Iwasawa decompositions of split Kac-Moody groups
aa r X i v : . [ m a t h . G R ] J un Iwasawa decompositions of split Kac–Moodygroups
Tom De Medts Ralf Gramlich ∗ Max HornOctober 28, 2018
In this article we characterize the fields over which connected split semisimple alge-braic groups and split Kac-Moody groups admit an Iwasawa decomposition.
The Iwasawa decomposition of a connected semisimple complex Lie group or a connected semisim-ple split real Lie group is one of the most fundamental observations of classical Lie theory. It impliesthat the geometry of a connected semisimple complex resp. split real Lie group G is controlledby any maximal compact subgroup K . Examples are Weyl’s unitarian trick in the representationtheory of Lie groups, or the transitive action of K on the Tits building G/B . In the case of the con-nected semisimple split real Lie group of type G the latter implies the existence of an interestingepimorphism from the real building of type G , the real split Cayley hexagon, onto the real build-ing of type A , the real projective plane, by means of the epimorphism SO ( R ) → SO ( R ) (see[20]). This epimorphism cannot be described using the group of type G , because it is quasisimple.To be able to transfer these ideas to a broader class of groups that includes the class of splitKac–Moody groups over arbitrary fields, we extend the notion of an Iwasawa decomposition inthe following way, which is inspired by the fact that a maximal compact subgroup of a semisimplesplit Lie group is centralized by an involution: Definition 1.1.
A group G with a twin BN -pair (cf. Definition 2.2) admits an Iwasawa decom-position , if there exist an involution θ ∈ Aut( G ) such that(i) B θ + = B − and(ii) G = G θ B + where G θ := Fix G ( θ ).Our interest in Iwasawa decompositions stems from geometric group theory: the group G θ actswith a fundamental domain on the flag complex of the building G/B + , which is simply connectedby [1, Theorem 4.127], [46, Theorem 13.32], if its rank is sufficiently large. Hence Tits’ Lemma[36, Lemma 5], [47, Corollary 1] yields a presentation of G θ by generators and relations.Since G θ need not be finitely generated, this presentation is usually formulated as a universalenveloping result of an amalgam. The following theorem specifies the presentation we have inmind. Refer to Section 2 for a definition of a root group datum and the construction of the twin BN -pair resulting from it. Theorem 1.
Let G be a group with a root group datum { U α } α ∈ Φ , and assume that G = G θ B + isan Iwasawa decomposition of G with respect to an involution θ . Furthermore, let Π be a systemof fundamental roots of Φ and, for { α, β } ⊆ Π , let X α,β := h U α , U − α , U β , U − β i . ∗ The second author gratefully acknowledges a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft. hen θ induces an involution on each X α,β , and G θ is the universal enveloping group of theamalgam (( X α,β ) θ ) { α,β }⊆ Π of the subgroups of the θ -fixed elements of the groups X α,β . The proof of Theorem 1 can be adapted as follows from what has been done in [18] and [21]for compact real forms of complex Lie groups and of complex Kac-Moody groups. The involution θ induces an involution of each group X α,β by Corollary 3.6. By the Iwasawa decompositionthe group G θ acts with a fundamental domain on the flag complex ∆ associated to the building G/B + . Choose F to be a fundamental domain of ∆ stabilized by the standard torus T := B + ∩ N of G arising from the positive BN -pair of G . The stabilizers of the simplices of F of dimensionzero and one with respect to the natural action of G on ∆ are exactly the groups ( X α ) θ T and( X αβ ) θ T , α, β ∈ Π. By the simple connectedness of building geometries of rank at least three (cf.[1, Theorem 4.127] or [46, Theorem 13.32]) plus Tits’ Lemma (cf. [36, Lemma 5], [47, Corollary1]) the group G θ equals the universal enveloping group of the amalgam (( X αβ ) θ T ) α,β ∈ Π . Finally,the torus T equals the universal enveloping group of the amalgam ( T αβ ) α,β ∈ Π , where T αβ denotesthe maximal torus T ∩ X αβ of X αβ , and so by [19, Lemma 29.3] the group G actually equals theuniversal enveloping group of the amalgam (( X αβ ) θ ) α,β ∈ Π . Remark 1.2.
For a group G with a 3-spherical root group datum and an involution θ of G that interchanges B + and B − , the local-to-global criterion proposed in [14] allows one to studyamalgam presentations of the subgroup G θ of θ -fixed elements even if G does not admit an Iwasawadecomposition with respect to θ . The main obstruction encountered when using that approach isestablishing that certain two-dimensional simplicial complexes are simply connected. We refer theinterested reader to [15] and [24] for an application of the criterion from [14].From Theorem 1 rises the question which groups actually admit an Iwasawa decomposition. Inthe literature one can find a lot of information on Iwasawa decompositions for prescribed groundfields, usually over the complex numbers C , the real numbers R , or real closed fields, cf. [4], [5],[25], [26], [31], [37]. In [27] on the other hand, no fixed ground field is chosen. Instead, it is shownthat, under the assumption that the ground field F is infinite and that each square is a fourthpower, the Iwasawa decomposition, the polar decomposition and the KAK -decomposition of thesubgroup of F -rational points of a connected reductive algebraic F -group are equivalent.Neither do we choose a fixed field in this paper. Instead, in Theorem 2 below we characterizethe fields F over which a group with an F -locally split root group datum (Definition 2.3) admitsan Iwasawa decomposition. We point out that this class of groups contains the class of groups of F -rational points of a connected split semisimple algebraic group defined over F (cf. [43]) and theclass of split Kac–Moody groups over F (cf. [39], [48]). Not every group that admits an Iwasawadecomposition admits a polar or a KAK -decomposition.For the next definition recall that any Cartan–Chevalley involution of (P)SL ( F ) is given, resp.induced by the transpose-inverse automorphism with respect to the choice of a basis of the naturalSL ( F )-module F . Definition 1.3.
Let F be a field, let σ be an automorphism of F of order 1 or 2, and let G be agroup with an F -locally split root group datum { U α } α ∈ Φ . We call an automorphism θ ∈ Aut( G )a σ -twisted Chevalley involution of G , if it satisfies the following for all α ∈ Φ:(i) θ = id,(ii) U θα = U − α , and(iii) θ ◦ σ induces a Cartan–Chevalley involution (resp. its image under the canonical projection)on X α := h U α , U − α i ∼ = (P)SL ( F ).A σ -twisted Chevalley involution of a split Kac-Moody group can be constructed by taking theproduct of a sign automorphism and the field automorphism σ , see [9, Section 8.2]. The sameis true for all split semisimple algebraic groups. Moreover, by Lemma 3.9 below a group witha 2-spherical F -locally split root group datum with | F | ≥ σ -twisted Chevalleyinvolution. Hence the following theorem applies to these three classes of groups.2 heorem 2. Let F be a field and let G be a group with an F -locally split root group datum. Thegroup G admits an Iwasawa decomposition if and only if F admits an automorphism σ of order or such that (i) − is not a norm, (ii) (a) if there exists a rank one subgroup h U α , U − α i of G isomorphic to SL ( F ) , then a sumof norms is a norm, or (b) if each rank one subgroup h U α , U − α i of G is isomorphic to PSL ( F ) , then a sum ofnorms is ± times a norm,with respect to the norm map N : F → Fix F ( σ ) : x xx σ , and (iii) G admits a σ -twisted Chevalley involution. Remark 1.4.
We emphasize that the above theorem does not state that each Iwasawa decompo-sition is realized by a σ -twisted Chevalley involution. For example, any involution σ of C yieldsan Iwasawa decomposition SL n ( C ) = SU n ( C , θ σ ) B + with respect to the σ -twisted Chevalley in-volution θ σ by the preceding theorem, because Fix C ( σ ) is real closed. Hence, given two distinctinvolutions σ , σ ∈ C , the involution ( θ σ , θ σ ) yields an Iwasawa decomposition of the productSL n ( C ) × SL n ( C ) although it is not a σ -twisted Chevalley involution.In a finite field F q of order q ≡ − § F q -locally split root group datum where allrank one subgroups are isomorphic to PSL ( F q ) admits an Iwasawa decomposition with respect tothe standard Chevalley involution. We point out that it follows from inspection of the Chevalleygroups of rank two (see [44]) that the diagram of a group with an F -locally split root groupdatum must be right angled (i.e., any edge of the diagram is labelled with infinity), if all itsrank one groups are isomorphic to PSL ( F ). Nontrivial such examples can be obtained using freeconstructions (see [8, Example 2.8], [40]).As mentioned before, among the most prominent groups that admit σ -twisted Chevalley invo-lutions are the connected split semisimple algebraic groups and the split Kac-Moody groups. Weexplicitly re-state our characterization given in Theorem 2 for these two classes of groups. Corollary 3.
