aa r X i v : . [ m a t h . G R ] J a n PARABOLIC DELIGNE-LUSZTIG VARIETIES.
FRANC¸ OIS DIGNE AND JEAN MICHEL
Abstract.
Motivated by the Brou´e conjecture on blocks with abelian defectgroups for finite reductive groups, we study “parabolic” Deligne-Lusztig va-rieties and construct on those which occur in the Brou´e conjecture an actionof a braid monoid, whose action on their ℓ -adic cohomology will conjecturallyfactor through a cyclotomic Hecke algebra. In order to construct this action,we need to enlarge the set of varieties we consider to varieties attached to a“ribbon category”; this category has a Garside family , which plays an impor-tant role in our constructions, so we devote the first part of our paper to thenecessary background on categories with Garside families. Introduction
In this paper, we study “parabolic” Deligne-Lusztig varieties, one of the mainmotivations being the Brou´e conjecture on blocks with abelian defect groups forfinite reductive groups.Let G be a connected reductive algebraic group over an algebraic closure F p ofthe prime field F p of characteristic p . Let F be an isogeny on G such that somepower F δ is a Frobenius endomorphism attached to a split structure over the finitefield F q δ ; this defines a positive real number q such that q δ is an integral power of p . When G is quasi-simple, any isogeny F such that the group of fixed points G F is finite is of the above form; such a group G F is called a “finite reductive group”or a “finite group of Lie type”.Let L be an F -stable Levi subgroup of a (non necessarily F -stable) parabolicsubgroup P of G . Then, for ℓ a prime number different from p , Lusztig has con-structed a “cohomological induction” R GL which associates with any Q ℓ L F -modulea virtual Q ℓ G F -module. We study the particular case R GL (Id), which is given bythe alternating sum of the ℓ -adic cohomology groups of the variety X P = { g P ∈ G / P | g P ∩ F ( g P ) = ∅} on which G F acts on the left. We will construct a monoid of endomorphisms M of X P related to the braid group, which conjecturally will induce in some cases anaction of a cyclotomic Hecke algebra on the cohomology of X P . To construct M we need to enlarge the set of varieties we consider, to include varieties attached tomorphisms in a “ribbon category” — the “parabolic Deligne-Lusztig varieties” ofthis paper; M corresponds to the endomorphisms in the “conjugacy category” ofthis ribbon category of the object attached to X P .The relationship with Brou´e’s conjecture for the principal block comes as fol-lows: assume, for some prime number ℓ = p , that a Sylow ℓ -subgroup S of G F isabelian. Then Brou´e’s conjecture [Br1] predicts in this special case an equivalence This work was partially supported by the “Agence Nationale pour la Recherche” project“Th´eories de Garside” (number ANR-08-BLAN-0269-03). of derived categories between the principal block of Z ℓ G F and that of Z ℓ N G F ( S ).Now L := C G ( S ) is a Levi subgroup of a (non F -stable unless ℓ | q −
1) parabolicsubgroup P ; restricting to unipotent characters and discarding an eventual torsionby changing coefficients from Z ℓ to Q ℓ , this translates after refinement (see [BM])into conjectures about the cohomology of X P (see 9.1); these conjectures predictthat the image in the cohomology of our monoid M is a cyclotomic Hecke algebra.The main feature of the ribbon categories we consider is that they have Garsidefamilies . This concept has appeared in recent work to understand the ordinaryand dual monoids attached to the braid groups; in the first part of this paper, werecall its basic properties and go as far as computing the centralizers of “periodicelements”, which is what we need in the applications. The reader who wants toavoid the general theory of Garside families can try to read only Section 5 wherewe spell out the results in the case of Artin monoids.In the second part, we first define the parabolic Deligne-Lusztig varieties whichare the aim of our study, and then go on to establish their properties. We extendto this setting in particular all the material in [BM] and [BR2].We thank C´edric Bonnaf´e and Rapha¨el Rouquier for discussions and an initialinput which started this work, and Olivier Dudas for a careful reading and manysuggestions for improvement.After this paper was written, we received a preprint from Xuhua He and SianNie (see [HN]) where, amidst other interesting results, they also prove Theorem8.1.
I. Garside families
This part collects some prerequisites on categories with Garside families. It ismostly self-contained apart from the next section where the proofs are omitted; werefer for them to [DDM] or the book [DDGKM] in preparation.2.
Basic results on Garside families
Given a category C , we write f ∈ C to say that f is a morphism of C , and wewrite C ( x, y ) (resp. C ( x, -), resp. C (- , y )) for the set of morphisms from x ∈ Obj C to y ∈ Obj C (resp. the set of morphisms with source x , resp. the set of morphismswith target y ). We write f g for the composition of f ∈ C ( x, y ) and g ∈ C ( y, z ), and C ( x ) for C ( x, x ). By S ⊂ C we mean that S is a set of morphisms in C .Recall that a category is cancellative if each one of the relations hf = hg or f h = gh implies f = g ; equivalently every morphism is a monomorphism and anepimorphism. We say that f left-divides g , or equivalently that g is a right-multipleof f , written f g , if there exists h such that g = f h ; in this situation since thecategory is cancellative h is uniquely defined by g and f and we write h = f − g .Similarly we say that f right-divides g , or that g is a left-multiple of h and write g < f if there exists h such that g = hf .We denote by C × the set of invertible morphisms of C , and write f = × g if thereexists h ∈ C × such that f h = g (or equivalently there exists h ∈ C × such that f = gh ). Definition 2.1.
