aa r X i v : . [ m a t h . R T ] S e p On spherical twisted conjugacy classes,thirdversion
Giovanna CarnovaleDipartimento di Matematica Pura ed Applicatavia Trieste 63 - 35121 Padova - Italyemail: [email protected]
Abstract
Let G be a simple algebraic group over an algebraically closed field ofgood odd characteristic, and let θ be an automorphism of G arising from aninvolution of its Dynkin diagram. We show that the spherical θ -twisted con-jugacy classes are precisely those intersecting only Bruhat cells correspond-ing to twisted involutions in the Weyl group. We show how the analogueof this statement fails in the triality case. As a by-product, we obtain a di-mension formula for spherical twisted conjugacy classes that was originallyobtained by J-H. Lu in characteristic zero. Key-words: twisted conjugacy class; spherical G -space; twisted involution; Bruhatdecomposition MSC:
Twisted conjugacy classes were originally introduced by Gantmakher in [10] anddeveloped in [11], where they were viewed as orbits under the conjugacy actionof the identity component in a disconnected algebraic group. It is needless tomention that reductive disconnected groups frequently occur in the study of alge-braic groups, for example, as centralizers of semisimple elements in non-simply-connected semisimple groups. Twisted classes also occur in problems concerningconjugacy classes of rational forms. 1n recent years the attention to the twisted conjugacy classes of an algebraicgroup G has increased in different contexts of mathematics: for example the clo-sure of a twisted Steinberg fiber in the wonderful compactification of a simplelinear algebraic group has been computed in [12] and twisted conjugacy classeshave been shown to be Poisson submanifolds with respect to a natural Poissonstructure π θ induced by an automorphism θ of G ([16]). Moreover, conjugacyclasses in disconnected groups play a role in physics, due to their connection withbranes in the Wess-Zumino-Witten model (see, for instance [9], where they arecalled twined conjugacy classes). In a different context, twisted conjugacy classesin finite simple groups, so also in simple groups of Lie type, occur in the classifica-tion of racks, which is an important tool for the classification of finite dimensionalpointed Hopf algebras ([1]).Given an automorphism θ of G , the simplest example of a twisted conjugacyclass is the class G ∗ { gθ ( g ) − , g ∈ G } of the unit element in G . When θ isan involution, G ∗ has been extensively studied in [20, 22]. It provides a modelfor symmetric spaces, and it is shown in [22, 30, 8] that a Borel subgroup B of G acts on this class with finitely many orbits. Transitive G -varieties satisfyingthis property are called spherical . The combinatorics of the Zariski closures ofthe B -orbits in G ∗ has been described in [19] by means of a map from theset of B -orbits to the (set of twisted involutions in the) Weyl group. This mapis given by looking at which Bruhat cell contains the twisted B -orbit. In theuntwisted case, the analysis of the intersection of conjugacy classes and Bruhatcells is a powerful tool and it has been used in many different situations. Forspherical twisted conjugacy classes, this analysis is subject of current research.We summarize here the recent developments.A characterization of spherical θ -twisted conjugacy classes, when the auto-morphism θ is induced from an automorphism of the Dynkin diagram of G andthe characteristic of the base field is zero, is given in [15]. It is provided in terms ofa dimension formula involving the Weyl group element whose associated Bruhatcell intersects a class C densely. Such a Weyl group element, which we shall de-note by w C , is the maximum among all Weyl group elements σ for which C ∩ BσB is non-empty. The above mentioned characterization generalizes to the twistedcase a result described in [5, 6] with a more elegant proof, but it requires somerestrictions on the base field. The motivation for such a generalization lies in therelation of the element w C with the smallest dimension of symplectic leaves ofthe natural Poisson structure π θ on G . Besides, the dimension formula is relatedto the vanishing of π θ in a class C .The aim of the present paper is to provide another characterization of spherical2 -twisted conjugacy classes, when θ is induced from an involution of the Dynkindiagram, by means of their intersection with Bruhat cells. This has to be seenas a twisted analogue of some results in [6] and the main result in [7]. It can beformulated as follows, when we restrict to simply-connected groups. Theorem
Let G be a simply-connected simple algebraic group over an alge-braically closed field of good odd characteristic. Let θ be an automorphism of G induced by an involution of its Dynkin diagram. A θ -twisted conjugacy classis spherical if and only if it intersects only Bruhat cells corresponding to twistedinvolutions in the Weyl group of G . The triality case falls out of this picture. Indeed, we show that there are notwisted classes intersecting only Bruhat cells corresponding to twisted involutionsin the Weyl group, whereas it is shown in [8, 15] that there exists a sphericaltwisted conjugacy class. We expect that the combination of this characterizationwith the one in [15] can be exploited in order to obtain a complete classificationof spherical twisted conjugacy classes when θ is an involution of the Dynkin dia-gram. This is part of a forthcoming project.As a by-product of our results, we are able to prove Lu’s dimension formulawhen θ is an involution and k is of good, odd characteristic. This can be stated,for G simply-connected, as follows: Theorem
Let G be a simply-connected simple algebraic group over an alge-braically closed field of good odd characteristic. Let θ be an automorphism of G induced by an involution of its Dynkin diagram θ ′ . A θ -twisted conjugacy class C is spherical if and only if dim C = ℓ ( w C ) + rk(1 − w C θ ′ ) . Here ℓ denotes the length function in the Weyl group and rk denotes the rank ofthe operator in the geometric representation of the Weyl group.It was pointed out in [15, Remark 1.2] that the case of τ -twisted classes fora general automorphism τ of G can be reduced to the above setting as follows.For a fixed maximal torus T contained in B we have the equality ( τ ( B ) , τ ( T )) =( gBg − , gT g − ) for some g ∈ G . Multiplying g on the right by a suitable elementin T , we may choose g so that θ := Int( g − ) ◦ τ is the automorphism of G induced from an automorphism of the Dynkin diagram. Then, right translation by g induces a G -equivariant isomorphism between the τ -twisted conjugacy class ofan element x and the θ -twisted conjugacy class of xg .The paper is structured as follows. The basic notation and terminology, andthe first properties of twisted conjugacy classes are provided in Sections 2 and3. The first properties of spherical twisted conjugacy classes are dealt with inSection 4. Here, it is shown that if θ is an involution, then a spherical θ -twistedconjugacy class intersects only Bruhat cells associated with θ -twisted involutionsin the Weyl group of G . This result is obtained by induction on the length of apath in the set V of B -orbits which is constructed using the action on V , definedin [19], of a monoid associated with the Weyl group, and the Weyl group actionon V introduced in [14]. The approach is similar to that in [6] but the proof hasbeen shortened and simplified. In Section 5 we analyze the twisted conjugacyclasses intersecting only Bruhat cells corresponding to twisted involutions in theWeyl group (involutive classes). By a simple case-by-case analysis on the possiblemaximal elements w C ’s we get to a better understanding of a representative lyingin the Bruhat cell corresponding to w C . Here, we use the classification of allpossible w C ’s in [15], which holds under very mild restrictions on the base field.The case-by-case analysis is simpler here than in [6] because there are less cases tobe dealt with. In Section 6 we show that, except from the case in which w C = w and G is of type D n , if C is an involutive twisted conjugacy class, then thereare finitely many B -orbits in Bw C B . A simple topological argument shows that C is spherical. The strategy is similar to the strategy used in [7] in order to dealwith the case w C = w = − . However, it has been improved in order to beapplied to a wider range of cases, namely all but the one in which w C = w but w = − θ . The remaining case is dealt with in Section 7. Here we needto use a different argument. We show that, for a suitable representative x of aninvolutive class C , with stabilizer G x , the set BG x is dense in G . We do so byshowing that the intersection of G x B with U σB is dense in
