ), and that where w 0 denotes the longest element in W . For µ ∈ X + (T ), we write [V, L G (µ)] to denote the number of times the irreducible KG-module L G (µ) appears as a composition factor of V. We also use the notation to denote the T -root subgroup of G corresponding to the root α ∈ Φ (that is, x α : K → G is a morphism of algebraic groups inducing an isomorphism onto Im(x α ), such that tx α (c)t −1 = x α (α(t)c) for t ∈ T and c ∈ K). Finally, we fix a Chevalley basis B = {f α , h r , e α : α ∈ Φ + , 1 ≤ r ≤ ℓ} for the Lie algebra Lie(G) of G, compatible with our choice of T ⊂ B, where e α ∈ Lie(G) α , f α ∈ Lie(G) −α are root vectors for α ∈ Φ + and h r = [e αr , f αr ] for 1 ≤ r ≤ ℓ. The proof of the following result can be deduced from applying the Poincaré-Birkhoff-Witt Theorem [4] to [10, A. 6.4].
(i) The weight µ affords the highest weight of a composition factor of V G (λ).
Proof. Clearly (i) implies (ii). Conversely, the only dominant weights ν ∈ Λ + (V ) such that µ ≺ ν ≺ λ are λ − α 1 (if a > 1) and λ − α ℓ (if b > 1). These weights have multiplicity 1 in V G (λ) and hence none of them can afford the highest weight of a composition factor of V G (λ) by [22]. Consequently (ii) implies (i). Now an application of Freudenthal's formula if and only if the ℓ generators in (1) are linearly dependent. Therefore (ii) and (iii) are equivalent as well.
Finally, let (η r ) 1≤r≤ℓ ∈ K ℓ and set A straightforward calculation shows that Bw + = w + K if and only if p | a + b + ℓ − 1 and η ℓ = 0, in which case w + = 0 (since V is irreducible and w + / ∈ V λ ). This shows that (iii), (iv) and (v) are equivalent, thus completing the proof.

Proof of Theorem 1.2
Let K be an algebraically closed field having characteristic p ≥ 0 and let Y = Spin 2n+2 (K) be a simply connected, simple algebraic group of type D n+1 over K, with n ≥ 2. Let X ⊂ Y be the subgroup of type B n , embedded in the usual way, as the stabilizer of a non-singular 1-dimensional subspace of the natural (2n + 2)-dimensional module for Y. Fix T Y a maximal torus of Y and T X a maximal torus of X such that T X ⊂ T Y and let T Y ⊂ B Y , T X ⊂ B X denote Borel subgroups of Y, X respectively, with B X ⊂ B Y . Let Π(Y ) = {α 1 , . . . , α n+1 } be the corresponding base for the root system of Y, and denote by {λ 1 , . . . , λ n+1 } the corresponding set of fundamental weights for T Y , where the natural KY -module has highest weight λ 1 . Let σ be a graph automorphism of Y stabilizing T Y , and with X = Y σ , the group of σ-fixed points. Finally, let Π(X) = {β 1 , . . . , β n } be the base for the corresponding root system of X, associated with the choice of Borel subgroup B X , and denote by {ω 1 , . . . , ω n } the associated set of fundamental dominant weights for T X .

Preliminary considerations
For σ as above and for a KY -module V, Proof. Let λ ∈ X + (T Y ) be the highest weight of V. Then V * has highest weight −w 0 λ. If n + 1 is even, then −w 0 = 1 by [25, Exercise 78] and so V is self-dual as a KY -module, from which the desired result follows. If on the other hand n + 1 is odd, Section 8], we may assume that the B n -type subalgebra Lie(X) of Lie(Y ) is generated by the root vectors e βr = e αr for 1 ≤ r ≤ n − 1, (2) In particular, we get that and f n,n+1 = 0 by convention. In a similar fashion, for 1 ≤ k ≤ n, we set We will require the following relations in Lie(X).
Proof. We start by showing (i), arguing by induction on 0 ≤ s − r ≤ n − 2. If r = s, then the assertion immediately follows from (2), so we assume 1 ≤ r < s ≤ n − 1 in the remainder of the proof. We then successively get where the second equality follows from (2) and our induction assumption. Therefore (i) holds as desired. For the second assertion, we again argue by induction on 0 ≤ n − r ≤ n − 1. In the case where r = n, then the result holds by (2), hence we assume 1 ≤ r ≤ n − 1 in the remainder of the proof. We then successively get where again the second equality follows from (2) and our induction assumption. Therefore (ii) holds as well. Next we show the third assertion, letting 1 ≤ r ≤ n − 1 be fixed, and setting µ = β r + · · · + β n−1 + 2β n . The aforementioned root restrictions yield Lie(X) µ ⊂ Lie(Y ) αr+···+α n+1 , and since the latter T Y -weight space is 1-dimensional, we get that Lie(X) µ is at most 1-dimensional as well. Now as f βr+···+βn+2β n+1 ∈ Lie(X) µ and Lie(Y ) αr+···+α n+1 = f r,n+1 K , we get that f βr+···+βn+2β n+1 is a non-zero multiple of f r,n+1 , from which (iii) follows. Finally, the assertion (iv) can be dealt with in a similar fashion.
In the remainder of this section, we let The following result provides a necessary and sufficient condition for V to be irreducible in the case where a n = 0 = a n+1 . Lemma 3.3. Let V = L Y (λ) be as above, and assume a n = 0 = a n+1 . Then X acts irreducibly on V if and only if V = Lie(X)v + .
Proof. First assume X acts irreducibly on V, so that V | X ∼ = L X (ω), and observe that since a n a n+1 = 0, the T X -weight ω is p-restricted. Therefore the Lie algebra Lie(X) of X acts irreducibly on V by [10, Theorem 1], from which the desired assertion follows. Conversely, assume V = Lie(X)v + . Then V = Xv + and hence V | X has a quotient isomorphic to L X (ω) by [16 In view of Lemma 3.3, a necessary condition for X to act irreducibly on V = L Y (λ), with λ such that a n = 0 = a n+1 , is for f γ v + to belong to Lie(X)v + for every γ ∈ Φ + (Y ). We conclude this section by showing that V | X is irreducible if and only if f r,n v + ∈ Lie(X)v + for every 1 ≤ r ≤ n (see Proposition 3.5 below). We first need the following preliminary lemma.