In a cancellative category C a Garside family is a subset S ⊂ C such that; (i) S together with C × generates C , and C × S ⊂ SC × ∪ C × . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 3 (ii)
For every product f g with f, g ∈ S , we can write f g = f g with f , g ∈ S such that if for k ∈ C and h ∈ S we have h kf g then h kf . If item (ii) of the above definition holds we say that the 2-term sequence ( f , g )is an S -normal decomposition of f g . We extend this notion first to the case where f, g ∈ SC × ∪ C × by requiring the same condition but with f , g ∈ SC × ∪ C × ; weextend then S -normal decompositions to longer lengths by saying that ( x , . . . , x n )is an S -normal decomposition of x = x . . . x n if for each i the sequence ( x i , x i +1 )is an S -normal decomposition. We finally extend it to elements x ∈ SC × ∪ C × bysaying that ( x ) is an S -normal decomposition.In a cancellative category with a Garside family every element x admits an S -normal decomposition. We will just say “normal decomposition” if S is clear fromthe context. A normal decomposition ( x , . . . , x n ) is strict if no entry is invertibleand all entries excepted possibly x n are in S . In a cancellative category with a Gar-side family every non-invertible element admits a strict S -normal decomposition.Normal decompositions are unique up to invertible elements, precisely Lemma 2.2 ([DDM, 2.11]) . If ( x , . . . , x n ) and ( x ′ , . . . , x ′ n ′ ) with n ≤ n ′ are twonormal decompositions of x then for any i ≤ n we have x · · · x i = × x ′ · · · x ′ i andfor i > n we have x ′ i ∈ C × . Head functions.Definition 2.3.
Let C be a cancellative category and let S ⊂ C . Then we saythat a function
C − C × H −→ S is an S -head function if for any h ∈ S , we have h g ⇔ h H ( g ) . We say that a subset
S ⊂ C is closed under right-divisor if f < g with f ∈ S implies g ∈ S . We have the following criterion to be Garside: Proposition 2.4 (see [DDM, 3.10 and 3.34]) . Assume that C is a cancellativecategory and that S ⊂ C together with C × generates C . Consider the followingproperty for an S -head function: ( H ) ∀ f ∈ C , ∀ g ∈ C − C × , H ( f g ) = × H ( f H ( g )) . Then S is Garside if there exists an S -head function satisfying ( H ) or there existsan S -head function and SC × ∪ C × is closed under right-divisor. Conversely if S is Garside then SC × ∪ C × is closed under right-divisor and any S -head functionsatisfies ( H ). An S -head function H computes the first term of a normal decomposition in thesense that if ( x , . . . , x n ) is a normal decomposition of x ∈ C − C × then H ( x ) = × x . Further any x ∈ C − C × has a strict normal decomposition ( x , . . . , x n ) with H ( x ) = x .Let C be a cancellative category with a Garside family S . For f ∈ C we de-fine lg S ( f ) to be the minimum number k of morphisms s , . . . , s k ∈ S such that s · · · s k = × f , thus lg S ( f ) = 0 if f ∈ C × ; if f / ∈ C × then lg S ( f ) is also the numberof terms in a strict normal decomposition of f .The following shows that S “determines” C up to invertible elements; we say thata subset C of C is closed under right-quotient if an equality f = gh with f, g ∈ C implies h ∈ C . F. DIGNE AND J. MICHEL
Lemma 2.5 ([DDGKM, VII 2.13]) . Let C be a subcategory of C closed underright-quotient which contains S . Then C = C C × ∪ C × and S is a Garside familyin C . Categories with automorphism.
Most categories we want to consider will haveno non-trivial invertible element, which simplifies Definition 2.1(i) to “ S generates C ”. The only source of invertible elements will be the following construction.An automorphism of a category C is a functor F : C → C which has an inverse.Given such an automorphism we define
Definition 2.6.
The semi-direct product category C ⋊ h F i is the category whoseobjects are the objects of C and whose morphisms are the pairs ( g, F i ) , which will bedenoted by gF i , where g ∈ C and i is an integer. The source of gF i is source( g ) andthe target of gF i is F − i (target( g )) . The composition rule is given by gF i · hF j = gF i ( h ) F i + j when source( h ) = target( gF i ) . Note that we do not identify ( g, F i ) and ( g, F j ) even when F i − j is the identityfunctor — it will be convenient in our semi-direct products to have the cyclic groupgenerated by F to be infinite even though F acts via a finite order automorphism.The conventions on F are such that the composition rule is natural. However,they imply that the morphism (Id , F ) of the semi-direct product category representsthe functor F − : it is a morphism from the object F ( A ) to the object A and wehave the commutative diagram: F ( A ) F ( f ) / / F (cid:15) (cid:15) F ( B ) F (cid:15) (cid:15) A f / / B C embeds in C ⋊ h F i by identifying g and ( g, F ). Lemma 2.7 ([DDGKM, VIII 1.34 (ii)]) . If S is a Garside family in the cancellativecategory C , and F an automorphism of C preserving S , then S is also a Garsidefamily in C ⋊ h F i . If ( f , . . . f k ) is an S -normal decomposition of f ∈ C then ( f , . . . , f k F i ) is an S -normal decomposition of f F i ∈ C ⋊ h F i . Note that if C has no non-trivial invertibleelement, then the only invertible elements in C ⋊ h F i are { F i } i ∈ Z . In general, if a, b ∈ C then aF i bF j if and only if a b .We have the following property Proposition 2.8 ([DDGKM, VII 4.4]) . Assume that the cancellative category C has a Garside family S and has no non-trivial invertible morphisms. Let F bean automorphism of C preserving S . Then the subcategory of fixed objects andmorphisms C F has a Garside family which consists of the fixed points S F . Gcds and lcms, Noetherianity.
We call right-lcm of a family C ⊂ C a right-multiple f of all morphisms in C such that for any other common right-multiple f ′ we have f f ′ ; this corresponds to the categorical notion of a pullback. Similarly a left-gcd of the family C is a common left-divisor f such that for any other commonleft-divisor f ′ we have f ′ f ; it corresponds to the notion of a pushout. Left-lcmsand right-gcds are defined in the same way exchanging left and right. ARABOLIC DELIGNE-LUSZTIG VARIETIES. 5
The existence of left-gcds and right-lcms are related when the cancellative cat-egory C is right-Noetherian, which means that there is no infinite sequence f < f < · · · < f n < · · · where f i +1 is a proper right-divisor of f i , that is we do not have f i = × f i +1 . It means equivalently since C is cancellative that there is no infinitesequence g g · · · g n · · · g where g i is a proper left-divisor of g i +1 .The equivalence is obtained by f i = g − i g and g = f . In a right-Noetherian cate-gory any element is right-divisible by an atom , which is an element which cannot bewritten as the product of two non-invertible elements. If the category is Noetherian(that is, both left and right-Noetherian) we have: Proposition 2.9 ([DDGKM, II 2.64]) . A cancellative and Noetherian category isgenerated by its atoms and its invertible elements.
