Invariant forms on irreducible modules of simple algebraic groups
aa r X i v : . [ m a t h . G R ] M a r Invariant forms on irreducible modules ofsimple algebraic groups
Mikko Korhonen ∗†‡
November 5, 2018
Abstract
Let G be a simple linear algebraic group over an algebraically closedfield K of characteristic p ≥ and let V be an irreducible rational G -module with highest weight λ . When V is self-dual, a basic question toask is whether V has a non-degenerate G -invariant alternating bilinearform or a non-degenerate G -invariant quadratic form.If p = 2 , the answer is well known and easily described in terms of λ . In the case where p = 2 , we know that if V is self-dual, it alwayshas a non-degenerate G -invariant alternating bilinear form. However,determining when V has a non-degenerate G -invariant quadratic formis a classical problem that still remains open. We solve the problem inthe case where G is of classical type and λ is a fundamental highestweight ω i , and in the case where G is of type A l and λ = ω r + ω s for ≤ r < s ≤ l . We also give a solution in some specific cases when G is of exceptional type.As an application of our results, we refine Seitz’s descriptionof maximal subgroups of simple algebraic groups of classical type. Oneconsequence of this is the following result. If X < Y <
SL( V ) aresimple algebraic groups and V ↓ X is irreducible, then one of thefollowing holds: (1) V ↓ Y is not self-dual; (2) both or neither of themodules V ↓ Y and V ↓ X have a non-degenerate invariant quadraticform; (3) p = 2 , X = SO( V ) , and Y = Sp( V ) . ∗ Section de mathématiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lau-sanne, Switzerland † Email address: mikko.korhonen@epfl.ch ‡ The author was supported by a grant from the Swiss National Science Foundation(grant number _ ). ontents G -modules 63 Fundamental representations for type C l V ( ω r ) . . . . . . . . . . . . . . . . . . . . . . 133.3 Computation of a quadratic form Q on V ( ω r ) . . . . . . . . . 18 L ( ω r + ω s ) for type A l V ( ω r + ω n − r ) . . . . . . . . . . . . . . . . . . 235.3 Computation of a quadratic form Q on V ( ω r + ω n − r ) . . . . . 31 Introduction
Let V be a finite-dimensional vector space over an algebraically closed field K of characteristic p ≥ .A fundamental problem in the study of simple linear algebraic groupsover K is the determination of maximal closed connected subgroups of sim-ple groups of classical type ( SL( V ) , Sp( V ) and SO( V ) ). Seitz [Sei87] hasshown that up to a known list of examples, these are given by the im-ages of p -restricted, tensor-indecomposable irreducible rational representa-tions ϕ : G → GL( V ) of simple algebraic groups G over K .Then given such an irreducible representation ϕ , one should still deter-mine which of the groups SL( V ) , Sp( V ) and SO( V ) contain ϕ ( G ) . In mostcases the answer is known. • If V is not self-dual, then ϕ ( G ) is only contained in SL( V ) . Furthermore,we know when V is self-dual (see Section 2). • If p = 2 and V is self-dual, then ϕ ( G ) is contained in Sp( V ) or SO( V ) ,but not both [Ste68, Lemma 78, Lemma 79]. Furthermore, we know forwhich irreducible representations the image is contained in Sp( V ) andfor which the image is contained in SO( V ) (see Section 2). • If p = 2 and V is self-dual, then ϕ ( G ) is contained in Sp( V ) [Fon74,Lemma 1].Currently what is still missing is a method for determining in character-istic two when exactly ϕ ( G ) is contained in SO( V ) . This problem is the mainsubject of this paper, and we can state it equivalently as follows. Problem 1.1.
Assume that p = 2 and let L G ( λ ) be an irreducible G -modulewith highest weight λ . When does L G ( λ ) have a non-degenerate G -invariantquadratic form? This is a nontrivial open problem. There is some literature on the subject[Wil77], [SW91], [GW95], [GW97], [GN16], but currently only partial resultsare known. The main result of this paper is a solution to Problem 1.1 in thefollowing cases: • G is of classical type ( A l , B l , C l or D l ) and λ is a fundamental dominantweight ω r for some ≤ r ≤ l (Theorem 4.2). • G is of type A l and λ = ω r + ω s for ≤ r < s ≤ l (Theorem 5.1).3n the case where G is of exceptional type, we will give some partialresults in Section 6. For G of type G and F , we are able to give a completesolution (Proposition 6.1, Proposition 6.3). For types E , E , and E , wegive the answer for some specific λ (Table 6.1). In the final section of thispaper, we will give various applications of our results and describe some openproblems motivated by Problem 1.1.One particular application, given in subsection 7.3, is a refinement ofSeitz’s [Sei87] description of maximal subgroups of simple algebraic groupsof classical type. In [Sei87], Seitz gives a full list of all non-maximal irreduciblesubgroups of SL( V ) , but the question of which classical groups contain theimage of an irreducible representation is not considered. For example, it ispossible that we have a proper inclusion X < Y of irreducible subgroups of
SL( V ) such that X is a maximal subgroup of SO( V ) . In subsection 7.3, wego through the list given by Seitz and describe when exactly such inclusionsoccur. In particular, our results have the consequence (Theorem 7.9) that if X < Y <
SL( V ) are simple algebraic groups and V ↓ X is irreducible, thenone of the following holds:(i) The module V ↓ Y is not self-dual;(ii) Both V ↓ X and V ↓ Y have an invariant quadratic form;(iii) Neither of V ↓ X or V ↓ Y has an invariant quadratic form;(iv) p = 2 , X = SO( V ) and Y = Sp( V ) .The general approach for the proofs of our main results is as follows. Abasic method used throughout is Theorem 9.5. from [GN16] (recorded here inProposition 2.2), which allows one to determine whether L G ( λ ) is orthogonal(when p = 2 ) by computing within the Weyl module V G ( λ ) . For G of classicaltype and V irreducible with fundamental highest weight, we will first proveour result in the case where G is of type C l (Proposition 3.1). From thisthe result for other classical types is a fairly straightforward consequence(Theorem 4.2).In the case where G is of type C l and λ = ω r , and in the case where G is of type A l and λ = ω r + ω s , the proofs of our results are heavily based onvarious results from the literature on the representation theory of G . We willuse results about the submodule structure of the Weyl module V G ( λ ) found in[PS83], [Ada84] and [Ada86]. We will also need the first cohomology groups of L G ( λ ) which were computed in [KS99] and [KS01, Corollary 3.6]. One morekey ingredient in our proof will be the results of Baranov and Suprunenko in[BS00] and [BS05], which give the structure of the restrictions of L G ( λ ) tocertain subgroups defined in terms of the natural module of G .4 otation and terminology We fix the following notation and terminology. Throughout the whole text,let K be an algebraically closed field of characteristic p ≥ . All groupsthat we consider are linear algebraic groups over K , and by a subgroup wealways mean a closed subgroup. All modules and representations will befinite-dimensional and rational.Unless otherwise mentioned, G denotes a simply connected simple alge-braic group over K with l = rank G , and V will be a finite-dimensionalvector space over K . Throughout we will view G as its group of rationalpoints over K , and most of the time G will studied either as a Chevalleygroup constructed with the usual Chevalley construction (see e.g. [Ste68]),or as a classical group with its natural module (i.e. G = SL( V ) , G = Sp( V ) or G = SO( V ) ). We will occasionally denote G by its type, so notation suchas G = C l means that G is a simply connected simple algebraic group of type C l . We fix the following notation, as in [Jan03]. • T : a maximal torus of G , with character group X ( T ) . • X ( T ) + : the set of dominant weights for G , with respect to some systemof positive roots. • ch V : the character of a G -module V . Here ch V is an element of Z [ X ( T )] . • ω , ω , . . . , ω l : the fundamental dominant weights in X ( T ) + . We usethe standard Bourbaki labeling of the simple roots, as given in [Hum72,11.4, pg. 58]. • L ( λ ) , L G ( λ ) : the irreducible G -module with highest weight λ ∈ X ( T ) + . • V ( λ ) , V G ( λ ) : the Weyl module for G with highest weight λ ∈ X ( T ) + . • rad V ( λ ) : unique maximal submodule of V ( λ ) .For a dominant weight λ ∈ X ( T ) + , we can write λ = P li =1 m i ω i where m i ∈ Z ≥ . We say that λ is p -restricted if p = 0 , or if p > and ≤ m i ≤ p − for all ≤ i ≤ l . The irreducible representation L G ( λ ) is said to be p -restricted if λ is p -restricted.A bilinear form b is non-degenerate , if its radical rad b = { v ∈ V : b ( v, w ) = 0 for all w ∈ V } is zero. For a quadratic form Q : V → K on a vector space V , its polarization is the bilinear form b Q defined by5 Q ( v, w ) = Q ( v + w ) − Q ( v ) − Q ( w ) for all v, w ∈ V . We say that Q is non-degenerate , if its radical rad Q = { v ∈ rad b Q : Q ( v ) = 0 } is zero.For a KG -module V , a bilinear form ( − , − ) is G -invariant if ( gv, gw ) =( v, w ) for all g ∈ G and v, w ∈ V . A quadratic form Q : V → K is G -invariant if Q ( gv ) = Q ( v ) for all g ∈ G and v ∈ V . We say that V is symplectic if ithas a non-degenerate G -invariant alternating bilinear form, and we say that V is orthogonal if it has a non-degenerate G -invariant quadratic form.Note that if V has a G -invariant bilinear form, then for λ, µ ∈ X ( T ) theweight spaces V λ and V µ are orthogonal if λ = − µ . Thus to compute theform on V it is enough to work in the zero weight space of V and V λ ⊕ V − λ for nonzero λ ∈ X ( T ) . For a G -invariant quadratic form Q on V , we have Q ( v ) = 0 for any weight vector v ∈ V with non-zero weight.Given a morphism φ : G ′ → G of algebraic groups, we can twist represen-tations of G with φ . That is, if ρ : G → GL( V ) is a representation of G , then ρφ is a representation of G ′ . We denote the corresponding G ′ -module by V φ .When p > , we denote by F : G → G the Frobenius endomorphism inducedby the field automorphism x x p of K , see for example [Ste68, Lemma 76].When G is simply connected and λ ∈ X ( T ) + , we have L G ( pλ ) ∼ = L G ( λ ) F .If a representation V of G has composition series V = V ⊃ V ⊃ · · · ⊃ V t ⊃ V t +1 = 0 with composition factors W i ∼ = V i /V i +1 , we will occasionallydenote this by V = W /W / · · · /W t . Acknowledgements
I am very grateful to Prof. Donna Testerman for suggesting the problem, andfor her many helpful suggestions and comments on the earlier versions of thistext. I would also like to thank Prof. Gary Seitz for providing the argumentused in the proof of Lemma 4.1, and the two anonymous referees for theirhelpful comments for improvements. G -modules Let L ( λ ) be an irreducible representation of a simple algebraic group G withhighest weight λ = P li =1 m i ω i . Write d ( λ ) = P α> h λ, α ∨ i , where the sumruns over the positive roots α , where α ∨ is the coroot corresponding to α ,and h , i is the usual dual pairing between X ( T ) and the cocharacter group.We know that L ( λ ) is self-dual if and only if w ( λ ) = − λ , where w is the longest element in the Weyl group [Ste68, Lemma 78]. Furthermore,if L ( λ ) is self-dual and p = 2 , then L ( λ ) is orthogonal if d ( λ ) is even and6oot system When is λ = − w ( λ ) ? d ( λ ) mod 2 when λ = − w ( λ ) A l ( l ≥ ) iff m i = m l − i +1 for all i , when l is even l +12 · m l +12 , when l is odd B l ( l ≥ ) always , when l ≡ , m l , when l ≡ , . C l ( l ≥ ) always m + m + m + · · · D l ( l ≥ ) l even: always l odd: iff m l = m l − , when l m l + m l − , when l ≡ . G always F always E iff m = m and m = m E always m + m + m E always Table 2.1: Values of d ( λ ) modulo for a weight λ = P li =1 m i ω i symplectic if d ( λ ) is odd [Ste68, Lemma 79]. Hence in characteristic p =2 deciding whether an irreducible module is symplectic or orthogonal is astraightforward computation with roots and weights. In Table 2.1, we givethe value of d ( λ ) mod 2 (when λ = − w ( λ ) ) for each simple type, in termsof the coefficients m i .In characteristic , it turns out that each nontrivial, irreducible self-dualmodule is symplectic, as shown by the following lemma found in [Fon74]. Weinclude a proof for convenience. Lemma 2.1.
