aa r X i v : . [ m a t h . G R ] M a y Moufang sets of mixed type F Abstract
Moufang sets were introduced by Jacques Tits in order to under-stand isotropic linear algebraic groups of relative rank one, but thenotion is more general. We describe a new class of Moufang sets, aris-ing from so-called mixed groups of type F in characteristic 2, obtainedas the fixed point set under a suitable involution. Moufang sets were introduced by Jacques Tits in [19] as an axiomization ofthe isotropic simple algebraic groups of relative rank one, and they are, infact, the buildings corresponding to these algebraic groups, together withsome of the group structure (which comes from the root groups of the al-gebraic group). In this way, the Moufang sets are a powerful tool to studythese algebraic groups. On the other hand, the notion of a Moufang set ismore general, and includes many more examples that do not directly ariseby this procedure. In fact, it is still a wide open question whether everyMoufang set is, in some sense, of algebraic origin.In this paper, we are studying Moufang sets arising from so-called mixedgroups of type F . These groups exist only over fields of characteristic 2,and they are defined over a pair of fields ( k, ℓ ) such that ℓ ≤ k ≤ ℓ . Therehas been an increasing interest in a systematic study of these insepara-ble situations over non-perfect fields, most notably by the recent work on pseudo-reductive groups by B. Conrad, O. Gabber and G. Prasad [7].Formally speaking, a Moufang set is a set X together with a collectionof groups (cid:0) U x ≤ Sym( X ) (cid:1) x ∈ X , such that each U x acts regularly on X \ { x } ,and such that U ϕx = U x.ϕ for all ϕ ∈ G † := h U x | x ∈ X i . The groups U x are called the root groups of the Moufang set, and the group G † is called its little projective group . We will always write our permutation actions on the right. F have already beendescribed explicitly in [9]. The techniques used in that paper, however,rely heavily on the fact that the algebraic groups of type F arise as theautomorphism groups of certain 27-dimensional algebraic structures knownas Albert algebras, and such a description is not available for the mixedgroups of type F .We therefore take a completely different approach, inspired by [11] and[17], using the description of the corresponding Chevalley groups, and replac-ing the geometric ingredients of the algebraic approach (namely polaritiesof the octonion plane) by group theoretic ingredients (namely involutionsof the Chevalley groups). More specifically, we show how to construct asplit saturated BN-pair of rank one from a well-chosen involution. Such aBN-pair is essentially equivalent to a Moufang set.The paper is organized as follows. In section 2, we recall the necessarybasics about Chevalley groups, Moufang sets, and BN-pairs. Section 3 dealswith the basic theory of mixed groups as introduced by J. Tits. In section 4,we specifically look at mixed Chevalley groups of type F , and in section 5,we study involutions of these mixed groups such that the centralizer of theinvolution is a split BN-pair of rank one. These BN-pairs give rise to theMoufang sets we are interested in, and in section 6 we proceed to explicitlydescribe these Moufang sets. This culminates in our main result (Theo-rem 6.2). In the last section 7, we point out that in the algebraic case, werecover the known description of algebraic Moufang sets of type F as in [9]. Acknowledgments
It is our pleasure to thank Bernhard M¨uhlherr, whoprophetically predicted the existence of the mixed Moufang sets of type F ,and shared its mysteries with us. We are grateful to Hendrik Van Maldeghemfor the lively geometric discussions about this topic. Finally, we thank thereferee, for very carefully reading our paper, correcting some of our mistakesand making good suggestions to fill in certain details that were initiallymissing. In particular, the referee pointed out how to improve both theresult and the proof of Theorem 4.2 below; see also Remark 4.3. We briefly recall some basics about Chevalley groups that we will need inthe sequel. Our main reference is [6].2 .1.1 The Cartan decomposition of a complex simple Lie algebra
Let L be a Lie algebra over C , with Lie bracket [ · , · ]. A Cartan subalgebra H is a subalgebra of L which is nilpotent, and such that H is not containedas an ideal in any larger subalgebra of L , i.e. if x ∈ L is such that [ x, h ] ∈ H for all h ∈ H , then x ∈ H .Now if L is simple over C , then L can be decomposed into a direct sumof H with a number of one-dimensional H -invariant subspaces: L = H ⊕ L r ⊕ · · · ⊕ L r m . The one-dimensional subspaces L r i are called the root spaces of L (w.r.t. H ).In each one-dimensional subspace L r , we choose a non-zero element e r .Then for each h ∈ H , we have [ he r ] = r ( h ) e r (2.1)for some r ( h ) ∈ C . This defines a linear map r : H → C ; h r ( h ) . It can be shown that each element f of the dual space H ∗ of H is of theform f : H → C ; h ( x, h )for some unique x ∈ H , where ( · , · ) is the Killing form on H . In this way r corresponds to a unique element in H , which we also denote by r . We canrepeat this procedure for each root space and denote by Ψ the subset of H we obtain in this way.One can show that Ψ forms a root system in H R , where H R is the set oflinear combinations of elements of Ψ with real coefficients. As the Killingform induces an isomorphism between H and its dual space H ∗ , there is acorresponding root system Φ in H ∗ . The elements of H ∗ are called the roots of L (w.r.t. H ).We can rewrite the Cartan decomposition of L as L = H ⊕ M r ∈ Φ L r , in such a way that for any pair of roots r, s ∈ Φ, we have[ L r , L s ] = L r + s if r + s ∈ Φ , [ L r , L s ] = 0 if r + s Φ , r + s = 0 , [ L r , L − r ] = C r, [ H , L r ] = L r . r is any root, then theelement h r ∈ H corresponding to (2 r ) / h r, r i under the isomorphism is calledthe coroot of r . Now let Π be a set of fundamental roots for Φ; if we choose e − r ∈ L − r such that [ e r , e − r ] = h r for each r ∈ Φ then { h r | r ∈ Π } ∪ { e r | r ∈ Φ } (2.2)forms a basis for L , called a Chevalley basis , satisfying[ h r , h s ] = 0 , [ h r , e s ] = A rs e s , [ e r , e − r ] = h r , [ e r , e s ] = 0 if r + s Φ , r + s = 0 , [ e r , e s ] = N rs e r + s if r + s ∈ Φ . The constants A rs are easily determined by the root system, as is the abso-lute value of the constants N rs ; determining the sign of the N rs is much moredelicate, however. Since we will be working over fields of characteristic 2,we need not worry about these signs. Let L be a simple Lie algebra over C with Chevalley basis as in (2.2). Nowlet L Z be the subset of L of all integral linear combinations of the basiselements; then L Z becomes a Lie algebra over Z .Now let k be any field. Then we can form the tensor product of theadditive group of k with the additive group of L Z , and define L k = k ⊗ L Z , which is in a natural way a Lie algebra over k .We now introduce certain automorphisms of L k . For every root r ∈ Φand every element t ∈ k , we define an automorphism u r ( t ) as follows: u r ( t ) · e r = e r ,u r ( t ) · e − r = e − r + th r − t e r ,u r ( t ) · e s = q X i =0 M r,s,i t i e ir + s if r, s are linearly independent ,u r ( t ) · h s = h s − A sr te r for s ∈ Π , where in the rule for u r ( t ) · e s , q is the largest integer such that qr + s ∈ Φ,and where the constants M r,s,i are defined in terms of the structure constants N rz . 4he (adjoint) Chevalley group of type L over k is now defined as thegroup L ( k ) := h u r ( t ) | r ∈ Φ , t ∈ k i . It turns out that this group is independent of the choice of the Chevalleybasis; its isomorphism type depends only on L and k . Note that u r ( s ) u r ( t ) = u r ( s + t ) for all r ∈ Φ and all s, t ∈ k . The subgroups U r = { u r ( t ) | t ∈ k } are called the root subgroups of G = L ( k ). Remark 2.1. (i) If k = C , then u r ( t ) = exp( t ad e r ), and in fact, thisis where the definition of the automorphisms u r ( t ) in the general casecomes from.(ii) We have been following Chevalley’s original approach to constructChevalley groups. This construction has later been generalized to in-clude not only adjoint groups but also more general connected semisim-ple split linear algebraic groups. The corresponding building, and con-sequently also the Moufang set that we will construct, does not detectthis distinction (its little projective group always corresponds to theadjoint representation) so it is no loss of generality to restrict to theadjoint case.(iii) The relation between (not necessarily adjoint) Chevalley groups andlinear algebraic groups is as follows. Let k be an arbitrary field, and let K be its algebraic closure. Then a Chevalley group L ( K ) is a connectedsemisimple linear algebraic group G over K of type L , defined andsplit over k (and in fact over the prime subfield of k ). The Chevalleygroup L ( k ) is the commutator subgroup of the group G ( k ) of k -rationalpoints of G .An important feature of Chevalley groups and of linear algebraic groupsis that the root groups satisfy certain commutator relations . We will dis-cuss those relations (and the extension of this concept to mixed groups) insection 3 below. We introduce some notation for certain important subgroups of Chevalleygroups that we will need in the future.Let Φ be the root system associated to an arbitrary Chevalley group L ( k ), let Π be a fundamental root system of Φ, Φ + be the set of positiveroots and Φ − be the set of negative roots of Φ. One can associate to everyroot r ∈ Φ a reflection w r ; the group generated by all these reflections iscalled the Weyl group W of L ( k ). More information on Weyl groups can befound in [6, Chapter 2].Furthermore, every Chevalley group G has an associated BN-pair , i.e. apair of subgroups (
B, N ) of G such that the following conditions hold.51) G = h B, N i .(2) B ∩ N is a normal subgroup of N .(3) W = N/B ∩ N is generated by elements w i such that w i = 1, i ∈ I . Foreach i ∈ I , we choose a preimage n i ∈ N of w i .(4) For each i ∈ I and each n ∈ N , we have Bn i B · BnB ⊆ Bn i nB ∪ Bnb. (5) For each i ∈ I , we have n i Bn i = B .For each J ⊆ I , we define W J := h w j | j ∈ J i . A subgroup P of G is called a parabolic subgroup if P contains B or someconjugate of B . One can show that the parabolic subgroups containing B are of the form BN J B with N J the preimage of W J under the naturalprojection from N to W . A consequence of the axioms is that the group G has a so-called Bruhat decomposition G = BN B .In our setting of Chevalley groups, we define n r ( t ) := u r ( t ) u − r ( − t − ) u r ( t ), n r := n r (1) and h r ( t ) := n r ( t ) n r ( −
1) for all r ∈ Φ and all t ∈ k × . Let N := h n r ( t ) | r ∈ Φ , t ∈ k × i ,H := h h r ( t ) | r ∈ Φ , t ∈ k × i ,U := h u r ( t ) | r ∈ Φ + , t ∈ k i and B := U H.