We say that C admits conditional right-lcms if, whenever f and g have a commonright-multiple, they have a right-lcm. We then have: Proposition 2.10 ([DDGKM, II 2.41]) . If C is cancellative, right-Noetherian andadmits conditional right-lcms, then any family of morphisms of C with the samesource has a left-gcd. If C admits conditional right-lcms we say that a subset X ⊂ C is closed (resp.weakly closed) under right-lcm if whenever two elements of X have a right-lcm in C this lcm is in X (resp. in X C × ). If further X is closed under right-quotient anlcm in C which is in X is also an lcm in X . The following is proved in [DDM,Proposition 3.25] (where there is a Noetherianity assumption not used in the directpart of the proof). Lemma 2.11. If S is a Garside family in a category which admits conditionalright-lcms then SC × is closed under right-lcm. Here is a general situation when a Garside family of a subcategory can be deter-mined.
Lemma 2.12 ([DDGKM, VII 1.10]) . Let S be a Garside family in C assumedcancellative, right-Noetherian and having conditional right-lcms. Let S ⊂ S be asubfamily such that S C × ∪ C × is as a subset of SC × ∪ C × closed under right-lcmand right-quotient; then S is a Garside family in the subcategory C generated by S C × . Moreover C is a subcategory closed under right-quotient. Lemma 2.13 ([DDGKM, VII 1.18]) . Let M be a cancellative right-Noetherianmonoid which admits conditional right-lcms and let M ′ be a submonoid of M closedunder right-quotient and weakly closed under right-lcm. Then any u ∈ M has aunique (up to right-multiplication by M ′× ) maximal left-divisor in M ′ . Garside maps.
An important special case is when a Garside family S is attachedto a Garside map. A Garside map is a map Obj C ∆ −→ C where ∆( x ) ∈ C ( x, -) suchthat the map x target(∆( x )) is injective and such that SC × ∪ C × is both theset of elements that left-divide some ∆( x ) and the set of elements that right-dividesome ∆( x ).This definition of a Garside map agrees with [DDGKM, V 2.30] if we take inaccount that, using the notation of loc. cit., the fact that S is the set of left- andright-divisors of ∆ implies that the Garside family S is bounded.A Garside map allows to define a functor Φ, first on objects by taking for Φ( x )the target of ∆( x ), then on morphisms, first on morphisms s ∈ S by, if s ∈ C ( x, -) F. DIGNE AND J. MICHEL defining s ′ by ss ′ = ∆ (we omit the source of ∆ if it is clear from the context) andthen Φ( s ) by s ′ Φ( s ) = ∆. We then extend Φ by using normal decompositions; itcan be shown that this is well-defined and defines a functor such that for any f ∈ C we have f ∆ = ∆Φ( f ). It can also be shown that the cancellativity of C impliesthat Φ is an automorphism.The automorphism Φ is a typical automorphism of C preserving S that we willcall the Garside automorphism .If S is attached to a Garside map, we then have the following properties: Proposition 2.14. (i) If f g then lg S ( f ) ≤ lg S ( g ) . (ii) Assume f, g, h ∈ S and ( f, g ) is S -normal; then lg S ( f gh ) ≤ implies gh ∈ SC × . (iii) For f ∈ C ( x, - ) , the first term of an S -normal decomposition of x is aleft-gcd of f and ∆( x ) .Proof. (i) is [DDGKM, V 2.39 (v)], (iii) is [DDGKM, V 1.14]. (ii) is [DDGKM, IV1.38] using [DDGKM, 2.15] which says, with the notation as in loc. cit., that S est left-comultiple-closed. (cid:3) We will write ∆ p for the map which associates with an object x the morphism∆( x )∆(Φ( x )) · · · ∆(Φ p − ( x )). For any f ∈ C ( x, -) there exists p such that f ∆ p ( x ). Proposition 2.15 ([DDGKM, III 1.37 and V 2.14]) . If S is a Garside familyattached to a Garside map ∆ then for any positive integer p , ∆ p is a Garside mapand { f f · · · f p | f i ∈ S} is a Garside family attached to ∆ p . The conjugacy category
The context for this section is a cancellative category C . Definition 3.1.
Given a category C , we define the conjugacy category Conj C of C as the category whose objects are the endomorphisms of C and where, for w ∈ C ( A ) and w ′ ∈ C ( B ) we set Conj C ( w, w ′ ) = { x ∈ C ( A, B ) | xw ′ = wx } . We say that x conjugates w to w ′ and call centralizer of w the set Conj C ( w ) . The composition ofmorphisms in Conj C is given by the composition in C , which is compatible with thedefining relation for Conj C . Note that it is the formula for Conj C ( w, w ′ ) that forces the objects of Conj C tobe endomorphisms of C .Since C is cancellative, the data x and w determine w ′ (resp. x and w ′ determine w ). This allows us to write w x for w ′ (resp. x w ′ for w ); this illustrates that ourcategory Conj C is a right-conjugacy category; we call left-conjugacy category theopposed category.A proper notation for an element of Conj C ( w, -) is a triple w x −→ w x (that wewill abbreviate often to x w −→ -), since x by itself does not specify its source; butwe will use just x when the context makes clear which source w is meant (or whichtarget is meant). The forgetful functor which sends w ∈ Obj(Conj C ) to source( w )and w x −→ - to x is faithful, though not injective on objects; it allows us to identifyConj C ( w, -) with the subset { x ∈ C (source( w ) , -) | x wx } ; similarly we mayidentify Conj C (- , w ) with the subset { x ∈ C (- , source( w )) | xw < x } . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 7
It follows that the category Conj C inherits automatically from C properties suchas cancellativity or Noetherianity. The forgetful functor maps (Conj C ) × surjec-tively to C × , so in particular the subset Conj C ( w, -) of C (source( w ) , -) is closedunder multiplication by C × . In the proofs and statements which follow we iden-tify Conj C with a subset of C and (Conj C ) × to C × ; for the statements obtainedabout Conj C to make sense, the reader has to check that the sources and target ofmorphisms viewed as morphisms in Conj C make sense. Lemma 3.2. (i)
The subset
Conj C of C is closed under right-quotient. (ii) The subset
Conj C ( w, - ) of C (source( w ) , - ) is closed under right-lcm. Inparticular if C admits conditional right-lcms then so does Conj C .Similarly Conj C ( - , w ) is a subset of C ( - , source( w )) closed under left-lcm andleft-quotient.Proof. We show (i). If y = xz with y ∈ Conj C ( w, w ′ ), x ∈ Conj C ( w, -) and z ∈ C (- , source( w ′ )) we have x wx and yw ′ = wy . By cancellation, let us define w ′′ by xw ′′ = wx . Now since y = xz the equality yw ′ = wy gives xzw ′ = wxz = xw ′′ z which gives by cancellation that zw ′ = w ′′ z showing that z ∈ Conj C (- , w ′ ).We now show (ii). If x, y ∈ Conj C ( w, -) then x wx and y wy . Suppose nowthat x and y have a right-lcm z in C . Then x wz and y wz from which itfollows that z wz , that is z ∈ Conj C ( w, -), thus z is the image by the forgetfulfunctor of a right-lcm of x and y in Conj C .The proof of the second part is just a mirror symmetry of the above proof. (cid:3) Proposition 3.3.