Assume that char K = 2 . Let V be a nontrivial, irreducibleself-dual representation of a group G . Then V is symplectic for G .Proof. [Fon74] Since V is self-dual, there exists an isomorphism ϕ : V → V ∗ of G -modules, which induces a non-degenerate G -invariant bilinear form ( − , − ) defined by ( v, w ) = ϕ ( v )( w ) . Since ϕ t : V → V ∗ defined by ϕ t ( v )( w ) = ϕ ( w )( v ) is also an isomorphism of G -modules, by Schur’s lemma there existsa scalar c such that ( v, w ) = c ( w, v ) for all v, w ∈ V . Then ( v, w ) = c ( v, w ) ,so c = 1 because ( − , − ) is nonzero. Because we are in characteristic two, itfollows that c = 1 , so ( − , − ) is a symmetric form. Now { v ∈ V : ( v, v ) = 0 }
7s a submodule of G . Because V is nontrivial and irreducible, this submodulemust be all of V and so ( − , − ) is alternating.Lemma 2.1 above shows that the image of any irreducible self-dual rep-resentation lies in Sp( V ) . The following general result reduces determiningwhether L ( λ ) is orthogonal (in characteristic two) to a computation withinthe Weyl module V ( λ ) . Proposition 2.2.
Assume that char K = 2 . Let λ ∈ X ( T ) + be nonzero, λ = − w ( λ ) and suppose that λ = ω if G has type C l . Then (i) The Weyl module V ( λ ) has a nonzero G -invariant quadratic form Q ,unique up to scalar. (ii) The unique maximal submodule of V ( λ ) is equal to rad b Q . (iii) The irreducible module L ( λ ) has a nonzero, G -invariant quadratic formif and only if rad Q = rad b Q . If this is not the case, then rad Q is asubmodule of rad b Q with codimension , and H ( G, L ( λ )) = 0 . (iv) If V ( λ ) has no trivial composition factor, then L ( λ ) is orthogonal.Proof. See Theorem 9.5. and Proposition 10.1. in [GN16] for (i), (ii) and(iii). The claim in (iii) about H ( G, L ( λ )) can also be deduced from [Wil77,Satz 2.5]. The claim (iv) is a consequence of (iii), since H ( G, L ( λ )) ∼ =Ext G ( K, L ( λ )) ∼ = Hom G (rad V ( λ ) , K ) by [Jan03, II.2.14].In the case where G is of type C l and λ = ω , we have the following resultwhich is well known. We include a proof for completeness. Proposition 2.3.
Assume that char K = 2 and that G is of type C l . Then V = V ( ω ) = L ( ω ) has no nonzero G -invariant quadratic form.Proof. ([GN16, Example 8.4]) The claim follows from a more general resultthat any G -invariant rational map f : V → K is constant. Indeed, for such f we have f ( gv ) = f ( v ) for all g ∈ G , v ∈ V . Because G acts transitively onnonzero vectors in V , it follows that f ( w ) = f ( v ) for all w ∈ V − { } . Thus f ( w ) = f ( v ) for all w ∈ V since f is rational. Lemma 2.4.
Let V and W be G -modules. If V and W are both symplecticfor G , then V ⊗ W is orthogonal for G .Proof. See [SW91, Proposition 3.4], [GN16, Proposition 9.2], or [KL90, 4.4,pg. 126-127]. 8 emark 2.5.
Assume that char K = 2 . Then lemmas 2.1 and 2.4 showthat if V is a non-orthogonal irreducible G -module, then V must be tensorindecomposable. By Steinberg’s tensor product theorem, this implies that V is a Frobenius twist of L G ( λ ) for some -restricted weight λ ∈ X ( T ) + .Therefore to determine which irreducible representations of G are orthogonal,it suffices to consider V = L G ( λ ) with λ ∈ X ( T ) + a -restricted dominantweight. C l Throughout this section, assume that G is simply connected of type C l , l ≥ . In this section we determine when in characteristic a fundamentalirreducible representation L ( ω r ) , ≤ r ≤ l , of G has a nonzero G -invariantquadratic form. The answer is given by the following proposition, which wewill prove in what follows. Proposition 3.1.
Assume char K = 2 . Let ≤ r ≤ l . Then L ( ω r ) is notorthogonal if and only if r = 1 , or r = 2 i +1 for some i ≥ and l + 1 ≡ i +1 + 2 i + t mod 2 i +2 , where ≤ t < i . The following examples are immediate consequences of Proposition 3.1.
Example 3.2. If char K = 2 , then L ( ω ) is orthogonal if and only if l . Example 3.3. If char K = 2 , then L ( ω ) is orthogonal if and only if l , . Example 3.4. If char K = 2 , then L ( ω l ) is orthogonal if and only if l ≥ (this was also proven in [Gow97, Corollary 4.3]) and L ( ω l − ) is orthogonal ifand only if l = 3 , l = 4 or l ≥ .A rough outline for the proof of Proposition 3.1 is as follows. Variousresults from the literature about the representation theory of G will reducethe claim to specific r which must be considered. We will then study V ( ω r ) byusing a standard realization of it in the exterior algebra of the natural module V of G . Here we can explicitly describe a nonzero G -invariant quadratic form Q on V ( ω r ) . We will then find a vector γ ∈ rad V ( ω r ) such that L ( ω r ) isorthogonal if and only if Q ( γ ) = 0 . The proof is finished by computing Q ( γ ) .9 .1 Representation theory The composition factors of V ( ω r ) were determined in odd characteristic byPremet and Suprunenko in [PS83, Theorem 2]. Independently, the compo-sition factors and the submodule structure of V ( ω r ) were found in arbi-trary characteristic by Adamovich in [Ada84], [Ada86]. Using the results ofAdamovich, it was shown in [BS00, Corollary 2.9] that the result of Premetand Suprunenko also holds in characteristic two.To state the result about composition factors of V ( ω r ) , we need to make afew definitions first. Let a, b ∈ Z ≥ and write a = P i ≥ a i p i and b = P i ≥ b i p i for the expansions of a and b in base p . We say that a contains b to base p if for all i ≥ we have b i = a i or b i = 0 . For r ≥ , we define J p ( r ) to bethe set of integers ≤ j ≤ r such that j ≡ r mod 2 and l + 1 − j contains r − j to base p . The main result of [PS83], also valid in characteristic , canbe then described as follows. Here we set ω = 0 , so that L ( ω ) is the trivialirreducible module. Theorem 3.5.
Let ≤ r ≤ l . Then in the Weyl module V ( ω r ) , eachcomposition factor has multiplicity , and the set of composition factors is { L ( ω j ) : j ∈ J p ( r ) } . In view of Proposition 2.2 (iii), it will also be useful to know when thefirst cohomology group H ( G, L ( ω r )) is nonzero. This has been determinedby Kleshchev and Sheth in [KS99] [KS01, Corollary 3.6]. Theorem 3.6.
Let ≤ r ≤ l and write l + 1 − r = P i ≥ a i p i in base p . Then H ( G, L ( ω r )) = 0 if and only if r = 2( p − a i ) p i for some i such that a i > ,and either a i +1 < p − or r < p i +1 . In characteristic , the result becomes the following. Corollary 3.7.
Assume that char K = 2 . Let ≤ r ≤ l . Then H ( G, L ( ω r )) =0 if and only if r = 2 i +1 for some i ≥ , and l + 1 ≡ i + t mod 2 i +1 forsome ≤ t < i . Throughout this section we will consider subgroups C l ′ < C l = G , whichare embedded into G as follows. Consider G = Sp( V ) and let ( − , − ) be thenon-degenerate G -invariant alternating form ( − , − ) on V . Fix a symplec-tic basis e , . . . , e l , e − , . . . , e − l of V , where ( e i , e − i ) = 1 = − ( e − i , e i ) and ( e i , e j ) = 0 for i = − j . Then for ≤ l ′ < l , the embedding C l ′ < C l is Note that in [PS83] there is a typo, the definition on pg. 1313, line 9 should say “forevery i = 0 , , . . . , n . . . ” p( V ′ ) < Sp( V ) , where V ′ ⊆ V has basis e ± , . . . , e ± l ′ and Sp( V ′ ) fixes thebasis vectors e ± ( l ′ +1) , . . . , e ± l .The module structure of the restrictions L ( ω r ) ↓ C l − have been deter-mined by Baranov and Suprunenko in [BS00, Theorem 1.1 (i)]. We will onlyneed to know the composition factors which occur in such a restriction, andin this case the result is the following. Below we define L C l − ( ω r ) = 0 for r < . Theorem 3.8.
Let ≤ r ≤ l and assume that l ≥ . Set d = ν p ( l + 1 − r ) ,and ε = 0 if l + 1 − r ≡ − p d mod p d +1 and ε = 1 otherwise. Then thecharacter of L C l ( ω r ) ↓ C l − is given by ch L C l − ( ω r )+2 ch L C l − ( ω r − )+ d − X k =0 L C l − ( ω r − p k ) ! + ε ch L C l − ( ω r − p d ) where the sum in the brackets is zero if d = 0 . Above ν p denotes the p -adic valuation on Z , so for a ∈ Z + we have ν p ( a ) = d , where d ≥ is maximal such that p d divides a . Note that if d = ν ( l +1 − r ) , then l +1 − r ≡ d ≡ − d mod 2 d +1 . Therefore if char K = 2 ,we always have ε = 0 in Theorem 3.8. In particular, the composition factorsoccurring in L ( ω r ) ↓ C l − are L C l − ( ω r ) and L C l − ( ω r − k ) for ≤ k ≤ d .We will now give some applications of Theorem 3.8 and Theorem 3.5 incharacteristic two, which will be needed in our proof of Proposition 3.1. Lemma 3.9.
Assume that char K = 2 , and let l ≥ i +1 , where i ≥ . Supposethat l + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . Then for t + 1 ≤ j ≤ i +1 ,the following hold: (i) All composition factors of the restriction L ( ω j ) ↓ C l − have the form L ( ω j ′ ) for some l − ≥ j ′ ≥ t . (ii) L C l ( ω j ) ↓ C l − t has no trivial composition factors.Proof. If t = 0 there is nothing to prove, so suppose that t ≥ . It willbe enough to prove (i) as then (ii) will follow by induction on t . Let d = ν ( l + 1 − j ) . Suppose first that ≤ d < i + 1 . Now l + 1 − j ≡ t − j mod 2 i ,so then ν ( l + 1 − j ) = ν ( j − t ) . By Theorem 3.8, the composition factorsoccurring in L ( ω j ) ↓ C l − are L ( ω j ) and L ( ω j − k ) for ≤ k ≤ d , so the claimfollows since ν ( j − t ) = d and thus j − d ≥ t .Consider then the case where d ≥ i + 1 . Then l + 1 − j ≡ i + ( t − j ) ≡ i +1 , so j − t ≡ i mod 2 i +1 . On the other hand < j − t < i +1 , so j − t = 2 i . By Theorem 3.8, the composition factors occurring in L ( ω j ) ↓ C l − are L ( ω j ) and L ( ω j − k ) for ≤ k ≤ i (because j − k < for i + 1 ≤ k ≤ d ),so again the claim follows. 11 emma 3.10. Let x ≥ i +1 , where i ≥ . Suppose that x ≡ i mod 2 i +1 . If ≤ k ≤ i and x − k contains i − k to base , then k = 0 or k = 2 i .Proof. If i = 0 there is nothing to do, so suppose that i > . Replacing k by i − k , we see that it is equivalent to prove that if x + 2 k contains k to base , then k = 0 or k = 2 i .Suppose that ≤ k < i and that x + 2 k contains k to base . Considerfirst the case where ≤ k < i − . Here since x + 2 k ≡ k mod 2 i , we havethat k contains k to base , which can only happen if k = 0 .Consider then i − ≤ k < i and write k = 2 i − + k ′ , where ≤ k ′ < i − .Then x + 2 k ≡ k ′ mod 2 i , so k ′ contains k = 2 i − + k ′ to base . Butthen k ′ must also contain k ′ to base , so k ′ = 0 and k = 2 i − . In this case x + 2 k ≡ i + 2 i ≡ i +1 , so x + 2 k does not contain k to base ,contradiction. Lemma 3.11.