Proof. We proceed by induction on s ≥ 1. First take η ∈ Φ + (X) and consider If the latter equals zero, then we immediately get the desired result. So we may assume the contrary and thus we have f η is one of the following: Consequently f γ f η v + ∈ Lie(X)v + for γ = n r=i α r as desired. Arguing in a similar fashion, one shows that the same holds for γ = n−1 r=i α r + α n+1 and γ = α n+1 as well, where 1 ≤ i ≤ n − 1. Therefore the lemma holds in the situation where s = 1, and hence we assume s > 1 in the remainder of the proof. For η 1 , . . . , η s ∈ Φ + (X), we have We now establish the following necessary and sufficient condition for V | Lie(X) to be generated by v + as a Lie(X)-module (and hence for V | X to be irreducible by Lemma 3.3), where V has highest weight λ with a n = 0 = a n+1 .
Hence in particular f r,n v + ∈ Lie(X)v + for every 1 ≤ r ≤ n as desired. Conversely, suppose that f r,n v + ∈ Lie(X)v + for every 1 ≤ r ≤ n and notice that since Theorem 1]. Hence in order to show that V = Lie(X)v + , it suffices to show that f γ 1 · · · f γs v + ∈ Lie(X)v + for every s ∈ Z >0 and γ 1 , . . . , γ s ∈ Φ + (Y ). Assume for a contradiction that this is not the case and let m ∈ Z ≥0 be minimal such that there exists and so m ≥ 2. Now by minimality, we have f γ 2 · · · f γm v + ∈ Lie(X)v + and hence an application of Lemma 3.4 completes the proof.

Conclusion
Let V = L Y (λ) be an irreducible KY -module having p-restricted non-zero highest weight λ = n+1 r=1 a r λ r ∈ X + (T Y ), and set ω = λ| T X . In this section, we will complete the proof of Theorem 1.2. We first show that for certain weights λ, it is straightforward to see that V | X is reducible. Although the proof of the following proposition can be found in [23, Section 8], we include it here for completeness. Proposition 3.6. Let λ ∈ X + (T Y ) and V be as above. Then the following assertions hold.
(i) If a n a n+1 = 0, or if a n = a n+1 = 0, then V | X is reducible.
(iii) If V | X is irreducible, so a n a n+1 = 0 by (i), and λ is not as in (ii), then taking 1 ≤ k < n maximal such that a k = 0, we have p | (a k + a n + a n+1 + n − k).
Proof. For (i), first consider the case where a n a n+1 = 0. Here the T X -weight ω ′ = ω − β n has multiplicity at most 1 in L X (ω), while each of λ − α n and λ − α n+1 is a T Y -weight of V restricting to ω ′ . Therefore the latter occurs in a second KX-composition factor of V, showing that V | X is reducible as desired. Next assume a n = a n+1 = 0, and let 1 ≤ k ≤ n − 1 be maximal such that a k = 0. Then the T X -weight ω ′′ = ω − (β k + · · · + β n ) has multiplicity at most 1 in L X (ω) (since the corresponding weight space is generated by f β k +···+βn v + ), while each of λ − (α k + · · · + α n ) and λ − (α k + · · · + α n−1 + α n+1 ) restricts to ω ′′ . Consequently V | X is reducible in this case as well, and (i) holds as desired.
Now turn to (ii) and let λ = aλ n for some a ∈ Z >0 , and assume for a contradiction that V | X is reducible. By [23, 8.5], there exist 1 ≤ i ≤ n and a maximal vector w + ∈ V for B X such that 0 = e α i +···+αn w + ∈ v + K . In particular w + ∈ V λ−(α i +···+αn) , and one checks that the only thus yielding e β i w + = e α i w + = 0, a contradiction. Therefore (ii) holds as desired.
For (iii), we assume V | X is irreducible, and by (i), we suppose, without loss of generality, that a n = 0 = a n+1 . Assume as well that λ is not as in (ii), and seeking a contradiction, take k as in (iii), and suppose that p ∤ (a k + a n + n − k). Con- while an application of Proposition 2.2 yields m V (λ − (α k + · · · + α n )) = n − k + 1. (Recall that we assumed a n = 0 = a n+1 and p ∤ (a k + a n + n − k).) Now m V (λ − (α k + · · · + α n−1 + α n+1 )) = 1, as the latter weight is conjugate to λ − α k under the action of the Weyl group for Y. Therefore , yielding the existence of a second composition factor of V for X, a contradiction. The proof is complete.
In view of Proposition 3.6, we may and shall assume λ = n r=1 a r λ r , with a n = 0 throughout the rest of the section, as well as the existence of 1 ≤ k ≤ n − 1 maximal such that a k = 0. For 1 ≤ i ≤ n, set P (i, i) = ∅ and for 1 ≤ i < j ≤ n, set The following special case of [8,Theorem A.7], inspired by [13, Proposition 3.1], shall play a key role in the proof of Theorem 1.2.
Theorem 3.7. Let Y be a simple algebraic group of type D n+1 over K, and consider an irreducible KY -module V = L(λ) having p-restricted highest weight λ = n r=1 a r λ r , with a n = 0. Then f r,n v + ∈ V r,n for every Proof. The result follows from an application of [8,Theorem A.7] to the A n -Levi subgroup of Y corresponding to the simple roots α 1 , . . . , α n .
The proof of the main result of this section (namely, Theorem 3.9) relies on the following preliminary result.
Lemma 3.8. Adopt the notation introduced above and let 1 ≤ r ≤ k − 1 be such that f r,n+1 v + ∈V r,n+1 . Then f r,n v + ∈ V r,n .
Proof. Let 1 ≤ r ≤ k − 1 be as in the statement of the lemma. If a r = 0, then f αr v + = 0 and hence f r,n v + = ±f αr f r+1,n v + , so that f r,n v + ∈ V r,n , as claimed. Therefore we assume a r = 0 in the remainder of the proof. Asf r,n+1 v + ∈V r,n+1 and sincef s,n+1 v + = 0 for k < s ≤ n, we get the existence of (ξ s , ξ s Now observe that f αn e α n+1f t, Consequently successively applying e α n+1 , f αn to each side of (3) yields N 1 f r,n v + +N 2 f r,n−1 f αn v + ∈ V r,n for some N 1 = 0, N 2 ∈ {±1}, from which the desired result follows.