Assume that S is a Garside family in C ; then Conj
C ∩ S is aGarside family in
Conj C and S -normal decompositions of an element of Conj C are Conj
C ∩ S -normal decompositions.Proof.
We will use Proposition 2.4 by showing that (Conj
C ∩ S ) ∪ C × generatesConj C and exhibiting a Conj C ∩ S -head function H : Conj C − C × → Conj
C ∩ S satisfying ( H ).Let H be a S -head function in C . We first show that the restriction of H toConj C takes its values in Conj C ∩ S . Indeed if x wx then H ( x ) H ( wx ) = × H ( wH ( x )) wH ( x ) where the middle = × is by ( H ).We now deduce by induction on lg S that (Conj C ∩ S ) ∪ C × generates Conj C .The induction starts with elements of length 0 which are exactly the elementsof C × . Assume now that x ∈ Conj C is such that lg S ( x ) = n > x ′ by x = H ( x ) x ′ ; since H ( x ) can be taken as the first term of a strict normaldecomposition we have lg S ( x ′ ) = n −
1. Since we proved H ( x ) ∈ Conj C , we deduceby Lemma 3.2(i) that x ′ ∈ Conj C , whence the result by induction.It is straightforward that the restriction of H to Conj C − C × is still a headfunction satisfying ( H ), which proves that Conj C ∩ S is a Garside family. Theassertion about normal decompositions follows. (cid:3)
Simultaneous conjugacy.
A straightforward generalization of the conjugacy cat-egory is the “simultaneous conjugacy category”, where objects are families of mor-phisms w , . . . , w n with same source and target, and morphisms verify x w i x forall i . Most statements have a straightforward generalization to this case. F. DIGNE AND J. MICHEL F -conjugacy. We want to consider “twisted conjugation” by an automorphism,which will be useful for applications to Deligne-Lusztig varieties, but also for in-ternal applications, with the automorphism being the one induced by a Garsidemap.
Definition 3.4.
Let F be an automorphism of finite order of the category C . Wedefine the F -conjugacy category of C , denoted by F - Conj C , as the category whoseobjects are the morphisms in some C ( A, F ( A )) and where, for w ∈ C ( A, F ( A )) and w ′ ∈ C ( B, F ( B )) we set F - Conj C ( w, w ′ ) = { x ∈ C | xw ′ = wF ( x ) } . We say that x F -conjugates w to w ′ and we call F -centralizer of a morphism w of C the set F - Conj C ( w ) . Note that F -conjugacy specializes to conjugacy when F = Id; again, it is theformula for F - Conj C ( w, w ′ ) which forces the objects of F - Conj C to lie in some C ( A, F ( A )).The notion of F -conjugacy turns out to be a particular form of conjugacy inthe semi-direct product category C ⋊ h F i ; this is the same as the relation betweentwisted conjugacy classes in a group and conjugacy classes in cosets.Consider the application which sends w ∈ C ( A, F ( A )) ⊂ Obj( F - Conj C ) to wF ∈ ( C ⋊ h F i )( A ) ⊂ Obj(Conj( C ⋊ h F i )). Since x ( w ′ F ) = ( wF ) x is equivalent to xw ′ = wF ( x ), this extends to a functor ι from F - Conj C to Conj( C ⋊ h F i ). Thisfunctor is clearly an isomorphism onto its image.The image ι (Obj( F - Conj C )) is the subset of C ⋊ h F i which consists of endomor-phisms which lie in C F ; and ι ( F - Conj C ) identifies via the forgetful functor withthe subset Conj( C ⋊ h F i ) ∩ C of C ⋊ h F i .Remark that, since in Conj( C ⋊ h F i ) there is no morphism between gF i and g ′ F j when i = j , the full subcategory that we will denote by Conj( C F ) of Conj( C ⋊ h F i )whose objects are in C F is a union of connected components of Conj( C ⋊ h F i ); thusmany properties will transfer automatically from Conj( C ⋊ h F i ) to Conj( C F ).In particular, if C has a Garside family S and F is a Garside automorphism,then S is still a Garside family for C ⋊ h F i by 2.7, and by Proposition 3.3 and theabove remark gives rise to a Garside family S ∩
Conj( C F ) of Conj( C F ). The image ι ( F - Conj C ) is the subcategory of Conj( C F ) consisting (via the forgetful functor) ofthe morphisms in C , thus satisfies the assumptions of Lemma 2.5: it is closed underright-quotient, because in a relation f g = h if f and h do not involve F the samemust be true for g , and contains the Garside family S ∩
Conj( C F ) of Conj( C F ).This will allow to generally translate statements about conjugacy categories tostatements about F -conjugacy categories. For example, ι − ( S ∩
Conj( C F )) is aGarside family for F - Conj C ; this last family is just F - Conj C ∩ S when identifying F - Conj C with a subset of morphisms of C by the forgetful functor.The assumption that F acts through an automorphism of finite order is used asfollows: since ( xF ) x = F x = ( xF ) F − and the action of F has finite order, twomorphisms in C F are conjugate in C ⋊ h F i if and only if they are conjugate by amorphism of C . The cyclic conjugacy category.
A restricted form of conjugation called “cyclicconjugacy” will be important in applications. In particular, it turns out (a partic-ular case of Proposition 3.9) that two periodic braids are conjugate if and only ifthey are cyclically conjugate. The context for this subsection is again a cancellativecategory C . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 9
Definition 3.5.