Let x ≥ i +1 , where i ≥ . Suppose that x ≡ i + t mod 2 i +1 ,where ≤ t < i . If ≤ j ≤ t and x − j contains i − j to base , then j = 0 .Proof. We prove the claim by induction on i . If i = 0 or i = 1 , then theclaim is immediate since ≤ j ≤ t < . Suppose then that i > . Assumethat < j ≤ t and that x − j contains i − j to base . Now j ≤ t < i ,so < j < i − . Therefore i − must occur in the binary expansion of i − j = 2 i − + (2 i − − j ) , so by our assumption i − occurs in the binaryexpansion of x − j . Note that this also means that x − j contains i − − j to base .Now x − j ≡ i + ( t − j ) mod 2 i +1 and ≤ t − j < i , so it followsthat i − will occur in the binary expansion of t − j . Write t = 2 i − + t ′ ,where ≤ t ′ < i − . Here t ′ ≥ j because t − j ≥ i − . Finally, since x − j contains i − − j in base and x ≡ i − + t ′ mod 2 i , we have j = 0 byinduction.Now the following corollaries are immediate from Theorem 3.5 and lem-mas 3.10 and 3.11. Corollary 3.12.
Assume that char K = 2 , and let l ≥ i +1 , where i ≥ .Suppose that l + 1 ≡ i mod 2 i +1 . Then V ( ω i +1 ) = L ( ω i +1 ) /L (0) .Proof. For ≤ j ≤ i +1 , by Theorem 3.5 the irreducible L ( ω j ) is a composi-tion factor of V ( ω i +1 ) if and only if j = 2 j ′ and l + 1 − j ′ contains i − j ′ to base . By Lemma 3.10, this is equivalent to j ′ = 0 or j ′ = 2 i . Corollary 3.13.
Assume that char K = 2 , and let l ≥ i +1 , where i ≥ .Suppose that l + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . Then any nontrivialcomposition factor of V ( ω i +1 ) has the form L ( ω j ) , where i +1 ≥ j ≥ t + 1 . roof. For ≤ j ≤ i +1 , by Theorem 3.5 the irreducible L ( ω j ) is a composi-tion factor of V ( ω i +1 ) if and only if j = 2 j ′ and l + 1 − j ′ contains i − j ′ to base . If ≤ j ≤ t , then by Lemma 3.11 we have j = 0 . V ( ω r ) We now describe the well known construction of the Weyl modules V ( ω r ) for G using the exterior algebra of the natural module. We will consider ourgroup G as a Chevalley group constructed from a complex simple Lie algebraof type C l . For details of the Chevalley group construction see [Ste68].Let e , . . . , e l , e − l , . . . , e − be a basis for a complex vector space V C , and let V Z be the Z -lattice spanned by this basis. We have a non-degenerate alternat-ing form ( − , − ) on V C defined by ( e i , e − i ) = 1 = − ( e − i , e i ) and ( e i , e j ) = 0 for i = − j . Let sp ( V C ) be the Lie algebra formed by the linear endomorphisms X of V C satisfying ( Xv, w ) + ( v, Xw ) = 0 for all v, w ∈ V C . Then sp ( V C ) isa simple Lie algebra of type C l . Let h be the Cartan subalgebra formed bythe diagonal matrices in sp ( V C ) . Then h = { diag( h , . . . , h l , − h l , . . . , − h ) : h i ∈ C } . For ≤ i ≤ l , define maps ε i : h → C by ε i ( h ) = h i where h is a diagonal matrix with diagonal entries ( h , . . . , h l , − h l , . . . , − h ) . Now Φ = {± ( ε i ± ε j ) : 1 ≤ i < j ≤ l } ∪ {± ε i : 1 ≤ i ≤ l } is the root systemfor sp ( V C ) , Φ + = { ε i ± ε j : 1 ≤ i < j ≤ l } ∪ { ε i : 1 ≤ i ≤ l } is a system ofpositive roots, and ∆ = { ε i − ε i +1 : 1 ≤ i < l } ∪ { ε l } is a base for Φ .For any i, j let E i,j be the linear endomorphism on V C such that E i,j ( e j ) = e i and E i,j ( e k ) = 0 for k = j . Then a Chevalley basis for sp ( V C ) is given by X ε i − ε j = E i,j − E − j, − i for all i = j , by X ± ( ε i + ε j ) = E ± j, ∓ i + E ± i, ∓ j for all i = j ,by X ± ε i = E ± i, ∓ i for all i , and by H ε i − ε i +1 = E i,i − E − i, − i , H ε l = E l,l − E − l, − l .Let U Z be the Kostant Z -form with respect to this Chevalley basis of sp ( V C ) . That is, U Z is the subring of the universal enveloping algebra of sp ( V C ) generated by and all X kα k ! for α ∈ Φ and k ≥ .Now V Z is a U Z -invariant lattice in V C . We define V = V Z ⊗ Z K . Note that ( − , − ) also defines a non-degenerate alternating form on V . Then the simplyconnected Chevalley group of type C l induced by V is equal to the group G = Sp( V ) of invertible linear maps preserving ( − , − ) [Ree57, pg. 396-397].By abuse of notation we identify the basis ( e i ⊗ of V with ( e i ) .Note that for all ≤ k ≤ l , the Lie algebra sp ( V C ) acts naturally on ∧ k ( V C ) by X · ( v ∧ · · · ∧ v k ) = k X i =1 v ∧ · · · ∧ v i − ∧ Xv i ∧ v i +1 ∧ · · · ∧ v k for all X ∈ sp ( V C ) and v i ∈ V C . With this action, the Z -lattice ∧ k ( V Z ) is13nvariant under U Z and this induces an action of G on ∧ k ( V Z ) ⊗ Z K . One canshow that g · ( v ∧ · · · ∧ v k ) = gv ∧ · · · ∧ gv k for all g ∈ G and v i ∈ V , so wecan and will identify ∧ k ( V Z ) ⊗ Z K and ∧ k ( V ) as G -modules.The diagonal matrices in G form a maximal torus T . Then a basis ofweight vectors of ∧ k ( V ) is given by the elements e i ∧ · · · ∧ e i k , where − l ≤ i < · · · < i k ≤ l . The basis vector e ∧ · · · ∧ e k has weight ω k .The form on V induces a form on the exterior power ∧ k ( V ) by h v ∧ · · · ∧ v k , w ∧ · · · ∧ w k i = det(( v i , w j )) ≤ i,j ≤ k for all v i , w j ∈ V [Bou59, §1, Définition 12, pg. 30]. This form on ∧ k ( V ) isinvariant under the action of G since ( − , − ) is. Furthermore, let e i ∧ · · · ∧ e i k and e j ∧ · · · ∧ e j k be two basis elements of ∧ k ( V ) . Then h e i ∧ · · · ∧ e i k , e j ∧ · · · ∧ e j k i = ( ± , if { i , · · · , i k } = {− j , · · · , − j k } . , otherwise.Therefore it follows that the form h− , −i on ∧ k ( V ) is nondegenerate if ≤ k ≤ l . In precisely the same way we find a basis of weight vectors for ∧ k ( V Z ) and define a form h− , −i Z on ∧ k ( V Z ) . Note that h− , −i , h− , −i Z are alter-nating if k is odd and symmetric if k is even.It is well known that there is a unique submodule of ∧ k ( V ) isomorphic tothe Weyl module V ( ω k ) of G , as shown by the following lemma. The followinglemma is also a consequence of [AJ84, 4.9]. Lemma 3.14.
Let ≤ k ≤ l , and let W be the G -submodule of ∧ k ( V ) generated by e ∧ · · · ∧ e k . Then (i) W is equal to the subspace of ∧ k ( V ) spanned by all v ∧ · · · ∧ v k , where h v , · · · , v k i is a k -dimensional totally isotropic subspace of V . Further-more, dim W = (cid:0) lk (cid:1) − (cid:0) lk − (cid:1) . (ii) W is isomorphic to the Weyl module V ( ω k ) .Proof. (i) Since G acts transitively on the set of k -dimensional totallyisotropic subspaces of V , it follows that W is spanned by all v ∧· · ·∧ v k ,where h v , . . . , v k i is a k -dimensional totally isotropic subspace of V .Then the claim about the dimension of W follows from a result provenfor example in [DB09, Theorem 1.1], [Bro92, Theorem 1.1] or (in oddcharacteristic) [PS83, pg. 1337].14ii) Since e ∧ · · · ∧ e k is a maximal vector of weight ω k for G , the sub-module W generated by it is an image of V ( ω k ) [Jan03, II.2.13]. Now dim V ( ω k ) = (cid:0) lk (cid:1) − (cid:0) lk − (cid:1) [Bou75, Ch. VIII, 13.3, pg. 203], so by (i) W must be isomorphic to V ( ω k ) .In what follows we will identify V ( ω k ) as the submodule W of ∧ k ( V ) given by Lemma 3.14. Set V ( ω k ) Z = U Z ( e ∧ · · · ∧ e k ) . Note that now we can(and will) identify V ( ω k ) and V ( ω k ) Z ⊗ Z K .We will denote y i = e i ∧ e − i for all ≤ i ≤ l . Then if k = 2 s is even,a basis for the zero weight space of ∧ k ( V ) is given by vectors of the form y i ∧ · · · ∧ y i s , where ≤ i < · · · < i s ≤ l . There is also a description of abasis for the zero weight space of V ( ω k ) Z in [Jan73, Lemma 10, pg.43]. Forour purposes, we will only need a convenient set of generators given by thenext lemma. Lemma 3.15 ([Jan73, pg. 40, Lemma 6]) . Suppose that k is even, say k = 2 s ,where ≤ k ≤ l . Then the zero weight space of V ( ω k ) Z (thus also of V ( ω k ) )is spanned by vectors of the form ( y j − y k ) ∧ · · · ∧ ( y j s − y k s ) , where ≤ k r < j r ≤ l for all r and j r , k r = j r ′ , k r ′ for all r = r ′ . Lemma 3.16.