Let F be a field and let G be a connected split semisimple algebraic group definedover F or a split Kac-Moody group over F . The group G ( F ) admits an Iwasawa decomposition G ( F ) = G θ ( F ) B ( F ) if and only if F admits an automorphism σ of order or such that (i) − is not a norm, (ii) (a) if there exists a rank one subgroup h U α , U − α i of G isomorphic to SL ( F ) , then a sumof norms is a norm, or (b) if each rank one subgroup h U α , U − α i of G is isomorphic to PSL ( F ) , then a sum ofnorms is ± times a norm,with respect to the norm map N : F → Fix F ( σ ) : x xx σ . We remark that Corollary 3 also holds for split reductive algebraic groups, which in the settingwe chose are excluded by axiom (RGD0) of Definition 2.3. By using a slightly more general notionof a group with a root group datum (such as in [49]) one can generalize Corollary 3 to the reductivecase. 3 n application to graph theory
Another consequence of Theorem 2 is a combinatorial local characterization of certain graphssimilar to the main result of [3]. Before we can state that result we have to introduce someadditional terminology.
Definition 1.5.
Let F be a field admitting an automorphism σ of order 1 or 2 such that(i) − N : F → Fix F ( σ ) : x xx σ . Then the pair ( F , σ ) is called an Iwasawa pair .Let F be a field with an automorphism σ of order 2 such that both ( F , σ ) and (Fix F ( σ ) , id)are Iwasawa pairs. Furthermore, let V be a six-dimensional F -vector space endowed with ananisotropic σ -hermitian sesquilinear form. Define S ( V ) to be the graph with the 2-dimensionalsubspaces of V as vertices and adjacency given by orthogonality. Because of this definition itmakes sense to use the symbol ⊥ for adjacency.There exists a well known group-theoretical characterization of S ( V ) as follows. Let G ∼ = SL ( F ),so that G has an F -locally split root group datum { U α } α ∈ Φ of type A . Let θ be the σ -twistedChevalley involution of the group G , let K := Fix G ( θ ) and, for α ∈ Φ, denote by ( X α ) θ the fixedpoint subgroup of the rank one group X α = h U α , U − α i . Then S ( V ) is isomorphic to the graph onthe K -conjugates of ( X α ) θ with the commutation relation as adjacency. This setup allows us todefine the graph S for any connected split semisimple algebraic F -group, so that for instance itmakes sense to use the symbol S ( E ( F )). In this notation, we have S ( V ) ∼ = S ( A ( F )).The proof of [3, Theorem 4.1.2], which deals with the special case of F = C and σ complexconjugation, applies verbatim to the setting just introduced, so that we obtain the followingresult. Recall that, if ∆ is a graph, one says that a graph Γ is locally ∆, if for each vertex x ∈ Γthe subgraph of Γ induced on the neighbors of x is isomorphic to ∆. Theorem 4.
Let F be a field admitting an automorphism σ of order such that both ( F , σ ) and (Fix F ( σ ) , id) are Iwasawa pairs. Assume that N ( F ) is a subset of the squares of Fix F ( σ ) , where N : F → Fix F ( σ ) : x xx σ . Let Γ be a connected locally S ( A ( F )) graph satisfying that, for everychain x ⊥ w ⊥ y ⊥ z ⊥ x ⊥ y consisting of four distinct vertices of Γ , the vertices w , x , z have y as unique common neighbor if and only if the vertices w , y , z have x as unique common neighbor.Then Γ is a quotient of S ( A ( F )) or of S ( E ( F )) . An application to group theory
Finally, as explained in [3, Section 4.6], Theorem 4 implies the following group-theoretic statementvia a standard argument. All twisted groups are understood to be anisotropic forms of connectedsplit semisimple algebraic groups with respect to the Chevalley involution composed with theindicated field automorphism.
Theorem 5.
Let F be a field admitting an automorphism σ of order such that both ( F , σ ) and (Fix F ( σ ) , id) are Iwasawa pairs. Assume that N ( F ) is a subset of the squares of Fix F ( σ ) , where N : F → Fix F ( σ ) : x xx σ . Moreover, let G be a group containing an involution x and a subgroup K E C G ( x ) such that (i) K ∼ = SU ( F , θ σ ) ; (ii) C G ( K ) contains a subgroup X ∼ = SU ( F , θ σ ) with x = Z ( X ) ; (iii) there exists an involution g ∈ G such that Y := gXg is contained in K ; (iv) if V is a natural module for K , then the commutator [ Y, V ] := { yv − v ∈ V | y ∈ Y, v ∈ V } has F -dimension ; G = h K, gKg i ; moreover, there exists z ∈ K ∩ gKg which is a gKg -conjugate of x and a K -conjugate of gxg .Then G/Z ( G ) ∼ = PSU ( F , θ σ ) or G/Z ( G ) ∼ = E ( F , θ σ ) . Organization of the article
In Section 2 we quickly recall the definitions of a (twin) BN -pair and a root group datum. InSection 3 we collect basic facts about flips of groups with a root group datum and provide a local-to-global argument that reduces the proof of Theorem 2 to the study of flips of rank one groups.In Section 4 we thoroughly study flips of (P)SL ( F ) and in Section 5 we use Moufang sets in orderto generalize some of our results for (P)SL ( F ) to arbitrary rank one groups. In particular, westudy flips of (P)SL ( D ), where D is an arbitrary division ring. Acknowledgements
The authors thank Bernhard M¨uhlherr for several discussions on the topic of this article. More-over, the authors express their gratitude to Andreas Mars for a careful proof-reading and helpfulcomments on an earlier version of this paper. They also thank the referee for further remarks,comments and suggestions. BN -pairs and root group data Definition 2.1.
We call the tuple (
G, B, N, S ) consisting of a group G with subgroups B and N and of a subset S of the coset space N/ ( B ∩ N ), a Tits system or a BN -pair if the followingconditions are satisfied:(i) G = h B, N i ;(ii) T := B ∩ N is normal in N ;(iii) the elements of S have order 2 and generate the group W := N/T , called
Weyl group ;(iv)
BwBsB ⊂ BwsB ∪ BwB for all w ∈ W , s ∈ S ;(v) sBs B for all s ∈ S . Definition 2.2.