Then one can show that (
B, N ) forms a BN-pair for L ( k ) with B ∩ N = H and N/H = W with W the Weyl group of L ( k ).Finally, we introduce some standard notation. Let J be a subset of Π,then we define Φ J as Φ ∩ h J i and W J as the Weyl group generated by all w α with α ∈ J . We denote by w the longest element in W and similarly w J is the longest element in W J . We then define U J := h U r | r ∈ Φ + \ Φ J i ,L J := h H, U r | r ∈ Φ J i ,P J := U J L J . In this section, we recall some of the basics of Moufang sets, and we refer to[8] for more details.A
Moufang set M = (cid:0) X, ( U x ) x ∈ X (cid:1) is a set X together with a collectionof groups U x ≤ Sym( X ), such that for each x ∈ X :61) U x fixes x and acts sharply transitively on X \ { x } ;(2) U ϕx = U xϕ for all ϕ ∈ G := h U z | z ∈ X i .The group G is called the little projective group of the Moufang set.A typical example is given by the group G = PSL (2 , k ) acting on theprojective line X = P ( k ) = k ∪ {∞} . One of the main motivations (but certainly not the only one) for studyingMoufang sets, is that they provide a tool to understand linear algebraicgroups of relative rank one. We will briefly explain the connection.So suppose that G is an absolutely simple algebraic group defined over afield k , and assume that G has k -rank one. Let X be the set of all k -parabolicsubgroups of G . For each x ∈ X , we let U x be the root subgroup of the k -parabolic subgroup x (which coincides with the k -unipotent radical of x ).Then (cid:0) X, ( U x ) x ∈ X (cid:1) is a Moufang set, which we will denote by M ( G , k ).If we define G + ( k ) to be the subgroup of G ( k ) generated by all the rootsubgroups, then G + ( k ) modulo its center acts faithfully on X .As an example, we could consider groups of the form G = PSL (2 , D ),where D is a division algebra of degree d over a field k ; in this case, G is analgebraic group of type A d − of k -rank one. Note that PSL (2 , D ) still givesrise to a Moufang set if D is infinite-dimensional over its center, but in thiscase the Moufang set no longer arises from an algebraic group. We will now explain how any Moufang set can be reconstructed from a singleroot group together with one additional permutation [10].Let ( U, +) be a group, with identity 0, and where the operation + is notnecessarily commutative . Let X = U ∪ {∞} , where ∞ is a new symbol. Foreach a ∈ U , we define a map α a ∈ Sym( X ) by setting α a : ( ∞ 7→ ∞ x x + a for all a ∈ U . (2.3)Let U ∞ := { α a | a ∈ U } . Now let τ be a permutation of U ∗ . We extend τ to an element of Sym( X )(which we also denote by τ ) by setting 0 τ = ∞ and ∞ τ = 0. Next we set U := U τ ∞ and U a := U α a (2.4)7or all a ∈ U , where U ϕx denotes conjugation inside Sym( X ). Let M ( U, τ ) := (cid:0) X, ( U x ) x ∈ X (cid:1) (2.5)and let G := h U ∞ , U i = h U x | x ∈ X i . Then M ( U, τ ) is not always a Moufang set, but every Moufang set can beobtained in this way. The next lemma shows us how to do this.
Lemma 2.2.
Let M = ( X, ( U x ) x ∈ X ) be Moufang set. Pick two elements , ∞ ∈ X and define U as X \ {∞} .For every a ∈ U , define α a ∈ U ∞ as the unique element such that α a (0) = a .Let a + b := α b ( a ) for every a, b ∈ U and τ ∈ Sym( X ) be a permutationinterchanging and ∞ such that U τ ∞ = U . Then M = M ( U, τ ) .Proof. This is obvious from the above construction of M ( U, τ ).Note that, for a given Moufang set, the map τ is certainly not unique:different choices for τ can give rise to the same Moufang set. We introduce the notion of a saturated split BN-pair of rank one because ofthe correspondence with Moufang sets. In the context of Chevalley groups,it sometimes is more natural to work with BN-pairs. We show how toconstruct a Moufang set from these BN-pairs.A
BN-pair of rank one in a group G is a system ( B, N ) of two subgroups B and N of G such that the following axioms hold:(i) G = h B, N i .(ii) H := B ∩ N E N .(iii) There is an element ω ∈ N \ H with ω ∈ H such that N = h H, w i , G = B ∪ BωB and ωBω = B .We call such a pair split if additionally(iv) There exists a normal subgroup U of B such that B = U ⋊ H .holds, and saturated if additionally(v) H = B ∩ B ω . Lemma 2.3.
Let G be a group with a saturated split BN-pair of rank one,let X := { U g | g ∈ G } be the set of conjugates of U in G. Denote by V x , theelement x ∈ X viewed as a subgroup of G . Then ( X, ( V x ) x ∈ X ) is a Moufangset.Proof. For a proof, see [8, Proposition 2.1.3.].8sing the alternative definition of a Moufang set, we find that a repre-sentation of the Moufang set corresponding to this BN-pair is M ( U, ω ). F We will now give an easy description of the Moufang sets arising from alge-braic groups of type F using the method we have explained in the previoussection. Theorem 2.4.