In the next result, we show that in order to determine whether V | X is irreducible or not, it is enough to determine whether f r,n v + ∈ V r,n or not, this for every 1 ≤ r ≤ n − 1.
Theorem 3.9. Let λ = n r=1 a r λ r be a p-restricted dominant weight such that a k a n = 0. Let V = L Y (λ) be an irreducible KY -module having highest weight λ. Then V | X is irreducible if and only if f r,n v + ∈ V r,n for every 1 ≤ r ≤ n − 1.
Proof. First assume f r,n v + ∈ V r,n for every 1 ≤ r ≤ n − 1. By Proposition 3.5, in order to show that V | X is irreducible, it suffices to show that f s,n v + ∈ Lie(X)v + for every 1 ≤ s ≤ n. We proceed by induction on 0 ≤ n − s ≤ n − 1, starting by assuming s = n. Now since a n+1 = 0, we immediately get that f α n+1 v + = 0 and hence f βn v + = f αn v + by (2), that is, f αn v + ∈ Lie(X)v + . Next assume 1 ≤ s ≤ n − 1. Since f s,n v + ∈ V s,n by assumption, there exists ( By Lemma 3.2, f s,i ∈ Lie(X) and f (m) ∈ Lie(X) for every s ≤ i ≤ n − 1 and every (m) ∈ P (s, i). Furthermore, we also have f i+1,n v + ∈ Lie(X)v + for s ≤ i ≤ n − 1 by induction. Therefore f s,n v + ∈ Lie(X)v + as desired.
Conversely, assume V | X irreducible, and let 1 ≤ r ≤ n − 1 be fixed. If a r = 0, then f αr v + = 0 and hence f r,n v + = ±f αr f r+1,n v + , so that f r,n v + ∈ V r,n in this situation. In addition, observe that p | (a k + a n + n − k) by Proposition 3.6 (iii), thus yielding f k,n v + ∈ V k,n by Proposition 2.2. Therefore we assume a r = 0 in the remainder of the proof, and r < k. By Lemma 3.3, we have V = Lie(X)v + , in which case Proposition 3.5 applies, thus yielding f r,n v + ∈ Lie(X)v + . Since f r,n v + ∈ (Lie(X)v + ) ω−(βr+···+βn) , we get the existence of ξ ∈ K, x ∈ V r,n and y ∈V r,n+1 such that f r,n v + = ξf r,n v + ± ξf r,n+1 v + + x + y. Comparing T Y -weights yields (ξ − 1)f r,n v + ∈ V r,n and ± ξf r,n+1 v + ∈V r,n+1 .
If ξ = 1, then the assertion is immediate, while if on the other hand ξ = 1, then an application of Lemma 3.8 yields the desired result.
We are now able to complete the proof of Theorem 1.2.
Proof of Theorem 1.2. First assume X acts irreducibly on V. By Proposition 3.6(i), we then have that, up to a graph automorphism of Y , a n = 0 = a n+1 , and if λ = aλ n , then λ is as in the statement of the result. So assume the existence of 1 ≤ k ≤ n − 1 maximal with a k = 0. Then Theorem 3.9 applies, yielding f r,n v + ∈ V r,n for every 1 ≤ r ≤ n − 1. An application of Theorem 3.7 then implies the desired divisibility conditions. Conversely, assume λ ∈ X + (T Y ) satisfies the conditions in Theorem 1.2. In the case where λ = aλ n for some a ∈ Z >0 , then the result follows from Proposition 3.6(ii), hence we assume the existence of 1 ≤ k ≤ n − 1 maximal such that a k a n = 0, and assume moreover the divisibility conditions as in the theorem. Here an application of Theorem 3.7 yields f r,n v + ∈ V r,n for every 1 ≤ r ≤ n − 1, and hence Theorem 3.9 then shows that X acts irreducibly on V as desired.
Fix T Y a maximal torus of Y and let T Y ⊂ B Y denote a Borel subgroup of Y. Also let {λ 1 , . . . , λ n+1 } be the corresponding set of fundamental dominant weights for T Y . In this section, we give a proof of Proposition 1.5, starting with three results, proven in a more general setting in [23,Section 5]. For the convenience of the reader, and in order to render the current manuscript more self-contained, we include the proofs of these special cases here.
Proposition 4.1. Let Y be a simply connected simple algebraic group of type D n+1 with natural module W and let X be a semisimple, connected, proper, closed subgroup of Y acting irreducibly on a non-trivial, p-restricted, irreducible KY -module V = L Y (λ). Then one of the following holds.
Note that W 1 is not totally singular, else X lies in a proper parabolic subgroup of Y and so cannot act irreducibly on V . Therefore, we have p = 2; the set of singular vectors in W 1 being an X-invariant subspace of W 1 forces W 1 to be generated by a non-singular vector, and so (ii) holds with  Proof. To see that H acts irreducibly on the two half-spin modules for Y , we simply compute the restriction of the highest weight to a maximal torus of H, note that the restriction induces the tensor product of the spin modules for the two simple factors of H, and then a dimension comparison completes the proof.
We now assume H acts irreducibly on V , the irreducible KY -module of highest weight λ. We proceed by induction on n and first consider the case n = 2, so that Y is of type D 3 = A 3 and the image of H in SO 6 lies in the subgroup SO 3 × SO 3 ⊂ SO 6 . Since H acts irreducibly on V , H does not lie in a parabolic subgroup of Y and so acts irreducibly on the 4-dimensional p-restricted KY -modules, which correspond to the highest weights as in the statement of the result. Thus the preimage of SO 3 × SO 3 in SL 4 acts on the 4-dimensional module via the tensor product representation of A 1 × A 1 on E ⊗ E, where E is the natural 2-dimensional representation of SL 2 . To see that SL 2 × SL 2 acts reducibly (and hence H as well) on all other non-trivial, p-restricted, irreducibles for Y, we require a further argument (and some additional notation).