U σB for every σ inthe Weyl group. This concludes the proof when w C = w and G is of type D n .Here, the final strategy resembles the strategy used in [7, Section 5]. However, thetechniques used in Lemma 7.1 are specific of the twisted case and [7, Theorem5.7] is extended in Lemma 7.4 to a statement on a general transitive G -variety. Onthe other hand, the computational work in this paper is considerably simpler thanin [6, 7] because we do not need to consider doubly-laced root systems. Finally,in Section 8 we show how to apply the obtained results in order to retrieve Lu’sdimension formula in good odd characteristic, when θ is an involution. Unless otherwise stated, G is a simply-connected, simple algebraic group overan algebraically closed field k of zero or odd good characteristic. We recall that4he characteristic is good if it does not divide the coefficients in the expressionof the highest root as a linear combination of simple roots. Let T be a fixedmaximal torus of G , and let Φ be the associated root system. Let B ⊃ T be aBorel subgroup with unipotent radical U , let ∆ = { α , . . . , α n } be the basis of Φ relative to ( T, B ) , with numbering of the simple roots as in [2]. The set of positiveroots will be denoted by Φ + . The Weyl group of G will be denoted by W and thereflection with respect to α ∈ Φ will be denoted by s α . For w ∈ W we shalldenote by ˙ w a representative of w in N ( T ) . The symbol X α will denote the rootsubgroup corresponding to α . We will choose parametrisations of roots subgroups x α ( ξ ) and x − α ( ξ ) for α ∈ Φ + in such a way that ˙ s α = x α (1) x − α ( − x α (1) liesin N ( T ) ([24, Lemma 8.1.4]). Then, as in [27], for any root α ∈ Φ + and any ξ ∈ k ∗ we define h α ( ξ ) = x α ( ξ ) x − α ( − ξ − ) x α ( ξ ) ˙ s − α ∈ T . By θ we denote anon-trivial automorphism of the Dynkin diagram of G . By abuse of notation, theinduced automorphism of G will also be denoted by θ . We recall that for every α ∈ Φ there is ǫ α ∈ {± } such that θ ( x α ( ξ )) = x θα ( ǫ α ξ ) , with ǫ α = 1 for α ∈ ∆ ∪ ( − ∆) (see [27, Corollary to Theorem 29]). By [24, Lemma 8.1.4(iv)],we deduce that ǫ β = ǫ − β for every β ∈ Φ + so θ ( h β ( ξ )) = h θ ( β ) ( ξ ) . It wasobserved in [18, Proposition 2.1] that, unless Φ is of type A n , one may choosethe parametrisations x ± γ ( ξ ) in such a way that they also satisfy ǫ β = 1 for every β ∈ Φ . This is achieved by replacing some of the x ± γ ( ξ ) by x ′± γ ( ξ ) := x ± γ ( − ξ ) .We shall choose such a parametrisation.For a subset Π ⊂ ∆ we shall denote by Φ Π the root system generated by Π andby P Π the standard parabolic subgroup containing B associated with Π , i.e., suchthat its standard Levi subgroup L Π is generated by T and by the root subgroupscorresponding to roots in Φ Π . The intersection U ∩ L Π will be denoted by U Π .If α ∈ ∆ then we shall put P α to indicate P { α } . For any parabolic subgroup P of G we will denote by P u its unipotent radical. The parabolic subgroup of W generated by the simple reflections with respect to roots in Π ⊂ ∆ will be denotedby W Π .For a subgroup H of G we shall denote by Z ( H ) its center and by H ◦ itsidentity component. When an automorphism τ acts on an algebraic structure S (e.g. a subgroup of G or a root system) we shall indicate by S τ the substructureof τ -invariant elements of S . 5 Twisted conjugacy classes A θ -twisted conjugacy class in G is an orbit for the G -action on itself by g · θ x = gxθ ( g ) − . When there is no ambiguity on θ , we shall call it also a twistedconjugacy class and we shall use the simplified notation g ∗ x for g · θ x . The θ -stabilizer of x ∈ G in a subgroup H of G is the stabilizer for the ∗ -action and itwill be denoted by H x .Let C be a twisted conjugacy class of G . Since C is an irreducible varietythere exists a unique element in W for which C ∩ BwB is dense in C . We shalldenote this element by w C . We have C ⊂ C = C ∩ Bw C B ⊂ Bw C B = [ σ ≤ w C BσB so the element w C is the maximum among those w ∈ W for which BwB ∩ C isnon-empty (cfr. [5, Section 1]). The collection of B -orbits for the ∗ -action in C will be denoted by V . We will call maximal orbits the elements v in V lying in Bw C B and we shall denote by V max the set of maximal B -orbits in C . Definition 3.1
An element w ∈ W is called a θ -twisted involution if wθ ( w ) = 1 . If there is no ambiguity on the automorphism, we shall also say that w is atwisted involution. It is shown in [15] that w C is always a twisted involutionand a genuine involution in W , that it commutes with the automorphism θ of Φ and with the longest element w in W , and that it is of the form w w Π where Π is a suitable θ -invariant subset of ∆ and w Π is the longest element in W Π . Al-though the general assumption in the paper is that the base field is of characteristiczero, the arguments used in Section 3 from Lemma 3.1 until Proposition 3.7 arecharacteristic-free. The set Π is recovered from w C by the equality(3.1) Π = { α ∈ ∆ | w C θα = α } . The list of possible w C ’s for θ a non-trivial automorphism of the Dynkin diagramis provided in [15, Proposition 3.7]. We report here the list of possible pairs (Φ , Π) θ for completeness.(3.2) (Φ , ∅ ) for any Φ and any θ ; ( A n +1 , { α , α , . . . , α n +1 } ) θ = − w ;( D , { α } ) θ = 1;( D , { α , α i , θα i } ) θ = 1 and α i = θα i ; ( D n , { α l , α l +1 , . . . , α n − , α n } ) n > , ≤ l ≤ n − and θα n − = α n ;( D n +1 , { α l , α l +1 , . . . α n , α n +1 } ) n ≥ , ≤ l ≤ n , and θ = − w ; ( E , { α , α , α , α } ) θ = − w . By definition, maximal B -orbits in C are contained in Bw C B . By B ∗ conjugacy,we can make sure that every such orbit contains an element of the form ˙ w C v where v ∈ U and ˙ w C is a representative of w C in N ( T ) . We analyze now the possiblerepresentatives of a θ -twisted conjugacy class lying in a maximal B -orbit. Lemma 3.2
Let C be a θ -twisted conjugacy class and let w C = w w Π . Let x =˙ w C v ∈ C ∩ T w C U for some lift ˙ w C of w C in N ( T ) . Then v ∈ P uα for every α ∈ Π . Proof.
Let v = x α ( ξ )v ′ for some v ′ ∈ P uα and ξ ∈ k . Let ˙ s α ∈ N ( T ) be as inSection 2. We consider y = θ − ( ˙ s α ) ∗ x = θ − ( ˙ s α ) ˙ w C x α ( ξ )v ′ ˙ s − α = t ˙ w C ˙ s α x α ( ξ ) ˙ s − α v ′′ ∈ T ˙ w C x − α ( ηξ ) B for some t ∈ T , η ∈ k ∗ and v ′′ ∈ P uα . Here we have used (3.1).If ξ = 0 , then y ∈ Bw C Bs α B . Since w C α = θα is a positive root, then y ∈ C ∩ Bw C s α B with w C s α > w C in the Bruhat order, a contradiction. (cid:3) Lemma 3.3
Let C be a θ -twisted conjugacy class and let w C = w w Π . Let x =˙ w C v ∈ C ∩ T w C U for some lift ˙ w C of w C . Then v ∈ P u Π . Proof.
We will show by induction on the height of β ∈ Φ Π that once we fixan ordering of Φ + , the coefficient c β of x β in the expression of v as a productof elements in the root subgroups is trivial. We assume that the fixed orderingis compatible with the height of the roots. The basis of the induction is Lemma3.2. Let β be the first root in Φ Π for which c β = 0 and let its height be h . Then, v = v x β ( c β )v for some v ∈ P u Π and some v in a product of root subgroupsassociated with roots of height greater or equal than h and different from β . Since Φ is simply-laced, there exists w ∈ W Π such that ℓ ( w ) = h − and wβ = α ∈ Π .Let ˙ w be a lift of w in N ( T ) . We consider the following representative of C : y = θ − ( ˙ w ) ∗ x = θ − ( ˙ w ) x ˙ w − = θ − ( ˙ w ) ˙ w C v x β ( c β )v ˙ w − = t ˙ w C ( ˙ w v ˙ w − )( ˙ wx β ( c β ) ˙ w − )( ˙ w v ˙ w − ) = t ˙ w C v ′ x α ( ηc β )v ′ t ∈ T and some η ∈ k ∗ . The element ˙ w normalizes P u Π and by thehypothesis on the height also v ′ ∈ U , so v ′ , v ′ ∈ P uα . Hence, y ∈ T ˙ w C U . ByProposition 3.2 we necessarily have c β = 0 . (cid:3) Lemma 3.4
Let C be a θ -twisted conjugacy class and let α ∈ Φ + . Assume thatfor every x = ˙ w C v ∈ C ∩ T w C U the coefficient of x α in the expression of v as aproduct of elements in the root subgroups is zero for any ordering of the positiveroots. Then, for every such x the coefficient of x β in the expression of v is zero forevery β ∈ W Π α and for every ordering of the positive roots. Proof. If α ∈ Φ Π this is clear by Lemma 3.3 so we may assume α ∈ Φ + \ Φ Π .Let α = wβ with w ∈ W Π and let ˙ w be a lift of w in N ( T ) . We will write v = v x β ( c β )v for some v , v ∈ P u Π , products in root subgroups different from X β . We consider the element y = θ − ( ˙ w ) ∗ x = θ − ( ˙ w ) x ˙ w − = θ − ( ˙ w ) ˙ w C ˙ w − v ′ x α ( c ′ β )v ′ for some c ′ β ∈ k which is nonzero if and only if c β is nonzero. Since W Π δ ∈ Φ + for every δ ∈ Φ + \ Φ Π , we have v ′ , v ′ ∈ P u Π . Moreover, w C θγ = γ for every γ ∈ Φ Π so w − C θ − wθw C = w . Thus, y ∈ C ∩ T w C U and by the assumption c ′ β = 0 , whence the statement. (cid:3) Lemma 3.5
Let C be a θ -twisted conjugacy class and let x = ˙ w C v ∈ T w C U ∩ C .Then, [ L Π , L Π ] lies in the θ -stabilizer of ˙ w C . Proof.