Let (
G, B + , N, S ) and ( G, B − , N, S ) be Tits systems such that B + ∩ N = B − ∩ N ,i.e., with equal Weyl group W . Then ( G, B + , B − , N, S ) is called a twin Tits system or twin BN -pair if the following conditions are satisfied, cf. [49]:(i) B ε wB − ε sB − ε = B ε wsB − ε for ε = ± and all w ∈ W , s ∈ S such that l ( ws ) < l ( w );(ii) B + s ∩ B − = ∅ for all s ∈ S .The conjugates of B + and B − in G are called the Borel subgroups of G .The twin BN -pair is called saturated , if B + ∩ B − = B + ∩ N = B − ∩ N . From a geometric point ofview there is no loss in restricting one’s attention to saturated twin BN -pairs by [1, Lemma 6.85].A group G admitting a BN -pair satisfies G = F w ∈ W BwB , the
Bruhat decomposition of G . Foreach s ∈ S the set P s := B ∪ BsB is a subgroup of G . A Tits system ( G, B, N, S ) leads to a building whose set of chambers equals
G/B and whose distance function δ : G/B × G/B → W isgiven by δ ( gB, hB ) = w if and only if Bh − gB = BwB .A group G with a twin BN -pair hence yields two buildings G/B + and G/B − with distancefunctions δ + and δ − . Furthermore, it admits the Birkhoff decomposition G = F w ∈ W B + wB − fromwhich we can define the codistance function δ ∗ : ( G/B − × G/B + ) ∪ ( G/B + × G/B − ) → W via5 ∗ ( gB − , hB + ) = w if and only if B + h − gB − = B + wB − and δ ∗ ( hB + , gB − ) := ( δ ∗ ( gB − , hB + )) − .The tuple (( G/B + , δ + ) , ( G/B − , δ − ) , δ ∗ ) is called the twin building of G .In the present paper we are only interested in (twin) buildings coming from a group with a(twin) BN -pair. For detailed treatments of the theory of buildings we refer to [1], [7], [42], [46],[50]. Twin buildings are treated in [39], [49].Let ( W, S ) be a Coxeter system and let Φ be the set of its roots. Following [9] a root is a setof the form wα s , where w ∈ W and α s = { x ∈ W | l ( sx ) = l ( x ) + 1 } with length function l of W with respect to the generating set S . Moreover, let Π be a system of fundamental roots of Φ and,for ε = ± , let Φ ε denote the set of positive, resp. negative roots of Φ with respect to Π. For aroot α ∈ Φ, denote by s α the reflection of W which permutes α and − α . For each w ∈ W , defineΦ w := { α ∈ Φ + | wα ∈ Φ − } . A pair { α, β } of roots is called prenilpotent if α ∩ β and ( − α ) ∩ ( − β )are both non-empty. In that case denote by [ α, β ] the set of all roots γ of Φ such that α ∩ β ⊆ γ and ( − α ) ∩ ( − β ) ⊆ − γ , and set ] α, β [ := [ α, β ] \ { α, β } .The following definition of a root group datum is taken from [8]. Definition 2.3. A root group datum of type ( W, S ) for a group G is a family { U α } α ∈ Φ of subgroups(the root subgroups ) of G satisfying the following axioms, where U + := h U α | α ∈ Φ + i and U − := h U α | α ∈ Φ − i : (RGD0) For each α ∈ Φ, we have U α = 1; moreover, G = h U α | α ∈ Φ i . (RGD1) For each α ∈ Π we have U α U − . (RGD2) For each α ∈ Π and u ∈ U α \ { } , there exist elements u ′ , u ′′ of U − α such that theproduct µ ( u ) := u ′ uu ′′ conjugates U β onto U s α ( β ) for each β ∈ Φ. (RGD3) For each prenilpotent pair { α, β } ⊂ Φ, we have [ U α , U β ] ⊂ h U γ | γ ∈ ] α, β [ i . (RGD4) For each α ∈ Π there exists α ′ ∈ Φ s α such that U β ⊆ U α ′ for each β ∈ Φ s α .Defining T := h µ ( u ) µ ( v ) | u, v ∈ U α \{ } , α ∈ Π i ,N := h µ ( u ) | u ∈ U α \{ } , α ∈ Π i ,B + := T.U + ,B − := T.U − , we obtain a twin BN -pair of G , which by the above leads to a twin building on which G acts. Weset X α := h U α , U − α i and X α,β := h X α , X β i . A root group datum is called locally split if the group T is abelian and if for each α ∈ Φ there is a field F α such that the root group datum { U α , U − α } of X α of type A is isomorphic to the natural root group datum of SL ( F α ) or PSL ( F α ). A locallysplit root group datum is called F -locally split if F α ∼ = F for all α ∈ Φ. We start this section by discussing very basic properties of flips of twin buildings and of groupswith a twin BN -pair. For a more thorough treatment beyond what is needed in the present articlewe refer the reader to [23]. Definition 3.1.
Let G be a group with a saturated twin BN -pair B + , B − , N . An automorphism θ of G is called a BN -flip , if the following holds:(i) θ = id;(ii) B θ + = B − ; 6iii) θ induces a trivial action on the Weyl group N/T . Remark 3.2.
We stress that the key issue in item (iii) of the preceding definition is not that θ acts on the Weyl group, but that it centralizes it; cf. [23, Proposition 3.1, Remark 3.3].We now give the definition of a flip of a twin building and describe the correspondence betweena BN -flip and a flip of the twin building induced by the twin BN -pair. We refer to [23] and [28]for a more general treatment of flips. Originally, the concept of a flip of a twin building has beenintroduced in [6]. Definition 3.3.
Let B = ( B + , B − , δ ∗ ) = (( C + , δ + ) , ( C − , δ − ) , δ ∗ ) be a twin building. A buildingflip is an involutory permutation θ of C + ∪ C − with the following properties:(i) C θ + = C − ;(ii) θ flips the distances, i.e., for ε = ± and for all x, y ∈ C ε we have δ ε ( x, y ) = δ − ε ( x θ , y θ ); and(iii) θ preserves the codistance, i.e., for ε = ± and for all x ∈ C ε , y ∈ C − ε we have δ ∗ ( x, y ) = δ ∗ ( x θ , y θ ).If, additionally,(iv) there exists a chamber c ∈ C ± such that δ ∗ ( c, c θ ) = 1 W ,the building flip θ is called a Phan involution .The following proposition, which is is a special case of [23, Proposition 3.1], shows that a BN -flip induces a building flip (even a Phan involution) justifying the choice of name. Conversely, asBernhard M¨uhlherr pointed out to us, a building flip induces a BN -flip, if the group G admits aroot group datum and is center free (see [28, Theorem 2.2.2]). Proposition 3.4.
Let G be a group with a saturated twin BN -pair inducing the twin building B .Then any BN -flip θ of G induces a Phan involution of B .Proof. Recall from Definition 2.2 that B consists of the buildings G/B ε with distance functions δ ε : G/B ε × G/B ε → W satisfying δ ε ( gB ε , hB ε ) = w if and only if B ε g − hB ε = B ε wB ε for ε = ± .These buildings are twinned by the codistance function δ ∗ : ( G/B + × G/B − ) ∪ ( G/B − × G/B + ) → W satisfying δ ∗ ( gB ε , hB − ε ) = w if and only if B ε g − hB − ε = B ε wB − ε . By definition θ = id.Moreover, B θ + = B − implies that θ interchanges the two parts of the twin building. The image of g − h ∈ B ε wB ε under θ satisfies ( g − ) θ h θ ∈ B θε w θ B θε = B − ε wB − ε . Therefore δ ε ( gB ε , hB ε ) = w implies δ − ε ( g θ B − ε , h θ B − ε ) = w , whence θ flips the distances. For g − h ∈ B ε wB − ε we have( g − ) θ h θ ∈ B θε w θ B θ − ε = B − ε wB ε , so δ ∗ ( gB ε , hB − ε ) = w implies δ ∗ ( g θ B − ε , h θ B ε ) = w , whence θ preserves the codistance. Finally, the chamber B + is mapped onto its opposite chamber B − .Altogether, this implies that θ induces a Phan involution on the twin building B . Proposition 3.5.
Let G be a group with a saturated twin BN -pair B + , B − , N , and let θ be anautomorphism of G satisfying (i) θ = id ; and (ii) B θ + = B − ; moreover, every Borel subgroup B ′ of G containing T = B + ∩ N = B − ∩ N ismapped to an opposite Borel subgroup, i.e., B ′ ∩ θ ( B ′ ) = T .Then θ centralizes the Weyl group N/T . In particular, θ is a BN -flip.Proof. For each s ∈ S the set P s := B + ∪ B + sB + is a rank one parabolic subgroup of positivesign of G . Let n s be a representative of s in N . Then P θs = B θ + ∪ B θ + n θs B θ + = B − ∪ B − n θs B − is aparabolic subgroup of negative sign of G , as it is the image of a subgroup of G under the groupautomorphism θ and contains B − . It consists of precisely two Bruhat double cosets, implying itmust be a rank one parabolic subgroup. Hence n θs is a representative of some s ′ n s ∈ S . As s ′ is7ndependent of the choice of n s , the map θ induces a permutation of S by setting s θ := s ′ n s . Since θ maps every Borel group containing T to an opposite one, for every s ∈ S the chamber sB + isopposite to the chamber s θ B − in the associated twin building B . That is, they have codistance1 W , whence B + s − s θ B − = B + B − which by the uniqueness of the Birkhoff decomposition yieldsthat s θ = s . Hence θ centralizes W = h S i . Corollary 3.6.
Let G be a group with a root group datum { U α } α ∈ Φ , let B + , B − , N be the inducedsaturated twin BN -pair, and let θ be an automorphism of G satisfying (i) θ = id ; and (ii) B θ + = B − ; moreover, every Borel subgroup B ′ of G containing T = B + ∩ N = B − ∩ N ismapped to an opposite Borel subgroup, i.e., B ′ ∩ θ ( B ′ ) = T .Then θ centralizes the Weyl group N/T . In particular, θ is a BN -flip. Moreover, θ normalizes h U α , U − α i for each (simple) root α .Proof. In view of the preceding proposition it remains to note that the final statement of thecorollary follows from the main result of [10], since θ maps bounded subgroups of G to boundedsubgroups of G , because it acts on the building. Remark 3.7.
If in Corollary 3.6 the root system Φ does not admit a direct summand of type A ,then [1, Lemma 8.17] also implies that θ preserves root subgroups and hence normalizes h U α , U − α i for each (simple) root α , because θ acts on the building. Proposition 3.8.