Let k be an arbitrary field. Every Moufang set of type F over k is determined by an octonion division algebra O /k . More precisely,if O is such an octonion division algebra, then we define U := { ( a, b ) ∈ O × O | N( a ) + T( b ) = 0 } , where N and T are the standard norm and trace maps from O to k . Wedefine the (non-abelian) group operation + on U by setting ( a, b ) + ( c, d ) := ( a + c, b + d − ca ) for all ( a, b ) , ( c, d ) ∈ U . Finally, we define a permutation τ ∈ Sym( U ∗ ) bysetting ( a, b ) τ := (cid:0) − ab − , b − (cid:1) for all ( a, b ) ∈ U . Then the corresponding Moufang set of type F is equalto M ( U, τ ) .Proof. See [9].The goal of this paper is to extend this result to so-called mixed groupsof type F (which we will define in the next section). We will see, however,that we will not only have to use very different methods, but that alsothe resulting description will not simply be of the same form as the nicedescription that we have in Theorem 2.4; see Theorem 6.2 below. In this section, we recall some basic facts about mixed groups. Our mainreference is [18, Section 10.3].Let G be an adjoint split simple algebraic group of type X defined over afield k of characteristic p , where either X = B n , C n , F and p = 2, or X = G and p = 3. Assume moreover that ℓ is a field such that ℓ p ≤ k ≤ ℓ .Let T be a maximal k -split torus, let N = N G ( T ) be the normalizer of T in G , let B be a Borel subgroup of G containing T , and let Φ be a root system9f type X corresponding to the maximal torus T . Since X is not simplylaced, Φ consists of long and short roots, and we write Φ = Φ ℓ ∪ Φ s , whereΦ ℓ and Φ s denote the sets of long and short roots, respectively. For eachroot r ∈ Φ, we have a corresponding root group U r , i.e. a one-dimensional k -unipotent subgroup of G acted upon by T . In the algebraic group G , allroot groups are isomorphic to the additive group G a . For each r ∈ Φ, wechoose an isomorphism u r from G a to U r . We also define Φ + to be the set ofpositive roots of Φ, i.e. the roots r ∈ Φ such that U r ⊆ B ; correspondingly,we write Φ + ℓ := Φ ℓ ∩ Φ + and Φ + s := Φ s ∩ Φ + .Now let T ( k, ℓ ) := (cid:26) t ∈ T (cid:12)(cid:12)(cid:12)(cid:12) r ( t ) ∈ k for all r ∈ Φ l and r ( t ) ∈ ℓ for all r ∈ Φ s (cid:27) ,N ( k, ℓ ) := N ( k ) T ( k, ℓ ) ,B ( k, ℓ ) := h T ( k, ℓ ) ∪ { U r ( k ) | r ∈ Φ + l } ∪ { U r ( ℓ ) | r ∈ Φ + s }i , and finally G ( k, ℓ ) := h T ( k, ℓ ) ∪ { U r ( k ) | r ∈ Φ l } ∪ { U r ( ℓ ) | r ∈ Φ s }i . The group G ( k, ℓ ) is the mixed group of type X corresponding to the pairof fields ( k, ℓ ), and it is also denoted by X ( k, ℓ ), particularly when X isspecified. One can show that the pair ( B ( k, ℓ ) , N ( k, ℓ )) forms a BN-pair of G ( k, ℓ ). Example 3.1 ([18, p. 204]) . Let ( k, ℓ ) be a pair of fields of characteristic 2with ℓ ≤ k ≤ ℓ , and let q be the “mixed quadratic form” from k n × ℓ to k given by q ( x , x , . . . , x n − , x n − , x n ) = x x + · · · + x n − x n − + x n . Then the mixed group B n ( k, ℓ ) is isomorphic to the group PGO ( q ), i.e. thequotient of the group of all invertible similitudes of q by the subgroup k × .This group is also isomorphic to the mixed group C n ( ℓ , k ).When we are considering the corresponding building, i.e. the “mixedquadric” consisting of the isotropic vectors of q , it will often be convenientto drop the last coordinate x n , since it is uniquely determined from theother coordinates by the equation q ( x , . . . , x n − , x n ) = 0. Thus, themixed quadric will then consist of points in PG (2 n − , k ) with (projective)coordinates ( X , . . . , X n − ) satisfying the condition X X + · · · + X n − X n − ∈ ℓ , (3.1)and the higher-dimensional objects of the building are now simply the sub-spaces of the underlying projective space lying on this mixed quadric.10or algebraic groups, it is well known that the root groups satisfy certaincommutator relations depending on the root system. More precisely, it ispossible to renormalize the parametrizations u r in such a way that there areconstants c λ,r,µ,s ∈ {± , ± , ± } , called the structure constants , such that[ u r ( x ) , u s ( y )] = Y λ,µ ∈ Z > λr + µs ∈ Φ u λr + µs (cid:16) c λ,r,µ,s · x λ y µ (cid:17) (3.2)for all r, s ∈ Φ and all x, y ∈ k ; see, for example, [14, Propositions 9.2.5and 9.5.3], or [6, Theorem 5.2.2] for the analogous statement for Chevalleygroups.This goes through for mixed groups without any change: we get the samecommutator relations (3.2), but this time for all r, s ∈ Φ and all x, y ∈ k or ℓ depending on whether the corresponding roots r and s are long orshort, respectively. Observe that the condition ℓ p ≤ k ≤ ℓ is exactly thecondition which is needed for these commutator relations to make sense, i.e.the elements x λ y µ belong to k whenever the root λr + µs is a long root.In the case p = char( k ) = 2, which will be the only case we will bedealing with in this paper, the constants c λ,r,µ,s are all equal to 0 or 1, soequation (3.2) simplifies further. In the case p = 2 and X = F , whichis the case that we are interested in in this paper, we can summarize thecommutator relations as follows; see, for instance, [12, (2.2)–(2.5)]:[ u r ( x ) , u s ( y )] = 1 if r, s ∈ Φ but r + s Φ , [ u r ( x ) , u s ( y )] = u r + s ( xy ) if r, s ∈ Φ s and r + s ∈ Φ s , [ u r ( x ) , u s ( y )] = 1 if r, s ∈ Φ s and r + s ∈ Φ ℓ , [ u r ( x ) , u s ( y )] = u r + s ( xy ) if r, s ∈ Φ ℓ and r + s ∈ Φ ℓ , [ u r ( x ) , u s ( y )] = u r + s ( xy ) u r + s ( x y )if r ∈ Φ s , s ∈ Φ ℓ and r + s ∈ Φ s , r + s ∈ Φ ℓ , (3.3)for all x, y ∈ k or ℓ depending on whether the corresponding roots r and s are long or short, respectively. Note that this list is exhaustive: if r and s are long roots with r + s ∈ Φ, then r + s ∈ Φ ℓ ; and if r is a short root and s a long root with r + s ∈ Φ, then 2 r + s ∈ Φ as well and r + s is short and2 r + s is long. See [12, (1.2) and (1.3)]. F Let k and ℓ be fields of characteristic 2 such that ℓ ≤ k ≤ ℓ . Assumethat δ ∈ k is such that the polynomial x + x + δ is irreducible over k .Let γ be a solution of x + x = δ , and let K = k ( γ ) and L = ℓ ( γ ). Then11 ≤ K = h k, L i ≤ L , and K and L are separable quadratic extensions of k and ℓ , respectively. We denote the standard involution on both L and K corresponding to γ by x x .Let Φ be a root system of type F with fundamental system Π := { α , α , α , α } . We can represent the fundamental roots with respect toan orthonormal basis { e , e , e , e } of R as α = ( − e − e − e + e ), α = e , α = e − e , α = e − e and the full system of roots is given byΦ = ± e i ± e j for 1 ≤ i < j ≤ , ± e i for 1 ≤ i ≤ , ( ± e ± e ± e ± e ) . We define the mixed Chevalley group F ( K, L ) of type F as the mixedgroup that can be obtained from the ordinary Chevalley group F ( L ) of type F . For this, we remark (using the definitions introduced in section 2.1.3)that H is a maximal K -split torus, N = N F ( L ) ( H ) is the normalizer of H in F ( L ) and B is a Borel subgroup of F ( L ). Then F ( K, L ) = D { u r ( s ) | r ∈ Φ ℓ , s ∈ K } ∪ { u r ( t ) | r ∈ Φ s , t ∈ L } ∪ T ( K, L ) E is the mixed group of type F corresponding to the pair of fields ( K, L ) of F ( L ). Using the same procedure, we can construct mixed Chevalley groupsof type B n , C n and G . In general, we denote a mixed Chevalley group by X ( K, L ).Theorem 4.2 below shows that we can omit the subgroup T ( K, L ) in thegenerating set for X ( K, L ) if X ( K, L ) is of type G , F or B n with n odd.We will need the following observation. Lemma 4.1.
Let Φ be a root system of type B n , C n , F or G , and let p = 3 in the case of G and p = 2 otherwise. Let Π = { α , . . . , α n } be a set offundamental roots for Φ , and let Π s be the subset of Π of short fundamentalroots. If r ∈ Φ is a long root, and r = P ni =1 n i α i , then each coefficient n i corresponding to a short fundamental root α i ∈ Π s is divisible by p .Proof. This can easily be checked by a case by case analysis; see, for example,[6, section 3.6].
Theorem 4.2.
Let X ( K, L ) be a mixed Chevalley group of type B n , C n , F or G , with L p ⊆ K ( L , where p = 3 in the case of G and p = 2 otherwise.Let T ( K, L ) = (cid:8) h ∈ T ( L ) | r ( h ) ∈ K for all r ∈ Φ l (cid:9) as before. Then T ( K, L ) = D { h r ( t ) | r ∈ Φ l , t ∈ K × } ∪ { h r ( t ) | r ∈ Φ s , t ∈ L × } E (4.1)12 f and only if X ( K, L ) has type G , F or B n with n odd. In this case, wehave, in particular, X ( K, L ) = D { u r ( s ) | r ∈ Φ ℓ , s ∈ K } ∪ { u r ( t ) | r ∈ Φ s , t ∈ L } E . Proof.