Assume now that n ≥ 3 and that the result holds for Y = D ℓ with ℓ < n + 1. Assume as well that p > 2; we will treat the case p = 2 at the end of the proof. Let W 1 and W 2 be as in the previous proof so that B k acts on W 1 and B n−k acts on W 2 . Let U 1 , respectively U 2 , be maximal totally isotropic subspaces of W 1 , resp. W 2 . Then U 0 = U 1 ⊕ U 2 is an n-dimensional totally isotropic subspace of W . Let P ⊂ Y be the preimage of the stabilizer in Isom(W 1 ) ′ × Isom(W 2 ) ′ of U 0 and R the preimage in Y of the stabilizer in Isom(W ) of U 0 . Then P ⊂ R, R u (P ) ⊂ R u (R), (P/R u (P )) ′ ∼ = SL(U 1 ) × SL(U 2 ), while (R/R u (R)) ′ ∼ = SL(U 0 ) = SL n . In particular, the image of the Levi factor of P in R/R u (R) stabilizes U 1 and so lies in a proper parabolic subgroup of R/R u (R), and hence can act irreducibly on no non-trivial (R/R u (R)) ′ -module. Then the above remarks and an application of the main proposition of [24] shows that λ = aλ n + bλ n+1 , for some a, b.
Let P 1 be the stabilizer in SO(W 1 ) ′ of a singular 1-space. Then P 1 × SO(W 2 ) ′ is a proper parabolic subgroup of SO(W 1 ) ′ ×SO(W 2 ) ′ and is contained in the image (under the natural projection Y → SO(W )) of the stabilizer in Y of this 1-space. As P 1 = B k−1 R u (P 1 ), we have the Levi factor B k−1 B n−k projecting into the Levi factor of type D n . Another application of [24] and the induction hypothesis yield the result.
Turn now to the case n > 2 and p = 2. In this case we have that the subgroup B k × B n−k stabilizes a non-singular 1-space and so lies in a subgroup of type B n . Now the subgroup B k × B n−k is a maximal rank subgroup of B n and we can appeal to [23,Theorem 4.1] applied to the pair (B k × B n−k , B n ) to see that the irreducible KB n -module V | Bn is a twist of the spin module and hence we deduce that λ = λ n + λ n+1 , λ n , or λ n+1 . However, L Y (λ n + λ n+1 )| Bn is not irreducible, as the highest weight affords a twist of the spin module for B n and this is of dimension strictly less than the dimension of L Y (λ n + λ n+1 ). So λ = λ n or λ n+1 as claimed.
Propositions 4.1 and 4.2 show that under the hypotheses of Proposition 1.5, either Proposition 1.5(a) holds, or X acts irreducibly on the natural KY -module W , or X ⊂ B n is as in Proposition 4.1(ii) with k = 0. Theorem 1.2 handles the latter situation in case X = B n . The resolution of this case will follow from induction; see the end of this section. Now for the case where X acts irreducibly on W , we first show that X must act tensor indecomposably. Proposition 4.3. Let Y, X, V be as in Proposition 1.5, and let W be the natural (2n + 2)dimensional KY -module. If W | X is irreducible, then W | X is tensor indecomposable.

Proof. Suppose the contrary. Then write
. . , f t are distinct p-powers, and t ≥ 2. Then the criterion on the highest weight of self-dual modules shows that each W i carries an X-invariant nondegenerate bilinear form. Write W = D ⊗ F , where each of D and F is a KX-module and the image of X in Isom(W ) ′ lies in Isom(D) ′ • Isom(F ) ′ . Set dim D = d and dim F = f and assume d ≥ f ≥ 2, (and so at least one of d and f is even).
Extracting part of the argument given on page 76 of [23], we will now show that Y contains a semisimple group inducing Isom(D) ′ • Isom(F ) ′ . As pointed out above, X stabilizes the product bilinear form on W . But then the irreducibility of X on W forces this to be a scalar multiple of the form defining Y . Adjusting the form on D if necessary, we may assume that the product form is precisely the form defining Y . This establishes the claim unless p = 2. In this case, we have Isom(D) ′ • Isom(F ) ′ ⊂ Sp(D) • Sp(F ). The latter group preserves a quadratic form Q on W , such that Q has the same polarization as the bilinear form on W and also such that Q(x ⊗ y) = 0 for all x ∈ D and y ∈ F (see [17,Section 4.4]). By [2, 4.9], X fixes a unique quadratic form with prescribed bilinear form, and so this must necessarily be the form Q and we again have the claim. Hence we now have that X lies in a closed subgroup J of Y which is the preimage in Y of the group Isom(D) ′ • Isom(F ) ′ . In particular J acts irreducibly on V as well. Let D 1 be an isotropic 1-space in D; Set P J to be the stabilizer in J of the above flag, and P Y the stabilizer in Y of this flag. Then P J is the product of the preimages of the parabolic subgroups P D and P F of Isom(D) and Isom(F ) which are the stabilizers of the isotropic 1-spaces D 1 and F 1 , respectively. So R u (P D ) acts trivially on D 1 , D ⊥ 1 /D 1 and D/D ⊥ 1 , and similarly for R u (P F ). On the other hand, R u (P Y ) is precisely the subgroup of Y which acts trivially on V j+1 /V j for 0 ≤ j ≤ 4. One then sees that Assume for the moment that f > 2; then V 3 /V 2 is an orthogonal space of dimension at least 4. We note that the subspace ((D 1 ⊗ F 3 ) + V 2 )/V 2 is a non-zero singular subspace in V 3 /V 2 left invariant by (Isom(D ⊥ 1 /D 1 )) ′ • (Isom(F ⊥ 1 /F 1 )) ′ and so the projection of this latter group in Isom(V 3 /V 2 ) is contained in a proper parabolic. It then follows from an application of [24] to the irreducible KY -module V and the subgroup J that the portion of the Dynkin diagram for Y corresponding to the subgroup Isom(V 3 /V 2 ) has zero labels, when representing λ by a labeled diagram. But this contradicts our assumption on λ.