Let α ∈ Π and let β = θα ∈ Π . We have θx α ( ξ ) = x β ( ξ ) . By Lemma 3.2we know that v ∈ P uα . We consider the following representative of C : y = x α ( ξ ) xθ ( x α ( − ξ )) = x α ( ξ ) ˙ w C v x β ( − ξ ) = ˙ w C x β ( ηξ )v x β ( − ξ ) for some η ∈ k ∗ . Here we have used that w C θα = α . By Lemma 3.2, wehave x β ( ηξ )v x β ( − ξ ) ∈ P uβ and this is possible only if η = 1 , that is, if the rootsubgroup X α lies in the θ -stabilizer of ˙ w C .Let us now consider − α and − β . Again we have θx − α ( ξ ) = x − β ( ξ ) . Weconsider the following representative of C : z = x − α ( ξ ) xx − β ( − ξ ) = x − α ( ξ ) ˙ w C v x − β ( − ξ )= ˙ w C x − β ( ηξ )v x − β ( − ξ ) ∈ ˙ w C x − β ( ηξ − ξ ) P uβ η ∈ k ∗ . If we had η = 1 we would have z ∈ ˙ w C Bs β B ⊂ Bw C s β B because w C β = θ − β ∈ Φ + . This would contradict maximality of w C , hence η = 1 and the root subgroup X − α lies in the θ -stabilizer of ˙ w C . (cid:3) When θ = 1 , the class G ∗ { gθ ( g ) − | g ∈ G } , extensively studiedin [20, 22, 19], intersects only Bruhat cells corresponding to twisted involutionsin W because θ ( gθ ( g ) − ) = ( gθ ( g ) − ) − . We are going to study all θ -twistedconjugacy classes sharing this property. Definition 3.6
Let C be a θ -twisted conjugacy class. We will say that C is invo-lutive if C ∩ BwB = ∅ only when w is a twisted involution. Remark 3.7 If Φ is of type D and θ is the automorphism of order mapping α to α , then there exist no involutive θ -twisted conjugacy classes. Indeed, givena representative x = ˙ w C v ∈ C ∩ T w C U , if w C = w s then the representative y = ˙ s ∗ x lies in Bs w s s B ∪ Bs w s B and both Weyl group elements arenot twisted involutions. If instead w C = w , then ˙ s ∗ x ∈ Bs w s B ∪ Bs w B and we conclude as above. In this section we will introduce spherical G -spaces and we will show that if θ isan involution, then every spherical θ -twisted conjugacy class is involutive. Definition 4.1
A transitive G -variety is called spherical if it has a dense B -orbit. The dense B -orbit is necessarily unique. It has been shown in [3, 29, 14] thata G -variety is spherical if and only if B acts on it with finitely many orbits.The following Lemma is a θ -twisted analogue of [7, Lemma 3.1]. We reportthe proof to keep the paper self contained. Lemma 4.2
Let C be a θ -twisted conjugacy class. The following are equivalent1. C is spherical.2. V max is a finite set. roof. One implication is immediate from the above remarks. We have C = C ∩ Bw C B = [ v ∈V max v. If V max is finite, then irreducibility of C forces v = C for some v ∈ V max . (cid:3) Let M ( W ) denote the monoid generated by the symbols r α for α ∈ ∆ subjectto the braid relations and the relation r α = r α . Given a spherical G -variety, thereare an M ( W ) -action and a W -action on the set of its B -orbits V . These actionshave been introduced in [19] and [14], respectively, and they have been furtheranalyzed and applied in [4], [17, § v ∈ V , the B -orbit r α ( v ) is thedense B -orbit in P α v . In order to introduce the W -action we need to provide morebackground information.Let v ∈ V . Then, the action of P α on P α /B ∼ = P defines a group morphism ψ : P α → P GL ( k ) whose kernel is Ker( α ) P uα . The stabilizer ( P α ) y of a point y in v acts on P α /B with finitely many orbits. The image H of ( P α ) y in P GL ( k ) is of one of the following types: P GL ( k ) ; solvable and contains a connectednontrivial unipotent subgroup; a torus; the normalizer of a torus. More precisely,we fall in one of the following cases:I P α v = v and H = P GL ( k ) ;II P α v = v ∪ v ′ = P α v ′ where v ′ = r α ( v ) or v = r α ( v ′ ) , with | dim v − dim v ′ | = 1 and H is solvable containing a nontrivial unipotent subgroup.III P α v = v ∪ v ′ ∪ v ′′ = P α v ′ = P α v ′′ , with v, v ′ , v ′′ distinct, where either v = r α ( v ′ ) = r α ( v ′′ ) and dim v = dim v ′ + 1 = dim v ′′ + 1 , or v ′ = r α ( v ) = r α ( v ′′ ) and dim v ′ = dim v + 1 = dim v ′′ + 1 , and H is a torus.IV P α v = v ∪ v ′ = P α v ′ where v ′ = r α ( v ) or v = r α ( v ′ ) , with | dim v − dim v ′ | = 1 and H is the normalizer of a torus.The W -action on V can be defined as follows ([14], [17, § s α interchanges the two B -orbits in case II; it interchanges thetwo non-dense orbits in case III and it fixes all B -orbits in types I and IV and thedense B -orbit in type III. The action of s α on v will be denoted by s α .v .We will now show that every v ∈ V can be reached from a closed one bymeans of a path in which each step is given either by the action of s α ∈ W or theaction of r β ∈ M ( W ) . This is formalized as follows.10 efinition 4.3 ([23, § X be a spherical G -variety and let V be its setof B -orbits. A reduced decomposition of v ∈ V is a pair ( v , s ) with v =( v (0) , v (1) , . . . , v ( r )) a sequence of elements in V and s = ( s i , . . . , s i r ) a se-quence of simple reflections such that: v (0) is closed; v ( j ) = r i j ( v ( j − for ≤ j ≤ r ; dim( v ( j )) = dim( v ( j − and v ( r ) = v . Remark 4.4
Every B -orbit v in a symmetric space admits a reduced decomposi-tion by [19, § G -variety X . On the other hand, for every closed B -orbit v in X , there exists a reduced decomposition of the dense B -orbit v with v (0) = v . Indeed, if r α v ′ = v ′ for v ′ ∈ V , then dim r α v ′ = dim v ′ + 1 , sowe may inductively construct a sequence ( v (0) , v (1) , . . . , v ( r )) with v (0) = v and v ( j ) = r i j ( v ( j − for ≤ j ≤ r . We can choose s i j so that it satisfies dim( v ( j )) = dim( v ( j − provided there is α ∈ ∆ such that r α v ( j − = v ( j − . The procedure will stop at some B -orbit v ( r ) such that r α ( v ( r )) = v ( r ) for every α ∈ ∆ . Then, v ( r ) = P α v ( r ) = P α v ( r ) where we have adapted the argument in [24, Exercise 6.2.11(5)]. Thus, X = G · v ( r ) = v ( r ) and therefore v ( r ) = v . The same argument shows that anysequence ( v (0) , v (1) , . . . , v ( r )) with v ( j ) = r i j ( v ( j − for ≤ j ≤ r and dim( v ( j )) = dim( v ( j − can be completed to a reduced decomposition ofthe unique dense B -orbit.A weaker notion of reduced decomposition exists for every v ∈ V . Definition 4.5 ([23, § subexpression of a reduced decomposition ( v , s ) =(( v (0) , . . . , v ( r )) , ( s i , . . . , s i r )) of v ∈ V is a sequence x = ( v ′ (0) , v ′ (1) , . . . , v ′ ( r )) of elements in V with v ′ (0) = v (0) and such that for ≤ j ≤ r only one of thefollowing alternatives occurs:(a) v ′ ( j −
1) = v ′ ( j ) ;(b) dim v ′ ( j −
1) = dim v ′ ( j ) − and v ′ ( j ) = r i j ( v ′ ( j − ;(c) v ′ ( j − = v ′ ( j ) , dim v ′ ( j −
1) = dim v ′ ( j ) and v ′ ( j ) = s i j .v ′ ( j − .The element v ′ ( r ) is called the final term of the subexpression.