We define the cyclic conjugacy category cyc C of C as the subcat-egory of Conj C generated by S ′ = ∪ w { x ∈ Conj C ( w, - ) | x w } . That is, cyc C has the same objects as Conj C but contains only the products ofelementary conjugations of the form w = xy x −→ yx . Note that since C is cancellative ∪ w { x ∈ Conj C ( w, w ′ ) | x w } = { x ∈ Conj C (- , w ′ ) | w ′ < x } so cyclic conjugacy“from the left” and “from the right” are the same. To be more precise, the functorwhich is the identity on objects, and when w = xy and w ′ = yx , sends x ∈ cyc C ( w, w ′ ) to y ∈ cyc C ( w ′ , w ), is an isomorphism between cyc C and its opposedcategory. Proposition 3.6.
Assume C is right-Noetherian and admits conditional right-lcms;if S is a Garside family in C then S ′ ∩ S is a Garside family in cyc C .Proof. Set S = S ′ ∩ S . We first observe that S C × ∪ C × generates cyc C . Indeed if x w and we choose a decomposition x = s · · · s n as a product of morphisms in SC × ∪ C × it is clear that each s i is in cyc C , so is in S C × ∪ C × .The proposition then results from Lemma 2.12, which applies to cyc C since S C × ∪ C × is closed under right-divisor and right-lcm; this is obvious for right-divisor and for right-lcm results from the facts that SC × ∪ C × is closed underright-lcm by Lemma 2.11 and that a right-lcm of two divisors of w is a divisor of w . (cid:3) We see by Lemma 2.12 that cyc C is closed under right-quotient in Conj C .We now prove that S ′ — which does not depend on the existence of a Garsidefamily S in C — is a Garside family attached to a Garside map; S ′ is usually largerthan the Garside family S ′ ∩ S of Proposition 3.6, since it contains all left-divisorsof w even if w is not in S . Proposition 3.7.
Assume C is right-Noetherian and admits conditional right-lcms;then S ′ is a Garside family in cyc C attached to the Garside map ∆ such that ∆( w ) = w ∈ cyc C ( w ) ; the corresponding Garside automorphism Φ is the identityfunctor.Proof. The set S ′ generates cyc C by definition of cyc C . It is closed under right-divisors since xy w implies x w so that w x is defined and y w x ; since C isright-Noetherian and admits conditional right-lcms, any two morphisms of C withsame source have a gcd by Proposition 2.10. We define a function H : cyc C − C × →S ′ by letting H ( x ) be an arbitrarily chosen left-gcd of x and w if x ∈ cyc C ( w, -). Itis readily checked that H is an S ′ -head function. We conclude by Proposition 2.4that S ′ is a Garside family for cyc C . The set S ′ ( w, -) is the set of left-divisors of w = ∆( w ); similarly S ′ (- , w ) is the set of right-divisors of w = ∆( w ). Hence ∆ is aGarside map in cyc C . The equation xw x = wx shows that Φ is the identity. (cid:3) We say that a subset X ⊂ C is closed under left-gcd if whenever two elements of X have a left-gcd in C this gcd is in X . Proposition 3.8.
Assume C is right-Noetherian and admits conditional right-lcms;then the subcategory cyc C of Conj C is closed under left-gcd.Proof. Let ( x , . . . , x n ) and ( y , . . . , y m ) be S ′ -normal decompositions respectivelyof x ∈ cyc C ( w, -) and y ∈ cyc C ( w, -).We first prove that if gcd( x , y ) ∈ C × then gcd( x, y ) ∈ C × (here we considerleft-gcds in Conj C ). We proceed by induction on inf { m, n } . We write ∆ for ∆( w ) when there is no ambiguity on the source w . Since x n and y m divide ∆, we getthat gcd( x, y ) dividesgcd( x · · · x n − ∆ , y · · · y m − ∆) = × gcd(∆ x · · · x n − , ∆ y · · · y m − )= × ∆ gcd( x · · · x n − , y · · · y m − ) = × ∆ = w, where the first equality uses that Φ is the identity and the third results from theinduction hypothesis. So we get that gcd( x, y ) divides w thus is in S ′ ; by theproperty of normal decompositions it thus divides x and y , thus is in C × .We now prove the proposition. If gcd( x , y ) ∈ C × then gcd( x, y ) ∈ C × thus is incyc C and we are done. Otherwise let d be a gcd of x and y and let x (1) , y (1) bedefined by x = d x (1) , y = d y (1) . Similarly let d be a gcd of the first terms of anormal decomposition of x (1) , y (1) and let x (2) , y (2) be the remainders, etc. . . Since C is right-Noetherian the sequence d , d d , . . . of increasing divisors of x muststabilize at some stage k , which means that the corresponding remainders x ( k ) and y ( k ) have first terms of their normal decomposition coprime, so by the first part arethemselves coprime. Thus gcd( x, y ) = × d · · · d k ∈ cyc C . (cid:3) We now give a quite general context where cyclic conjugacy coincides with con-jugacy.
Proposition 3.9.