Consider now the case where f = 2, so dim W = 2n + 2 and d = n + 1. Since we are assuming Y = D n+1 with n ≥ 4, we have d ≥ 5. Now Isom(D 2 ) • Isom(F 2 ) stabilizes the image of D 1 ⊗F ⊥ 1 in V 2 /V 1 and so lies in a proper parabolic subgroup of Isom(V 2 /V 1 ). Note that dim(V 2 /V 1 ) = n − 1, and arguing as above we deduce that the nodes corresponding to SL(V 2 /V 1 ) in the Dynkin diagram are labelled zero. So now we have λ = a 1 λ 1 +a n λ n , with a n = 0. Moreover, the image of X in Isom(W ) ′ lies in Isom(D) ′ •Isom(F ) ′ = Isom(D) ′ •Sp 2 , which then implies that D is even-dimensional and the latter group is Sp(D) • Sp 2 , and acts irreducibly on V . The factor Sp(D) lies in the derived subgroup of an A n -type Levi factor L of Y , indeed is the naturally embedded Sp n+1 subgroup of A n . Moreover, Sp(D) acts homogeneously on V . Now λ and λ − α n , or λ − α n−1 − α n − α n+1 , depending on whether the root system of L contains α n+1 or α n , afford the highest weights of irreducible summands of V | L . It is then straightforward to see that the restrictions of these weights to the subgroup Sp(D) provide non-isomorphic composition factors. This provides the final contradiction in case f = 2 and completes the proof of the result.
For the proof of Proposition 1.5, we continue with our consideration of the case where X acts irreducibly and, by the previous result, tensor-indecomposably on the natural KYmodule W . In particular, we may now assume that X is a simple, proper, closed subgroup of Y . The hypotheses of Proposition 1.5 then imply that X is of type B m or of type F 4 . Moreover, we have the full set of hypotheses on the embedding of a parabolic subgroup of X in a parabolic subgroup of Y , in particular, with respect to the restriction of the highest weight λ to a Levi factor. We will in fact show that W | X irreducible and the hypotheses of Proposition 1.5 are incompatible.
In each case, we will require some detailed information about the commutator series of an irreducible KX-module with respect to a fixed maximal parabolic subgroup. We start by considering: Fix a maximal torus T X of X. Let Π(X) = {β 1 , . . . , β m } be a base of the root system of X, B X the corresponding Borel subgroup containing T X , and {ω 1 , . . . , ω m } a set of fundamental dominant weights chosen with respect to the fixed base Π(X). Set s i to be the reflection corresponding to the root β i for 1 ≤ i ≤ m. For a torus S of X, write X(S) for the group of rational characters of S. Let P X be a maximal parabolic subgroup of X corresponding to the subset {β i : 2 ≤ i ≤ m}, and containing the opposite Borel subgroup B − X . Let Q X = R u (P X ) and P X = L X Q X for a Levi factor L X of P X . Let ω = m i=1 d i ω i be a p-restricted dominant weight in X(T X ); set M = L X (ω) and assume X preserves a non-degenerate quadratic from on M ; let us denote this by Q and the associated bilinear form by ( , ) : M × M → K.
For a unipotent group J and a KJ-module N , we recall the standard notation [N, J] for the subspace spanned by the set of vectors n − xn, where x ∈ J and n ∈ N . We introduce an additional notation: set [N, (1) We will say a T X -weight ν in M has level i if ν = ω − iβ 1 − m j=2 c j β j for some integers i, c j ≥ 0.
If the form is non-degenerate on this quotient space, then there exists ν of level i such that −ν is also of level i. In particular, in this case, d m is even.
Proof. We start by proving (i), arguing by induction on i, the case i = 0 being trivial by definition. Let i ≥ 1, and fix an ordering on Φ + (X) such that any root containing β 1 is smaller than the others. By [15, Section 27], we clearly have that [M, Q i X ] ⊆ M ν , where the sum ranges over all T X -weights ν of level at least i, and so it remains to show that for all such T X -weights ν. It is enough to show the latter for T X -weights of level exactly i. Let then ν be a T X -weight of level i in M, and let γ 1 ≤ . . . ≤ γ k be roots where v + is a maximal vector in M. (This will be enough by Lemma 2.1.) Set w = f γ 2 · · · f γ k v + . Then w has level i − 1, as γ 1 involves β 1 thanks to our choice of ordering on Φ + , and hence w ∈ [M, Q i−1 X ] by induction. Also, we have and hence x −γ 1 (1)w − w has non-zero coefficient of f γ 1 · · · f γ k v + , and all other terms in the sum are weight vectors of weights different from ν. One then deduces that [M, Q i X ] must contain f γ 1 · · · f γ k v + as desired, showing that (i) holds.
The statement of (ii) now follows by induction on i.
For (iii), set w 0 to be the longest word of the Weyl group of X. Writing the ω i in terms of the simple roots β j , we see that w 0 (ω) = −ω = ω − 2(d 1 + d 2 + · · · + d m−1 +  For (viii), we note that for i ≥ (e/2), [M, Q i+1 X ] is totally singular and so the given quadratic form induces an L ′ X -invariant form on the quotient. Now if p = 2, the second statement follows directly from (iv). If p = 2, and [M, is odd-dimensional, so that the bilinear form has a 1-dimensional non-singular radical, then (iv) implies that the radical is generated by a vector of weight 0. So the weight ν = 0 satisfies the given condition. For the claim about the parity of d m , we recall that for ν of level j, −ν is of level e − j. So we deduce that e is even, and so d m as well.