Any σ -twisted Chevalley involution θ of a group G is a BN -flip.Proof. By definition, θ is an involution. Furthermore, the Borel subgroup B + is generated by T and the set of root groups associated to the positive root system Φ + ⊂ Φ. More precisely, B + = T. h U α | α ∈ Φ + i . Since T = T α ∈ Φ N G ( U α ) by [8, Corollary 5.3], the involution θ stabilizes T and maps B + to B − = T. h U − α | α ∈ Φ + i . Finally, θ acts trivially on W = N/T as each root α of the root lattice of W is mapped onto its negative − α , which means that the reflection given by α is mapped onto the reflection given by − α , which is identical to the reflection given by α . Lemma 3.9.
Let F be a field with at least four elements, let σ be an automorphism of F of order or , let G be a group with a -spherical F -locally split root group datum. Then G admits a σ -twisted Chevalley involution.Proof. By [2] (and also by the unpublished manuscript [34]) the group G is a universal envelopinggroup of the amalgam S α,β ∈ Π X α,β for a system Π of fundamental roots of Φ. This meansthat any automorphism of S α,β ∈ Π X α,β induces an automorphism of G . For each pair α, β ∈ Π the σ -twisted Chevalley involution of the split semisimple algebraic group X α,β induces σ -twisted Chevalley involutions θ α on X α and θ β on X β . Therefore there exists an involution ofthe amalgam S α,β ∈ Π X α,β inducing θ α on X α . Consequently there exists an involution θ onits universal enveloping group G inducing θ α on each subgroup X α . This involution θ of G byconstruction is a σ -twisted Chevalley involution of G .We now turn our attention to a local-to-global analysis of Iwasawa decompositions. It is well-known that an adjacency-preserving action of a group G on a connected chamber system C over I is transitive if and only if there exists a chamber c ∈ C such that for each i ∈ I the normalizer N G ([ c ] i ) acts transitively on the i -panel [ c ] i of C containing c . It is implied by the followingobservation concerning permutation groups, whose proof is left to the reader. Proposition 3.10.
Let X be a set endowed with a family of equivalence relations ( ∼ i ) i ∈ I ⊆ X × X such that the transitive hull of S i ∈ I ∼ i equals X × X . Moreover, let G be a group acting on X as permutations preserving each equivalence relation. If there exists a point p ∈ X such that thenormalizer G i := G [ p ] i of [ p ] i in G acts transitively on [ p ] i , then G acts transitively on X . orollary 3.11. Let C be a connected chamber system and let G be a group of automorphisms of C . The group G acts transitively on C if and only if there exists a chamber c ∈ C , such that foreach panel P of C containing c the stabilizer G P acts transitively on P . Corollary 3.12.
Let B be a twin building obtained from a group G with a twin BN -pair. Let θ be a BN -flip of G , let K := G θ be the group of all elements of G centralized by θ . The group K acts transitively on the positive/negative half of B if and only if there exists a chamber c opposite c θ such that for each panel P at c the stabilizer K P of that panel in K acts transitively on P . We finally have assembled all tools required to prove our main result up to the rank one analysisconducted in the remainder of this article.
Proof of Theorem 2.
Assume the existence of an Iwasawa decomposition of G . By definition thereexists an involution θ of G such that G = G θ B + . Hence any Borel subgroup of G is mappedonto an opposite one, so that by Corollary 3.6 the involution θ centralizes the Weyl group N/T and, for any simple root α , normalizes the group X α := h U α , U − α i , which by F -local splitness isisomorphic to (P)SL ( F ). In particular the restriction θ | X α of θ to X α is a BN -flip.We now argue that this restricted BN -flip induces an Iwasawa decomposition of X α . Let P α be the panel of the building corresponding to the root α . By Corollary 3.12 we know that( G P α ) θ = G P α ∩ G θ acts transitively on P α , and it remains to show that this is also the casefor ( X α ) θ = X α ∩ G θ . First observe that P − α = θ ( P α ) and hence ( G P α ) θ also stabilizes thepanel P − α . For, if g ∈ ( G P α ) θ , then g.P − α = g.θ ( P α ) = θ ( g.P α ) = θ ( P α ) = P − α and so g ∈ ( G P α ) θ = G P α ∩ G P − α ∩ G θ . If x ∈ ( G P α ) θ stabilizes the chamber B + in P α , then x.B − = x.θ ( B + ) = θ ( x.B + ) = θ ( B + ) = B − . We conclude that x ∈ B + ∩ B − = T . Moreover, the group U α < X α stabilizes B + and acts transitively on P − α . Thus, in fact ( G P α ) θ = ( X α T ) ∩ G θ . Any t ∈ T \ X α acts trivially on P α . Hence, since ( G P α ) θ acts transitively on P α , so does ( X α ) θ .Accordingly X α admits an Iwasawa decomposition.Therefore, by Corollary 4.6 below, the field F admits an automorphism σ with the requiredproperties.For the converse implication, let θ be the σ -twisted Chevalley involution of G . For each α ∈ Φthe involution θ induces a BN -flip θ α on X α . By Proposition 4.5 below, these induced flips aretransitive. Hence by Corollary 3.12, we have G = G θ B + , proving that G admits an Iwasawadecomposition. As we have seen in the preceding section, the key to understanding Iwasawa decompositions liesin the structure of the rank one groups. Since we are interested in groups with a locally split rootgroup datum, we can restrict our attention to flip automorphisms of SL ( F ) and PSL ( F ) where F is an arbitrary field.In Section 4.1 we classify all suitable flip automorphisms of these two groups. We use thatall automorphisms of PSL ( F ) are induced via projection from automorphisms of SL ( F ), whichfollows from the fact that SL is perfect, if | F | ≥
4, and is easily verified over the fields of two andthree elements. Alternatively, one can use the classification of endomorphisms of Steinberg groupsor apply the results in [38]. Hence it suffices here to study flips of SL ( F ).In Section 4.2 we compute the fixed point groups of these flips and give a geometric interpretationfor a rank one Iwasawa decomposition. This finally enables us to give a nice sufficient and necessaryalgebraic criterion for such a local Iwasawa decomposition in Section 4.3.Furthermore, in Section 5, we study flips of general Moufang sets which correspond to arbitraryrank one groups. Our aim is to show that it seems feasible to extend the theory to groups beyond F -locally split ones. As a first step we present some results for SL ( D ) for a division ring D .9 .1 Flip automorphisms of G = SL ( F ) In order to be able to understand flips of G = SL ( F ) we need to specify a suitable root groupdatum of G . To this end consider SL ( F ) as a matrix group acting on its natural module and let T denote the subgroup of diagonal matrices, which is a maximal torus of G . Let U + and U − denotethe subgroups of upper resp. lower triangular unipotent matrices, which are the root subgroupswith respect to the root system of type A associated to T . The standard Borel subgroups of G then are the groups B + := T.U + , B − := T.U − . Finally, set N := N G ( T ) to obtain a BN -pair.Consider a BN -flip θ with respect to this BN -pair, i.e. an involutory automorphism θ of G whichinterchanges B + and B − and centralizes N/T . It follows that θ stabilizes T and interchanges U + and U − .Let K := C G ( θ ), the fixed point group of θ . Then θ induces an Iwasawa decomposition G = KB + if and only if K acts transitively on the projective line P ( F ) = G/B + . In this case θ iscalled transitive . Since θ interchanges U + and U − and since the root subgroups are isomorphic to( F , +), there must exist a group automorphism φ ∈ Aut( F , +) such that the equalities θ (cid:18)(cid:18) x (cid:19)(cid:19) = (cid:18) φ ( x ) 1 (cid:19) and θ (cid:18)(cid:18) y (cid:19)(cid:19) = (cid:18) φ − ( y )0 1 (cid:19) (4.1)hold. This weak assumption implies much stronger properties of φ , as the next lemma shows. Lemma 4.1.