11y [23, § v ′ ∈ V has a reduced decomposition ( v ′ , s ) , thenfor any v ∈ V which is contained in the closure of v ′ , there exists a subexpres-sion of ( v ′ , s ) with end-point v . In particular, for every v ∈ V , there exists asubexpression of any reduced decomposition of the dense B -orbit v admitting v as final term. The statement is given in characteristic zero but the proof holds alsoin positive odd characteristic.The following theorem is a generalization of [6, Theorem 2.7], from the un-twisted case. The argument has been shortened and simplified and it also worksfor θ trivial. Theorem 4.6
Let θ be an involution of the Dynkin diagram of G and let C be aspherical θ -twisted conjugacy class. Then C is involutive. Proof.
By [28, Lemma 7.3] we may choose a representative y ∈ B for C . Hence, B ∗ y ⊂ B and its closure contains a closed B -orbit x (0) lying in B . By Remark4.4 there is a reduced decomposition ( v , s ) of the dense B -orbit v with initialpoint x (0) . Let v ∈ V . By [23, § x = ( v ′ (0) , v ′ (1) , . . . , v ′ ( r )) of ( v , s ) with initial point v ′ (0) = x (0) and finalpoint v ′ ( r ) = v .We will show by induction on j that v ′ ( j ) lies in the Bruhat cell correspondingto a twisted involution. For j = 0 this is immediate. Let us assume that v ′ ( j − ⊂ Bw j − B with w j − a twisted involution. We consider the step from v ′ ( j − to v ′ ( j ) . If we are in case (a) of Definition 4.5 there is nothing to prove. If we arein case (b) let α = α i j . Then P α ∗ v ′ ( j − ⊂ Bw j − B ∪ Bs α Bw j − Bs θα B. According to [22, Lemma 3.2] there are three possibilities: • ℓ ( s α w j − s θα ) = ℓ ( w j − ) + 2 so P α ∗ v ′ ( j − ⊂ Bw j − B ∪ Bs α w j − s θα B and both Weyl group elements involved are twisted involutions; • s α w j − = w j − s θα so P α ∗ v ′ ( j − ⊂ Bw j − B ∪ Bs α w j − B and bothWeyl group elements involved are twisted involutions; • ℓ ( s α w j − s θα ) = ℓ ( w j − ) − so P α ∗ v ′ ( j − ⊂ Bw j − B ∪ Bs α w j − B ∪ Bw j − s θα B ∪ Bs α w j − s θα B. Since v ′ ( j ) = r α ( v ′ ( j − is dense in P α ∗ v ′ ( j − , it lies in a cellcorresponding to a σ ∈ W with σ ≥ w j − in the Bruhat order. Thus, v ′ ( j ) ⊂ Bw j − B . 12f we are in case (c) then we are necessarily in situation III in the description of the W -action on V and v ′ ( j − , v ′ ( j ) are the non-dense B -orbits in P α ∗ v ′ ( j −
1) = P α ∗ v ′ ( j ) . If ℓ ( s α w j − s θ ( α ) ) = ℓ ( w j − ) + 2 or if s α w j − = w j − s θα we mayproceed as in case (b). Let us assume that ℓ ( s α w j − s θα ) = ℓ ( w j − ) − so P α ∗ v ′ ( j − ⊂ Bw j − B ∪ Bs α w j − B ∪ Bw j − s θα B ∪ Bs α w j − s θα B. Let x ∈ T w j − U ∩ v ′ ( j − , and x ∈ U w j − T ∩ v ′ ( j − . We have y := ˙ s α ∗ x ∈ P α ∗ v ′ ( j − ∩ ( Bs α w j − B ∪ Bs α w j − s θα B ) y := ˙ s α ∗ x ∈ P α ∗ v ′ ( j − ∩ ( Bs α w j − s θα B ∪ Bw j − s θα B ) . Thus y , y ∈ v ′ ( j ) because there are only three B -orbits in P α ∗ v ′ ( j − andby the discussion of case (b) we have r α ( v ′ ( j − ⊂ Bw j − B . Hence, v ′ ( j ) ⊂ Bs α w j − s θα B and s α w j − s θα is a twisted involution. (cid:3) Remark 4.7
Theorem 4.6 fails if we drop the assumption on θ to be an involution.Indeed, in the triality case it has been shown in [15, Example 3.9] and [8, Section4.5] that the class G ∗ is spherical. However, it is not involutive by Remark 3.7. This section is devoted to the understanding of involutive θ -twisted conjugacyclasses so we shall assume that θ is an involution. We aim at getting some controlon the representatives of C in maximal B -orbits. Lemma 5.1
Let C be an involutive θ -twisted conjugacy class, let w C = w w Π with Π = ∅ and let x = ˙ w C v ∈ C ∩ T w C U . Then v ∈ P uα for every α ∈ ∆ suchthat α Π . Proof. If α ∈ Π this is Lemma 3.2. Let α ∈ ∆ \ Π , let v = x α ( c )v ′ with v ′ ∈ P uα .We consider y = θ − ( ˙ s α ) ∗ x = θ − ( ˙ s α ) x ˙ s − α = θ − ( ˙ s α ) ˙ w C ˙ s − α x − α ( c ′ )v ′′ for some v ′′ ∈ P uα and some c ′ ∈ k which is nonzero if and only if c is nonzero.If c ′ = 0 then y lies in T s θ − α w C s α Bs α B ∩ C . It follows from a straightforwardverification that if α Π we have w Π α ∈ Φ + \ ∆ , so β = w C α ∈ − (Φ + \ ∆) .13hus, s θ − α w C s α α ∈ Φ + and y ∈ C ∩ Bs θ − α w C B . However, s θ − α w C is not atwisted involution. Indeed, s θ − α w C θ ( s θ − α w C ) = s θ − α ( w C s α w C ) = s θ − α s β = 1 where we have used that w C is θ -invariant. Hence, c ′ = c = 0 and we have thestatement. (cid:3) In the spirit of [22] we define the following subsets of roots for w = w C = w w Π ∈ W a Weyl group element in the list (3.2). C w = { α ∈ Φ + | wθα ∈ − Φ + and wθα = − α } ,I w = { α ∈ Φ + | wθα = α } ,R w = { α ∈ Φ + | wθα = − α } . Such sets are called the set of complex, imaginary and real roots relative to w ,respectively.Since for α ∈ Φ + we have w Π α ∈ − Φ + if and only if α ∈ Φ Π , we have I w = Φ Π ∩ Φ + . Besides, Φ + is the disjoint union of I w , R w , and C w .The set R w is contained in the ( − -eigenspace of the orthogonal map wθ ,therefore it lies in I ⊥ w = Π ⊥ so, for every α ∈ R w we have w θα = − α . On theother hand, if β ∈ Π ⊥ ∩ Φ + and w θβ = − β then wθβ = θw w Π β = θw β = − β .Hence, we have R w = (cid:26) Π ⊥ ∩ Φ + if w = − θ , (Π ⊥ ∩ Φ + ) θ if w = − .The union R = R w ∪ ( − R w ) is a root subsystem of Φ and we may considerthe reductive subgroup G R = h T, X α | α ∈ R i . We may choose a set ∆ R = { γ , . . . , γ r } ⊂ Φ + of simple roots in R w so that B ∩ G R is a Borel subgroup of G R and U R = U ∩ G R is its unipotent radical. The subset ∆ R need not be a subsetof ∆ .The Weyl group W R of G R is generated by some reflections in W so it is asubgroup of W . Remark 5.2
Since the root system R of G R is Q -closed, it follows from [21, Sec-tion 3.5] that G R is the Levi factor of a parabolic subgroup of G . Hence, its derivedsubgroup is simply-connected and W R is conjugate to a parabolic subgroup of W . Proposition 5.3