Let H = h h r ( λ ) | r ∈ Φ , λ ∈ L × i , i.e. H is the torus T ( L ) of theChevalley group X ( L ) (as defined in section 2.1.3). Let Π = { α , . . . , α n } be the set of fundamental roots of Φ, and let Π s and Π l be the subsets of Πof short and long fundamental roots, respectively. We claim that H = T ( L ) = n Y i =1 h α i ( L × ) . (4.2)Indeed, if r ∈ Φ is any root, then h r : L × → T ( L ) is precisely the coroot r ∨ of r , and hence we can write r ∨ as an integral linear combination r ∨ = ± P ni =1 n i α ∨ i of the coroots α ∨ i corresponding to the roots α i . Hence h r ( t ) = Q ni =1 h α i ( t ± n i ) for all t ∈ L × , and the claim (4.2) follows.Notice that by the same argument, the equality (4.1) is equivalent with T ( K, L ) = Y r ∈ Π l h r ( K × ) · Y r ∈ Π s h r ( L × ) , which is, in view of (4.2), also equivalent with the implication Y r ∈ Π l h r ( λ r ) ∈ T ( K, L ) = ⇒ λ r ∈ K × for all r ∈ Π l ; (4.3)so our goal is to show that the implication (4.3) holds if and only if X ( K, L )has type G , F or B n with n odd.Next, we claim that T ( K, L ) = { h ∈ T ( L ) | r ( h ) ∈ K for all r ∈ Π l } . (4.4)Indeed, assume that h ∈ T ( L ) satisfies the condition that r ( h ) ∈ K for all r ∈ Π l , and let r ∈ Φ l be arbitrary. Write r = P ni =1 n i α i ; hence r ( h ) = Q ni =1 α i ( h ) n i . By Lemma 4.1, each coefficient n i corresponding to a shortfundamental root α i ∈ Π s is divisible by p . If α i is a long fundamentalroot, then α i ( h ) ∈ K by assumption; if α i is a short fundamental root,then α i ( h ) n i ∈ L n i ⊆ L p ⊆ K , and we conclude that r ( h ) ∈ K , provingclaim (4.4).Notice that for each of the types B n , C n , F or G , the subset Π l oflong fundamental roots corresponds to a subdiagram of the Dynkin diagramof type A m , with m equal to n − , , l = { α , . . . , α m } accordingly, where we number the fundamental roots in13he canonical way. Hence we will rewrite an element h = Q r ∈ Π l h r ( λ r ) as h = Q mi =1 α ∨ i ( λ i ), and by (4.4), such an h belongs to T ( K, L ) if and only if m Y i =1 α j (cid:0) α ∨ i ( λ i ) (cid:1) ∈ K for all j ∈ { , . . . , m } . (4.5)We now do a case by case analysis. • If X ( K, L ) is of type C n , then m = 1, and condition (4.5) says that α ( α ∨ ( λ )) = λ ∈ K . This is satisfied for any element λ ∈ L , sosince K = L , the implication (4.3) is false. • If X ( K, L ) is of type G , then m = 1, and condition (4.5) says that α ( α ∨ ( λ )) = λ ∈ K . Since L ⊆ K , this implies λ = λ − λ ∈ K ,and hence the implication (4.3) is true. • Assume that X ( K, L ) is of type B or of type F . Then m = 2, andcondition (4.5) says that α ( α ∨ ( λ )) α ( α ∨ ( λ )) = λ λ − ∈ K , and α ( α ∨ ( λ )) α ( α ∨ ( λ )) = λ − λ ∈ K . Since L ⊆ K , this is equivalentwith λ ∈ K and λ ∈ K , and hence the implication (4.3) is true. • Assume finally that X ( K, L ) is of type B n with n ≥
4. Then m = n − λ λ − ∈ K, λ − i − λ i λ − i +1 ∈ K for i ∈ { , . . . , m − } , λ − m − λ m ∈ K. Since L ⊆ K , this is equivalent with λ ∈ K, λ m − ∈ K, and λ i − ∈ K ⇐⇒ λ i +1 ∈ K for i ∈ { , . . . , m − } . If n is odd, then also m − λ , λ , . . . , λ n − ∈ K and λ m − , λ m − , . . . , λ ∈ K ; hence the implication (4.3) is true in this case. If n is even, how-ever, then condition (4.5) is equivalent to the fact that λ i ∈ K for alleven values of i , without any conditions on the other λ i . Since K = L ,it follows that the implication (4.3) is false in this case. Remark 4.3.
The original version of this paper only contained a proof forthe positive statement in the case of groups of type F and B n , n odd, andour proof was more involved. The referee pointed out how we could simplifythe proof, and simultaneously get a complete answer for all possible mixedChevalley groups. We thank him for sharing his insight with us.The previous lemma will allow us to transfer known facts about BN-pairs of (ordinary) Chevalley groups to BN-pairs of mixed Chevalley groups.Indeed, when X ( K, L ) is a mixed group, the subgroups N ( K, L ) := N ( K ) T ( K, L ) and B ( K, L ) := (cid:10) T ( K, L ) ∪ { U r ( K ) | r ∈ Φ + l } ∪ { U r ( L ) | r ∈ Φ + s } (cid:11) X ( K, L ). Using Theorem 4.2, we actually get B ( K, L ) = B ( L ) ∩ X ( K, L ) and N ( K, L ) = N ( L ) ∩ X ( K, L )if X ( K, L ) is of the appropriate type, where ( B ( L ) , N ( L )) is the naturalBN-pair of X ( L ). This implies (using the general properties of a BN-pair)that X ( K, L ) = B ( K, L ) N ( K, L ) B ( K, L )and that all parabolic subgroups containing B ( K, L ) are of the form P J ( K, L ) := B ( K, L ) N J ( K, L ) B ( K, L ) = P J ∩ X ( K, L ) . (4.6)Notice that N ( K, L ) /T ( K, L ) is also isomorphic to the Weyl group of X ( L ).So N J ( K, L ) is the preimage of W J under the canonical epimorphism from N ( K, L ) to W .We end this section with a unique decomposition lemma for mixedChevalley groups. Lemma 4.4.
Let X ( K, L ) be a mixed Chevalley group of type F or of type B n with n odd. If g ∈ X ( K, L ) is such that P J ( K, L ) gP J ( K, L ) = P J ( K, L ) nP J ( K, L ) (4.7) with n ∈ N ( K, L ) such that nT ( K, L ) = w ∈ Stab(Φ J ) , then g has aunique decomposition g = ulnu ′ with u ∈ U J ( K, L ) := U J ∩ X ( K, L ) , l ∈ L J ( K, L ) := L J ∩ X ( K, L ) and u ′ ∈ U − w,J , where U − w,J := (cid:10) U r | r ∈ Φ + \ Φ J , w ( r ) ∈ Φ − (cid:11) , with U r := ( { u r ( s ) | s ∈ K } if r ∈ Φ ℓ , { u r ( t ) | t ∈ L } if r ∈ Φ s . Proof.
From the equality (4.7), we find that g = p np for some p , p ∈ P J ∩ X ( K, L ). As ordinary Chevalley groups have a Levi decomposition P J = L J · U J , it follows, using (4.6), that there is a corresponding Levidecomposition P J ( K, L ) = L J ( K, L ) · U J ( K, L ) . Assume p = l ′ u ′ for some l ′ ∈ L J ( K, L ) and u ′ ∈ U J ( K, L ), then (as nH ∈ Stab(Φ J )), we can switch l ′ to the left of n ; moreover, we can also switch thefactors of u ′ belonging to some U r with r ∈ Φ + \ Φ J and w ( r ) ∈ Φ + , to theleft of n , so that we are left with an element u ′ ∈ U − w,J . We find that indeed g = pnu ′ = ulnu ′ for some u ∈ U J ( K, L ), l ∈ L J ( K, L ), and u ′ ∈ U − w,J .Suppose that g = u l nu ′ = u l nu ′ , then nu ′ u ′ − n − = ( u l ) − u l ∈ U − J ∩ P J = 1, so uniqueness follows. 15 Construction of a split BN-pair of rank one
In this section we construct a split saturated BN-pair out of an involution on F ( K, L ). For Chevalley groups there exists a general procedure to constructa BN-pair from an involution σ satisfying certain conditions, as carried outin [17].Using a similar procedure, we show that we can construct a split sat-urated BN-pair of rank one from a suitable involution on F ( K, L ). Froma geometric point of view, we actually have constructed, starting from amixed building of type F , a new type of Moufang set; these Moufang setswill be called Moufang sets of mixed type F . F ( K, L ) We follow the ideas from [17], but in order to deal with the situation of mixedChevalley groups, we impose slightly adjusted conditions on the involution σ on F ( K, L ). More precisely, we fix a set J ( Π, and we choose σ in sucha way that(1) σ permutes root groups and N ( K, L ) is invariant under σ .(2) If P is a parabolic subgroup of L J ( K, L ) = L J ( L ) ∩ F ( K, L ), invariantunder σ , then P = L J ( K, L ).(3) h{ U r ( K ) | r ∈ Φ − l \ Φ J } ∪ { U r ( L ) | r ∈ Φ − s \ Φ J }i ∩ Fix( σ ) = 1.In order to take care of the first condition, we consider an involution σ of F ( K, L ) with the following action on the generators of the mixed group(where we denote the corresponding action on the root system also by σ ): σ : F ( K, L ) → F ( K, L ) : u r ( t ) u σ ( r ) ( c r t ) . In analogy with the situation in the algebraic case, we will choose the actionof σ on the root system so that the corresponding Tits index is as follows: α α α α Notice that this is the only admissable Tits index of relative rank one ofabsolute type F , and in fact, every linear algebraic group of absolute type F is either anisotropic (i.e. has relative rank 0), or split (i.e. has relative rank 4),or has the above Tits index. (In the mixed case, however, an additional Titsindex of relative rank 2 can arise; see [11].)If we now look at the F -building corresponding to F ( K, L ), our goalis to construct the involution σ on F ( K, L ) in such a way that the corre-sponding fixbuilding has only points of the first type (these are the pointscorresponding to α ). Therefore, we choose J to be the subset { α , α , α }
16f Π. In particular, the action of σ on the root system Φ is given by thelongest element w J in the Weyl group W J generated by w α , w α and w α .Then σ is an involution fixing e and inverting e , e and e , implying thatthe action of σ on Π is given by α ( e + e + e + e ) = α + 3 α + 2 α + α α
7→ − α α
7→ − α α