The group automorphism φ ∈ Aut( F , +) induces an involution θ ∈ Aut(SL ( F )) if and only if φ ( x ) = εx σ for some field automorphism σ ∈ Aut( F ) of order or and some ε ∈ Fix F ( σ ) .Proof. The key here is to use the following equation derived from the Steinberg relations forChevalley groups of rank one for various values s, t ∈ F × , u ∈ F : (cid:18) s (cid:19) (cid:18) − s (cid:19) (cid:18) s − t + u (cid:19) (cid:18) t (cid:19) (cid:18) − t (cid:19) = (cid:18) st − ust ts (cid:19) . (4.2)Assume θ is an involution induced by some group automorphism φ of ( F , +) as described in (4.1).In case s = t we can thus apply θ to (4.2) and obtain the equality (cid:18) φ ( t ) 1 (cid:19) (cid:18) − φ − ( t )0 1 (cid:19) (cid:18) φ ( u ) 1 (cid:19) (cid:18) φ − ( t )0 1 (cid:19) (cid:18) − φ ( t ) 1 (cid:19) = (cid:18) − φ − ( ut )0 1 (cid:19) . (4.3)Reducing (4.3) further by setting u = t and expanding the left side, we arrive at (cid:18) − φ ( t ) φ − ( t ) (cid:0) − φ ( t ) φ − ( t ) (cid:1) − φ ( t ) φ − ( t ) φ ( t ) (cid:0) − φ ( t ) φ − ( t ) (cid:1) φ ( t ) φ − ( t ) (cid:0) − φ ( t ) φ − ( t ) (cid:1)(cid:19) = (cid:18) − φ − ( t )0 1 (cid:19) which readily implies φ − ( t ) = φ ( t ) − for all t ∈ F × . Defining ε := φ (1) and σ ( x ) := φ ( x ) ε (notethat ε = 0 since φ is an automorphism of the group ( F , +)), we again use (4.2), this time for u = 0and arbitrary s and t . We obtain θ (cid:18) st ts (cid:19) = φ ( t ) φ ( s ) φ ( s ) φ ( t ) ! = σ ( t ) σ ( s ) σ ( s ) σ ( t ) ! . We use this equality twice, once substituting ( y, xy ) for ( s, t ) and once (1 , x ) for ( s, t ) in order toobtain our final equality σ ( xy ) σ ( y ) σ ( y ) σ ( xy ) ! = θ (cid:18) x − x (cid:19) = σ ( x ) σ (1) σ (1) σ ( x ) ! = (cid:18) σ ( x ) 00 σ ( x ) (cid:19) . This allows us to conclude that σ ( xy ) = σ ( x ) σ ( y ). We already know that σ ( x + y ) = σ ( x ) + σ ( y ), σ (0) = 0, σ (1) = 1, and hence σ ∈ Aut( F ) as required. Furthermore, σ ( ε ) = ε since1 = σ ( εε − ) = σ ( ε ) σ ( ε − ) = σ ( ε ) ε − φ ( ε − ) = ε − σ ( ε ) φ − ( ε ) − = ε − σ ( ε ) . σ = id, since σ − ( x ) = φ − ( εx ) = 1 φ ( ε − x − ) = 1 εσ ( ε − x − ) = 1 σ ( x − ) = σ ( x ) . The converse implication, deriving a group automorphism θ of SL ( F ) from a given group auto-morphism φ of ( F , +), results from the fact that the following automorphism restricts as requiredto U + resp. U − : θ : SL ( F ) → SL ( F ) : X θ ( X ) = (cid:18) ε (cid:19) X σ (cid:18) ε − (cid:19) . For an alternative proof see [28, Section 3.1.1].
Definition 4.2.
For a field automorphism σ of F of order 1 or 2 and δ ∈ Fix F ( σ ) define θ δ,σ : SL ( F ) → SL ( F ) : X θ δ,σ ( X ) = (cid:18) − δ − (cid:19) X σ (cid:18) − δ (cid:19) . By slight abuse of notation, we will use the same symbol θ δ,σ to denote the induced flip onPSL ( F ). We now turn our attention to the centralizers of a given flip θ . It is easy to verify that K δ,σ := C SL ( F ) ( θ δ,σ ) = (cid:26)(cid:18) u σ δv σ − v u (cid:19) | uu σ + δvv σ = 1 (cid:27) , which is precisely the group preserving the σ -sesquilinear form f ( x, y ) := x T (cid:18) δ (cid:19) y σ on the vector space F and its associated unitary form q ( x ) := f ( x, x ). This alternative charac-terization will turn out to be quite useful.For PSL ( F ), the situation is slightly different. Let Z denote the center of SL ( F ). By definitionPSL ( F ) = SL ( F ) /Z , so that the centralizer of θ in PSL ( F ) is C PSL ( F ) ( θ ) = { gZ ∈ PSL ( F ) | ( gZ ) θ = gZ } . We are mainly interested in the action of this centralizer on P ( F ). Since the actionof PSL ( F ) is induced by that of SL ( F ), this means studying the preimage of the centralizer inSL ( F ). This suggests the following definition: Definition 4.3.
Let θ be an automorphism of SL ( F ). We define the projective centralizer of θ in SL ( F ) as the group P C SL ( F ) ( θ ) := { g ∈ SL ( F ) | g θ ∈ gZ } , which is the preimage of C PSL ( θ )in SL ( F ) under the canonical projection π : SL → PSL .We compute P K δ,σ := P C SL ( F ) ( θ δ,σ ) = (cid:26)(cid:18) εu σ δεv σ − v u (cid:19) | uu σ + δvv σ = ε, ε ∈ {− , +1 } (cid:27) . While K δ,σ preserves the σ -sesquilinear form f ( x, y ) and its associated unitary form q ( x ), thegroup P K δ,σ preserves these forms up to sign. 11 .3 Transitivity of centralizers of flips
The observations of the previous sections allows us to characterize when θ δ,σ is transitive. Set N : F → F : a aa σ . Note that q ( ab ) = N ( a ) + δN ( b ). All results are written with PSL ( F )in mind. The corresponding results for SL ( F ) can be obtained by replacing P K δ,σ by K δ,σ andsubstituting 1 for ε . Lemma 4.4.
If the involution θ δ,σ is transitive, then it is conjugate to θ ,σ by an element of GL ( F ) normalizing both B + and B − . In case char( F ) = 2 , the involution θ δ,σ cannot be transitive.Proof. Due to the transitivity of
P K δ,σ on P ( F ), there exists g ∈ P K δ,σ such that g ( ) = ( x )for some x = 0. Thus1 = q ( ) = εq ( g ( )) = εq ( x ) = εq ( ) N ( x ) = εδN ( x ) , with ε ∈ {− , +1 } , whence δ = εN ( x − ). If ε = −
1, then q would be isotropic, as q ( x ) = 0,contradicting transitivity. (This in particular implies that in case char( F ) = 2, the involution θ δ,σ cannot be transitive.) Thus δ = N ( x − ). Let Y := (cid:0) x − (cid:1) , and denote by Inn Y ( g ) the innerautomorphism of G induced by Y . Then conjugating θ ,σ by Inn Y ( g ) yields θ δ,σ .Because of the preceding lemma it remains to determine when exactly θ ,σ is transitive. Proposition 4.5.
The involution θ ,σ is transitive if and only if − is not a norm, and the sumof two norms is ε times a norm, where ε ∈ { +1 , − } .Proof. Assume θ ,σ is transitive. Take an arbitrary nonzero vector ( ab ) ∈ F . Due to transitivity,there exists g ∈ P K ,σ such that g ( ab ) = ( x ) for some nonzero x ∈ F . Consequently, N ( a ) + N ( b ) = q ( ab ) = εq ( g ( ab )) = εq ( x ) = εN ( x ) ∈ N ( F ) , ε ∈ {− , +1 } , proving that a sum of two norms is a norm or − − x ∈ F with N ( x ) = − x )contradicting transitivity.For the converse implication assume that the sum of two norms is a norm or − − P K ,σ that mapsan arbitrary non-trivial vector ( ab ) ∈ F onto some non-trivial vector ( x ). Choose x such that εN ( x ) = N ( a ) + N ( b ) for ε ∈ {− , +1 } , and note that x = 0, as else N ( a ) = − N ( b ) contradictingthat − (cid:18) ε (cid:0) ax (cid:1) σ ε (cid:0) bx (cid:1) σ − bx ax (cid:19) (cid:18) ab (cid:19) = (cid:18) x (cid:19) finishes the proof, since the given matrix is clearly in P K ,σ . Corollary 4.6.