Let C be a right-Noetherian category with a Garside map ∆ , andlet x be an endomorphism of C such that for n large enough we have ∆ x n . Thenwe have cyc C ( x, - ) = Conj C ( x, - ) .Proof. We first show that the property ∃ n, ∆ x n is stable by conjugacy. Indeed,if u ∈ Conj C ( x, -) then there exists k such that u ∆ k . Since ∆ k +1 x n ( k +1) ,we have u − ∆ k · ∆ u − x n ( k +1) . If Φ is the Garside automorphism attached to∆, we have u − ∆ k · ∆ = ∆ · Φ( u − ∆ k ) thus ∆ u − x n ( k +1) . We deduce that( x u ) n ( k +1) = ( u − x · u ) n ( k +1) = u − x n ( k +1) · u is divisible by ∆.We prove then by Noetherian induction on f that f ∈ Conj C ( x, -) implies f ∈ cyc C ( x, -). This is true if f is invertible. Otherwise, write f = u f with u =gcd( f, x ); then u ∈ cyc C ( x, x u ). If we can prove that if f ∈ Conj C ( x, -), f / ∈ C × ,then gcd( f, x ) / ∈ C × , we will be done by Noetherian induction since we can writesimilarly f = u f , . . . and the sequence u , u , . . . has to exhaust f .Since as observed any u ∈ Conj C ( x, -) divides some power of x ( x nk if u ∆ k ) itis enough to show that if u ∈ Conj C ( x, -), u / ∈ C × and u x n , then gcd( u, x ) / ∈ C × .We do this by induction on n . From u ∈ Conj C ( x, -) we have u xu , and from u x n we deduce u x gcd( u, x n − ). If gcd( u, x n − ) ∈ C × then u x and we aredone: gcd( x, u ) = u . Otherwise let u = gcd( u, x n − ). We have u xu , u / ∈ C × and u x n − thus we are done by induction. (cid:3) The F -cyclic conjugacy. Let F be a finite order automorphism of the cate-gory C . We define F - cyc C as the subcategory of F - Conj C generated by ∪ w { x ∈ F - Conj C ( w, -) | x w } , or equivalently, since C is cancellative, by ∪ w ′ { x ∈ Conj C (- , w ′ ) | w ′ < F ( x ) } . By the functor ι , the morphisms in F - cyc C ( w, w ′ )identify with the morphisms in cyc( C ⋊ h F i )( wF, w ′ F ) which lie in C . To sim-plify notation, we will denote by cyc C ( wF, w ′ F ) this last set of morphisms. If C is right-Noetherian and admits conditional right-lcms, then so does C ⋊ h F i .If S is a Garside family in C and F is an automorphism preserving S , and we ARABOLIC DELIGNE-LUSZTIG VARIETIES. 11 translate Proposition 3.6 to the image of ι and then to F - cyc C , we get that ∪ w { x ∈ F - Conj C ( w, -) | x w and x ∈ S} is a Garside family in F - cyc C .Similarly Proposition 3.7 says that the set ∪ w { x ∈ F - Conj C ( w, -) | x w } is aGarside family in F - cyc C attached to the Garside map ∆ which sends the object w to the morphism w ∈ F - cyc C ( w, F ( w )); the associated Garside automorphismis the functor F .Finally Proposition 3.8 says that under the assumptions of Proposition 3.7 thesubcategory F - cyc C of F - Conj C is closed under left-gcd. Periodic elements.Definition 3.10.
Let C be a cancellative category with a Garside family S attachedto Garside map ∆ ; then an endomorphism f of C is said to be ( d, p )-periodic if f d ∈ ∆ p C × for some positive integers d, p . Note that if f is ( d, p )-periodic it is also ( nd, np )-periodic for any non-zero integer n ; conversely, up to cyclic conjugacy, a ( nd, np )-periodic element is ( d, p )-periodic,see 3.12. We call d/p the period of f . If the Garside automorphism Φ given by ∆ isof finite order, then a conjugate of a periodic element is periodic of the same period,though the minimal pair ( d, p ) may change. Our interest in periodic elements comesmainly from the fact that one can describe their centralizers, which is related tothe fact that by Proposition 3.9 two periodic morphisms are conjugate if and onlyif they are cyclically conjugate.We deal first with the case p = 2, where we show by elementary computationsthe following: Lemma 3.11.
Let f be a ( d, -periodic element of C and let e = ⌊ d ⌋ . Then f iscyclically conjugate to a ( d, -periodic element g such that g e ∈ SC × .Further, if g is a ( d, -periodic element such that g e ∈ SC × , then • if d is even g is ( d/ , -periodic. • if d is odd and we define h ∈ SC × by g e h = ∆ and ε ∈ C × by g d = ∆ ε then g = h Φ( h ) ε .Proof. We will prove by increasing induction on i that for i ≤ d/ v ∈ cyc C such that ( f v ) i ∈ SC × ∪ C × and ( f v ) d ∈ ∆ C × . We start the inductionwith i = 0 where the result holds trivially with v = 1.We consider now the general step: assuming the result for i such that i +1 ≤ d/ i + 1. We thus have a v for step i , thus replacing f by f v wemay assume that f i ∈ SC × ∪ C × and f d ∈ ∆ C × ; we will conclude by finding v ∈ S such that v f , ( f v ) i +1 ∈ SC × and ( f v ) d ∈ ∆ C × . If f i +1 ∆ we havethe desired result with v = 1. We may thus assume that lg S ( f i +1 ) ≥
2. Since f i +1 ∆ we have actually lg S ( f i +1 ) = 2 by Proposition 2.14(i); since f i is in SC × and divides f i +1 , a normal decomposition of f i +1 can be written ( f i v, w )with f i v, w ∈ SC × . As f i vw · f i v f i vw · f i vw = f i +1) f d = × ∆ , we stillhave 2 = lg S ( f i v · w · f i v ) = lg S ( f i v · w ). By Proposition 2.14(ii) we thus have w · f i v ∈ SC × . Then SC × ∋ w · f i v = w ( vw ) i v = ( f v ) i +1 and v f .So v will do if ( f v ) d ∈ ∆ C × . Write f d = ∆ ε with ε ∈ C × ; then f com-mutes with ∆ ε , thus f i +1 also, which can be written Φ ( f i +1 ) ε = εf i +1 orequivalently Φ ( f i v )Φ ( w ) ε = εf i vw . Now since Φ preserves normal decomposi-tions (Φ ( f i v ) , Φ ( w ) ε ) is a normal decomposition thus comparing with ( f i v, w ) byLemma 2.2 there exists ε ′ ∈ C × such that Φ ( f i v ) ε ′ = εf i v . Thus f i ∆ Φ ( v ) ε ′ = ∆ Φ ( f i v ) ε ′ = ∆ εf i v = f i ∆ εv , the last equality using again that f commuteswith ∆ ε . Canceling f i ∆ we get Φ ( v ) ε ′ = εv , whence v ( f v ) d = f d v = ∆ εv =∆ Φ ( v ) ε ′ = v ∆ ε ′ whence the result by canceling v .We prove now the second part. Since g e ∈ SC × the element h defined by g e h = ∆is in SC × ∪ C × . Defining ε ∈ C × by g d = ∆ ε we get g e h ∆ ε = ∆ ε = g d , whenceby cancellation h ∆ ε = g e g a with a = 1 if d is odd and a = 0 if d is even. Using h ∆ ε = ∆Φ( h ) ε = g e h Φ( h ) ε and canceling g e we get h Φ( h ) ε = g a .If d is odd we get the statement of the lemma, and if d is even we get h Φ( h ) ∈ C × ,so h ∈ C × , so g e ∈ ∆ C × . (cid:3) We will need at one stage the following more general statement (see [DDGKM,VIII, 3.33]) whose proof uses an interpretation by Bestvina of normal decomposi-tions as geodesics.