Suppose that k is even, say k = 2 s , where ≤ k ≤ l . Thenthe vector γ = X ≤ i < ···
Let l ≥ i +1 , where i ≥ . Suppose that l + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . Define the vector γ ∈ ∧ i +1 ( V ) to be equal to X ≤ i < ··· be the number of j r such that j r ≤ l − t .The vector δ can be written as X f s ∈{ j s ,k s } y f ∧ · · · ∧ y f i . Now h y f ∧ · · · ∧ y f i , y g ∧ · · · ∧ y g i i = ( , if { f , . . . , f i } = { g , . . . , g i } . , otherwise.and so h δ, γ i is an integer, equal to the number of y f ∧ · · · ∧ y f i in thesum such that f s ≤ l − t for all ≤ s ≤ i . Thus if j r ≥ k r ≥ l − t + 1 for some r , then h γ, δ i = 0 . If k r ≤ l − t for all r , then it follows that h δ, γ i = 2 q = 0 since q > .(iii) By Lemma 3.7 we have H ( G, L ( ω i +1 )) = 0 and so there exists a non-split extension of L ( ω i +1 ) by the trivial module K . We can find thisextension as an image of the Weyl module V ( ω i +1 ) [Jan03, II.2.13,II.2.14], so rad V ( ω i +1 ) /M ∼ = K for some submodule M of rad V ( ω i +1 ) .Since each composition factor of V ( ω i +1 ) occurs with multiplicity one(Theorem 3.5), each composition factor of M is nontrivial. Then byCorollary 3.13 and Lemma 3.9 (ii), the restriction M ↓ C l − t has notrivial composition factors. But by (i) γ is a fixed point for C l − t , so itfollows that γ M and then rad V ( ω i +1 ) = h γ i ⊕ M as C l − t -modules.Now let Q be a nonzero G -invariant quadratic form on V ( ω i +1 ) . Sincefor the polarization b Q of Q we have rad b Q = rad V ( ω i +1 ) (Proposition2.2 (ii)), composing Q with the square root map K → K defines a mor-phism rad V ( ω i +1 ) → K of G -modules. Therefore Q must vanish on M , since M has no trivial composition factors. Thus for all m ∈ M andscalars c we have Q ( cγ + m ) = c Q ( γ ) , so Q vanishes on rad V ( ω i +1 ) ifand only if Q ( γ ) = 0 . Hence by Proposition 2.2 (iii) L ( ω i +1 ) is orthog-onal if and only if Q ( γ ) = 0 . 17 .3 Computation of a quadratic form Q on V ( ω r ) To finish the proof of Proposition 3.1 we still have to compute Q ( γ ) for thevector γ from Lemma 3.17.We retain the notation from the previous subsection and keep the as-sumption that char K = 2 . Let r be even, say r = 2 s , where ≤ r ≤ l .Now the form h− , −i Z on ∧ r ( V Z ) induces a quadratic form q Z on ∧ r ( V Z ) by q Z ( x ) = h x, x i . We will use this form to find a nonzero G -invariant quadraticform on V ( ω r ) = V ( ω r ) Z ⊗ Z K . Lemma 3.18.
We have q Z ( V ( ω r ) Z ) ⊆ Z and q Z ( V ( ω r ) Z ) Z .Proof. Let α = e ∧ · · · ∧ e r and β = e − ∧ · · · ∧ e − r . Now α, β ∈ V ( ω r ) Z and h α, α i = h β, β i = 0 and h α, β i = 1 , giving q Z ( α + β ) = 2 and so q Z ( V ( ω r ) Z ) Z . If we had q Z ( V ( ω r ) Z ) Z , then Q = q Z ⊗ Z K definesa nonzero G -invariant quadratic form on V ( ω r ) . But then the polarizationof Q is equal to h− , −i = 0 , which by Proposition 2.2 (i) and (ii) is notpossible.Therefore Q = q Z ⊗ Z K defines a nonzero G -invariant quadratic formon V ( ω r ) with polarization h− , −i . A similar construction when r is odd isdiscussed in [GN16, Proposition 8.1].Now if we consider a zero weight vector of the form X { i , ··· ,i s }∈ I y i ∧ · · · ∧ y i s in V ( ω r ) , the value of Q for this vector is equal to | I | since h y f ∧ · · · ∧ y f k , y g ∧ · · · ∧ y g k i = ( , if { f , . . . , f k } = { g , . . . , g k } . , otherwise.Now we can compute the value of Q ( γ ) from Lemma 3.17. Since there are (cid:0) l − t i (cid:1) terms occurring in the sum that defines γ , we have Q ( γ ) = (cid:0) l − t i (cid:1) . Thus Q ( γ ) = 0 if and only if (cid:0) l − t i (cid:1) is divisible by . Now the proof of Proposition3.1 is finished with the following lemma. Lemma 3.19.
Let l + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . The integer (cid:0) l − t i (cid:1) is divisible by if and only if l + 1 ≡ i + t mod 2 i +2 .Proof. According to Kummer’s theorem, if p is prime and d ≥ is maximalsuch that p d divides (cid:0) xy (cid:1) ( x ≥ y ≥ ), then d is the number of carries that18ccur when adding y to x − y in base p . Now ( l − t ) − i ≡ − ≡ i + · · · +2+1mod 2 i +1 . If l − t − i ≡ i + · · · + 2 + 1 mod 2 i +2 , then adding i to l − t − i in binary results in just one carry. The otherpossibility is that l − t − i ≡ i +1 + 2 i + · · · + 2 + 1 mod 2 i +2 , and in this case there are ≥ carries. Therefore (cid:0) l − t i (cid:1) is divisible by if andonly if l − t − i ≡ i +1 + 2 i + · · · + 2 + 1 ≡ − i +2 , which is equivalent to l + 1 ≡ i + t mod 2 i +2 . With a bit more work, we can use Proposition 3.1 to determine for all classicaltypes the fundamental irreducible representations that are orthogonal. In thissection assume that char K = 2 .For a groups of type A l ( l ≥ ), the only self-dual fundamental irreduciblerepresentations are those of form L ( ω l +12 ) , where l is odd. Furthermore, allfundamental representations are minuscule, so V ( ω l +12 ) = L ( ω l +12 ) and thusby Proposition 2.2 (iv) the representation L ( ω l +12 ) is orthogonal.Now for type B l , there exists an exceptional isogeny ϕ : B l → C l betweensimply connected groups of type B l and C l [Ste68, Theorem 28]. Then irre-ducible representations of C l induce irreducible representations B l by twist-ing with the isogeny ϕ . For fundamental irreducible representations, we have L C l ( ω r ) ϕ ∼ = L B l ( ω r ) if ≤ r ≤ l − , and L C l ( ω l ) ϕ is a Frobenius twist of L B l ( ω l ) . Therefore for all ≤ r ≤ l , the representation L C l ( ω r ) is orthogonalif and only if L B l ( ω r ) is orthogonal.Consider then type D l ( l ≥ ). First note that the natural representation L D l ( ω ) of D l is orthogonal. Now since we are working in characteristic two,there is an embedding D l < C l as a subsystem subgroup generated by theshort root subgroups. Then if ≤ r ≤ l − , we have L C l ( ω r ) ↓ D l ∼ = L D l ( ω r ) for ≤ r ≤ l − by [Sei87, Theorem 4.1]. By combining this fact with thelemma below, we see for ≤ r ≤ l − that L C l ( ω r ) is orthogonal if L D l ( ω r ) is orthogonal. Lemma 4.1.
Let G be simple of type C l and consider H < G of type D l as the subsystem subgroup generated by short root subgroups. Suppose that V is a nontrivial irreducible -restricted representation of G and V = L G ( ω ) . hen if V ↓ H is -restricted irreducible, the representation V is orthogonalfor G if and only if V is orthogonal for H .Proof (G. Seitz). If V is an orthogonal G -module, it is clear that it is anorthogonal H -module as well. Suppose then that V is not orthogonal for G .Since V is not the natural module for G , by Proposition 2.2 (iii) there existsa nonsplit extension → h w i → M → V → of G -modules, where w ∈ M . Furthermore, there exists a nonzero G -invariantquadratic form Q on M such that Q ( w ) = 0 .We claim that M ↓ H is also a nonsplit extension. If this is not the case,then M ↓ H = W ⊕ h w i for some H -submodule W of M . We will showthat W is invariant under G , which is a contradiction since M is nonsplitfor G . Now W is -restricted irreducible for H , so by a theorem of Curtis[Bor70, Theorem 6.4] the module W is also an irreducible representation of Lie( H ) . Since Lie( H ) is an ideal of Lie( G ) that is invariant under the adjointaction of G , it follows that gW is Lie( H ) -invariant for all g ∈ G . But as a Lie( H ) -module M is the sum of a trivial module and W , so we must have gW = W for all g ∈ G .Thus if V ↓ H = L H ( λ ) , then there exists a surjection π : V H ( λ ) → M of H -modules [Jan03, II.2.13]. Now the quadratic form Q induces via π anonzero, H -invariant quadratic form on V H ( λ ) which does not vanish onthe radical of V H ( λ ) . By Proposition 2.2 (iii) the representation V is notorthogonal for H .Finally, the half-spin representations of D l are minuscule representations,so L D l ( ω l ) = V D l ( ω l ) and L D l ( ω l − ) = V D l ( ω l − ) . As before, by Proposition2.2 (iv) it follows that L D l ( ω l ) and L D l ( ω l − ) are orthogonal if they are self-dual. Therefore we can conclude that for l = 4 and l = 5 all self-dual L D l ( ω i ) are orthogonal.Note that if l ≥ , then L C l ( ω l ) and L C l ( ω l − ) are also orthogonal (Exam-ple 3.4). Thus for l ≥ we have for all ≤ r ≤ l that L C l ( ω r ) is orthogonalif L D l ( ω r ) is orthogonal.Taking all of this together, Proposition 3.1 is improved to the following. Theorem 4.2.
Assume that char K = 2 . Let G be simple of type A l ( l ≥ ), B l ( l ≥ ), C l ( l ≥ ) or D l ( l ≥ ). Suppose ≤ r ≤ l and ω r = − w ( ω r ) .Then L ( ω r ) is not orthogonal if and only if one of the following holds: • G is of type B l ( l ≥ ) or C l ( l ≥ ) and r = 1 . • G is of type B l ( l ≥ ), C l ( l ≥ or D l ( l ≥ ) and r = 2 i +1 for some i ≥ such that l + 1 ≡ i +1 + 2 i + t mod 2 i +2 , where ≤ t < i . Representations L ( ω r + ω s ) for type A l Assume that G is simply connected of type A l , l ≥ . Set n = l + 1 .In this section, we determine when in characteristic the irreduciblerepresentation L ( ω r + ω s ) , ≤ r < s ≤ l , of G has a nonzero G -invariantquadratic form. Now L ( ω r + ω s ) is not orthogonal if it is not self-dual, so itwill be enough to consider L ( ω r + ω n − r ) , where ≤ r < n − r ≤ l (see Table2.1). In this case, the answer and the methods to prove it are very similar tothose found in Section 3. The result is the following theorem, which we willprove in what follows. Theorem 5.1.
Assume char K = 2 . Let ≤ r < n − r ≤ l . Then L ( ω r + ω n − r ) is not orthogonal if and only if r = 2 i for some i ≥ and n + 1 ≡ i +1 + 2 i + t mod 2 i +2 , where ≤ t < i . The following examples follow easily from Theorem 5.1 (cf. examples 3.2and 3.3).
Example 5.2. If char K = 2 , then L ( ω + ω l ) is orthogonal if and only if n . This result was also proven in [GW95, Theorem 3.4 (b)]. Example 5.3. If char K = 2 , then L ( ω + ω l − ) is orthogonal if and only if n , . The composition factors and the submodule structure of the Weyl modules V ( ω r + ω s ) , ≤ r < s ≤ l , were determined by Adamovich [Ada92]. Usingher result, Baranov and Suprunenko have given in [BS05, Theorem 2.3] adescription of the set of composition factors, similarly to Theorem 3.5. For ≤ r < s ≤ l , define J p ( r, s ) be the set of pairs ( r − k, s + k ) , where n − s, r ≥ k ≥ and s − r + 1 + 2 k contains k to base p . Here we willdefine ω = 0 and ω n = 0 , so then L ( ω ) = L ( ω n ) = L ( ω + ω n ) is thetrivial irreducible module and L ( ω + ω r ) = L ( ω r + ω n ) = L ( ω r ) . Now [BS05,Theorem 2.3] gives the following . Theorem 5.4.
Let ≤ r < s ≤ l . Then in the Weyl module V ( ω r + ω s ) ,each composition factor has multiplicity , and the set of composition factorsis { L ( ω j + ω j ′ ) : ( j, j ′ ) ∈ J p ( r, s ) } . Baranov and Suprunenko give the result in terms of π i,j = L ( ω ( j − i +1) / + ω ( i + j − / ) ,but from π y − x +1 ,x + y = L ( ω x + ω y ) we get the formulation in Theorem 5.4. A l gives the following (cf.Corollary 3.7). Theorem 5.5.