With the above notation, let C be an involutive θ -twisted conju-gacy class in G and let x = ˙ w C v ∈ T w C U . Then v lies in U R . roof. If θ = − w and w C = w (i.e. Π = ∅ ) this condition is empty.The basic idea of the proof for all non-trivial cases is to exhibit, for β ∈ C w C ,a Weyl group element σ satisfying the following properties:(1) α = σβ ∈ ∆ ;(2) ˙ σ v ˙ σ − ∈ U for a representative ˙ σ ∈ N ( T ) ;(3) the root γ = θ − ( σ ) w C β lies in − (Φ + \ { θ − α } ) . Then, the element y = θ − ( ˙ σ ) ∗ x lies in T θ − ( σ ) w C σ − x α ( ηc β ) P uα for some η ∈ k ∗ . Hence, if the coefficient c β of x β in the expression of v is non-zero wehave z = θ − ( ˙ s α ) ∗ y ∈ T s θ − α θ − ( σ ) w C σ − s − α Bs α B. As γ lies in − Φ + and it is different from − θ − α , we have, for τ = s θ − α θ − ( σ ) w C σ − the inequality τ > τ s α so z lies in Bτ B ∩ C . However, τ θ ( τ ) = s θ − α θ − ( σ ) w C σ − s α σw C θ ( σ − )= s θ − α ( θ − ( σ ) w C s β w − C θ ( σ − ))= s θ − α s γ = 1 . Therefore, if c β is non-zero then C is not involutive.We discuss the different cases separately, according to the classification of the w C ’s in (3.2). Case ( A n +1 , { α , α , . . . , α n +1 } ) . We will show that C is the twisted conjugacyclass of a lift of w C . Let x = ˙ w C v ∈ T w C U ∩ C and let us assume that v = Q x γ ( c γ ) with c γ = 0 for γ of height smaller than h . Then h ≥ by Lemma 3.2and Lemma 5.1. Let β = α i + · · · + α j be a root of minimal height for which c β =0 . If j or i were odd then we could apply Lemma 3.4 obtaining a contradiction,so i and j are even. Then the Weyl group element σ = s j − s j − · · · s i satisfiesproperties (1), (2), (3) and we have the statement in this case. Case ( D n , { α j } j ≥ l ) , for n ≥ and ≤ l ≤ (cid:2) n − (cid:3) . Let θ be the automorphismof the Dynkin diagram of type D n interchanging α n and α n − . If n is even then w = − whereas if n is odd w = − θ . If C is an involutive θ -twisted conjugacyclass with w C = w w Π , for Π = { α j } j ≥ l with ≤ l ≤ m − if n = 2 m and ≤ l ≤ m if n = 2 m + 1 , we shall show that the coefficient of x β in theexpression of v is trivial for every β which is not orthogonal to Π . If β is simplethis is Lemma 5.1. By Lemma 3.4 it is enough to prove the statement for the rootsof the form β j = α j + · · · + α l − . Let β = β i be the root of minimum height15mong the β j ’s for which the coefficient is non-zero. Then β is not simple and σ = s i +1 · · · s l − satisfies properties (1), (2), (3), so c β j = 0 for every j . Theremaining cases in type D follow by symmetry. Case ( D n , ∅ ) . In this case the positive roots that are not θ -invariant are of theform β i = α i + · · · + α n − + α n − and θβ i for i ≤ n − . Let β = β j be theroot of minimal height of this form for which the coefficient in the expression of v is non-zero. Then β is not simple and σ = s j +1 · · · s n − s n − satisfies properties(1), (2) and (3). Hence, the coefficient of β i is zero for every i . The case of θβ j istreated similarly. Case ( E , { α , α , α , α } ) . In this case C is represented by a lift of w C in N ( T ) .Indeed, it follows from Lemma 3.2, Lemma 5.1 and Lemma 3.4 that v can beexpressed as a product in the root subgroups associated with the positive rootsoutside Φ Π , W Π α and W Π α , that is, the positive roots in the orbit W Π β for β = α + α + α + α + α . All positive roots in W Π β \ { β } have height strictlygreater than . Then, the Weyl group element σ = s s s s satisfies properties(1), (2) and (3) for the root β so v = 1 . This concludes the proof of Proposition5.3. (cid:3) An element in W is called a twisted-identity ([13]) if it is of the form wθ ( w ) − for some w ∈ W . A θ -twisted conjugacy class is called θ -semisimple if it has arepresentative in T ([25]). Corollary 5.4
Let C be an involutive θ -twisted conjugacy class such that w C fallsin one of the following cases: ( A n +1 , { α , α , . . . , α n +1 } ) , ( D n , { α , . . . , α n } ) , ( E , { α , α , α , α } ) . Then C is θ -semisimple. Proof.
In these cases C may be represented by some ˙ w C ∈ w C T . It is enough toshow that w C is a twisted identity because if w C = wθ ( w − ) for some w ∈ W ,then we have ˙ w − ∗ ˙ w C ∈ T ∩ C for every lift ˙ w of w in N ( T ) .In type A n +1 the element w C = w w Π is the permutation on n +2 letters withcyclic decomposition (1 2 n +1)(2 2 n +2)(3 2 n − n ) · · · ( n n +2)( n +1 n +3) for n odd and (1 2 n + 1)(2 2 n + 2)(3 2 n − n ) · · · ( n − n + 3)( n n + 4) with n + 1 and n + 2 fixed for n even. In both expressions, each transposition ( a b ) is followed by θ (( a b )) and since all transpositions in these expressions commute, w C = wθ ( w ) − is always a twisted identity. In type D n one may verify that w C = wθ ( w ) − for w = s α + α + ··· + α n − . In type E we have w C = wθ ( w ) − for w = s α + α + α +2 α +2 α + α . (cid:3) The intersections C ∩ ˙ w C U The aim of this section is to show that, unless we are in type D n and w C = w ,every involutive θ -twisted conjugacy class is spherical. The crucial step is Lemma6.1, where we conclude that it is enough to show that that the number of suitablerepresentatives of a maximal B -orbit in an involutive class is nonzero and finite. Lemma 6.1
Let C be a θ -twisted conjugacy class and let ˙ w C ∈ N ( T ) be suchthat C ∩ ˙ w C U = ∅ . If1. for every v ∈ V max there is z ∈ Z ( G ) such that v ∩ ˙ w C zU = ∅ ,2. | C ∩ ˙ w C zU | is finite for every z ∈ Z ( G ) then C is spherical. Proof.
Under the above assumptions we have: |V max | = P v ∈V max ≤ P v ∈V max P z ∈ Z ( G ) | v ∩ ˙ w C zU | = P z ∈ Z ( G ) | S v ∈V max v ∩ ˙ w C zU | = P z ∈ Z ( G ) | C ∩ ˙ w C zU | < ∞ where we used that Z ( G ) is finite. We conclude by using Lemma 4.2. (cid:3) Let C be an involutive θ -twisted conjugacy class and let ˙ w C ∈ N ( T ) be suchthat C ∩ ˙ w C U = ∅ . Then, for every v ∈ V max there is x ∈ ˙ w C tU ∩ v for some t ∈ T . It follows from Lemma 3.5 that [ L Π , L Π ] fixes ˙ w C and ˙ w C t under the θ -action, and it is easy to conclude that it centralizes t . Proposition 6.2
Let C be a θ -twisted conjugacy class and let ˙ w C U ∩ C = ∅ . Letus assume that, for w C , we are not in case ( D m , ∅ ) . Then, condition 1 in Lemma6.1 is satisfied. Proof.
Let us consider the morphism ψ : T → Ts ( ˙ w − C s ˙ w C ) θ ( s − ) . Then ψ is a group morphism so its image is closed ([24, Proposition 2.2.5]) andit lies in Z ( L Π ) by (3.1). It is also connected, so it lies in Z ( L Π ) ◦ . We recallthat dim Z ( L Π ) ◦ = rkG − | Π | . On the other hand, Ker ψ = T w C θ . By a sim-ple direct computation, we see that, for all cases except from ( D m , ∅ ) , we have dim Z ( L Π ) ◦ = dim T − dim T w C θ so in all those cases Im ψ = Z ( L Π ) ◦ .17et v ∈ V max and let x = ˙ w C t v ∈ v . Then for every s ∈ T we have s ∗ x = ˙ w C tψ ( s ) θ ( s )v θ ( s ) − ∈ v ∩ ˙ w C tψ ( s ) U. Thus, for every r ∈ Im( ψ ) = Z ( L Π ) ◦ we have v ∩ ˙ w C trU = ∅ . In the adjointquotient of G the center of any Levi factor of a parabolic subgroup is connected,so in G we have Z ( L Π ) = Z ( G ) Z ( L Π ) ◦ and t lies in zZ ( L Π ) ◦ = z Im( ψ ) forsome z ∈ Z ( G ) , whence the statement. (cid:3) In the following Lemmas we shall prove that if C is involutive then | C ∩ ˙ w C U | is finite for any ˙ w C ∈ N ( T ) . Lemma 6.3
Let C be an involutive twisted conjugacy class. Let ˙ w C be a rep-resentative of w C for which ˙ w C U ∩ C = ∅ . Let x = ˙ w C v ∈ C ∩ ˙ w C U , with v = Q γ ∈ R wC x γ ( c γ ) in a fixed ordering of R w C . Let α and β be adjacent simpleroots in ∆ R . Then, the number of possibilities for c α and c β is finite. Moreover,there is a k α,β in k depending only on the fixed ordering of the roots, on the struc-ture constants of G , and on ˙ w C , such that c α + β = k α,β c α c β . Proof.