The group (P)SL ( F ) admits an Iwasawa decomposition if and only if F admitsan automorphism σ of order or such that (i) − is not a norm, and (ii) (a) either a sum of norms is a norm (in the SL ( F ) case) (b) or a sum of norms is ε times a norm, where ε ∈ { +1 , − } (in the PSL ( F ) case),with respect to the norm map N : F → Fix F ( σ ) : x xx σ .Proof. Assume we have an Iwasawa decomposition of G . Then we have an involution θ whichinterchanges U + and U − and satisfies G = G θ B + , whence θ is transitive. Then by Lemmas 4.1and 4.4 plus Proposition 4.5 the claim for F follows. If on the other hand F is as described, thenagain by Proposition 4.5 the map θ ,σ induces an Iwasawa decomposition.12 .4 Fields permitting Iwasawa decompositions Besides the real closed fields and the field of complex numbers there exist lots of fields admittingautomorphisms that satisfy the conditions of Corollary 4.6. For instance any pythagorean formallyreal field F satisfies the conditions of Corollary 4.6 with respect to the identity automorphism asdoes F [ √−
1] with respect to the non-trivial Galois automorphism. Such fields have been studiedvery thoroughly, cf. [32], [33], [41]. In the PSL ( F ) case, the finite fields F q with q ≡ G = G θ B + of a group G with an F -locallysplit root group datum with respect to an involution θ involving a non-trivial field automorphism σ : F → F implies the existence of an Iwasawa decomposition over the field Fix F ( σ ) with respect toan involution involving the trivial field automorphism on Fix F ( σ ). The following example showsthat this in general is not the case. Example 4.7.
Let F be a formally real field which is not pythagorean and admits four squareclasses. Such fields exist, see for example [45]. This means exactly two square classes containabsolutely positive elements, so that there exists a unique ordering. Choose a positive non-squareelement w ∈ F . Set α := √− w and ˜ F := F [ α ]. Then N ( x + αx ) + N ( y + αy ) = x + wx + y + wy which is a non-negative number, hence either a square or a square multiple of w . Hence thereexist z and z in F such that N ( x + αx ) + N ( y + αy ) = x + wx + y + wy = z + wz = N ( z + αz )and thus the field ˜ F together with the non-trivial Galois automorphism satisfies the conditions ofCorollary 4.6, while F together with the identity does not, because F is not pythagorean. In this last section, we turn our attention to the study of Moufang sets. Our motivation is Corollary3.6 which states that in order to understand flips of arbitrary non-split groups with a root groupdatum one needs to understand flips of arbitrary rank one subgroups. Moufang sets are essentiallyequivalent to these and seem to be the natural setting to study flips. For a concise introductionto Moufang sets, we refer the interested reader to [12].Maybe Moufang sets, or rather Moufang sets with transitive flips (cf. Sections 5.2 and 5.3),can be used for a better understanding of certain anisotropic forms of reductive algebraic groups.Indeed, the machinery developed in this paper implies that an anisotropic form K of a reductivealgebraic group G acts transitively on the building B of G if and only if there exists a chamber c of B such that, for each i -panel P containing c , the stabilizer K P acts transitively on P , cf.Corollary 3.12. Therefore any result about Moufang sets with transitive flips has immediateconsequences for certain classes of anisotropic algebraic groups, namely those which act transitivelyon a building. In order to be consistent with the standard notation used in the theory of Moufang sets we willalways denote the action of a permutation on a set on the right, i.e. we will write aϕ rather than ϕ ( a ). Definition 5.1. A Moufang set is a set X together with a collection of subgroups ( U x ) x ∈ X , suchthat each U x is a subgroup of Sym( X ) fixing x and acting regularly (i.e. sharply transitively)on X \ { x } , and such that each U x permutes the set { U y | y ∈ X } by conjugation. The group G := h U x | x ∈ X i is called the little projective group of the Moufang set; the groups U x are called root groups . 13ur approach to Moufang sets is taken from [13]. Let M = ( X, ( U x ) x ∈ X ) be an arbitraryMoufang set, and assume that two of the elements of X are called 0 and ∞ . Let U := X \ {∞} .Each α ∈ U ∞ is uniquely determined by the image of 0 under α . If 0 α = a , we write α =: α a .Hence U ∞ = { α a | a ∈ U } . We make U into a (not necessarily abelian) group with composition+ and identity 0, by setting a + b := aα b . (5.1)Clearly, U ∼ = U ∞ . Now let τ be an element of G interchanging 0 and ∞ . (Such an element alwaysexists, since G is doubly transitive on X .) By the definition of a Moufang set, we have U = U τ ∞ and U a = U α a (5.2)for all a ∈ U . In particular, the Moufang set M is completely determined by the group U and thepermutation τ ; we will denote it by M = M ( U, τ ). Remark 5.2.
In view of equation (5.1), it makes sense to use the convention that a + ∞ = ∞ + a = ∞ for all a ∈ U . Definition 5.3.
For each a ∈ U , we define γ a := α τa , i.e. xγ a = ( xτ − + a ) τ for all x ∈ X .Consequently, U = { γ a | a ∈ U } .The following two definitions may appear technical at first sight, but it turns out that they playa key role in the theory of Moufang sets. Definition 5.4.
For each a ∈ U ∗ = U \{ } , we define a Hua map to be h a := τ α a τ − α − ( aτ − ) τ α − ( − ( aτ − )) τ ∈ Sym( X );if we use the convention of Remark 5.2, then we can write this explicitly as h a : X → X : x (cid:0) ( xτ + a ) τ − − aτ − (cid:1) τ − (cid:0) − ( aτ − ) (cid:1) τ . Observe that each h a fixes the elements 0 and ∞ . Wedefine the Hua subgroup of M as H := h h a | a ∈ U ∗ i . By [13, Theorem 3.1], the group H equals G , ∞ := Stab G (0 , ∞ ), and by [13, Theorem 3.2], the restriction of each Hua map to U is additive,i.e. H ≤ Aut( U ). Definition 5.5.
For each a ∈ U ∗ , we define a µ -map µ a := γ ( − a ) τ − α a γ − aτ − . Lemma 5.6.
For each a ∈ U ∗ , we have (i) µ a is the unique element in the set U ∗ α a U ∗ interchanging and ∞ ; (ii) µ − a = µ − a ; (iii) µ a = τ − h a .Proof. See [11, Section 3].
Definition 5.7.
Let ( X, ( U x ) x ∈ X ) and ( Y, ( V y ) y ∈ Y ) be two Moufang sets. A bijection β from X to Y is called an isomorphism of Moufang sets, if the induced map χ β : Sym( X ) → Sym( Y ) : g β − gβ maps each root group U x isomorphically onto the corresponding root group V xβ .An automorphism of M = ( X, ( U x ) x ∈ X ) is an isomorphism from M to itself. The group of allautomorphisms of M will be denoted by Aut( M ).Now we introduce pointed Moufang sets, which will be Moufang sets with a fixed identityelement. We will then, in analogy with the theory of Jordan algebras, introduce the notions of anisotope of a pointed Moufang set, and we will define Jordan isomorphisms between Moufang sets. Definition 5.8. A pointed Moufang set is a pair ( M , e ), where M = M ( U, ρ ) is a Moufang set and e is an arbitrary element of U ∗ . The τ -map of this pointed Moufang set is τ := µ − e = µ − e , andthe Hua maps are the maps h a := τ µ a = µ − e µ a for all a ∈ U ∗ . We also define the opposite Huamaps g a := τ − µ a = µ e µ a for all a ∈ U ∗ . Clearly, M = M ( U, τ ) = M ( U, τ − ).14ote that, in contrast with Moufang sets which are not pointed, the maps τ , h a and g a arecompletely determined by the data ( M , e ). On the other hand, there can be many differentelements f for which ( M , e ) = ( M , f ), namely all those for which µ e = µ f . Definition 5.9.
Let ( M , e ) and ( M ′ , f ) be two pointed Moufang sets, with M = M ( U, ρ ) and M ′ = M ( U ′ , ρ ′ ). A pointed isomorphism from ( M , e ) to ( M ′ , f ) is an isomorphism from U to U ′ mapping e to f and extending to a Moufang set isomorphism from M to M ′ (by mapping ∞ to ∞ ′ ). A pointed isomorphism from ( M , e ) to itself is called a pointed automorphism of ( M , e ), andthe group of all pointed automorphisms is denoted by Aut( M , e ).Observe that G ∩ Aut( M , e ) = C H ( e ). Definition 5.10.