In case e is even, we will consider a certain T X -weight at level (e/2), namely the weight Note that Lemma 4.6. Assume d m is even. For each 0 ≤ j ≤ d m , there exists a unique weight In addition, each ν j is of multiplicity 1 and if ν ∈ X(T X ) with ν| T X ∩L ′ X = µ| T X ∩L ′ X , then ν = ν j for some j. In particular, there exists an odd number of weights ν such that ν| T X ∩L ′ Solving these equations leads to the relations c j = c m + m−1 i=j d i , for 1 ≤ j ≤ m − 1. Now note as well that for ν as given, s m−1 s m−2 · · · s 1 ν = ω − c m β m . Hence ν has multiplicity at most 1 and has multiplicity 1 if and only if c m ≤ d m . So for all 0 ≤ j ≤ d m , there exists a weight ν j , of multiplicity 1, and of level j Proof. Suppose the contrary, that is, let W be the natural KY -module and suppose that W | X is irreducible. Then by Proposition 4.3, W | X is tensor indecomposable. If W | X is not p-restricted, then we replace W | X by a p-restricted representation ρ of a simply-connected coverX of X, and consider W | ρ(X) . For the purposes of the argument, we may replace X by ρ(X). Let W = L X (ω) for ω = m i=1 d i ω i , a p-restricted dominant weight. As in the hypotheses of Proposition 1.5, let P Y be a parabolic subgroup of Y containing P X with Q X ⊂ R u (P Y ) = Q Y . Fix a maximal torus T Y of P Y with T X ⊂ T Y . We choose a base Π(Y ) = {α 1 , . . . , α n+1 } of the root system of Y so that P Y contains the opposite Borel B − Y subgroup with respect to this base. Now let {0} = W 0 W 1 · · · W t−1 be a flag of totally singular subspaces, such that P Y is the full stabilizer in Y of this flag; so P Y is the stabilizer of the flag of which P Y is the full stabilizer. By our hypotheses on P Y , we have that W t /W t−1 is an orthogonal space of dimension at least 8 (so W t−1 = W t ) and we have L t = (Isom(W t /W t−1 )) ′ , a group of type D m for m ≥ 4. Note that Recall that by hypothesis, L ′ X is of type B m−1 . If m > 4, L ′ X is the stablizer in L t of a non-singular 1-space of W t /W t−1 . Note that if m = 4, so L ′ X is of type B 3 and (Isom(W t /W t−1 )) ′ = D 4 , it may be that L ′ X acts irreducibly on W t /W t−1 as a spin module. We will take care to consider the latter possibility in what follows.
We need to see how the two flags which are stabilized by P X are related, i.e. the flag of subspaces {[W, Q j X ]} j≥0 and the flag of subspaces {W i } i≥0 . We first show that e (as defined in Lemma 4.5 (iii)) is even. Indeed, suppose the contrary. Then P X lies in the stabilizer in Y of [W, Q e+1 2 X ], a totally singular subspace of dimension 1 2 dim W (using Lemma 4.5 (vii)). Set P = QL to be this stabilizer, a maximal parabolic subgroup of Y with Levi factor L of type A n and Q = R u (P ). Consider now the action of L ′ X and L ′ on the commutator quotient V /[V, Q]. By [24], each group acts irreducibly. But given that the restriction of λ to T Y ∩ L ′ has at least two non-zero labels, we see that there are no compatible configurations coming from Table 1 of [23]. (Here we are invoking Hypothesis 1.4 (ii).) Hence e is even as claimed. In particular, the weight µ of (4) exists and Lemma 4.6 applies.
Choose s minimal such that [W, Q We will treat the case p = 2 separately below.
Assume for now that p > 2.
Let P X be the maximal parabolic subgroup of X corresponding to the simple root β 1 , containing the opposite Borel subgroup. We extend the notion of "level" to this setting and rely on [23, 2.3]. Note that the maximum level of weights with respect to this parabolic is e := 2(2d 1 + 3d 2 + 2d 3 + d 4 ). Consider the flag of totally singular subspaces Then P X lies in the stabilizer of this flag, which is the stabilizer of the full flag ] is a non-degenerate subspace and we claim that it is non-trivial of dimension at least 6, which then implies that the Levi factor L ′ X of type C 3 projects non-trivially into Isom([W, Q (e/2) X ]/[W, Q (e/2)+1 X ]), a group of type D ℓ with ℓ ≥ 4 (as there is no non-trivial morphism from C 3 into D ℓ for ℓ ≤ 3. Indeed, consider the weight This weight is of level (e/2), occurs in W by [22], and affords the dominant ]) ≥ 6 as claimed. Now consider the action of X on the irreducible module V = L Y (λ) where λ satisfies the congruence relations. Since C 3 projects non-trivially into the D ℓ factor of P Y , we must have that C 3 acts irreducibly on the D ℓ module with the highest weight as given. But there is no such example in [23, Table 1], giving the desired contradiction. (Here we invoked Hypothesis 1.4 (iii).) Finally, consider the case where p = 2. Since W | X is tensor-indecomposable, [23, 1.6] implies that ω = d 1 ω 1 + d 2 ω 2 or d 3 ω 3 + d 4 ω 4 . Moreover, using the graph automorphism of F 4 , it suffices to consider the second situation, in which case [23, 2.3] still applies. Using [19], we see that the weight lattice of W | X is as in characteristic 0, in particular, we can exhibit as above in each case a non-zero dominant weight of level e/2 and the above argument goes through. If ω = ω 3 , when e = 4, we take ω − 2β 1 − 2β 2 − 3β 3 − β 4 ; if ω = ω 4 and e = 2, take ω − β 1 − β 2 − β 3 − β 4 ; if ω = ω 3 + ω 4 and e = 6, take Proof of Proposition 1.5. We can now complete the proof of Proposition 1.5. Let X, Y and V be as in the statement of the proposition and assume Hypothesis 1.4 for all embeddings H ⊂ G such that rank G < rank Y. By Propositions 4.1, 4.2, 4.3, 4.7, 4.8, we have that either Proposition 1.5(a) holds, or X acts reducibly on W and X ⊂ B n ⊂ Y . If X = B n , Proposition 1.5(b) holds; so assume X B n . The restriction of V to B n affords a p-restricted irreducible KB n -module with highest weight having at least three nonzero coefficients when expressed in terms of the fundamental dominant weights for B n . Moreover, X has a Levi subgroup of type B m , by hypothesis. We now refer to [23, Table 1], invoking Hypothesis 1.4 (i), to see that there are no such irreducible triples (X, B n , λ| T Bn ), where T Bn is a maximal torus of B n lying in the maximal torus T Y . This completes the proof of the result.

Proof of Theorem 1.7
Let Y be a simply connected, simple algebraic group of type E n , n = 6, 7, 8, defined over K. We start by considering a maximal, closed, connected semisimple subgroup X of Y , satisfying the hypotheses of Theorem 1.7, namely, X has a proper parabolic subgroup whose Levi factor is of type B m , for some m ≥ 3. Referring to [18,Theorem 1], we see that we must consider the following pairs (X, Y ): We start by dealing with the two latter possibilities.