Assume that char K = 2 . Let ≤ r < s ≤ l . Then H ( G, L ( ω r + ω s )) = 0 if and only if r = 2 i , s = n − i for some i ≥ , and n + 1 ≡ i + t mod 2 i +1 for some ≤ t < i . Throughout this section we will consider subgroups A l ′ < A l = G , whichare embedded into G as follows. We consider G = SL( V ) , where V has basis e , e , . . . , e l +1 . Then for ≤ l ′ < l , the embedding A l ′ < A l is SL( V ′ ) < SL( V ) , where V ′ ⊆ V has basis e , . . . , e l ′ +1 and SL( V ′ ) fixes the basis vectors e l ′ +2 , . . . , e l +1 .Baranov and Suprunenko have determined the submodule structure ofthe restrictions L ( ω r + ω s ) ↓ A l − for all ≤ r ≤ s ≤ n in their article[BS05, Theorem 1.1]. As in Section 3, for our purposes it will be enough toknow which composition factors occur in the restriction. To state the resultof Baranov and Suprunenko, we will denote π lr,s = L A l ( ω r + ω l +1 − s ) for all ≤ r ≤ l + 1 − s ≤ l + 1 . We will define π lr,s = 0 if r < , s < or r + s > l + 1 .Now the main result of [BS05] gives the following (cf. Theorem 3.8). Theorem 5.6.
Let ≤ r ≤ n − s ≤ n and assume that n ≥ . Set d = ν p ( n + 1 − ( r + s )) , and ε = 0 if n + 1 − ( r + s ) ≡ − p d mod p d +1 and ε = 1 otherwise. Then the character of π lr,s ↓ A l − is given by ch π l − r,s − + ch π l − r − ,s + ch π l − r,s + d − X k =0 π l − r − p k ,s − p k ! + ε ch π l − r − p d ,s − p d where the sum in the brackets is zero if d = 0 . As with Theorem 3.8, note that when char K = 2 , we always have ε =0 in Theorem 5.6. The following applications of theorems 5.4 and 5.5 incharacteristic two will be needed later. Lemma 5.7.
Assume that char K = 2 , and let n ≥ i +1 , where i ≥ .Suppose that n + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . Let ≤ x ≤ i and ≤ y ≤ i be such that i +1 ≥ x + y ≥ t + 1 . Then (i) All composition factors of the restriction π lx,y ↓ A l − have the form π l − x ′ ,y ′ for some ≤ x ′ , y ′ ≤ i such that x ′ + y ′ ≥ t . Baranov and Supruneko give their result in terms of L li,j = L A l ( ω i + ω j ) for ≤ i ≤ j ≤ n , but replacing j by n − j gives the formulation in Theorem 5.6. π lx,y ↓ A l − t has no trivial composition factors.Proof. (cf. Lemma 3.9) If t = 0 there is nothing to prove, so suppose that t ≥ . It will be enough to prove (i) as then (ii) will follow by inductionon t . Let d = ν ( n + 1 − ( x + y )) . Suppose first that ≤ d < i + 1 . Then n + 1 − ( x + y ) ≡ t − ( x + y ) mod 2 i , so ν ( n + 1 − ( x + y )) = ν (( x + y ) − t ) .By Theorem 5.6, the composition factors occurring in π lx,y ↓ A l − are π l − x,y − , π l − x − ,y , π l − x,y , and π l − x − k ,y − k for ≤ k ≤ d − . Therefore the claim followssince ν (( x + y ) − t ) = d and thus x + y − d ≥ t .Consider then the case where d ≥ i + 1 . Then n + 1 − ( x + y ) ≡ i + t − ( x + y ) ≡ i +1 , so ( x + y ) − t ≡ i mod 2 i +1 . On the other hand ≤ ( x + y ) − t < i +1 , so ( x + y ) − t = 2 i . By Theorem 5.6 the compositionfactors occurring in π lx,y ↓ A l − are π l − x,y − , π l − x − ,y , π l − x,y , and π l − x − k ,y − k for ≤ k ≤ i − (since x − k < or y − k < for i ≤ k ≤ d − ), so again theclaim follows.As a consequence of Theorem 5.4 and lemmas 3.10 and 3.11, we get thefollowing (cf. corollaries 3.12 and 3.13). Corollary 5.8.
Assume that char K = 2 and let n > i +1 , where i ≥ . Sup-pose that n + 1 ≡ i mod 2 i +1 . Then V ( ω i + ω n − i ) = L ( ω i + ω n − i ) /L (0) .Proof. According to Theorem 5.4, the composition factors of V ( ω i + ω n − i ) are L ( ω i − k + ω n − i + k ) , where ≤ k ≤ i and n + 1 − i +1 + 2 k contains k to base . We can replace k by i − k , and then the condition is equivalentto n + 1 − k containing i − k to base , which implies k = 0 or k = 2 i byLemma 3.10. Corollary 5.9.
Assume that char K = 2 and let n > i +1 , where i ≥ .Suppose that n + 1 ≡ i + t mod 2 i +1 , where i ≥ and ≤ t < i . Then anynontrivial composition factor of V ( ω i + ω n − i ) has the form L ( ω x + ω n − y ) for some ≤ x ≤ n − y ≤ n and x + y ≥ t + 1 .Proof. According to Theorem 5.4, the composition factors of V ( ω i + ω n − i ) are L ( ω i − k + ω n − i + k ) , where ≤ k ≤ i and n + 1 − i +1 + 2 k contains k to base . Setting k ′ = 2 i − k , the composition factors are L ( ω k ′ + ω n − k ′ ) ,where n + 1 − k ′ contains i − k ′ to base . By Lemma 3.11 we have k ′ = 0 or k ′ ≥ t + 1 , which proves the claim. V ( ω r + ω n − r ) We now describe a construction of V ( ω r + ω n − r ) , in many ways similar tothat of V C l ( ω r ) described in Section 3.2. We will consider our group G as aChevalley group constructed from a complex simple Lie algebra of type A l .23et e , e , . . . , e n be a basis for a complex vector space V C , and let V Z bethe Z -lattice spanned by this basis. Let sl ( V C ) be the Lie algebra formed bythe linear endomorphisms of V C with trace zero. Then sl ( V C ) is a simple Liealgebra of type A l . Let h be the Cartan subalgebra formed by the diagonalmatrices in sl ( V C ) (with respect to the basis ( e i ) ). For ≤ i ≤ n , define maps ε i : h → C by ε i ( h ) = h i where h is a diagonal matrix with diagonal entries ( h , h , . . . , h n ) . Now Φ = { ε i − ε j : i = j } is the root system for sl ( V C ) and Φ + = { ε i − ε j : i < j } is a system of positive roots, and ∆ = { ε i − ε i +1 : 1 ≤ i ≤ l } is a base for Φ .For any i, j let E i,j be the linear endomorphism on V C such that E i,j ( e j ) = e i and E i,j ( e k ) = 0 for k = j . Now a Chevalley basis for sl ( V C ) is given by X ε i − ε j = E i,j for i = j and H ε i − ε i +1 = E i,i − E i +1 ,i +1 for ≤ i ≤ l . Let U Z be the Kostant Z -form with respect to this Chevalley basis of sl ( V C ) . Thatis, U Z is the subring of the universal enveloping algebra of sl ( V C ) generatedby and all X kα k ! for α ∈ Φ and k ≥ .Now V Z is a U Z -invariant lattice in V C . We define V = V Z ⊗ Z K . Thenthe simply connected Chevalley group of type A l induced by V is equal to G = SL( V ) .Let e ∗ , e ∗ , . . . , e ∗ n be a basis for V ∗ C , dual to the basis ( e , e , . . . , e n ) of V C (so here e ∗ i ( e j ) = δ ij ). Denote the Z -lattice spanned by e ∗ , e ∗ , . . . , e ∗ n by V ∗ Z .Then V ∗ Z is U Z -invariant and we can identify V ∗ Z ⊗ Z K and V ∗ as G -modules.Here the action of G on V ∗ is given by ( g · f )( v ) = f ( g − v ) for all g ∈ G , f ∈ V ∗ and v ∈ V .By abuse of notation we identify the basis ( e i ⊗ of V with ( e i ) , and thebasis ( e ∗ i ⊗ of V ∗ with ( e ∗ i ) .Let ≤ k < n − k ≤ l . Now the Lie algebra sl ( V C ) acts naturally on ∧ k ( V C ) by X · ( v ∧ · · · ∧ v k ) = k X i =1 v ∧ · · · ∧ v i − ∧ Xv i ∧ v i +1 ∧ · · · ∧ v k for all X ∈ sl ( V C ) and v i ∈ V C . Similarly we have an action of sl ( V C ) on ∧ k ( V ∗ C ) . Furthermore, sl ( V C ) acts on ∧ k ( V C ) ⊗ ∧ k ( V ∗ C ) by X · ( v ⊗ w ) = Xv ⊗ w + v ⊗ Xw for all X ∈ sl ( V C ) , v ∈ ∧ k ( V C ) and w ∈ ∧ k ( V ∗ C ) . Here ∧ k ( V Z ) ⊗ ∧ k ( V ∗ Z ) is an U Z -invariant lattice in ∧ k ( V C ) ⊗ ∧ k ( V ∗ C ) , and we canand will identify ∧ k ( V Z ) ⊗ ∧ k ( V ∗ Z ) ⊗ Z K and ∧ k ( V ) ⊗ ∧ k ( V ∗ ) as G -modules.The diagonal matrices in G form a maximal torus T . Then a basis ofweight vectors of ∧ k ( V ) ⊗ ∧ k ( V ∗ ) is given by the elements ( e i ∧ · · · ∧ e i k ) ⊗ ( e ∗ j ∧ · · · ∧ e ∗ j k ) , where ≤ i < · · · < i k ≤ n and ≤ j < · · · < j k ≤ n . Thebasis vector ( e ∧ · · · ∧ e k ) ⊗ ( e ∗ n ∧ e ∗ n − ∧ · · · ∧ e ∗ n − k +1 ) has weight ω k + ω n − k .24he natural dual pairing between ∧ k ( V ) and ∧ k ( V ∗ ) (see for example[FH91, B.3, pg.475-476]) induces a G -invariant symmetric form h− , −i on ∧ k ( V ) ⊗ ∧ k ( V ∗ ) . If x = ( v ∧ · · · ∧ v k ) ⊗ ( f ∧ · · · ∧ f k ) and y = ( w ∧ · · · ∧ w k ) ⊗ ( g ∧ · · · ∧ g k ) , with v i , w j ∈ V and f i , g j ∈ V ∗ , we define h x, y i = det( f i ( w j )) ≤ i,j ≤ k det( g i ( v j )) ≤ i,j ≤ k . Let b = ( e i ∧· · ·∧ e i k ) ⊗ ( e ∗ j ∧· · ·∧ e ∗ j k ) and b ′ = ( e i ′ ∧· · ·∧ e i ′ k ) ⊗ ( e ∗ j ′ ∧· · ·∧ e ∗ j ′ k ) be two basis elements of ∧ k ( V ) ⊗ ∧ k ( V ∗ ) . Then h b, b ′ i = ( ± , if { i , · · · , i k } = { j ′ , · · · , j ′ k } and { i ′ , · · · , i ′ k } = { j , · · · , j k } . , otherwise.Therefore the form h− , −i on ∧ k ( V ) ⊗ ∧ k ( V ∗ ) is non-degenerate. In pre-cisely the same way we can find a basis of weight vectors for ∧ k ( V Z ) ⊗ ∧ k ( V ∗ Z ) and define a symmetric form h− , −i Z on ∧ k ( V Z ) ⊗ ∧ k ( V ∗ Z ) .We can find the Weyl module V ( ω k + ω n − k ) as a submodule of ∧ k ( V ) ⊗∧ k ( V ∗ ) , as shown by the following lemma (cf. Lemma 3.14). Lemma 5.10.
Let ≤ k < n − k ≤ l , and let W be the G -submodule of ∧ k ( V ) ⊗ ∧ k ( V ∗ ) generated by v + = ( e ∧ · · · ∧ e k ) ⊗ ( e ∗ n ∧ e ∗ n − ∧ · · · ∧ e ∗ n − k +1 ) .Then W is isomorphic to the Weyl module V ( ω k + ω n − k ) .Proof. It is a general fact about Weyl modules that V ( λ ) ⊗ V ( µ ) always has V ( λ + µ ) as a submodule. For simple groups of classical type (in particular,for our G of type A l ) this follows from results proven first by Lakshmibai et.al. [LMS79, Theorem 2 (b)] or from a more general result of Wang [Wan82,Theorem B, Lemma 3.1]. For other types, the fact is a consequence of resultsdue to Donkin [Don85] (all types except E and E in characteristic two) orMathieu [Mat90] (in general). In any case, now the weight λ + µ occurs withmultiplicity in V ( λ ) ⊗ V ( µ ) , so any vector of weight λ + µ in V ( λ ) ⊗ V ( µ ) will generate a submodule isomorphic to V ( λ + µ ) .To prove our lemma, note that ∧ k ( V ) = L ( ω k ) and ∧ k ( V ∗ ) = L ( ω n − k ) .Furthermore, ω k and ω n − k are minuscule weights, so L ( ω k ) = V ( ω k ) and L ( ω n − k ) = V ( ω n − k ) . Here v + is a vector of weight ω k + ω n − k in ∧ k ( V ) ⊗∧ k ( V ∗ ) , so the claim follows from the result in the previous paragraph.For all ≤ k < n − k ≤ l , we will identify V ( ω k + ω n − k ) with thesubmodule W from Lemma 5.10. Set V ( ω k + ω n − k ) Z = U Z v + where v + isas in Lemma 5.10. Then we can and will identify V ( ω k + ω n − k ) Z ⊗ Z K and V ( ω k + ω n − k ) as G -modules.Note that a basis for the zero weight space of ∧ k ( V ) ⊗ ∧ k ( V ∗ ) is given byvectors of the form ( e i ∧· · ·∧ e i k ) ⊗ ( e ∗ i ∧· · ·∧ e ∗ i k ) , where ≤ i < · · · < i k ≤ n .25e will need the following lemma, which gives a set of generators for the zeroweight space of V ( ω k + ω n − k ) (cf. Lemma 3.15). Lemma 5.11.
Suppose that ≤ k < n − k ≤ l . Then the zero weight spaceof V ( ω k + ω n − k ) Z (thus also of V ( ω k + ω n − k ) ) is spanned by vectors of theform X f s ∈{ j s ,k s } ( − |{ s : f s = j s }| ( e f ∧ · · · ∧ e f k ) ⊗ ( e ∗ f ∧ · · · ∧ e ∗ f k ) , where ( k , . . . , k k ) and ( j , . . . , j k ) are sequences such that ≤ k r < j r ≤ n for all r , and j r , k r = j r ′ , k r ′ for all r = r ′ .Proof. We give a proof somewhat similar to that of Lemma 3.15 given in[Jan73, pg. 40, Lemma 6]. The zero weight space of V ( ω k + ω n − k ) Z is generatedby elements of the form Y α ∈ Φ + X k α − α k α ! v + where k α are non-negative integers, P α ∈ Φ + k α α = ω k + ω n − k and the productis taken with respect to some fixed ordering of the positive roots. For α ∈ Φ + such that X − α v + = 0 , we can assume k α = 0 by choosing a suitable orderingof Φ + . Therefore we will assume that if k α > , then α is of one of thefollowing types.(I) α = ε i − ε j for ≤ i ≤ k and k + 1 ≤ j ≤ n − k .(II) α = ε i − ε j for k + 1 ≤ i ≤ n − k and n − k + 1 ≤ j ≤ n .(III) α = ε i − ε j for ≤ i ≤ k and n − k + 1 ≤ j ≤ n .Note that the X − α with α of type (I) commute with each other. The sameis also true for types (II) and (III).Writing ω k + ω n − k in terms of the simple roots, we see that ω k + ω n − k isequal to α +2 α + · · · +( k − α k − + kα k + · · · + kα n − k +( k − α n − k +1 + · · · + α n − (*)Then from the fact that P α ∈ Φ + k α α = ω k + ω n − k we will deduce the following.(1) For any ≤ i ≤ k , there exists a unique α ∈ Φ + such that k α = 1 and α = ε i − ε j ′ for some k + 1 ≤ j ′ ≤ n .(2) For any n − k + 1 ≤ j ≤ n there exists a unique α ∈ Φ + such that k α = 1 and α = ε i ′ − ε j for some ≤ i ′ ≤ n − k .26or i = 1 and j = n these claims are clear, since α and α n − occur onlyonce in the expression (*) of ω k + ω n − k as a sum of simple roots. For i > claim(1) follows by induction, since ε i − ε j ′ contributes α i + α i +1 + · · · + α k + · · · + α j ′ − to the expression (*) of ω k + ω n − k as a sum of simple roots. Claim (2) followssimilarly for j < n .In particular, it follows from claims (1) and (2) that k α ∈ { , } for all α ∈ Φ + . Let A , A and A be the sets of α ∈ Φ + of type (I), (II) and (III)respectively such that k α = 1 . It follows from claim (1) that | A | + | A | = k and from claim (2) that | A | + | A | = k , so then | A | = | A | = k ′ for some ≤ k ′ ≤ k . Thus we can write A = { ε i − ε w , . . . , ε i k ′ − ε w k ′ } A = { ε z − ε j , . . . , ε z k ′ − ε j k ′ } A = { ε i k ′ +1 − ε j k ′ +1 , . . . , ε i k − ε j k } where ≤ i r ≤ k and n − k + 1 ≤ j r ≤ n for all ≤ r ≤ k , and k + 1 ≤ w r , z r ≤ n − k for all ≤ r ≤ k ′ . Furthermore, { i , . . . , i k } = { , , . . . , k } and { j , . . . , j k } = { n − k + 1 , . . . , n − , n } .We choose the ordering of Φ + so that Y α ∈ Φ + X k α − α k α ! = Y α ∈ A X − α Y α ∈ A X − α Y α ∈ A X − α . It is another consequence of P α ∈ Φ + k α α = ω k + ω n − k that { w , . . . , w k ′ } = { z , . . . , z k ′ } . Indeed, in the expression (*) of ω k + ω n − k as a sum of simpleroots, for any k + 1 ≤ r ≤ n − k the simple root α r occurs k times. On theother hand, the α of types (I), (II), (III) that contribute to α r in the sum areprecisely those of type (I) or (III) with j > r (total of k − |{ r ′ : w r ′ ≤ r }| ),and those of type (II) with j ≤ r (total of |{ r ′ : z r ′ ≤ r }| ).Therefore in the sum P α ∈ Φ + k α α , the contribution to α r is equal to k −|{ r ′ : w r ′ ≤ r }| + |{ r ′ : z r ′ ≤ r }| . Since this has to be equal to k , we get |{ r ′ : z r ′ ≤ r }| = |{ r ′ : w r ′ ≤ r }| for all k + 1 ≤ r ≤ n − k , which implies { w , . . . , w k ′ } = { z , . . . , z k ′ } .Then since the X − α with α of type (II) commute with each other, wemay assume that z r = w r for all ≤ r ≤ k ′ . Denote w = Q k ′ r =1 E w r ,i r v + . Astraightforward computation shows that w = ( e π (1) ∧ · · · ∧ e π ( k ) ) ⊗ ( e ∗ n ∧ e ∗ n − ∧ · · ∧ e ∗ n − k +1 ) , where π ( r ) = w r ′ if r = i r ′ and π ( r ) = r otherwise. Now Y α ∈ Φ + X k α − α k α ! v + = k Y r = k ′ +1 E j r ,i r k ′ Y r =1 E j r ,w r k ′ Y r =1 E w r ,i r v + = k Y r = k ′ +1 E j r ,i r k ′ Y r =1 E j r ,w r w = k Y r =1 E j r ,k r w where ( k , . . . , k k ) = ( w , . . . , w k ′ , i k ′ +1 , . . . , i k ) . In the last equality we justcombine the terms, and this makes sense since X − α of type (III) commutewith those of type (II).Computing the expression Q kr =1 E j r ,k r w , we see that it is equal to a sumof k distinct elements of ∧ k ( V Z ) ⊗ ∧ k ( V ∗ Z ) , with each summand being equalto w transformed in the following way: • For all ≤ s ≤ k ′ , replace e w s by e j s , or replace e ∗ j s by − e ∗ w s . • For all k ′ + 1 ≤ s ≤ k , replace e i s by e j s , or replace e ∗ j s by − e ∗ i s .For this we conclude that up to a sign, Q α ∈ Φ + X kα − α k α ! v + is as in the statementof the lemma, with sequences ( k , . . . , k k ) and ( j , . . . , j k ) as defined here. Lemma 5.12.
Suppose that ≤ k < n − k ≤ l . Then the vector X ≤ i < ···
Let n > i +1 , where i ≥ . Suppose that n + 1 ≡ i + t mod 2 i +1 , where ≤ t < i . Define the vector γ ∈ ∧ i ( V ) ⊗ ∧ i ( V ∗ ) to beequal to X ≤ i < ···
We have q Z ( V ( ω k + ω n − k ) Z ) ⊆ Z and q Z ( V ( ω k + ω n − k ) Z ) Z .Proof. Same as Lemma 3.18, but with α = ( e ∧ · · · ∧ e k ) ⊗ ( e ∗ n ∧ e ∗ n − ∧ · · · ∧ e ∗ n − k +1 ) and β = ( e n ∧ e n − ∧ · · · ∧ e n − k +1 ) ⊗ ( e ∗ ∧ · · · ∧ e ∗ k ) .Therefore Q = q Z ⊗ Z K defines a nonzero G -invariant quadratic form on V ( ω k + ω n − k ) with polarization h− , −i . As in Section 3.3, we have Q ( γ ) = (cid:0) n − t i (cid:1) for the vector γ ∈ ∧ i ( V ) ⊗∧ i ( V ∗ ) from Lemma 5.13. Finally applyingLemma 3.19 completes the proof Theorem 5.1. In this section, let G be a simple group of exceptional type and assume that char K = 2 . We will give some results about the orthogonality of irreduciblerepresentations of G . For G of type G or F we give a complete answer. Fortypes E , E , and E we only have results for some specific representations,given in Table 6.1 below and proven at the end of this section. For irreduciblerepresentations occurring in the adjoint representation of G , answers weregiven earlier by Gow and Willems in [GW95, Section 3]. Proposition 6.1.
Let G = G and let V be a non-trivial irreducible repre-sentation of G . Then V is not orthogonal if and only if V is a Frobenius twistof L G ( ω ) .Proof. In view of Remark 2.5, it will be enough to consider V = L G ( λ ) with λ ∈ X ( T ) + a -restricted dominant weight. If λ = ω or λ = ω + ω , then V G ( λ ) = L G ( λ ) and so V is orthogonal by Proposition 2.2.31hat remains is to show that V = L G ( ω ) not orthogonal. There areseveral ways to see this, for example since dim V = 6 this could be doneby a direct computation. Alternatively, note that the composition factors of ∧ ( V ) are L G ( ω ) and L G (0) [LS96, Proposition 2.10], so H ( G, ∧ ( V )) = 0 .Then by [SW91, Proposition 2.7], the module V is not orthogonal. For athird proof, note that the action of a regular unipotent u ∈ G on V hasa single Jordan block [Sup95, Theorem 1.9], but no such element exists in SO( V ) [LS12, Proposition 6.22].The following lemma will be useful throughout this section to show thatcertain representations are orthogonal. Lemma 6.2.