Let ( M , e ) be a pointed Moufang set, and let a ∈ U ∗ be arbitrary. Then ( M , a )is called the a -isotope of ( M , e ), or simply an isotope if one does not want to specify the element a . The τ -map and the Hua maps of ( M , a ) will be denoted by τ ( a ) and h ( a ) b , respectively. Observethat τ ( a ) = µ − a and h ( a ) b = µ − a µ b = h − a h b (5.3)for all a, b ∈ U ∗ .Our notion of an a -isotope is, in a certain sense, the inverse of the usual notion of an a -isotopein (quadratic) Jordan algebras, where our a -isotope would be called the a − -isotope (where a − denotes the inverse in the Jordan algebra) and where h ( a ) b := h a h b . It is, in the general contextof Moufang sets, not natural to try to be compatible with this convention, because h − a is ingeneral not of the form h b for some b ∈ U ∗ . In fact, we have h − a = g aτ for all a ∈ U ∗ ; see [13,Lemma 3.8(i)]. Definition 5.11.
Let ( M , e ) and ( M ′ , f ) be two pointed Moufang sets with M = M ( U, ρ ) and M ′ = M ( U ′ , ρ ′ ), and with Hua maps h a and k a , respectively. An isomorphism ϕ from U to U ′ iscalled a Jordan isomorphism if ( bh a ) ϕ = ( bϕ ) k aϕ for all a, b ∈ U ∗ . If ( M ′ , f ) is an isotope ( M , a )of ( M , e ), then a Jordan isomorphism from ( M , e ) to ( M , a ) is called an isotopy from ( M , e ) to its a -isotope. Explicitly, a map ϕ ∈ Aut( U ) is an isotopy if and only if h a ϕ = ϕh ( eϕ ) aϕ (5.4)for all a ∈ U ∗ . The group of all isotopies from ( M , e ) to an isotope is called the structure group of( M , e ), and is denoted by Str( M , e ). Note that it is not clear whether Str( M , e ) ≤ Aut( M ). Alsoobserve that G ∩ Str( M , e ) = H ; we call H the inner structure group of ( M , e ). Our goal in this section is to determine all involutions θ ∈ Aut( G ) interchanging U ∞ and U . Suchan involution θ maps each α a to some γ aϕ and each γ b to some α bψ . Since θ ∈ Aut( G ), we have ϕ, ψ ∈ Aut( U ). Moreover, θ = id implies ψ = ϕ − . In particular, θ is competely determined by ϕ . More precisely, for each ϕ ∈ Aut( U ), we define θ ϕ : U ∞ ∪ U → U ∪ U ∞ : ( α a γ aϕ γ a α aϕ − ;the question is when θ ϕ extends to an automorphism of G . Observe that if θ ϕ extends, then thisextension is unique and is involutory, since θ is involutory on U ∞ ∪ U and G = h U ∞ ∪ U i . Proposition 5.12.
Let ϕ ∈ Aut( U ) . Then θ ϕ extends to an (involutory) automorphism of G ifand only if ( ϕτ ) = id . Moreover, if this is the case, then ϕ ∈ Aut( M ) . roof. Let θ := θ ϕ and β := ϕτ . Assume first that θ extends to an automorphism χ of G . Then χ ( U a ) = χ ( U α a ) = χ ( U ) χ ( α a ) = U γ aϕ ∞ = U aϕτ = U aβ (5.5)for all a ∈ U . Since θ is the identity on U ∞ ∪ U and since G = h U ∞ , U i , this implies that χ = 1and hence β = 1.Conversely, assume that β = 1, and let χ β be as in Definition 5.7. Then for all a ∈ U , χ β ( α a ) = α ϕτa = α τaϕ = γ aϕ ,χ β ( γ a ) = γ ϕτa = γ τ − ϕ − a = α ϕ − a = α aϕ − ;hence χ β and θ coincide on U ∞ ∪ U . Note that χ β is an (inner) automorphism of Sym( X ), andhence the same calculation as in equation (5.5) (with χ β in place of χ ) shows that β ∈ Aut( M ).Hence the restriction of χ β to G is an automorphism of G ; this is the (unique) extension of θ toan element of Aut( G ).Finally, since we have just shown that β ∈ Aut( M ) and since obviously τ ∈ Aut( M ), we concludethat ϕ ∈ Aut( M ) as well. Definition 5.13.
An automorphism ϕ ∈ Aut( U ) with the property that ( ϕτ ) = 1 will be calleda flip automorphism of M .The following theorem gives important information about such flip automorphisms. Theorem 5.14.
Let M be a Moufang set, and let ϕ be a flip automorphism of M . Then g aϕ = ϕ · h a · ϕ for all a ∈ U ∗ . Moreover, if e is an identity element of M , i.e. τ = µ − e , then ϕ ∈ Str( M , e ) ∩ Aut( M ) .Proof. For each a ∈ U ∗ , the map g a is the Hua map of a with τ replaced by τ − , and hence g aϕ = τ − α aϕ τ α − aϕτ τ − α − ( − ( − aϕτ )) τ − for all a ∈ U ∗ . Using the facts that α ϕa = α aϕ , ϕτ = τ − ϕ − and ( − a ) ϕ = − aϕ several times, we get ϕ − g aϕ = τ α a τ − α − aτ − τ α − ( − ( − aτ − )) τ ϕ = h a ϕ . Inparticular, if e is an identity element of M , then h e = 1 and hence ϕ − g eϕ = ϕ . It follows that ϕg − eϕ g aϕ = h a ϕ for all a ∈ U ∗ . However, g − eϕ g aϕ = ( µ e µ eϕ ) − ( µ e µ aϕ ) = ( µ − e µ eϕ ) − ( µ − e µ aϕ ) = h − eϕ h aϕ = h ( eϕ ) aϕ and hence h a ϕ = ϕh ( eϕ ) aϕ for all a ∈ U ∗ , proving that ϕ ∈ Str( M , e ). The fact that ϕ ∈ Aut( M ) was shown in Proposition 5.12 above.We will now illustrate the strength of Theorem 5.14 by explicitly determining all flips of PSL ( D ),where D is a field or a skew field. This can be considered as a natural extension of the resultsfrom Lemma 4.1 to the non-commutative case. Proposition 5.15.
Let D be an arbitrary field or skew field, and let M = M ( D ) be the correspond-ing Moufang set, i.e. the Moufang set M = M ( U, τ ) where U := ( D , +) and τ : D ∗ → D ∗ : x x − . (i) Let ϕ be a flip automorphism of M . Then there exists an automorphism or anti-automorphism σ of D and an element ε ∈ Fix D ( σ ) such that xϕ = εσ ( x ) for all x ∈ D . If σ is an automor-phism, then σ ( x ) = ε − xε for all x ∈ D ; if σ is an anti-automorphism, then σ = 1 . (ii) Conversely, suppose that either σ is an anti-automorphism of order and ε ∈ Fix D ( σ ) isarbitrary, or σ is an automorphism such that σ ( x ) = ε − xε for some ε ∈ Fix D ( σ ) . Thenthe map ϕ : D → D : x εσ ( x ) is a flip automorphism of M .Proof. (i) Observe that 1 ∈ D ∗ is an identity element of M ; also note that τ = id. For all a, b ∈ U ∗ , we have bh a = aba . The condition ( ϕτ ) = 1 translates to( a − ) ϕ = ( aϕ − ) − (5.6)16or all a ∈ D ∗ . Let ε := 1 ϕ ; then bh (1 ϕ ) a = bh − ϕ h a = aε − bε − a for all a, b ∈ U ∗ . ByTheorem 5.14, ϕ ∈ Str( M , e ), which means that bh a ϕ = bϕh (1 ϕ ) aϕ for all a, b ∈ U ∗ , orexplicitly, ( aba ) ϕ = aϕ · ε − · bϕ · ε − · aϕ for all a, b ∈ D ∗ . Now let σ ( a ) := ε − · aϕ forall a ∈ D . Then σ ∈ Aut( U ), and the previous equation can be rewritten as σ ( aba ) = σ ( a ) σ ( b ) σ ( a ) for all a, b ∈ D , i.e. σ is a Jordan automorphism of D . It is a well known resultby Jacobson and Rickart [30] (see also [29, page 2]), which simply amounts to calculatingthat (cid:0) σ ( ab ) − σ ( a ) σ ( b ) (cid:1) · (cid:0) σ ( ab ) − σ ( b ) σ ( a ) (cid:1) = 0, that σ is either an automorphism or an anti-automorphism of D . Now by equation (5.6), we have ( ε − ) ϕ = ( εϕ − ) − = 1 − = 1, andhence σ ( ε − ) = ε − ; since σ is an automorphism or anti-automorphism, it follows that σ ( ε ) = ε . Finally, again by equation (5.6), we obtain σ ( εσ ( a )) = σ ( aϕ ) = σ (cid:0) (( a − ) ϕ − ) − (cid:1) = σ (( a − ) ϕ − ) − = ( ε − a − ) − = aε for all a ∈ D ∗ . If σ is an automorphism, then thiscan be rewritten as εσ ( a ) = aε , or σ ( a ) = ε − aε ; if σ is an anti-automorphism, we get σ ( a ) ε = aε , i.e. σ = 1.(ii) It suffices to check that equation (5.6) holds. This amounts to checking that εσ ( a − ) =( σ − ( ε − a )) − for all a ∈ D . It is straightforward to check that this is valid in both cases.By [38] the flips of SL ( D ) are just the lifts of the flips of PSL ( D ). Definition 5.16. If τ = id, then ϕ = 1 is a flip automorphism. We will call the correspondingautomorphism θ of G (as defined in the beginning of this section) the obvious flip . Observe that θ is just conjugation by τ . Definition 5.17.