Proposition 5.1. The maximal subgroup A 1 F 4 ⊂ E 7 acts reducibly on all non-trivial irreducible KE 7 -modules. The maximal subgroup G 2 F 4 ⊂ E 8 acts reducibly on all nontrivial irreducible KE 8 -modules.
Proof. We assume the contrary. Let X = M 1 F 4 ⊂ Y be a maximal subgroup of Y = E n , for M 1 = A 1 , respectively G 2 and n = 7, respectively 8. The factor F 4 is embedded in the usual way as a maximal subgroup of an E 6 Levi factor of Y . Assume that X acts irreducibly on V = L Y (λ) where λ is a non-zero p-restricted dominant weight. Adopting the usual notation as in previous results, we set λ = n i=1 a i λ i , where {λ 1 , . . . , λ n } are the fundamental dominant weights with respect to a fixed choice of base for the root system of Y . Since X acts irreducibly on V , F 4 acts homogeneously. Let {ω 1 , ω 2 , ω 3 , ω 4 } be a set of fundamental dominant weights for F 4 again with respect to a fixed choice of base of the root system which is compatible with the given choice for Y . By [24] we have one F 4 composition factor of V with highest weight ω = a 2 ω 1 + a 4 ω 2 + (a 3 + a 5 )ω 3 + (a 1 + a 6 )ω 4 . The homogeneity of V | F 4 implies that a 7 = 0, and a 8 = 0 if n = 8, as otherwise λ − α 7 (resp. λ − α 7 − α 8 ) would afford the highest weight of a composition factor different from L X (ω). Now let i 0 be maximal such that a i 0 = 0 (such exists since λ = 0). If i 0 = 2, then λ − α 2 − α 4 − α 5 − α 6 − α 7 affords an F 4 composition factor of V not isomorphic to that already given. Hence i 0 = 2. If i 0 = 1, then λ − α 1 − α 3 − α 4 − α 5 − α 6 − α 7 has the same property. For all other cases, we take λ − 7 i=i 0 α i , which again affords an F 4 -composition factor with highest weight different from ω. This is the final contradiction.
The remainder of this section is devoted to the proof of the following result, first proven by Testerman [26,Theorem 5.0 (i)] under the assumption that [23, Table 1, IV1, IV1 ′ ] formed a complete family of irreducible triples for the usual embedding B n ⊂ D n+1 .
Theorem 5.2. Let Y be a simply connected, simple algebraic group of type E 6 over K and let X be the subgroup of type F 4 , embedded in Y in the usual way. Also let V = L Y (λ) be a non-trivial, irreducible KY -module having p-restricted highest weight λ ∈ X + (T Y ). Assume Hypothesis 1.4 for all embeddings H ⊂ G with rank(G) < rank(Y ). Then X acts irreducibly on V if and only if one of the following holds, where we give λ up to graph automorphisms.
The proof relies on some detailed knowledge of the structure of certain Weyl modules for E 6 and F 4 . Section 5.1 below provides some results based on the Jantzen p-sum formula. In Section 5.2, we apply the methods from Section 5.1 to various Weyl modules in order to obtain insight on their structure, as well as information on certain weight multiplicities in the corresponding irreducible quotient. These results shall then prove useful in Section 5.3, in which we conclude by showing that Theorem 5.2 holds. Finally, at the end of the section, we see how these results lead to a proof of Theorem 1.7.

Understanding Weyl modules
Let G be a semisimple algebraic group over K, let B be a Borel subgroup of G containing a fixed maximal torus T. Let Π = {α 1 , . . . , α n } denote the corresponding base for the root system Φ = Φ + ⊔ Φ − of G, and let λ 1 , . . . , λ n denote the corresponding fundamental dominant weights for T. Let ρ denote the half-sum of all positive roots in Φ, or equivalently, the sum of all fundamental dominant weights. Also for λ, µ ∈ X + (T ) such that µ ≺ λ, define The following corollary to the strong linkage principle [1] gives a necessary condition for µ to afford the highest weight of a KG-composition factor of V G (λ), in the case where p > 2 and G is not of type G 2 . We refer the reader to [23, 6.2] for a proof. Proposition 5.3. Assume p > 2 and let G be a simple algebraic group of type different from G 2 . Also let λ and µ be as above, and assume the inner product on ZΦ is normalized so that long roots have length 1. If µ affords the highest weight of a composition factor of V G (λ), then 2d(λ, µ) ∈ pZ.
Let {e µ } µ∈X(T ) denote the standard basis of the group ring Z[X(T )] over Z. The Weyl group W of G acts on Z[X(T )] by we µ = e wµ , w ∈ W , µ ∈ X(T ), and we write Z[X(T )] W to denote the set of fixed points. The formal character of a given KG-module V is the linear polynomial ch V ∈ Z[X(T )] W defined by Also, for λ ∈ X + (T ), we write [16,II,2.13], for instance). The Jantzen p-sum formula [16,II,Proposition 8.19] yields the existence of a filtration V G (λ) Consider a T -weight µ ∈ X(T ) with µ ≺ λ. Following [7, Section 3.2], we now define a "truncated" version of ν c (T λ ), which shall prove useful in computations, by setting where the a ν (ν ∈ X + (T )) are as in (5), and where for every ν ∈ X + (T ), we have Finally, the latter decomposition yields for some b ξ ∈ Z. The following proposition shows how (7) can be used in order to determine the possible composition factors of V G (λ), together with an upper bound for their multiplicity. We refer the reader to [7, Proposition 3.6] for a proof.
Proposition 5.4. Let λ ∈ X + (T ) and consider a T -weight µ ≺ λ. Also let ξ ∈ X + (T ) be a dominant weight such that µ ξ ≺ λ. Then ξ affords the highest weight of a composition factor of V G (λ) if and only if b ξ = 0 in (7).
For ν ∈ X + (T ), we call the coefficient a ν in (6) the contribution of ν to ν c µ (T λ ), and we say that ν contributes to ν c µ (T λ ) if its contribution is non-zero. Now applying Proposition 5.4 for specific weights µ, ξ with µ ξ ≺ λ requires the knowledge of the contribution of ν to ν c µ (T λ ) for each dominant T -weight µ ν ≺ λ. In certain cases, knowing whether or not a given T -weight contributes to ν c µ (T λ ) can be easily determined, as the following result shows. We refer the reader to [7, Lemma 3.7] for a proof.