Let V be a nontrivial, self-dual and irreducible G -module. Sup-pose that one of the following holds: (i) dim V ≡ , and ∧ ( V ) has exactly one trivial composition factoras a G -module. (ii) dim V ≡ , and ∧ ( V ) has exactly two trivial composition fac-tors as a G -module.Then any nontrivial composition factor of ∧ ( V ) occuring with odd mul-tiplicity is an orthogonal G -module.Proof. Since V is nontrivial, we can assume G <
Sp( V ) by Lemma 2.1. If (i)holds, then by applying results in Section 3.2 (or [McN98, Lemma 4.8.2]) wecan find a vector γ ∈ ∧ ( V ) such that ∧ ( V ) = Z ⊕ h γ i as an Sp( V ) -module.Here Z is irreducible of highest weight ω for Sp( V ) , so by Proposition 3.1(see Example 3.2) the module Z is orthogonal for Sp( V ) . Therefore Z is anorthogonal G -module with no trivial composition factors. From this [GW95,Lemma 1.3] shows that any composition factor of Z with odd multiplicity isan orthogonal G -module.In case (ii), the assumption on dim V implies (for example by [McN98,Lemma 4.8.2]) that there exist Sp( V ) -submodules Z ′ ⊆ Z ⊆ ∧ ( V ) suchthat dim Z ′ = dim ∧ ( V ) /Z = 1 . Furthermore, Z/Z ′ is an irreducible Sp( V ) -module with highest weight ω , so by Proposition 3.1 (see Example 3.2)the module Z/Z ′ is orthogonal for Sp( V ) . Therefore Z/Z ′ is an orthogonal G -module with no trivial composition factors, so by [GW95, Lemma 1.3]any composition factor of Z/Z ′ with odd multiplicity is an orthogonal G -module. Proposition 6.3.
Let G = F and let V be a non-trivial irreducible repre-sentation of G . Then V is orthogonal. roof. Let τ : G → G be the exceptional isogeny of G as given in [Ste68,Theorem 28]. Then L G ( a ω + a ω + a ω + a ω ) τ ∼ = L G ( a ω + a ω +2 a ω + 2 a ω ) , and by Steinberg’s tensor product theorem this is isomorphicto L G ( a ω + a ω ) ⊗ L ( a ω + a ω ) F . Thus by lemmas 2.1 and 2.4, it isenough to prove the claim in the case where V = L G ( λ ) with λ = a ω + a ω a -restricted dominant weight. Now for λ = ω and λ = ω + ω , we have V G ( λ ) = L G ( λ ) = V and so V is orthogonal by Proposition 2.2 (iv) .What remains is to show that L G ( ω ) is orthogonal. Let W = L G ( ω ) .Now dim W = 26 , so by Lemma 6.2 (i), it will be enough to prove that ∧ ( W ) has exactly one trivial composition factor and that L G ( ω ) occurs in ∧ ( W ) with odd multiplicity.We have V G ( ω ) = L G ( ω ) and then by a computation with Magma[BCP97] (or [Don85, 7.4.3, pg. 98]) the G -character of ∧ ( W ) is given by ch ∧ ( W ) = ch V G ( ω )+ch V G ( ω ) . Furthermore, from the data in [Lüb17], wecan deduce V G ( ω ) = L G ( ω ) /L G ( ω ) and V G ( ω ) = L G ( ω ) /L G ( ω ) /L G (0) .Therefore as a G -module ∧ ( W ) has composition factors L G ( ω ) , L G ( ω ) , L G ( ω ) , L G ( ω ) , and L G (0) . G λ L G ( λ ) orthogonal? E ω yes E ω yes E ω + ω yes E ω no E ω yes E ω yes E ω yes E ω yes E ω yes E ω yes E ω yesTable 6.1: Orthogonality of some L G ( λ ) for G of type E , E and E .We finish this section by verifying the information given in Table 6.1.Suppose that G is of type E . We have V G ( ω ) = L G ( ω ) and V G ( ω + ω ) = L ( ω + ω ) /L ( ω ) by [Lüb17], so L G ( ω ) and L G ( ω + ω ) are orthogonalby Proposition 2.2 (iv). We show next that L ( ω ) is orthogonal. Now W = One can also construct a non-degenerate G -invariant quadratic form on L G ( ω ) ex-plicitly by realizing it as the space of trace zero elements in the Albert algebra. The detailsof this construction can be found in [Wil09, 4.8.4, pg. 151-152]. G ( ω ) is self-dual and dim W = 78 , so by Lemma 6.2 (ii) it will be enough toprove that ∧ ( W ) has exactly one trivial composition factor and that L G ( ω ) occurs in ∧ ( W ) with odd multiplicity.Now V G ( ω ) = L G ( ω ) , and then a computation with Magma [BCP97] (or[Don85, 8.12, pg. 136]) shows that ch ∧ ( W ) = ch V G ( ω ) + ch V G ( ω ) . From[Lüb17], we can deduce that the composition factors of V G ( ω ) are L G ( ω ) , L G ( ω + ω ) , L G ( ω + ω ) , L G ( ω ) and L G (0) . Thus L G (0) and L G ( ω ) bothoccur exactly once as a composition factor of ∧ ( W ) .Consider next G of type E . We can assume that G is simply con-nected. Then the Weyl module V G ( ω ) is the Lie algebra of G , and L G ( ω ) is not orthogonal by [GW95, Theorem 3.4 (a)]. We have V G ( ω ) = L G ( ω ) , V G ( ω ) = L G ( ω ) /L G ( ω + ω ) and V G ( ω ) = L G ( ω ) by the data in [Lüb17].Therefore L G ( ω ) , L G ( ω ) and L G ( ω ) are orthogonal by Proposition 2.2 (iv).We show that L G ( ω ) is orthogonal. Now for W = L G ( ω ) we have dim W = 56 , so by Lemma 6.2 (ii), it will be enough to prove that ∧ ( W ) hasexactly two trivial composition factors and that L G ( ω ) occurs in ∧ ( W ) withodd multiplicity. Now V G ( ω ) = L G ( ω ) , so by a computation with Magma[BCP97] we see ch ∧ ( W ) = ch V G ( ω ) + ch V G (0) . From [Lüb17], we see that V G ( ω ) has composition factors L G ( ω ) , L G ( ω ) , L G ( ω ) and L G (0) . There-fore ∧ ( W ) has exactly two trivial composition factors and L G ( ω ) occursexactly once as a composition factor.For G of type E , we have V G ( ω ) = L G ( ω ) and so L G ( ω ) is orthogonalby 2.2 (iv). Finally, we show that L G ( ω ) and L G ( ω ) are orthogonal. For W = L G ( ω ) we have dim W = 248 , so by Lemma 6.2 (ii), it will be enough to provethat ∧ ( W ) has exactly two trivial composition factors and that L G ( ω ) and L G ( ω ) occur in ∧ ( W ) with odd multiplicity. By a computation with Magma[BCP97] we see that ch ∧ ( W ) = ch V G ( ω ) + ch V G ( ω ) . From [Lüb17], wesee that V G ( ω ) has composition factors L G ( ω ) , L G ( ω ) , L G ( ω ) , L G (0) , and L G (0) . Therefore ∧ ( W ) has exactly two trivial composition factors and both L G ( ω ) and L G ( ω ) occur with multiplicity one. In this section, we describe consequences of some of our findings and proposesome questions motivated by Problem 1.1. Unless otherwise mentioned, welet G be a simply connected algebraic group over K and we assume that char K = 2 . 34 .1 Connection with representations of the symmetricgroup Denote the symmetric group on n letters by Σ n . We will describe a connectionbetween orthogonality of certain irreducible K [Σ n ] -representations and theirreducible representations L ( ω r ) of Sp l ( K ) . This is done by an applicationof Proposition 3.1 and various results from the literature. The result is nottoo surprising, since the representation theory of the symmetric group playsa key role in the representation theory of the modules L ( ω r ) of Sp l ( K ) . Forexample, many of the results that we applied in the proof of Proposition 3.1above are based on studying certain K [Σ n ] -representations associated with V ( ω r ) .It is well known that there exists an embedding Σ l +1 < Sp l ( K ) = G for all l ≥ (see e.g. [GK99] or [Tay92, Theorem 8.9]). Therefore if a rep-resentation V of G is orthogonal, it is clear that the same is true for therestriction V ↓ Σ l +1 . We will proceed to show that the converse is also truewhen V = L ( ω r ) for ≤ r ≤ l , which does not seem to be a priori obvious.First of all, the following result due to Gow and Kleshchev [GK99, The-orem 1.11] gives the structure of L ( ω r ) ↓ Σ l +1 . Theorem 7.1.
Let ≤ r ≤ l . Then the restriction L ( ω r ) ↓ Σ l +1 is ir-reducible, and it is isomorphic to the irreducible K [Σ n ] -module D (2 l +1 − r,r ) labeled by the partition (2 l + 1 − r, r ) of l + 1 . Now Gow and Quill have determined in [GQ04] when the irreducible K [Σ n ] -modules D ( n − r,r ) are orthogonal. Their result is the following. Theorem 7.2.
Let ≤ r ≤ n . Then the K [Σ n ] -module D ( n − r,r ) is notorthogonal if and only if r = 2 j , j ≥ and n ≡ k mod 2 j +2 for some j +1 + 2 j − ≤ k ≤ j +2 − . In the case where n = 2 l + 1 , one can express the result in the following way. Corollary 7.3.
Let n = 2 l + 1 and ≤ r ≤ l . Then K [Σ n ] -module D ( n − r,r ) is not orthogonal if and only if r = 2 i +1 , i ≥ and l + 1 ≡ i +1 + 2 i + t mod 2 i +2 for some ≤ t < i .Proof. By Theorem 7.2, the module D ( n − , is not orthogonal if and onlyif l + 1 ≡ , which never happens. Therefore D ( n − , is alwaysorthogonal, as desired.Consider then r > . According to Theorem 7.2, if D ( n − r,r ) is not orthog-onal, then r = 2 j for some j > . In this case D ( n − r,r ) is not orthogonal ifand only if l + 1 ≡ k mod 2 j +2 for some j +1 + 2 j − ≤ k ≤ j +2 − . This35s equivalent to saying that l + 1) ≡ k mod 2 j +2 for some j +1 + 2 j ≤ k ≤ j +2 − . Now this condition is equivalent to l + 1 ≡ k mod 2 j +1 for some j + 2 j − ≤ k ≤ j +1 − , giving the claim.Finally combining Theorem 7.1, Proposition 3.1 and Corollary 7.3 gives thefollowing result. Proposition 7.4.
Let ≤ r ≤ l . Then L ( ω r ) is orthogonal for Sp l ( k ) ifand only if L ( ω r ) ↓ Σ l +1 is orthogonal for Σ l +1 . To determine which irreducible G -modules are orthogonal, it is enough toconsider L G ( λ ) with λ ∈ X ( T ) + a -restricted dominant weight (Remark2.5). For groups of exceptional type, this leaves finitely many λ to consider.For groups of classical type, we can further reduce the question to G of type A l and type C l . This follows from the next two lemmas. Note that in Lemma7.5, we identify the fundamental dominant weights of B l and C l by abuse ofnotation. Lemma 7.5.
Let λ = P li =1 a i ω i , where l ≥ and a i ∈ { , } for all ≤ i ≤ l . (i) Let G be of type B l or C l . If a l = 1 , then V = L G ( λ ) is orthogonal,except when l = 2 and λ = ω . (ii) The irreducible B l -representation L B l ( λ ) is orthogonal if and only if theirreducible C l -representation L C l ( λ ) is orthogonal.Proof. (i) Let ϕ : B l → C l be the usual exceptional isogeny between simplyconnected groups of type B l and C l [Ste68, Theorem 28]. Then L C l ( λ ) ϕ ∼ = L B l ( l − X i =1 a i ω i + 2 a l ω l ) ∼ = L B l ( l − X i =1 a i ω i ) ⊗ L B l ( a l ω l ) F where the last equality follows by Steinberg’s tensor product theorem.Assume that a l = 1 . Note that a C l -module V is orthogonal if and onlyif V ϕ is an orthogonal B l -module. Thus it follows from Lemma 2.4 that L C l ( λ ) is orthogonal, except possibly when λ = ω l . Finally, we knowthat L C l ( ω l ) is orthogonal if and only if l ≥ by Example 3.4. Thisproves the claim for G of type C l .36or G of type B l , let τ : C l → B l be the usual exceptional isogenybetween simply connected groups of type C l and B l . Then L B l ( λ ) τ ∼ = L C l ( l − X i =1 a i ω i + a l ω l ) ∼ = L C l ( l − X i =1 a i ω i ) F ⊗ L C l ( a l ω l ) , and now the claim follows as in the type C l case.(ii) If a l = 1 , the claim follows from (i). If a l = 0 , then L C l ( λ ) ϕ ∼ = L B l ( λ ) and the claim follows since a C l -module V is orthogonal if and only if V ϕ is an orthogonal B l -module. Lemma 7.6.