Let P = P { α,β } be the standard parabolic subgroup of G R with unipo-tent radical P u . Let us assume that α precedes β in the ordering of the rootsin R w C . We may write: x = ˙ w C v ∈ ˙ w C x α ( c α ) x β ( c β ) x α + β ( c α + β ) P u . Let ˙ s γ = x γ (1) x − γ ( − x γ (1) as in Section 2, for γ ∈ { α, β } .The strategy of the proof is as follows: first we will show that for two precisevalues h and h of h ∈ k , depending on c α , the ordering, and the structureconstants of G , the element y ( h ) := θ − ( ˙ s α x α ( h )) ∗ x lies in Bw C s α B . Then,we will consider the elements θ − ( ˙ s β ) ∗ y ( h i ) for i = 1 , and we will detectthe Bruhat double cosets containing them. Imposing that this corresponds to atwisted involution will provide alternative necessary conditions on c α + β , c β , h and h . Then we will repeat the procedure interchanging the role of α and β ,obtaining new alternative necessary conditions. Combining all of them will yieldthe statement.We recall that for every ξ ∈ k ∗ (6.3) x α ( ξ ) x − α ( − ξ − ) x α ( ξ ) ∈ s α T. For h ∈ k we consider the family of representatives of C given by y ( h ) := θ − ( ˙ s α x α ( h )) ∗ x . Then, for some structure constants η , η , η , d αβ that are alwaysnon-zero in good characteristic, and for some t ∈ T we have: y ( h ) ∈ t ˙ w C ˙ s α x − α ( η h ) x α ( c α − h ) x α ( h ) x β ( c β ) x α + β ( c α + β ) x α ( − h ) ˙ s − α P u = t ˙ w C x α ( η h ) x − α ( η ( c α − h )) ˙ s α x β ( c β ) x α + β ( c α + β + hc β d αβ ) ˙ s − α P u . h and h be the solutions of X ( η η ) − c α η η X − so that − ( η h i ) − = ( c α − h i ) η and we may apply (6.3). The elements corre-sponding to h and h satisfy y ( h i ) ∈ C ∩ ˙ w C t ′ ˙ s α x β ( η ( c α + β + h i c β d αβ )) P uβ ⊂ C ∩ Bw C s α B for some t ′ ∈ T and some nonzero structure constant η . Here, P β denotes theminimal parabolic subgroup of G R associated with β .We let now θ − ( ˙ s β ) act on y ( h i ) for i = 1 , . We have, for some non-zero η ∈ k : θ − ( ˙ s β ) ∗ y ( h i ) ∈ Bw C s β s α s β x − β ( η ( c α + β + h i c β d αβ )) B. Moreover, w C s β s α s β β = θα holds because α, β ∈ R w C . Therefore, if we had c α + β + h i c β d αβ = 0 we would have θ − ( ˙ s β ) ∗ y ( h i ) ∈ C ∩ Bw C s β s α B , contra-dicting the assumption on C to be involutive. Thus(6.4) c α + β + h i c β d αβ = 0 . This condition must hold for both i = 1 , thus we have either h = h so that ∆ α = η η c α + 4 η η = 0 , or(6.5) c β = c α + β = 0 . (6.6)Let us now interchange the roles of α and β . We consider, for l ∈ k , the familyof elements z ( l ) = θ − ( ˙ s β x β ( l )) xx β ( − l ) ˙ s − β ∈ θ − ( ˙ s β x β ( l )) ˙ w C x β ( c β ) x α ( c α ) x α + β ( c α + β + c α c β d αβ ) x β ( − l ) ˙ s − β P u . Using the same procedure as above with β and α interchanged we see thatthere are nonzero structure constants ξ , ξ , such that if l and l are the solutionsof ξ X − c β ξ X − then z ( l j ) ∈ C ∩ ˙ w C ˙ s β T x α ( ξ ( c α + β + c α c β d αβ − l j c α d αβ )) P uα , j = 1 , , where P uα is as usual and d αβ is the structure constant occurring in(6.4).The action of θ − ( ˙ s α ) on z ( l j ) for j = 1 , would yield an element in C ∩ Bw C s α s β B unless(6.7) c α + β + c α c β d αβ − l j c α d αβ = 0 for both j = 1 , . This forces either l = l and therefore ∆ β = ξ c β + 4 ξ = 0 , or(6.8) c α = c α + β = 0 . (6.9)If c α = 0 then (6.5) does not hold so c α = c β = c α + β = 0 . In this situation any k α,β will do.If c α = 0 then (6.8) must hold so we have at most two choices for c β , and c β = 0 . Thus, (6.5) must hold and we have a finite number of possibilities for c α , too. In this case, by (6.4), we have c α + β = − c α c β d αβ so we may take k α,β = − d αβ . (cid:3) Lemma 6.4
Let C be an involutive θ -twisted conjugacy class and let ˙ w C ∈ N ( T ) be such that C ∩ ˙ w C U = ∅ . Let ∆ R = { γ , . . . , γ r } and let x = ˙ w C v ∈ C ∩ ˙ w C U .Then, for every γ = P rj =1 n j γ j ∈ R w C there is a polynomial p ˙ w C ,γ ( X ) ∈ k [ X j | n j = 0] without constant term, depending only on γ , ˙ w C , the fixed or-dering of the positive roots in R w C , and the structure constants of G , such that thecoefficient c γ of x γ in the expression of v is the evaluation of p ˙ w C ,γ ( X ) at X j = c γ j for every j = 1 , . . . , r in the support of γ . In particular, we have | C ∩ mU | < ∞ for every m ∈ w C T . Proof.
Without loss of generality we may assume that the ordering of the positiveroots is with increasing height. We shall proceed by induction on the height h ofthe root γ with respect to ∆ R . Let us assume that the claim holds for all γ with ht γ ≤ h − . Let ν ∈ R w C with ht ν = h . By Lemma 6.3 the statement holdsfor h ≤ , so we will assume that h is greater than . There exists β ∈ ∆ R forwhich ht s β ν = h − . The strategy will be to consider y = θ − ( ˙ s β ) ∗ x and tofind an element z in X θ − ( β ) ∗ y lying in w C B . The induction step will be obtainedby comparing the coefficient of x ν in the expression of v with the coefficient of x s β ( ν ) in the expression of u , for z = ˙ w C tu . We have(6.10) y = ( θ − ˙ s β ) ∗ x = ( θ − ˙ s β ) ˙ w C ˙ s − β ( ˙ s β v ˙ s − β ) = ˙ w C t Y γ ∈ R wC x s β γ ( η γ c γ ) η γ and some t ∈ T depending on ˙ s β and ˙ w C . Here, the product respects the fixed ordering of the γ ’s and not of the s β γ ’s.We have: y = ˙ w C t v x − β ( η β c β )v for some v , v ∈ P u , the unipotent radicalof the minimal standard parabolic subgroup P of G R associated with β ∈ ∆ R .Since the ordering is increasing in height, v is a product of elements of the form x s β γ ( η γ c γ ) for γ ∈ ∆ R .Let η ∈ k be such that x θ − β ( ηc β ) ˙ w C t = ˙ w C tx − β ( − η β c β ) and let us considerthe element z = x θ − β ( ηc β ) ∗ y . Then z = ˙ w C t ( x − β ( − η β c β )v x − β ( η β c β ))v x β ( − ηc β ) = ˙ w C tu = ˙ w C t Q γ ∈ R wC x γ ( d γ ) ∈ ˙ w C tU ∩ C where the product is taken according to the ordering of the positive roots in R w C . Here, the expression x − β ( − η β c β )v x − β ( η β c β ) is a product of terms ofthe form x s β γ ( η γ c γ ) for γ ∈ ∆ R such that s β γ = γ and terms of the form x s β γ ′ ( η γ ′ c γ ′ ) x s β γ ′ − β ( η γ ′ ,β c γ ′ c β ) with η γ ′ ,β a nonzero structure constant for γ ′ ∈ ∆ R such that s β γ ′ = γ ′ + β .By the induction hypothesis applied to z and s β ν , the coefficient d s β ν is eval-uation at the d α for α in the support of s β ν of a polynomial p ˙ w C t,s β ν ( X ) with-out constant term. Besides, each d µ differs from η s β µ c s β µ by a (possibly trivial)sum of monomials in the c µ ′ , c β , the structure constants η µ ′ , η β , and the structureconstants coming from application of Chevalley’s formula [24, Proposition 8.2.3]when reordering root subgroups. More precisely, we have(6.11) d µ = η s β µ c s β µ + X C j ,...,j q i ,...,i p ,j ( p Y l =1 c i l ν l ) c jβ q Y r =1 ( c γ ′ r c β ) j r where C j ,...,j q i ,...,i p ,j denotes a coefficient depending on the structure constants. Thesum is taken over the possible decompositions µ = p X l =1 i l s β ν l + jβ + q X r =1 j r ( s β γ ′ r − β ) for i l > , j r > and j ≥ and γ ′ r ∈ ∆ R such that s β γ ′ r = γ ′ r + β . Contributionto d s β ν as in (6.11) may occur only when(6.12) s β ν = p X l =1 i l s β ν l + jβ + q X r =1 j r ( s β γ ′ r − β ) i l , j r > and j ≥ . Then, ht s β ν l < ht s β ν = h − and s β γ ′ r − β = γ ′ r .Applying s β to (6.12) we have(6.13) ν + jβ = p X l =1 i l ν l + q X r =1 j r ( γ ′ r + β ) so the support of ν contains γ ′ r and the support of ν l . Since Φ is simply-laced, ht ν l ≤ ht s β ν l + 1 < h for every l . We may thus apply the induction hypothesisto c ν l . So c ν = η − ν p ˙ w C t,s β ν ( d γ i ) − η − ν X C j ,...,j q i ,...,i p ,j p Y l =1 ( p ˙ w C ,ν l ( c γ i )) i l ! c jβ q Y r =1 ( c γ ′ r c β ) j r . The statement is proved if we show that d γ i is a monomial in c γ i and possibly c β multiplied by a structure constant. If s β ( γ i ) = γ i then the coefficient d γ i of x γ i in u is equal to η γ i c γ i . If, instead, s β ( γ i ) = γ i + β and γ i follows β in the orderingof the positive roots, then the coefficient d γ i of x γ i in u equals η γ i + β c γ i + β , whichis as required by Lemma 6.3. Finally, if s β ( γ i ) = γ i + β and γ i precedes β inthe ordering of the positive roots, then the coefficient d γ i of x γ i in u is the sumof η γ i + β c γ i + β with the correction term obtained from moving x − β ( η β c β ) from theright of v to the left. By Chevalley’s commutator formula, this correction termequals Kc γ i c β for some product K of structure constants. Thus, c ν is evaluation ofa polynomial without constant term depending only on the structure constants, onthe choice of ˙ w C , and the fixed ordering of the roots. The last statement followsfrom Lemma 6.3 because when w = − , if R is non-empty, then it is alwaysirreducible of rank greater than . (cid:3) Remark 6.5
It follows from Lemma 6.4 that if v ∈ P uα for every α ∈ ∆ R then v = 1 so x = ˙ w C . In particular, by the proof of Lemma 6.3, this condition holdsif v ∈ P uα for some α ∈ ∆ R because R is either. empty or irreducible of rankgreater than .Combining Lemma 6.1, Proposition 6.2, and Lemma 6.4 we have the follow-ing result. Theorem 6.6
Let C be an involutive θ -twisted conjugacy class in G . If for w C = w w Π we have (Φ , Π) = ( D n , ∅ ) , then C is spherical. The case of ( D n , ∅ ) Let us now consider the case of involutive θ -twisted conjugacy classes C in type D n with w C = w . In this case ∆ R = { α , . . . , α n − , α n − + α n − + α n } whereas C w ∩ ∆ = { α n − , α n } . Let G ′ R = [ G R , G R ] where G R is as in Section5. The automorphism θ stabilizes G ′ R and it acts trivially on it. Lemma 7.1
Let C be an involutive θ -twisted conjugacy class in type D n with w C = w and let x = ˙ w − v ∈ T w B ∩ C . Then, the G ′ R -orbit G ′ R · θ x isspherical. Proof.
By Remark 5.2 the semisimple group G ′ R is simply-connected. It is in factsimple of type D n − . Let B R = B ∩ G ′ R and let T R be the maximal torus of G ′ R , generated by the elements of the form h γ ( ξ ) for γ ∈ ∆ R (see Section 2). ByProposition 5.3 we have x = ˙ w − v ∈ T w U R ∩ C . We may choose a representa-tive ˙ w R of the longest element of the Weyl group of G R in N ( T ) ∩ G ′ R . Conjuga-tion by ˙ w R ˙ w stabilizes T , B R and T R and it induces a non-trivial automorphismof R . Thus, for some t ∈ T R , conjugation by t ˙ w R ˙ w is the automorphism inducedby the non-trivial involution τ of the Dynkin diagram of G ′ R . Let g ∈ G ′ R . Wehave θ ( g ) = g so g · θ x = gxg − and the morphism f : G ′ R · θ x → G ′ R · τ ( t ˙ w R v) − z ( t ˙ w R ˙ w z ) − is a G ′ R -equivariant isomorphism. So, it is enough to show that the G ′ R -variety G ′ R · τ ( t ˙ w R v) − is spherical. We shall show that G ′ R · τ ( t ˙ w R v) − is involutive. Thestatement will follow from Theorem 6.6. Let σ be an element in the Weyl group of G ′ R such that G ′ R · τ ( t ˙ w R v) − ∩ B R σB R = ∅ and let y ∈ G ′ R · τ ( t ˙ w R v) − ∩ B R σB R .Then f − ( y ) = ˙ w − ˙ w − R t − y − ∈ ˙ w − ˙ w − R t − B R σ − B R = B R ˙ w − ˙ w − R σ − B R ⊂ Bw − w − R σ − B ∩ C. Since C is θ -involutive and w R and σ are all θ -invariant because they are productsof θ -invariant reflections, we have w − w − R σ − w − w − R σ − = 1 . The involu-tions w and w R commute because w acts trivially on each reflection in W ∆ R so ( w R w σ − w − w − R ) σ − = 1 and στ ( σ ) = 1 . (cid:3) Lemma 7.2
Let C be an involutive θ -twisted conjugacy class in type D n with w C = w and let x = ˙ w v ∈ T w B ∩ C . Let α ∈ ∆ R . Then for all but finitelymany ξ ∈ k the set x α ( ξ ) s α B ∩ G x is non-empty. roof. We have θα = α and θ ( x α ( ξ )) = x α ( ξ ) . Let v = x α ( c )v ′ ∈ x α ( c ) P uα andlet us consider the following representatives of C , for ξ ∈ k : y ξ = x α ( ξ ) · θ x = x α ( ξ ) ˙ w v x α ( − ξ ) = ˙ w x − α ( ηξ ) x α ( c − ξ )v ′′ for some nonzero structure constant η and some v ′′ ∈ P uα , and z ξ = ˙ s α · θ y ξ = ˙ s α ˙ w x − α ( ηξ ) x α ( c − ξ )v ′′ ˙ s − α ∈ ˙ w tx α ( η ′ ξ ) x − α ( η ′′ ( c − ξ )) P uα for some nonzero structure constants η ′ , η ′′ and some t ∈ T . It follows from (6.3)that if η ′ ξη ′′ ( c − ξ ) = − then z ξ ∈ Bw B .Let v be the dense B R -orbit in G ′ R · θ x and let BσB be the Bruhat double cosetin G containing it. Then G ′ R · θ x = v ⊂ ( S ω ≤ σ BωB ) . Since Bw B ∩ G ′ R · θ x isnon-empty we necessarily have σ = w .Moreover, using uniqueness of the Bruhat decomposition as in [15, Lemma2.1] , [8, Theorem 4.1] or [5, Theorem 5], we may show that the θ -centralizer in B R ⊂ U R T of an element in w B is finite. This shows that every B R -orbit v in G ′ R · θ x which is contained in Bw B has the same dimension as the dense one, thatis dim v = dim B R = dim v . Therefore, v must coincide with v . Thus, z ξ and x lie in v and there is b ξ ∈ B R such that b ξ · θ z ξ = x . In other words, for every ξ butfinitely many there is an element in G x ∩ B R ˙ s α x α ( ξ ) ⊂ G x ∩ B ˙ s α x α ( ξ ) . Takinginverses we have the statement. (cid:3) Lemma 7.3
Let C be an involutive θ -twisted conjugacy class in type D n with w C = w and let x = ˙ w v ∈ w T U ∩ C . Let α ∈ ∆ ∩ C w . Then for every ξ ∈ k the set x − α ( ξ ) U ∩ G x = ∅ . Proof.
The simple root α is either α n − or α n so α ± θα Φ . Let us considerthe following representatives of C for ξ ∈ k : y ξ = x θα ( ξ ) xx α ( − ξ ) = ˙ w x − θα ( ηξ )v x α ( − ξ ) for some non-zero structure constant η and z ξ = x − α ( ηξ ) y ξ x − θα ( − ηξ ) = ˙ w x α ( η ′ ξ ) x − θα ( ηξ )v x − θα ( − ηξ ) x α ( − ξ ) for some nonzero structure constant η ′ .The element x α ( η ′ ξ ) x − θα ( ηξ )v x − θα ( − ηξ ) x α ( − ξ ) lies in U because v lies in P uθα by Proposition 5.3. Applying the Proposition once more to z ξ ∈ C ∩ ˙ w U wesee that η ′ = 1 . 24et us fix an ordering of the positive roots so that all θ -invariant roots precedethe non-invariant ones. We recall that the non-invariant positive roots are of theform β i = α i + · · · + α n − + α or θβ i . It follows from Chevalley’s commutatorformula [24, Proposition 8.2.3] that x − θα ( ηξ )v x − θα ( − ηξ ) = vv ′ where v ′ lies inthe abelian subgroup Y α of U generated by the root subgroups associated with the β i ’s. Besides, x α ( ξ )vv ′ x α ( − ξ ) = vv ′ v ′′ where v ′′ lies again in Y α . By Proposition5.3 we conclude that v ′ v ′′ = 1 so z ξ = x and x − α ( ηξ ) x θα ( ξ ) ∈ G x . Since η is afixed non-zero structure constant and the statement holds for every ξ , we have thestatement. (cid:3) Lemma 7.4
Let X be a transitive G -variety and let x ∈ X . If, for every α ∈ ∆ we have x α ( ξ ) s α B ∩ G x = ∅ for all but finitely many ξ ∈ k , then the space X isspherical and B · x is the dense B -orbit. Proof.