7→ − α . Our next step will be to determine the coefficients c r so that σ does indeedgive rise to a Moufang set. We will first focus on the second conditionfor σ ; since this condition only concerns the subgroup L J ( K, L ) of F ( K, L ),we will achieve this by looking at the subgroup B ( K, L ) ≤ F ( K, L ) (thisis the subgroup of F ( K, L ) generated by the root groups U α ( L ), U α ( K )and U α ( K )). Once we will have constructed an involution σ such that thesecond condition is satisfied, we will see it is not very hard to check thatalso the third condition for σ holds.In the non-mixed case, we know that the action of σ on the B -buildinghas to be chosen in such a way that the group fixed under σ is isomorphic tothe projective orthogonal group of an anistropic quadratic form of dimension7 with trivial Hasse invariant; see [13, Section 3.4] for more details. One canshow that every such a quadratic form can be obtained as the restriction tothe trace zero part of an 8-dimensional norm form of an octonion divisionalgebra (i.e. of a 3-fold Pfister form). In a completely similar way, we obtainthat the fixed point set of the involution σ on the mixed B -subbuildinghas to be isomorphic to PGO ( q ) with q the trace zero part of the ‘mixed’norm form of an octonion division algebra. In the next subsection, wedetermine explicitly what this action should be and deduce in this way thecoefficients c r . B -subbuilding As we have seen in Example 3.1, we can identify B ( K, L ) with the group
PGO ( R ) where R is the mixed quadratic form R : L ⊕ K → K ; ( x , x , x − , x , x − , x , x − ) x + x x − + x x − + x x − with respect to a well chosen hyperbolic basis C ; this group consists of all( K, L )-linear maps ϕ (modulo scalars) such that R ( ϕ ( v )) = R ( v ) for all v ∈ L ⊕ K . There is a bijective correspondence between ( K, L )-linear maps17 and the invertible 7 by 7 matrices A such that the first row consistsof elements in L , while the others consist of elements in K and all theelements in the first column, except the first one, are zero. The condition R ( ϕ ( v )) = R ( v ) translates into [ R ] t A [ R ] = A where [ R ] is the matrix corresponding to the quadratic form R .First, we will determine the isomorphism between B ( K, L ) and
PGO ( R )explicitly, because this will allow us to describe the action of σ on the B -subbuilding entirely in terms of matrices. B ( K, L ) and PGO ( R )We use the correspondence between B ( L ) and PGO ( ˜ R ) (with ˜ R the uniqueextension of R to a quadratic form on L ) mentioned in [6, Section 11.3] todetermine a matrix representation for the elements of B ( K, L ). Thereforewe return to the original definition of B ( L ) as the group generated by someautomorphisms on L ⊗ L Z . We know L := L C has a Cartan decomposition L = H ⊕ M r ∈ Φ B L r = H ⊕ M C e r , where e r runs through the list ([6, p. 180]) E i,j − E − j, − i E i, − j − E j, − i E i, − E , − i − E − i, − j + E j,i − E − i,j + E − j,i − E − i, + E ,i for 0 < i < j ≤
3. The matrices E i,j are the 7 by 7 matrices with a 1 on the( i, j )-th position, with rows and columns indexed by { , , − , , − , , − } .Next, we want to find an explicit correspondence between the roots ofΦ B and the root spaces C e r of L . Therefore, it suffices to find an identifica-tion between the fundamental roots of H ∗ and those of Φ B . According to [6,p. 180], the elements of H are of the form diag(0 , λ , − λ , λ , − λ , λ , − λ ), λ i ∈ C . We find a fundamental system { ˜ α , ˜ α , ˜ α } for H ∗ with˜ α : H → C ; diag(0 , λ , − λ , λ , − λ , λ , − λ ) λ , ˜ α : H → C ; diag(0 , λ , − λ , λ , − λ , λ , − λ ) λ − λ , ˜ α : H → C ; diag(0 , λ , − λ , λ , − λ , λ , − λ ) λ − λ . With the use of equation (2.1), we obtain that the elements ˜ α , ˜ α , ˜ α of H ∗ correspond to the elements 2 E , − E , − , E , − E − , − and E , − E − , − of We will always use left multiplication by matrices on column spaces. When we talk about the matrix of a quadratic form, we mean the unique upper-triangular matrix representing this quadratic form w.r.t. the given basis. H ∗ and those of Φ B is˜ α ↔ α = e ˜ α ↔ α = e − e ˜ α ↔ α = e − e . We can now identify the elements u r ( t ) with matrices. This can be doneusing the epimorphism G → L ( L ) : exp( te r ) u r ( t ) , with exp( te r ) being the matrices described on [6, p.183] and G being thegroup generated by all these matrices. The kernel of this map turns out tobe the center of G .In this way we can identify the following elements for all i, j ∈ { , . . . , } : u e i − e j ( λ ) ↔ I + λ ( E i,j + E − j, − i ) u e i + e j ( λ ) ↔ I + λ ( E i, − j + E j, − i ) u − e i − e j ( λ ) ↔ I + λ ( E − i,j + E − j,i ) u e i ( λ ) ↔ I + λE , − i + λ E i, − i u − e i ( λ ) ↔ I + λE ,i + λ E − i,i . for all λ ∈ L , where I is the 7 by 7 identity matrix. σ on PGO ( R )As mentioned in the previous subsection, we wish to construct σ in such away that the fixed points form a group isomorphic to PGO ( q ) with q , thetrace zero part of a mixed norm form of an octonion division algebra. Sucha quadratic form is defined over the fields k and ℓ and is of the formN := N L ⊥ α N K ⊥ β N K ⊥ αβ N K : L ⊕ K ⊕ K ⊕ K → ℓ ;( y , y , y , y ) y y + αy y + βy y + αβy y , where α, β are constants in k × . Remark 5.1.
Denote by N ℓ the extension of N to the octonion algebra O ℓ = L ⊕ L ⊕ L ⊕ L . Although the norm on O ℓ is uniquely determined,there is no canonical way to define the product of two octonions (in terms ofthe decomposition O ℓ = L ), although all of these algebras are isomorphic.The most common way to define such a multiplication uses the fact that19very composition algebra of dimension d > d/ x = ( x , x , x , x ) and y = ( y , y , y , y ) be two arbitrary elementsof O ℓ , then we define the product x · y to be equal to (cid:0) x y + αx y + βx y + αβx y , x y + x y + βx y + βx y ,x y + x y + αx y + αx y , x y + x y + x y + x y (cid:1) , and we define the conjugate x to be equal to x = (cid:0) x , x , x , x (cid:1) ;this makes O ℓ = L into an octonion algebra with norm N ℓ , and N ℓ ( x ) = x · x = x · x for all x ∈ O ℓ .The restriction q of N to the subspace of trace zero elements is then ofthe following form: q : ℓ ⊕ K ⊕ K ⊕ K → k ; ( y , y , y , y ) y + αy y + βy y + αβy y . We will now show how to construct such an involution σ .Viewing the mixed quadratic form q as a form over ( K, L ) in the obviousway (we denote this extended form by Q := q ⊗ k,ℓ ( K, L )), the quadraticforms Q and R are isometric. Indeed, looking at the matrix representations[ Q ] B and [ R ] C with respect to the standard bases B and C we see that[ Q ] B = α α αδ β β βδ αβ αβ αβδ and [ R ] C = . A change of the base of B using the transition matrix S := γα αγ γβ βγ γαβ αβγ S t [ R ] C S is not equal to the matrix[ Q ] B , but it represents the same quadratic form.)It follows that PGO ( Q ) S − = PGO ( R ), and since PGO ( q ) is equal to { A ∈ PGO ( Q ) | A = A } we obtain PGO ( q ) S − = { SAS − | A ∈ PGO ( Q ) and A = A } = { B ∈ PGO ( R ) | S − BS = S − BS } = { B ∈ PGO ( R ) | B = ( SS − ) B ( SS − ) } = { B ∈ PGO ( R ) | B = M − BM } with M = SS − = α − α β − β α − β − αβ . We conclude that we can describe the restriction of the involution σ (using the isomorphism between PGO ( R ) and B ( K, L )) as σ | B K,L ) : PGO ( R ) → PGO ( R ); x M − xM. (5.1)We will from now on identify PGO ( R ) and B ( K, L ) without explicitly men-tioning the isomorphism. c r Using the identification between
PGO ( R ) and B ( K, L ), we can now writethe involution σ as σ ( u r ( t )) = M − u r ( t ) M = u σ ( r ) ( c r t ) . for all r ∈ Φ B . Using (5.1) we find for the generators α , α and α of Φ B that c α = αβ , c α = α − and c α = αβ − .It remains to determine the coefficient c α because then all c r follow usingthe Chevalley commutator relations. Since the anisotropic subbuilding is ofthe right form, the only thing we still have to express is that σ should be aninvolution. This is fulfilled if c α c σ ( α ) = 1.We would like to deduce all coefficients c r with r ∈ Φ + arbitrary andconsequently c σ ( α ) . By applying σ on the non-trivial relations from (3.3),21e see that c r c s = c r + s when r, s ∈ Φ s and r + s ∈ Φ s ,c r c s = c r + s when r, s ∈ Φ ℓ and r + s ∈ Φ ℓ ,c r c s = c r + s and c r c s = c r + s when r ∈ Φ s , s ∈ Φ ℓ and r + s ∈ Φ s . Now let r be any positive root, and write r = P i λ i α i , with all λ i positiveintegers. By [6, Lemma 3.6.2] (or [4, Chapter VI, section 1.6, Proposi-tion 19]), we can obtain r by adding one fundamental root α i at a time,and therefore we inductively obtain c r = Q c λ i α i . When we apply this on σ ( α ) = α + 3 α + 2 α + α , we find that c α c σ ( α ) = 1 for c α = α − β − .The other coefficients (belonging to negative roots) can be found usingthe relation c r c σ ( r ) = 1 for every r ∈ Φ. As before, this relation follows fromthe fact that σ is an involution. We show in this section that G = h U , V i with U := U J ∩ Fix( σ ) V := U − J ∩ Fix( σ )has a split saturated BN-pair of rank one.We verify that σ satisfies condition (2) on page 16: Lemma 5.2.