Let λ = P li =1 a i ω i , where l ≥ and a i ∈ { , } for all ≤ i ≤ l . (i) If a l − = a l , then L D l ( λ ) is orthogonal if it is self-dual. (ii) If a l − = a l , then L D l ( λ ) is orthogonal if and only if L C l ( P l − i =1 a i ω i ) isorthogonal, except when λ = ω .Proof. (i) If a l − = a l , then for example by [Hum72, Table 13.1] we see thatfor G of type D l , the weight λ is not a sum of roots. Therefore L G (0) cannot be a composition factor of V G ( λ ) , and thus L G ( λ ) is orthogonalby Proposition 2.2 (iv).(ii) Suppose that a l − = a l . Considering D l < C l as the subsystem subgroupgenerated by long roots, we have L C l ( P l − i =1 a i ω i ) ↓ D l ∼ = L D l ( λ ) by[Sei87, Theorem 4.1]. Now the claim follows from Lemma 4.1. In this subsection only, we allow char K to be arbitrary.As mentioned in the introduction, one motivation for Problem 1.1 is in thestudy of maximal closed connected subgroups of classical groups. Let Cl( V ) be a classical simple algebraic group, that is, Cl( V ) = SL( V ) , Cl( V ) = Sp( V ) ,or Cl( V ) = SO( V ) . Finding maximal closed connected subgroups of Cl( V ) can be reduced to the representation theory of simple algebraic groups. Weproceed to explain how this is done. For more details, see [LS98] and [Sei87].In [LS98], certain collections C , . . . , C of geometric subgroups were de-fined in terms of the natural module V and its geometry. A reduction theo-rem due to Liebeck and Seitz [LS98, Theorem 1] implies that for a positive-dimensional maximal closed subgroup X of Cl( V ) one of the following holds:37i) X belongs to some C i ,(ii) The connected component X ◦ is simple, and V ↓ X ◦ is irreducible andtensor-indecomposable.In particular, the reduction theorem implies the following. Theorem 7.7.
Let
X <
Cl( V ) be a subgroup maximal among the closedconnected subgroups of Cl( V ) . Then one of the following holds: (i) X is contained in a member of some C i , (ii) X is simple, and V ↓ X is irreducible and tensor-indecomposable. The maximal closed connected subgroups in case (i) of Theorem 7.7 arewell understood [Sei87, Theorem 3]. Furthermore, the maximal closed con-nected subgroups occurring in case (ii) of Theorem 7.7 can also be described.These were essentially determined by Seitz [Sei87] and Testerman [Tes88].The result can be stated in the following theorem, which tells when an irre-ducible tensor-indecomposable subgroup is not maximal.
Theorem 7.8.
Let Y be a simple algebraic group and let V be a non-trivialirreducible tensor-indecomposable p -restricted and rational Y -module. If X isa closed proper connected subgroup of Y such that X is simple and V ↓ X isirreducible, then ( X, Y, V ) occurs in [Sei87, Table 1]. To refine the characterization of maximal closed connected subgroups of
Cl( V ) given in [Sei87, Theorem 3], one should determine which of SL( V ) , Sp( V ) and SO( V ) contain X and Y in Theorem 7.8.For example, let Y be simple of type D and let X < Y be simple oftype B embedded in the usual way. Then for V = L Y ( ω ) we have V ↓ X = L X ( ω ) ; this situation corresponds to entry IV in [Sei87, Table 1]. Here V isnot self-dual as a Y -module, so Y <
SL( V ) only. However, V ↓ X is self-dualand X <
SO( V ) if p = 2 , and X <
SO( V ) < Sp( V ) if p = 2 (see Table2.1 and Theorem 4.2). In this situation Y is maximal in SL( V ) , while X ismaximal in SO( V ) .In fact, the results we have presented in this text allow one to determinefor almost all ( X, Y, V ) occurring in [Sei87, Table 1] whether V ↓ X and V ↓ Y are orthogonal, symplectic, both, or neither. If p = 2 , then this iseasily done using Table 2.1.For p = 2 , we list this information in Table 7.1, which is deduced asfollows. Entry IV is a consequence of Lemma 7.5, Example 3.4 and Lemma7.6. In entry S , we have V ↓ Y = L C ( ω ) which is orthogonal by Example38.2, and thus V ↓ X is also orthogonal. In entry S , we have V ↓ Y = L C ( ω + ω ) , which is orthogonal by Proposition 2.2 (iv) since V C ( ω + ω ) is irreducible. Entry S follows from Lemma 2.4 and Lemma 7.6. In entries S , S , and S , we have V ↓ X = L X ( λ ) and V X ( λ ) is irreducible, so V ↓ X is orthogonal by Proposition 2.2 (iv). Entries MR and MR follow fromProposition 6.3, which show that V ↓ Y is orthogonal. Entry MR is aconsequence of Lemma 7.5 (i).No. X < Y V ↓ X V ↓ Y IV B l < D l +1 orthogonal l + 1 even: orthogonal l + 1 odd: not self-dual S G < C orthogonal orthogonal S G < C orthogonal orthogonal S B n · · · B n k < D P n i orthogonal P n i even: orthogonal P n i odd: not self-dual S A < D orthogonal not self-dual S D < D orthogonal not self-dual S C < D orthogonal not self-dual MR D < F orthogonal orthogonal MR C < F orthogonal orthogonal MR D l < C l ? ? MR B n · · · B n k < B n + ··· + n k orthogonal orthogonalTable 7.1: Invariant forms on V ↓ X and V ↓ Y for ( X, Y, V ) occurring in[Sei87, Table 1] in the case p = 2 .What remains is the entry MR from [Sei87, Table 1]. Here X = D l ( l ≥ ) embedded in Y = C l as the subsystem subgroup of long roots, andwe have V = L Y ( P l − i =1 a i ω i ) with a i ∈ { , } , and V ↓ X = L X ( P l − i =1 a i ω i + a l − ( ω l − + ω l )) . In this situation we do not know in general whether V ↓ Y and V ↓ X are orthogonal, but we do know that except in the case where P l − i =1 a i ω i = ω , it is true that V ↓ Y is orthogonal if and only if V ↓ X isorthogonal (Lemma 7.6). Using this fact and the information in Table 7.1,we can deduce the following result. Theorem 7.9.
Let Y be a simple algebraic group and let V be a non-trivial ir-reducible tensor-indecomposable p -restricted Y -module. If X is a closed properconnected subgroup of Y such that X is simple and V ↓ X is irreducible, thenone of the following holds. V ↓ Y is not self-dual. (ii) Both V ↓ Y and V ↓ X are orthogonal. (iii) Neither of V ↓ Y or V ↓ X is orthogonal. (iv) p = 2 , X is of type D l , Y is of type C l and V is the natural module of Y . Among the irreducible self-dual G -modules that are not orthogonal, so farthe only ones that we know of are in some sense minimal among the self-dualirreducible modules of G . We make this more precise in what follows, andpose the question whether any other examples can be found.Recall that L G ( λ ) is self-dual if and only if λ = − w ( λ ) , where w isthe longest element in the Weyl group. We know that any dominant weight λ ∈ X ( T ) + can be written uniquely as a sum of fundamental dominantweights, that is, λ = P li =1 a i ω i for unique integers a i ≥ . Now similarly,there exists a collection µ , . . . , µ t ∈ X ( T ) + such that µ i = − w ( µ i ) for all i ,and such that any λ ∈ X ( T ) + with λ = − w ( λ ) can be written uniquely as P ti =1 a i µ i with a i ≥ . For each simple type, these µ i are listed below. • Type A l ( l odd): µ i = ω i + ω l +1 − i for ≤ i ≤ l − , and µ l +12 = ω l +12 . • Type A l ( l even): µ i = ω i + ω l +1 − i for ≤ i ≤ l . • Types B l , C l , D l ( l even), G , F , E , and E : µ i = ω i for ≤ i ≤ rank G . • Type D l ( l odd): µ i = ω i for ≤ i ≤ l − , and µ l − = ω l − + ω l . • Type E : µ = ω + ω , µ = ω , µ = ω + ω , and µ = ω .Currently the only known examples of non-trivial irreducible modules L G ( λ ) that are self-dual and not orthogonal are of the form L G ( µ i ) . Arethere any others? Problem 7.10.
Let λ ∈ X ( T ) + be -restricted and suppose that λ = λ + λ ,where λ i ∈ X ( T ) + are nonzero and − w ( λ i ) = λ i . Is L G ( λ ) orthogonal? If the answer to Problem 7.10 is yes, then our results would settle Problem1.1 almost completely. Indeed, a positive answer to Problem 7.10 would showthat any non-orthogonal self-dual irreducible representation of G must be40qual to a Frobenius twist of L G ( µ i ) for some i . Our results determine theorthogonality of L G ( µ i ) when G is of classical type. The non-orthogonal onesfor type A l are the L A l ( ω i + ω l +1 − i ) described in Theorem 5.1. For G of type B l , C l , or D l , the non-orthogonal ones are L G ( ω i ) as described in Theorem4.2, with the unique exception of L G ( ω + ω ) for G of type D (arising fromrestriction of L C ( ω ) to G ).Then a handful of µ i still remain for exceptional types. For G simpleof exceptional type, the irreducibles L G ( µ i ) whose orthogonality was notdecided in Section 6 are as follows. • L G ( ω + ω ) for G of type E , • L G ( ω ) and L G ( ω ) for G of type E , • L G ( ω i ) for ≤ i ≤ for G of type E .In any case, a natural next step towards solving Problem 1.1 should bedetermining an answer to Problem 7.10. The methods we have used in thispaper to solve Problem 1.1 for certain families of L G ( λ ) rely heavily on de-tailed information about the structure of the Weyl module V G ( λ ) , which isnot known in general. For small-dimensional representations the compositionfactors of V G ( λ ) can be found using the results of Lübeck given in [Lüb01]and [Lüb17]. However, in general this sort of information is not available,and in characteristic the composition factors of V G ( λ ) are known only in arelatively few cases. For example, for G of type E we do not even know thedimension of L G ( ω i ) for all i in characteristic . We finish by a question about a possible orthogonality criterion for irreduciblerepresentations. Let ϕ : G → SL( V ) be a non-trivial irreducible representa-tion of G . Assume that V is self-dual, so that ϕ ( G ) < Sp( V ) (Lemma 2.1).If V is an orthogonal G -module, then ϕ ( G ) < O( V ) and so ϕ ( G ) < SO( V ) since G is connected. Then for any unipotent element u ∈ G , the number ofJordan blocks of ϕ ( u ) is even [LS12, Proposition 6.22]. In other words, for all u ∈ G we have that dim V u is even, where V u is the subspace of fixed pointsfor u . Does the converse hold? Problem 7.11.
Let V be a non-trivial irreducible self-dual representation of G . If V is not orthogonal, does there exist a unipotent element u ∈ G suchthat dim V u is odd?
41n Table 7.2, we have listed examples (without proof) of some non-orthogonal representations V of G for which the answer to Problem 7.11is yes. If the answer to Problem 7.11 turns out to be yes, we would have aninteresting criterion for an irreducible representation V of G to be orthogo-nal. A positive answer would show that the orthogonality of an irreduciblerepresentation can be decided from the properties of individual elements of G . Type of G V
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