It is enough to show that BG x or, alternatively, G x B , is dense in G . Wewill do so by showing that G x B ∩ Bw B is dense in Bw B .Let U w be the subgroup generated by the root subgroups associated with rootsin Φ w = { α ∈ Φ + | w − α ∈ − Φ + } . Then BwB = U w wB and, once we havefixed an ordering of the roots in Φ w , we may identify U w wB/B ⊂ G/B withthe affine space A ℓ ( w ) through the map uwB = Q γ ∈ Φ w x γ ( c γ ) wB ( c γ ) γ ∈ Φ w .We will show by induction on the length ℓ ( w ) of w that the set U w of elements u in U w for which uwB ∩ G x is non-empty contains the complement of the unionof finitely many hyperplanes in U w ∼ = A ℓ ( w ) . For w = 1 there is nothing to say.Suppose that the claim holds for ℓ ( w ) = l . We consider ω ∈ W with ℓ ( ω ) = l + 1 .Then ω = σs α for some σ ∈ W with ℓ ( σ ) = l and some α ∈ ∆ with σα ∈ Φ + .Besides, Φ ω = Φ σ ∪ { σα } so U ω = U σ X σα .By the hypothesis, for all but finitely many ξ ∈ k and for every u ∈ U σ thereis a b ∈ B for which ( u ˙ σb )( x α ( ξ ) ˙ s α B ) ∩ G x = ∅ . Let b = x α ( r )v for r ∈ k and v ∈ P uα . Then for some nonzero structure constant η we have ( u ˙ σb )( x α ( ξ ) s α B ) = u ˙ σx α ( r + ξ ) s α B = ux σα ( η ( r + ξ )) σs α B and ux σα ( η ( r + ξ )) σs α B ∩ G x = ∅ . Since all but finitely many ξ were allowedand η = 0 the intersection G x ∩ BωB contains U σ x σα ( ξ ) ωB for all but finitelymany ξ , thus U ω contains the complement of finitely many hyperplanes in A ℓ ( ω ) . (cid:3) Proposition 7.5
Let C be an involutive θ -twisted conjugacy class in type D n with w C = w . Then C is spherical. roof. We have ∆ = (∆ ∩ R w ) ∪ (∆ ∩ C w ) . By Lemmas 7.2, 7.3 and formula(6.3) the hypotheses of Lemma 7.4 are satisfied. (cid:3) Remark 7.6
Let π : G → H be a central isogeny of simple groups with G simply-connected and suppose that the automorphism θ of G preserves Ker( π ) . Thenit preserves the character group of the maximal torus T H = π ( T ) of H . Theautomorphism θ induces an automorphism θ of H such that θ ◦ π = π ◦ θ . Thus,the θ -twisted conjugacy classes of G are mapped onto θ -twisted conjugacy classesof H . Clearly, the Bruhat cells they intersect correspond to the same Weyl groupelements. Moreover, it is not hard to verify that C is spherical if and only if π ( C ) is so. This allows the generalization of the obtained results from simply-connectedgroups to a more general setting.Combining Theorem 4.6, Theorem 6.6, Proposition 7.5 and Remark 7.6 weobtain our main result. Theorem 7.7
Let G be a simple algebraic group over an algebraically closedfield of good odd characteristic. Let B be a Borel subgroup of G and T a maximaltorus in B . Let θ be an involution of the Dynkin diagram of G preserving thecharacter group of T . A θ -twisted conjugacy class C is spherical if and only if itlies in S wθ ( w )=1 BwB . Remark 7.8
The above theorem can be viewed as an analogue of [7, Theorem5.7] for non-connected semisimple groups with simple identity component . In this section we will show how we get [15, Theorem 1.1] in good, odd charac-teristic, for θ a non-trivial involution of the Dynkin diagram, as a by-product ofthe results in the previous sections. Proposition 8.1
Let B ∗ x be a maximal B -orbit in an involutive θ -twisted con-jugacy class C . Then (8.14) dim B ∗ x = ℓ ( w C ) + rk(1 − w C θ ) . roof. Let us choose x = ˙ w C v ∈ C ∩ T w C U . We recall that Φ +Π is the set of rootswhose positivity is preserved by w C so dim U Π = | Φ + | − ℓ ( w C ) . It follows fromuniqueness of the Bruhat decomposition (see [8, Theorem 4.1] or [15, Lemma2.1]) that the θ -stabilizer of x in B = U w C U Π T , with notation as in the proof ofLemma 7.4, is contained in U Π T w C θ . Hence, dim B ∗ x ≥ ℓ ( w C ) + rk(1 − w C θ ) .Let α ∈ Π . Then x α ( t ) ∗ ˙ w C v = ( x α ( t ) ∗ ˙ w C )( x θα ( t )v x θα ( − t )) . By Lemma 3.5 we have x α ( t ) ∗ ˙ w C = ˙ w C and by Proposition 5.3 we have x θα ( t )v x θα ( − t ) = v , so x α ( t ) lies in the θ -stabilizer of x , and therefore the sameholds for all elements in U Π . Moreover, the maximal torus T Π of [ L Π , L Π ] gener-ated by the h α ( ζ ) for α ∈ Π is contained in ( T w C θ ) ◦ . It is not hard to verify by a di-mensional argument that ( T w C θ ) ◦ is equal to T Π for all choices for w C except from ( D n , ∅ ) . Since Π ⊥ R , Lemma 3.5 and Proposition 5.3 imply B ◦ x = ( T w C θ ) ◦ U Π and the statement.Let us now consider the case ( D n , ∅ ) . In this case B x ⊂ T w C θ and ( T w C θ ) ◦ is the -dimensional torus of the elements h ξ = h α n − ( ξ ) h α n ( ξ − ) , for ξ ∈ k ∗ .These elements certainly lie in the θ -stabilizer of ˙ w C . We have h ξ ∗ x = ( h ξ ∗ ˙ w C ) θ ( h ξ )v θ ( h ξ ) − = ˙ w C h − ξ v h ξ . By Proposition 5.3 the element h ξ centralizes v because the roots in R w are orthogonal to the − eigenspace of θ . We have ( B x ) ◦ = ( T w C θ ) ◦ and the statement. (cid:3) The main result of this section follows:
Theorem 8.2
Let G be a simple group over an algebraically closed field of good,odd characteristic. Let θ be an involution of its Dynkin diagram and let us assumethat the character group of T is θ -invariant. Then, a θ -twisted conjugacy class C is spherical if and only if dim C = ℓ ( w C ) + rk(1 − w C θ ) . Proof.
Let us assume first that G is simply-connected. If C is spherical thenits dense B -orbit v is necessarily maximal so dim C = dim v . Moreover, C isinvolutive by Theorem 4.6 so Proposition 8.1 yields the statement in this case. Forthe general case we use Remark 7.6.If dim C = ℓ ( w C ) + rk(1 − w C θ ) we argue as in [5],[8] or [15]. The idea isto take a representative x of C in w C B and use uniqueness of the Bruhat decom-position in order to show that B x is contained in T w C θ U Π . This implies that thedimension of the B -orbit of x equals the dimension of C . (cid:3) emark 8.3 The dimension formula in [15] is stated in characteristic zero andit generalizes to θ non-trivial the dimension formula in [5, 6]. The proof worksalso in positive characteristic provided that some requirements on the base fieldlisted in [15, Remark 2.3] hold. The present proof covers also the case in whichthe characteristic of k is not very good , i.e., char k divides n + 1 in type A n . Inthis case, the orbit map to a twisted conjugacy class is not necessarily separable([18, Page 380]), so the requirements in [15, Remark 2.3] are not satisfied. On theother hand, [15] covers the triality case, whereas the present approach does notreach the case of triality with w C = w s . The dimension formula in the trialitycase when w C = w easily follows from the fact that ( T w C θ ) ◦ = 1 in this case, sothe argument in [5, Theorem 5] already shows that the dimension of a B -orbit in Bw B is equal to the dimension of B , which is equal to ℓ ( w ) + rk(1 − w θ ) . I wish to thank Kei Yuen Chan for pointing out an inaccuracy in a first versionof the paper, Jiang-Hua Lu for answering several questions, and the referees forcareful reading, pointing out critical points in the proof of Lemma 6.4 and manyother useful suggestions, corrections and remarks. This paper was typed on acomputer supported by Project CPDA071244 of the University of Padova.
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