No parabolic subgroups of L J ( K, L ) are fixed.Proof. We prove that no parabolic subgroups of B ( K, L ) are fixed. Thisis enough since if L J ( K, L ) has a parabolic subgroup P fixed by σ , then P ∩ B ( K, L ) is a fixed parabolic subgroup of B ( K, L ).We have a closer look at the building corresponding to a general group G with BN-pair ( B, N ). By [1, Section 6.2.6], the parabolic subgroups of G ordered by the opposite of the inclusion relation form the simplicial complexof the building. In particular, the chambers (i.e. maximal flags) correspondto conjugates under G of B . Also, the group B is exactly the stabilizer in G of the chamber corresponding to B . Furthermore, the parabolic subgroups P J = BN J B are exactly the stabilizers in G of their corresponding flags.The group B ( K, L ) has a BN-pair ( B ( K, L ) ∩ B ( K, L ) , B ( K, L ) ∩ N ( K, L )). Since
PGO ( R ) is isomorphic to B ( K, L ), we know from thetheory of buildings that the building we obtain is the mixed quadric corre-sponding to R . More specifically, the quadric corresponding to R has points,lines and planes (since R has Witt index 3). The flags of this quadricare then exactly the flags of the building of PGO ( R ). So conjugates of22 ( K, L ) ∩ B ( K, L ) correspond to triples ( p, L, π ) with p a point of L and L a line on the plane π , while maximal parabolic subgroups correspond topoints, lines or planes of the mixed quadric. From now on, we will assumethat ( p, L, π ) is the maximal flag corresponding to the standard minimalparabolic subgroup B ( K, L ) ∩ B ( K, L ).Let C = ( x, x , y , x , y , x , y ) be a hyperbolic basis for the mixedquadratic form R , i.e. a basis for the K -vectorspace L ⊕ K such that R ( x ) = 1 , h x, x i = h x, x i i = h x, y i i = h x i , x j i = h y i , y j i = 0 , h x i , y j i = δ ij , where h· , ·i is the bilinear form corresponding to R .We observe that ( h x i , h x , x i , h x , x , x i ) forms a chamber in the build-ing of PGO ( R ). We claim that this chamber is precisely the chamber ( p, L, π )corresponding to the standard minimal parabolic B ( K, L ) ∩ B ( K, L ) underthe isomorphism between B ( K, L ) and the matrix group corresponding to
PGO ( R ) constructed in section 5.2.1. To prove our claim, we have to showthat all generators of B ( K, L ) ∩ B ( K, L ) fix the subspaces h x i , h x , x i and h x , x , x i . Consider the generators of the form x e − e ( t ) with t ∈ K . Theseelements correspond to matrices A = I + t ( E , + E − , − ), so we need tocheck that A (0 , λ , , , , , t ∈ h x i A (0 , λ , , λ , , , t ∈ h x , x i A (0 , λ , , λ , , λ , t ∈ h x , x , x i for all λ , λ , λ ∈ K , which is easily verified. The other generators can betreated similarly, and this proves our claim.Now suppose that a parabolic subgroup S of B ( K, L ) is fixed by σ ;our goal is to derive a contradiction. As σ is type-preserving, we knowthat if a flag is fixed, then certainly a point, line or plane must be fixed.As all parabolic subgroups are conjugate, there is some g ∈ B ( K, L ) suchthat S = P g , where P is a standard parabolic subgroup, i.e. P contains( B ( K, L ) ∩ B ( K, L )). So S corresponds to a flag contained in the chamber( g ( p ) , g ( L ) , g ( π )), and hence one of the subspaces h g ( x ) i , h g ( x ) , g ( x ) i or h g ( x ) , g ( x ) , g ( x ) i has to be fixed under σ .We claim that the involution σ maps g ( p ), g ( L ) and g ( π ) to h σ ( g )( y ) i , h σ ( g )( y ) , σ ( g )( y ) i and h σ ( g )( y ) , σ ( g )( y ) , σ ( g )( y ) i , respectively. Fromthis we deduce that in each of the 3 cases (fixed point, line or plane), σ ( g )( y ) ∈ h g ( x ) , g ( x ) , g ( x ) i . In particular, h g ( x ) i ⊥ h σ ( g )( y ) i .In order to prove our claim, we need to show that σ ( B g ), where B g isthe stabilizer of the flag (cid:0) g ( h x i ) , g ( h x , x i ) , g ( h x , x , x i ) (cid:1) , (cid:0) h σ ( g )( y ) i , h σ ( g )( y ) , σ ( g )( y ) i , h σ ( g )( y ) , σ ( g )( y ) , σ ( g )( y ) i (cid:1) . This is equivalent to showing that σ ( B ) is the stabilizer of the flag (cid:0) h y i , h y , y i , h y , y , y i (cid:1) . This last statement can again easily be checked on each of the generatorsof B , and this proves our claim.Suppose next that g ( x ) = ( z, z , a , z , a , z , a ), then z + P i z i a i = 0since R ( g ( x )) = R ( x ) = 0. Notice that g ( x ) is the second column ofthe matrix corresponding to g and that σ ( g )( y ) = M − gM ( y ) is the thirdcolumn of the matrix M − gM ; this implies, using the explicit descriptionof the matrix M , that σ ( g )( y ) = α − (cid:0) z, α − a , αz , β − a , βz , α − β − a , αβz (cid:1) . Since h g ( x ) i ⊥ h σ ( g )( y ) i , we get αz z + α − a a + βz z + β − a a + αβz z + α − β − a a = 0 . This is equivalent with( z + z ) + α ( z + α − a )( z + α − a ) + β ( z + β − a )( z + β − a )+ αβ ( z + α − β − a )( z + α − β − a ) = 0 . Since q is anistropic, this implies a = αz , a = βz and a = αβz . Finally,by expressing again that R ( g ( x )) = 0, we obtain that z + αz z + βz z + αβz z = 0 , and hence z = 0 and z i = 0 for all i , a contradiction. We conclude that noparabolic subgroup of B ( K, L ) is fixed.To proceed, we assemble a few lemmas about mixed BN-pairs. We write W for the Weyl group N ( K, L ) /T ( K, L ), which is isomorphic to the Weylgroup corresponding to a root system of type F , and we use the notation C W ( σ ) for the centralizer in W of σ , where we identify σ with the corre-sponding element w J in the Weyl group W J generated by w α , w α and w α (see section 5.1). Lemma 5.3.
Let g ∈ F ( K, L ) such that σ ( g ) ∈ P J ( K, L ) gP J ( K, L ) . If P J ( K, L ) gP J ( K, L ) = P J ( K, L ) nP J ( K, L ) for some n ∈ N ( K, L ) corre-sponding to the shortest element w in W J wW J . Then w ∈ C W ( σ ) . roof. See [17, Lemma 2.4]. Although the proof is not stated for mixedChevalley groups, it can be copied almost verbatim, by replacing P J and H by P J ( K, L ) and T ( K, L ), respectively.The next lemma is a mixed version of [17, Lemma 2.5]. We notice thatin the proof of this lemma we need the assumption (2) made in Section 5.1,page 16, which we proved in Lemma 5.2 above.
Lemma 5.4.
Let g ∈ F ( K, L ) with g P J ( K, L ) = gP J ( K, L ) g − invariantunder σ . If P J ( K, L ) gP J ( K, L ) = P J ( K, L ) nP J ( K, L ) , with n ∈ N ( K, L ) such that the corresponding element w of W is the shortest element in W J wW J , then w ∈ C W ( σ ) ∩ Stab(Φ J ) .Proof. Let g = pnp ′ with n ∈ N ( K, L ) and p, p ′ ∈ P J ( K, L ). Let I := J ∩ w ( J ); then P I ( K, L ) = U J ( K, L )( P J ( K, L ) ∩ n P J ( K, L )) . Hence p P I ( K, L ) = U J ( K, L )( P J ( K, L ) ∩ g P J ( K, L )) is σ -invariant. Further-more, if p = lu with l ∈ L J ( K, L ) and u ∈ U J ( K, L ), then l ( L J ∩ P I ( K, L )) = L J ( K, L ) ∩ p P I ( K, L )is a parabolic subgroup of L J ( K, L ). By Lemma 5.2, L J ( K, L ) ∩ p P I ( K, L ) = L J ( K, L ). We conclude that P J ( K, L ) ⊆ P I ( K, L ), so J = I and therefore w ( J ) = J . Lemma 5.5.
Let = w ∈ W with w (Φ + J ) = Φ + J . Then w J w = w .Proof. See [17, Lemma 2.6].In the next paragraph, we will prove that B := P J ( K, L ) ∩ G , to-gether with a suitable N (which we will construct on the way) forms a splitsaturated BN-pair for G . We let H := L J ( K, L ) ∩ G .We use the proof of [17, Lemma 2.7] to construct an element ˜ n ∈ ( n L J ) ∩ G with n an arbitrary element of N ( K, L ) such that n T ( K, L ) = w , thelongest element in W = N ( K, L ) /T ( K, L ). Lemma 5.6.