Lemma 5.5. Let λ, µ and ν be as above, with ν maximal, with respect to the partial order , such that ν contributes to ν c µ (T λ ). Then ν affords the highest weight of a composition factor of V G (λ).
Fix ν ∈ X + (T ), and recall from [3, Planches I-IV] the description of the simple roots and fundamental dominant weights for T in terms of a basis {ε 1 , . . . , ε d Φ } for a Euclidean space E of dimension d Φ . For α ∈ Φ + and r ∈ Z ≥0 such that 1 < r < λ + ρ, α , we write λ + ρ − rα = a 1 ε 1 + · · · + a d Φ ε d Φ , as well as ν + ρ = b 1 ε 1 + · · · + b d Φ ε d Φ , following the ideas of [21]. Also following [7, Section 3.2], we set The action of the Weyl group W of G on the basis {ε 1 , . . . , ε d Φ }, described in [3, Planches I-IV], extends to an action of W on Q d Φ in the obvious way. (We write w · A for w ∈ W , A ∈ Q d Φ .) In addition, define the support of an element z ∈ ZΦ to be the subset supp(z) of Π consisting of those simple roots α such that c α = 0 in the decomposition z = α∈Π c α α. Finally, for w ∈ W , we write det(w) for the determinant of w as an invertible linear transformation of X(T ) R = X(T ) ⊗ Z R. The following result is our main tool for determining the contribution of ν to ν c µ (T λ ), for each weight ν ∈ X + (T ) with µ ν ≺ λ. We refer the reader to [7, Theorem 3.8] for a proof.

Computing certain weight multiplicities
We now use the results introduced in the previous section to investigate the structure of certain Weyl modules, as well as to compute various weight multiplicities in certain irreducible modules. For λ ∈ X + (T ) and c 1 , . . . , c n ∈ Z ≥0 , we use the notation λ − c 1 c 2 · · · c n for the weight λ − n r=1 c r α r . Lemma 5.7. Let G be of type B 3 over K, and let a, b ∈ Z >0 be such a + b + 1 = p. Also let λ = aλ 1 + bλ 2 + aλ 3 ∈ X + (T ) and µ = λ − 111 ∈ Λ + (λ). Then µ affords the highest weight of a composition factor of V G (λ) if and only if (a, b) = (p − 2, 1).
Proof. We first deal with the situation where p = 3, a = b = 1, so that we have p = a + 2 in this case. Here the only dominant T -weights ν ∈ X + (T ) such that µ ν ≺ λ are λ − 110, λ − 011, and µ itself. Now an application of Proposition 5.3 shows that λ − 011 cannot afford the highest weight of a composition factor of V G (λ). On the other hand [V G (λ), L G (λ − 110)] = 1 by Proposition 2.2, but m L G (λ−110) (µ) = 0 (as λ − 110 = 3λ 3 ) and hence [20], from which we deduce the existence of a second composition factor of V G (λ). Now since no weight ν with µ ≺ ν ≺ λ affords the highest weight of a composition factor of V G (λ), the desired result follows in this case.
We will also require some knowledge about the structure of certain Weyl modules for a simple group of type A n over K. However, due to the complexity of the description of fundamental dominant weights in terms of an orthonormal basis of a Euclidean space for such G, it is more convenient to work in a group of type B n+1 , and then deduce the desired result for A n . Lemma 5.9. Let G be of type B n+1 (n ≥ 3) over K, and let 0 < a, b, c < p be positive integers. Also let λ = aλ 1 + bλ 2 + cλ n ∈ X + (T ), ν 1 = λ − α 1 − α 2 , ν 2 = λ − (α 2 + · · · + α n ), and µ = λ − (α 1 + · · · + α n ). Then µ affords the highest weight of a composition factor of V G (λ) if and only if p | (a + b + c + n − 1). Also if n = 3, then we have χ µ (λ) = ch L G (λ) + δ p,a+b+1 ch L G (ν 1 ) + δ p,b+c+1 ch L G (ν 2 ) + δ p,a+b+c+2 ch L G (µ).
Proposition 5.10. Let G, λ, and µ be as in Table 1, with a, b, c ∈ Z >0 such that a+b+1 = p. Then the multiplicity of µ in V G (λ), respectively L G (λ), is given in the fourth, resp. fifth, column of the table.  Table 1. Some weight multiplicities in various irreducibles. Here a, b, c ∈ Z >0 are such that a + b + 1 = p.
that if λ and p are as in (i) or (ii) of Theorem 1.7, then X acts irreducibly on V = L Y (λ) by [26, 5.6, 5.7]. By Lemma 5.11, in order to complete the proof, it only remains to show that V | X is reducible in the situation where λ = aλ 2 + aλ 3 + bλ 4 for some a, b ∈ Z >0 such that a + b + 1 = p, which we shall assume holds in the remainder of the proof. Here the restriction of λ to T X is given by ω = aω 1 + bω 2 + aω 3 ∈ X + (T X ). Consider the dominant T X -weight ω ′ = ω − β 1 − β 2 − 2β 3 − β 4 ∈ X + (T X ).
We now compute an upper bound for m L X (ω) (ω ′ ) by dealing with each of the following four possibilities separately.
Consequently, in each case, we get that m V | X (ω ′ ) > m L X (ω) (ω ′ ), showing the existence of a second composition factor of V for X. In particular V | X is reducible. This completes the proof of Theorem 5.2.
Proof of Theorem 1.7. Let X ⊂ Y = E n be as in the statement of Theorem 1.7.
We embed X in a maximal proper closed connected subgroup of Y and deduce by the remarks at the beginning of Section 5, Proposition 5.1, and Theorem 5.2, that X lies in the F 4 subgroup of Y = E 6 . So we must determine the irreducible configurations X ⊂ F 4 acting irreducibly on the irreducible F 4 -module with highest weight λ ∈ {(p − 3)λ 1 (p > 3), λ 1 +(p−2)λ 3 (p > 2)}; this is covered by [26,Main Theorem]. Then [26, Main Theorem (iii), (iv)] completes the proof of Theorem 1.7.