Theorem 3.12.
Let f be a ( d , k ) -periodic element of C ; let d = d / gcd( d , k ) and k = k / gcd( d , k ) ; then f is cyclically conjugate to a ( d, k ) -periodic element f . Further, write an equality dk ′ = 1 + kd ′ in positive integers. Then f is cyclicallyconjugate to a ( d, k ) -periodic element g such that g d ′ ∆ k ′ . If we then define g ∈ C by g d ′ g = ∆ k ′ then ( g Φ k ′ ) d = × ∆ and ( g Φ k ′ ) k = × g in C ⋊ h Φ i . F -periodic elements. Let us apply Lemma 3.11 to the case of a semi-direct prod-uct category C ⋊ h F i where C is a cancellative category with a Garside family S attached to a Garside map ∆ and F is an automorphism of finite order of C pre-serving S ; then S is still a Garside family of C ⋊ h F i . We assume further that C hasno non-trivial invertible elements. Then a morphism yF ∈ C F is ( d, p )-periodic ifand only if target( y ) = F (source( y )) and ( yF ) d = ∆ p F d .From Lemma 3.11 we can deduce: Corollary 3.13.
Let yF ∈ C F be ( d, -periodic and let e = ⌊ d ⌋ and Λ = Φ F − e .Then (i) If d is even, there exists an ( e, -periodic element xF ∈ C F cyclicallyconjugate to yF . The centralizer of xF in C identifies with cyc C ( xF ) .Further, we may compute this centralizer in the category of fixed points (cyc C ) Λ since the morphisms in cyc C ( xF ) are Λ -stable. (ii) If d is odd, there exists a ( d, -periodic element xF ∈ C F cyclicallyconjugate to yF such that ( xF ) e ∆ F e . The element s defined by ( xF ) e s = ∆ F e is such that, in the category C ⋊ h Λ i , we have x Λ = ( s Λ) and ( s Λ) d = ∆Λ d . The centralizer of xF in C identifies with the F d Φ − -fixed points of cyc C ( s Λ) . Note that 2.8 describes Garside families for the fixed point categories mentionedabove.
Proof.
Lemma 3.11 shows that yF is cyclically conjugate to a ( d, xF such that ( xF ) e ∈ S F e .If d is even Lemma 3.11 says that xF is ( e, xF is cyc C ( xF ). The elements of this centralizer, commutingto xF , commute to ( xF ) e = ∆ F e thus are Φ − F e -stable.If d is odd Lemma 3.11 says that if ( xF ) e h = ∆ then xF = h Φ( h ) F d . Since h = sF − e we get x = sF − e Φ( sF − e ) F d − = s Λ( s ). This can be rewritten x Λ =( s Λ) . Now since ∆ F e s − = ( xF ) e we get (∆ F e s − ) d = ∆ e F de which gives ARABOLIC DELIGNE-LUSZTIG VARIETIES. 13 (Λ − s − ) d ∆ d = ∆ e Λ − d and finally ( s Λ) d = ∆Λ d . The elements of Conj C ( xF )commute to ( xF ) e = ∆ F e s − thus commute to s Λ thus Conj C ( xF ) ⊂ Conj C ( s Λ).Note that the elements of Conj C ( xF ) commute to ( xF ) d thus to F d Φ − . Using x Λ = ( s Λ) we get Conj C ( s Λ) ⊂ Conj C ( x Λ ); but x Λ = xF ( F d Φ − ) − soConj C ( x Λ ) F d Φ − ⊂ Conj C ( xF ), whence the result using that by Proposition 3.9we have Conj C ( s Λ) = cyc C ( s Λ). (cid:3)
We will apply 3.12 in the following particular form
Corollary 3.14.
Assume that F is of finite order and that Φ = Id . Then anyperiodic element of C F is conjugate to a ( d, k ) -periodic element yF ∈ C F where k is prime to d . Further for any choice of positive integers d ′ and k ′ with dk ′ =1 + kd ′ , the element yF is cyclically conjugate to a ( d, k ) -periodic element xF satisfying ( xF ) d ′ ∆ k ′ . If we then define x ∈ C by ( xF ) d ′ x F − d ′ = ∆ k ′ then ( x F − d ′ ) d = ∆ F − dd ′ and ( x F − d ′ ) k = ( xF ) F − k ′ d . We have a partial converse:
Lemma 3.15.
Assume that F is of finite order and that Φ = Id . Let d, k, d ′ , k ′ bepositive integers such that dk ′ = 1 + kd ′ with d ′ prime to the order of F . If x ∈ C satisfies ( x F − d ′ ) d = ∆ F − dd ′ then the element xF ∈ C F defined by ( x F − d ′ ) k =( xF ) F − k ′ d satisfies ( xF ) d = ∆ k F d .Proof. The element x F − d ′ is F − dd ′ -stable since Φ = Id and ( x F − d ′ ) d = ∆ F − dd ′ .Since d ′ is prime to the order of F an element F − dd ′ -stable is F d -stable. Thus,raising the equality ( x F − d ′ ) k = ( xF ) F − k ′ d to the d -th power we get ( xF ) d =( x F − d ′ ) dk F k ′ d = (∆ F − dd ′ ) k F k ′ d = ∆ k F d . (cid:3) The following lemma shows that we can always choose d ′ satisfying the assump-tion of lemma 3.15. Lemma 3.16.
Given k and d coprime natural integers, and an integer δ , thereexists natural integers d ′ , k ′ such that dk ′ = 1 + kd ′ with d ′ prime to δ .Proof. k ′ and d ′ exist since k and d are coprime; we may change d ′ by any multipleof d . Thus it is sufficient to show that given coprime integers d and d ′ , we maychoose a such that d ′ + ad is prime to any given δ . Let p , . . . , p n be the prime factorsof δ . We have to choose a such that d ′ + ad is nonzero mod each p i . If p i | d this isautomatic. If p i is prime to d we have to avoid a ≡ − d ′ /d (mod p i ); by the Chineseremainder theorem we can choose a to avoid this finite set of congruences. (cid:3) An example: ribbon categories
An example of a category with a Garside family is a
Garside monoid , whichis just the case where C has one object. In this case we will say Garside elementinstead of Garside map. Example 4.1.