Let n ∈ N ( K, L ) be such that n T ( K, L ) = w , then n e ∈ n L J ( K, L ) ∩ G .Proof. We notice that u − e (1) ∈ G ∩ U Φ − \ Φ − J . Furthermore, P J u − e (1) P J = P J n e P J with w e the shortest element in W J w e W J . Lemma 5.4 showsthat w e ∈ C W ( σ ) ∩ Stab(Φ J ). This together with Lemma 5.5 allows us toconclude that w = w J w e , with w J being the longest element in Φ J .25ince W ∼ = N ( K, L ) /T ( K, L ), this yields that n e = n n J h for some h ∈ T ( K, L ), so n e ∈ n L J ( K, L ). Since n e = u e (1) u − e ( − u e (1), andeach of these three factors is fixed by σ , we also have n e ∈ G , proving thelemma.Next, we define N := h n , L J ( K, L ) i ∩ G . This group is certainly non-trivial since n e ∈ N .Similarly as with ordinary Chevalley groups, one can associate to theroot sytem Φ and corresponding vector space V a new root system ˜Φ andvector space ˜ V using the action of σ on V . Indeed, define ˜ V as C V ( σ ) ∩ J ⊥ ,then for every v ∈ V , ˜ v denotes the orthogonal projection of v in ˜ V . Onecan prove that ˜Φ := { ˜ r | r ∈ Φ \ Φ J } forms a (not necessarily reduced) rootsystem of ˜ V . We denote the Weyl group corresponding to ˜Φ by ˜ W . Lemma 5.7.
Let n ∈ N , then there exists an element n ∈ N ( K, L ) suchthat n L J ( K, L ) = nL J ( K, L ) where nT ( K, L ) corresponds to the shortestelement w in wW J . Then w ∈ C W ( σ ) ∩ Stab(Φ J ) and w | ˜ V ∈ ˜ W . The map φ : N /H → ˜ W ; n H w | ˜ V is an isomorphism.Proof. Again, the proof of [17, Lemma 2.9] holds almost verbatim.In the current setting, we simply have ˜ V = C V ( σ ) = J ⊥ = R e . Theset Φ \ Φ J consists precisely of those roots that have a non-zero coefficientfor e , and we find that ˜Φ is a root system of type BC ; in particular, thecorresponding Weyl group ˜ W has order 2. From this, we can immediatelyconclude that N = h n e i H .Before we can proceed with the actual proof of the existence of a splitsaturated BN-pair, we formulate a last important theorem, again inspiredby [17]. Theorem 5.8.
Every g ∈ G \ B can be written as g = uln e u ′ with u ∈ U J ( K, L ) ∩ Fix( σ ) , u ′ ∈ U − w e ,J ∩ Fix( σ ) and l ∈ L J ( K, L ) ∩ Fix( σ ) .Proof. Using Lemma 5.4 and Lemma 5.7, the proof can be taken over from[17, Theorem 2.10].We now have enough information to prove the existence of a split satu-rated BN-pair. 26 heorem 5.9. G together with ( B , N ) forms a saturated, split BN-pairof rank one.Proof. We check that all five conditions are satisfied.(i) We show that G = h B , N i . This follows immediately from Theo-rem 5.8.(ii) We first prove that H = B ∩ N . It is easy to see that H ⊆ B ∩ N .If on the other hand n e h ∈ B for some h ∈ H , this would implythat n e ∈ P J ( K, L ), a contradiction. From this it follows immediatelythat H E N since | ˜ W | = 2 and therefore [ N : H ] = 2.(iii) The element ω := n e ∈ N \ H with n e = e such that N = h H , n e i . We also have G = B ∪ B n e B since N = n e H ∪ H and n e B n e = B because x − e (1) / ∈ B ⊆ P J .(iv) The group U E B since U J ( K, L ) ∩ Fix( σ ) E P J ( K, L ) ∩ G . As B ⊆ P J ( K, L ) with P J ( K, L ) = U J ( K, L ) ⋊ L J ( K, L ), we find thatevery b ∈ B can be written uniquely as b = ul for some u ∈ U J ( K, L )and l ∈ L J ( K, L ). Now σ fixes U J ( K, L ) and L J ( K, L ) which meansthat u, l have to fixed by σ as well, so we get that u ∈ U and l ∈ H .This shows that B = U ⋊ H .(v) Because H E N , we obtain that n e H n e = H , so H ⊆ B ∩ n e B n e . Remains to check the direction B ∩ ( n e B n e ) ⊆ H . Wehave that x ∈ B ∩ ( n e B n e ) ⊆ B so the only thing left to checkis that x belongs to N . Since x is in the intersection of the abovegroups, we can write x = b = n e b n e = n e uhn e = n e un e h ′ forcertain b , b ∈ B , u ∈ U and h, h ′ ∈ H . This implies b h ′− ∈ V ∩ B = { e } or b = h ′ ∈ H . F From Lemma 2.3 we find that the set X := { ( U ) g | g ∈ G } together withthe set of subgroups { ( U ) g | g ∈ G } acting on X by conjugation, forms aMoufang set M = ( X, ( V x ) x ∈ X ).In general, every Moufang set of the form M = ( X, ( U x ) x ∈ X ) can bewritten as M ( U, τ ) for some group U and some permutation τ on X usingLemma 2.2. We use this lemma to find a representation of our Moufang setin terms of a group ( U, +) and a permutation τ .We choose ∞ := U and 0 := ( U ) n e = V , and we define U := { ( U ) g | g ∈ G } \ { U } . Notice that every g ∈ G \ B can be written in a unique way as g = bn e u with b ∈ B and u ∈ U ; equivalently, for any two elements g = bn e u and27 ′ = b ′ n e u ′ with b, b ′ ∈ B , u, u ′ ∈ U , we have ( U ) g = ( U ) g ′ if and onlyif u = u ′ . Therefore, the map ζ : U → U : u ( U ) n e u is a bijection. In particular, this makes U into a group which is isomorphicto U . Finally, we can set τ := n e , which acts on the set U ∗ of non-trivialelements of U by conjugation.We conclude that the corresponding Moufang set M is given by M = M ( U, n e ). In section 6.1, we will determine the group U ; in section 6.2, wewill determine the action of τ on U ∗ . U We determine what an arbitrary element of U looks like, and we will de-scribe the group structure of U ∼ = U .Since U = U J ∩ Fix( σ ), we find U := U r U r · · · U r ∩ Fix F ( K,L ) ( σ ) , with r = e r = e + e r = − e + e r = e + e r = − e + e r = e + e r = − e + e r = ( e + e − e + e ) r = ( − e − e + e + e ) r = ( e − e + e + e ) r = ( − e + e − e + e ) r = ( − e + e + e + e ) r = ( e − e + e + e ) r = ( − e − e − e + e ) r = ( e + e + e + e ) ,U r = { u r ( t ) | t ∈ L } if r ∈ Φ s and U r = { u r ( t ) | t ∈ K } if r ∈ Φ l .This implies that an arbitrary element x of U is of the form u r ( t ) u r ( t ′ ) u r ( t ′ ) · · · u r ( t ′ ) u r ( t ) u r ( t ) · · · u r ( t )with t i ∈ L and t ′ j ∈ K for all i, j and satisfies the relation x = σ ( x ).After rearranging some factors, using the commutator relations in (3.3),we find that σ ( x ) = u r ( c t + t t + t t + t t + t t ) u r ( c t ′ ) · · · u r ( c t )We now determine the values for each c r i , using the already known values28or c α , . . . , c α from paragraph 5.2.3 and the product formula. We get c r = 1 c r = α c r = α − c r = β c r = β − c r = αβ c r = α − β − c r = 1 c r = 1 c r = α c r = α − c r = β c r = β − c r = α − β − c r = αβ. and so the relation x = σ ( x ) implies the following relations: t + t + t · t + t · t + t · t + t · t = 0 ,t ′ = αt ′ , t ′ = βt ′ , t ′ = αβt ′ ,t = t , t = αt , t = βt , t = αβt . Replacing t ′ , t ′ , t ′ , t , t , t and t shows that an arbitrary element x of U is therefore of the form u r ( t ) u r ( t ′ ) u r ( αt ′ ) · · · u r ( αβt ) u r ( t )with t + t + t t + αt t + βt t + αβt t = 0.Now let x, y ∈ U be arbitrary, and write x = u r ( t ) u r ( t ′ ) u r ( t ′ ) · · · u r ( t ′ ) u r ( t ) u r ( t ) · · · u r ( t ) ,y = u r ( s ) u r ( s ′ ) u r ( s ′ ) · · · u r ( s ′ ) u r ( s ) u r ( s ) · · · u r ( s );then the product x · y is equal to u r ( t + s + t s + αt s + βt s + αβt s ) · u r ( t ′ + s ′ ) · u r ( t ′ + s ′ ) · · · u r ( t + s ) . To simplify the notation, we will identify an arbitrary element x = u r ( t ) u r ( t ′ ) u r ( t ′ ) · · · u r ( t ′ ) u r ( t ) u r ( t ) · · · u r ( t )with the element(( t , t , t , t ) , ( t , t ′ , t ′ , t ′ )) ∈ ( L ⊕ L ⊕ L ⊕ L ) ⊕ ( L ⊕ K ⊕ K ⊕ K ) . We conclude that U is isomorphic to the group ( U, +) with U := (cid:8) (( x , x , x , x ) , ( y , y , y , y )) ∈ ( L ) ⊕ ( L ⊕ K ) | x x + αx x + βx x + αβx x + T( y ) = 0 (cid:9) , (6.1)where the group addition + is given by the formula (cid:0) ( x , x , x , x ) , ( y , y , y , y ) (cid:1) + (cid:0) ( a , a , a , a ) , ( b , b , b , b ) (cid:1) = (cid:0) ( x + a , x + a , x + a , x + a ) , ( y + b + x a + αx a + βx