A flip automorphism ϕ ∈ Aut( U ) is called transitive if the group Fix G ( θ ϕ ) istransitive on X .Let M = M ( U, τ ) be a Moufang set with τ = id. Then the obvious flip θ is transitive if andonly if C G ( τ ) is transitive on X , because Fix G ( θ ) = C G ( τ ). Lemma 5.18.
Let M = M ( U, τ ) be a Moufang set with τ = id , and assume that the obvious flipis transitive. Then τ has no fixed points.Proof. Assume that aτ = a for some a ∈ U ∗ . Let g ∈ C G ( τ ) be such that 0 g = a . Then ∞ g = 0 τ g = 0 gτ = aτ = a = 0 g and hence ∞ = 0, a contradiction.We now examine the transitivity of the obvious flip for M ( D ) where D is an arbitrary skew field. Definition 5.19. If g = (cid:0) a bc d (cid:1) ∈ GL ( D ), then the Dieudonn´e determinant det( g ) ∈ D ∗ / [ D ∗ , D ∗ ]is defined asdet( g ) := (cid:12)(cid:12)(cid:12)(cid:12) a bc d (cid:12)(cid:12)(cid:12)(cid:12) := ( ad − aca − b if a = 0 ; − cb if a = 0 ;see [16]. Then SL ( D ) is precisely the kernel of the Dieudonn´e determinant, i.e. a matrix g ∈ GL ( D ) lies in SL ( D ) if and only if det( g ) ∈ [ D ∗ , D ∗ ]. Also observe that det( λg ) ≡ det( gλ ) ≡ λ det( g ) mod [ D ∗ , D ∗ ] for all λ ∈ D ∗ . Lemma 5.20.
Let G = SL ( D ) and let τ = (cid:0) − (cid:1) ∈ G . Then C G ( τ ) = (cid:26)(cid:18) a b − b a (cid:19) a + aba − b ∈ [ D ∗ , D ∗ ] if a = 0 b ∈ [ D ∗ , D ∗ ] if a = 0 (cid:27) ; P C G ( τ ) = (cid:26)(cid:18) ǫa ǫb − b a (cid:19) ǫ · ( a + aba − b ) ∈ [ D ∗ , D ∗ ] if a = 0 ǫ · b ∈ [ D ∗ , D ∗ ] if a = 0 where ǫ = ± (cid:27) . Proof.
This is a straightforward calculation. 17 roposition 5.21.
Let G = SL ( D ) and let τ = (cid:0) − (cid:1) ∈ G . Let X be the projective line over D , i.e. X = { ( ab ) D = 0 | a, b ∈ D } . Then the following are equivalent: (i) C G ( τ ) is transitive on X ; (ii) a + aba − b ∈ ( D ∗ ) [ D ∗ , D ∗ ] for all a, b ∈ D ∗ ; (iii) 1 + a ∈ ( D ∗ ) [ D ∗ , D ∗ ] for all a ∈ D ∗ .Proof. Since a + aba − b = a (1 + a − ba − b ), we have a + aba − b ∈ ( D ∗ ) [ D ∗ , D ∗ ] if and only if1 + a − ba − b ∈ ( D ∗ ) [ D ∗ , D ∗ ]. Equivalence between (ii) and (iii) follows by replacing a − b by a in the latter term.Assume now that (ii) holds. Let a, b ∈ D ∗ be arbitrary; we want to show that there existssome g ∈ C G ( τ ) mapping ( ab ) to ( z ) for some z ∈ D ∗ . By (ii), we know that there is some c ∈ D ∗ such that b − + b − a − ba − ≡ c − mod [ D ∗ , D ∗ ]. Let g := (cid:16) cb − ca − − ca − cb − (cid:17) . Then det( g ) ≡ c ( b − + b − a − ba − ) ≡
1, i.e. g ∈ G . Moreover, g ( ab ) = ( z ) for z = c ( b − a + a − b ), proving that C G ( τ ) acts transitively on X .Conversely, assume that C G ( τ ) acts transitively on X . Let a, b ∈ D ∗ be arbitrary; then thereexists some g ∈ C G ( τ ) mapping ( ) D to ( ab ) D , i.e. there is some z ∈ D ∗ such that g maps ( z )to ( ab ). By Lemma 5.20, we know that g has the form g = (cid:0) x y − y x (cid:1) with x + xyx − y ∈ [ D ∗ , D ∗ ].Then g ( z ) = ( xz − yz ), and hence a = xz and b = − yz . Hence a + aba − b = xzxz + xzyx − yz = xzx − · ( x + xyx − y ) · z , and since x + xyx − y ∈ [ D ∗ , D ∗ ], this implies a + aba − b ≡ xzx − z ≡ z mod [ D ∗ , D ∗ ]. Since a, b ∈ D ∗ were arbitrary, this proves (ii). Proposition 5.22.
Let G = PSL ( D ) , let τ = (cid:0) − (cid:1) ∈ SL ( D ) , and let ˜ τ be the image of τ in G .Let X be the projective line over D , i.e. X = { ( ab ) D | a, b ∈ D , not both zero } . Then the followingare equivalent: (i) C G (˜ τ ) is transitive on X ; (ii) P C G ( τ ) is transitive on X ; (iii) a + aba − b ∈ {± } · ( D ∗ ) [ D ∗ , D ∗ ] for all a, b ∈ D ∗ ; (iv) 1 + a ∈ {± } · ( D ∗ ) [ D ∗ , D ∗ ] for all a ∈ D ∗ .Proof. The equivalence between (i) and (ii) follows immediately from the definition of the pro-jective centralizer
P C G ( τ ). The other equivalences are shown exactly as in the proof of Proposi-tion 5.21 above. Corollary 5.23. (i)
Let G = SL ( D ) , and assume that for all a ∈ D ∗ , we have h a ∈ H .Then C G ( τ ) acts transitively on X . (ii) Let G = PSL ( D ) , and assume that for all a ∈ D ∗ , we have h a ∈ {± } · H . Then C G (˜ τ ) acts transitively on X .Proof. We only show (i). The proof of (ii) is completely similar. So let a ∈ D ∗ be arbitrary,and assume that 1 + h a = h ∈ H . Then 1 + 1 h a = 1 h , i.e. 1 + a = 1 h . Write h = h x · · · h x n with x , . . . , x n ∈ D ∗ . Then 1 h = x n · · · x · · x · · · x n ≡ ( x · · · x n ) mod [ D ∗ , D ∗ ], and hence1 + a = 1 h ∈ ( D ∗ ) [ D ∗ , D ∗ ]. So (iii) of Proposition 5.21 holds, and therefore the group C G ( τ )acts transitively on X .A natural extension of the study of the obvious flip would be to study its close relatives, the semi-obvious flips, which are obtained by composing the obvious flip with a field (anti-)automorphism.18 eferences [1] P. Abramenko, K.S. Brown, Buildings – theory and application , Springer, Berlin 2008.[2] P. Abramenko, B. M¨uhlherr, Pr´esentation des certaines BN -paires jumel´ees comme sommes amalgam´ees, C.R. Acad. Sci. Paris S´er. I Math. (1997), 701–706.[3] K. Altmann,
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Authors’ addresses:Tom De MedtsDepartment of Pure Mathematics and Computer AlgebraGhent UniversityKrijgslaan 281, S22B-9000 GentBelgiume-mail: [email protected]
Ralf Gramlich, Max HornTU DarmstadtFachbereich MathematikSchloßgartenstraße 764289 DarmstadtGermanye-mail: [email protected]@mathematik.tu-darmstadt.de
Second author’s alternative address:University of BirminghamSchool of MathematicsEdgbastonBirmingham 2015 2TTUnited Kingdome-mail: [email protected]@maths.bham.ac.uk