a + αβx a , y + b , y + b , y + b ) (cid:1) . .2 Description of the action of τ on U ∗ We present a way to calculate the image of τ = n e on an arbitrary non-trivial element u ∈ U . Using the isomorphism with U , we need to determinewhich element of U corresponds to the element ( U ) n e un e of U . As men-tioned in the previous section, this comes down to rewriting an arbitraryelement of the form g = n e un e with u ∈ U as g = bn e u ′ with b ∈ B and u ′ ∈ U . We have shown in the previous section that this can be done in aunique way and we therefore have τ ( u ) = u ′ .Bringing such an arbitrary element n e un e into the right form comesdown to quite long, but systematic calculations, as we will explain below.We implemented our algorithm in the computer algebra software package Sage [16]. We refer to [5] for the detailed implementation and the output ofthe program.We briefly describe the methods we use to rewrite n e un e as bn e u ′ forsome b ∈ B and u ′ ∈ U . Suppose u = x r ( t ) x r ( t ) · · · x r ( t ) x r ( t ),then using the relations n s x r ( t ) n s = x w s ( r ) ( t ) and n r ( t ) = h r ( t ) n r for all r, s ∈ Φ and all t ∈ L , we find that n e x r ( t ) x r ( t ) x r ( t ) · · · x r ( t ) x r ( t ) n e = x − r ( t ) x − r ( t ) x − r ( t ) · · · x − r ( t ) x − r ( t )or that this element has the same action on U as the element x = n e ( t ) x r ( t − ) x − r ( t ) x − r ( t ) · · · x − r ( t ) x − r ( t ) , provided t = 0. Remark 6.1. If t = 0, then the norm condition implies that t , . . . , t are also equal to zero. Again, we have to distinguish between t = 0 and t = 0 to proceed, and a similar distinction has to be made for t and t ,but in each case, the process is very similar (but gets easier as more andmore elements become zero), and we omit the details.We now describe our algorithm to rewrite x in the required form, i.e. inthe form x = bn e u ′ with b ∈ B and u ′ ∈ U . Our strategy is to try to swap all “bad rootelements”, i.e. all root elements x r ( t ) not belonging to U , and all “Huaelements” h r ( t ), in x , from the right side of n e to the left, in such a mannerthat the elements that we get at the left of n e all belong to P J . At the end,we will then indeed have rewritten x as bn e u ′ . As n e and u ′ are in G atthe end of the algorithm, so is b , and therefore b ∈ B as required.30 tep 1. We always start with the leftmost element on the right side of n e .This element will be of one of the following types:(1) a Hua element h r ( t ),(2) a root element x r ( t ), with r containing no term in e ,(3) a root element x r ( t ), with r containing a negative term in e ,(4) a root element x r ( t ), with r containing a positive term in e .We point out that elements of the the first three types can be swapped tothe left side of n e without any problem. Only if we encounter an element x r ( t ) = x r i ( t ) of the fourth type, we have a look at the element next to x r i ( t ). Step 2.
Depending on the form of this second element, we can distinguisha few cases.(a) This second element is a Hua element h s ( t ′ ). In this case, we can usethe conjugation relation x r ( t ) h s ( t ′ ) = h s ( t ′ ) x r (cid:0) tλ − h r,s i / h s,s i (cid:1) to reverse the order of x r ( t ) and h s ( t ′ ) in the product.(b) This second element is of the form x r ( t ′ ). In this case we simply combineboth elements to the single root element x r ( t + t ′ ).(c) This second element is of the form x s ( t ′ ) with either(i) s = r j , with j > i , or(ii) s = r j , with j < i , or(iii) s contains no positive term in e .In case (i), there is nothing to do, and we proceed to the next elementin the product, i.e. the element x r j ( t ′ ) now plays the role of x r ( t ), andwe apply Step 2 on this element. In cases (ii) and (iii), we distinguishbetween the case s = − r (i.e. s and r are opposite roots) and s = − r . If s = − r , we use the commutator relations to switch the roots x r ( t ) and x s ( t ′ ), and we add the possible new element(s) to the right of x s ( t ′ ) x r ( t ).If on the other hand s = − r , we use the equality x r ( t ) x − r ( t ′ ) = x − r (cid:16) t ′ tt ′ + 1 (cid:17) h r ( tt ′ + 1) x r (cid:16) ttt ′ + 1 (cid:17) (when tt ′ = −
1) to proceed. In both cases, we then return to step 1.By repeating these steps, we end up with an element in U on the right sideof n e , as we wanted.Applying the algorithm on an arbitrary element of U we get, using our Sage program [5], that the corresponding map τ : U → U (which maps u to u ′ as explained in the beginning of this section), expressed in terms of31he isomorphic group ( U, +) as in (6.1) on page 29, is explicitly given by τ : U → U :( a, b ) (cid:0) a · (cid:0) b + f ( a ) (cid:1) − , (cid:0) b + f ( a ) (cid:1) − + f (cid:0) a · ( b + f ( a )) − (cid:1)(cid:1) , where f : L ⊕ L ⊕ L ⊕ L → L ⊕ L ⊕ L ⊕ L :( a , a , a , a ) ( a a + αa a + βa a + αβa a ,a a + βa a , a a + αa a , a a + a a ) , and where the multiplication of the elements in L is the octonion multipli-cation described in Remark 5.1. We now summarize our results.
Theorem 6.2.
Let ( k, ℓ ) be a pair of fields of characteristic such that ℓ ≤ k ≤ ℓ . Let O be an octonion division algebra over k , with norm N , andlet O ℓ = O ⊗ k ℓ , with norm N ℓ . Let K and L be separable quadratic fieldextensions of k and ℓ , respectively, such that L ≤ K ≤ L ≤ O ℓ , and identify O ℓ with L . Under this identification, we define a subspace O mixed := L ⊕ K of O ℓ . Assume that the restriction of N ℓ to O mixed is anisotropic.There exist constants α, β ∈ k × such that the norm N is given by N : O ℓ = L → K : ( a , a , a , a ) a a + αa a + βa a + αβa a . Let f : O ℓ → O ℓ : ( a , a , a , a ) (cid:0) N( a , a , a , a ) ,a a + βa a , a a + αa a , a a + a a (cid:1) , and let g : O ℓ × O ℓ → L : (cid:0) ( a , a , a , a ) , ( b , b , b , b ) (cid:1) a b + αa b + βa b + αβa b . Define U := (cid:8) ( a, b ) ∈ O ℓ ⊕ O mixed | N( a ) + T( b ) = 0 (cid:9) , and make U into a group by setting ( a, b ) + ( c, d ) := ( a + c, b + d + g ( a, c )) for all ( a, b ) , ( c, d ) ∈ U . Define a permutation τ on U ∗ by τ ( a, b ) := (cid:0) a · (cid:0) b + f ( a ) (cid:1) − , (cid:0) b + f ( a ) (cid:1) − + f (cid:0) a · ( b + f ( a )) − (cid:1)(cid:1) for all ( a, b ) ∈ U . Then M ( U, τ ) is a Moufang set corresponding to a mixedgroup of type F . Algebraic Moufang sets of type F in character-istic We conclude with having a closer look at what happens to the above resultswhen K = L , which is in fact the algebraic case in characteristic 2. Moreprecisely, we show that every Moufang set M ( U, τ ) we obtained in section 6in this case is isomorphic to an algebraic Moufang set of type F (as weexpect).To see this, we apply the transformation ϕ on U = (cid:8) ( a, b ) ∈ O ℓ ⊕ O ℓ | N( a ) + T( b ) = 0 (cid:9) with ϕ : U → U : ( a, b ) ( a, b + f ( a ))Then ϕ (( a, b ) + ( c, d )) = ϕ ( a, b ) ˜+ ϕ ( c, d ) with( x , y ) ˜+ ( x , y ) = ( x + x , y + y + x · x )for all ( x , y ) , ( x , y ) ∈ U . Furthermore, ϕ ( τ (( x, y )) = ˜ τ ( x, y ) for all( x, y ) ∈ U with ˜ τ : U ∗ → U ∗ : ( x, y ) ( x · y − , y − ) . We find that M ( U, τ ) is isomorphic to M ( U, ˜ τ ), which is indeed a Moufangset of type F ; see [9, Theorem 2.1]. Remark 7.1.
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Groups,combinatorics & geometry (Durham, 1990) , 249–286, London Math.Soc. Lecture Note Ser. , Cambridge Univ. Press, Cambridge, 1992.34 lizabeth CallensGhent University, Dept. of MathematicsKrijgslaan 281 (S22), B-9000 Gent, Belgium [email protected]
Tom De MedtsGhent University, Dept. of MathematicsKrijgslaan 281 (S22), B-9000 Gent, Belgium [email protected]@cage.UGent.be