A classical example is given by the
Artin monoid ( B + , S ) associatedwith a Coxeter system ( W, S ). If the presentation of W is W = h S | s = 1 , sts · · · | {z } m s,t = tst · · · | {z } m s,t for s, t ∈ S i then B + is defined by the presentation B + = h S | sts · · · | {z } m s,t = tst · · · | {z } m s,t for s , t ∈ S i where S is a copy of S ; the group with the same presentation is the Artin group B . There is an obvious quotient B + → W since the relations of B + hold in W .Matsumoto’s lemma stating that two reduced expressions for an element of W canbe related by using only braid relations implies that there is a well-defined section W W of the quotient B + → W which maps a reduced expression s · · · s n tothe product s · · · s n ∈ B + . The monoid B + is cancellative, Noetherian, admitsconditional left-lcms and right-lcms; the set S is the set of atoms of B + and W isa Garside family in B + (for details, see [DDM, 6.27]). The Garside family W isattached to a Garside element if and only if W is finite. In this case we call B + spherical . The Garside element is the lift to W of the longest element w of W ; itwill be written w or ∆ depending on the context.Finally, an automorphism φ of ( W, S ) (that is, an automorphism of W whichpreserves S ) extends naturally to an automorphism of ( B + , S ) given by s φ ( s )which preserves the Garside family W . Example 4.2.
Another example, attached to the same Artin braid group B as theabove example, is the dual braid monoid introduced by David Bessis (see [B1]),whose construction can be extended to well-generated finite complex reflectiongroups.The constructions of this section apply to the study, in the semi-direct productof an Artin monoid ( B + , S ) by an automorphism stabilizing S , of the conjugatesand normalizer of a “parabolic” submonoid — the submonoid generated by a subsetof the atoms S . The “ribbon category” that we consider occurs, when the auto-morphism is the identity, in the work of Paris [Pa] and Godelle [G] on this topic.In Section 7 we will attach parabolic Deligne-Lusztig varieties to elements of theribbon category and endomorphisms of these varieties to elements of the conjugacycategory of this ribbon category.The next proposition gives a list of properties that spherical Artin monoidssatisfy; the rest of the section describes ribbons in an arbitrary monoid satisfyingthe same properties, which includes the case of the dual braid monoid; this is amotivation for giving the results in a more general context. Before stating thisproposition, we need a definition. Definition 4.3.
We say that a set I of atoms of a cancellative monoid M is parabolic if the submonoid M I of M generated by I is closed under right-quotientand weakly closed under right-lcm. Note that a monoid generated by a set I of atoms has no non-trivial invertibleelements, since such an element would be a product of atoms and an atom is notinvertible. Similarly, since an atom cannot be a product of several atoms, we seethat I is the whole set of atoms of the monoid. Proposition 4.4.
Let M = B + ⋊ h φ i be the semi-direct product of a sphericalArtin monoid by a diagram automorphism (see 4.1); then (i) M is cancellative, right-Noetherian and admits conditional right-lcms. (ii) There exists a finite set S ⊂ M which is a transversal of the = × -classes ofatoms in M , and together with M × generates M . (iii) Any conjugate in M of an element of S is in S . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 15 (iv) M has a Garside family S attached to a Garside element ∆ . (v) For any parabolic subset I of S , the maximal divisor ∆ I of ∆ given byLemma 2.13 (which is unique since M × I = { } ) is a Garside element in M I , and S ∩ M I is a Garside family attached to ∆ I . (vi) For any parabolic subset I ⊂ S and any s ∈ S − I there exists a parabolicsubset J such that ∆ J is the right-lcm of s and ∆ I .Proof. Let us prove (i). The monoid M is cancellative since it embeds in the semi-direct product of the Artin group by φ . Similarly it inherits from B + Noetherianityand the Garside family W , which implies that it admits conditional right-lcms.We prove (ii). Take for S the set of atoms of M . An invertible element must havelength 0, hence the powers of φ are the only invertible elements. The atoms are theelements of length 1 that is the elements of S h φ i , thus S is indeed a transversal ofthe atoms.For (iii), we have to check that if we have s f = f t with s ∈ S and f and t in M then t ∈ S . Taking lengths we see that the length of t is 1 so that t = s ′ φ k forsome integer k and some s ′ ∈ S . Looking then at the powers of φ on both sides weget k = 0.For (iv), take ∆ = w . We have seen in Example 4.1 that (using the notation ofloc. cit.) the lift w to W of the longest element w of W is a Garside element in B + . Hence ∆ = w is a Garside element in M by Lemma 2.7. We take S = W ; itis a Garside family attached to ∆.For (v) we notice first that M I , being generated by atoms, has no non-trivialinvertible elements.Before proving the rest, let us state the following (the fact that this fails in dualbraid monoids is a motivation for defining parabolic subsets). Lemma 4.5.
Any subset of S is parabolic.Proof. We show that M I is closed under right-quotient. Since both sides of eachdefining relation for an Artin monoid involve the same elements of S , two equivalentwords for an element v ∈ M involve the same subset of the generating set S ; wecall this subset the support of v . Hence if xy = z with x, z ∈ M I then the power of φ in y is 0 and the support of y is a subset of that of z , thus a subset of I , thus y is in M I .We now show that M I is weakly closed under right-lcms. Keeping the notationsof 4.1, B + is associated to the Coxeter system ( W, S ). Since M I is a spherical Artinmonoid associated with the Coxeter subgroup W I of W generated by the image in W of I (see for example [Pa, 3.1]) two elements of M I have a right-lcm in M I . Thisright-lcm is left-divisible by any of their right-lcms in M , so has to be equal to oneof these lcms since M I is obviously stable by left-divisor. (cid:3) Since by M I is a spherical Artin monoid it has a Garside element w I , the lift ofthe longest element of W I . The corresponding Garside family is W I = W ∩ M I ,that is the set of divisors in M I of ∆ which by definition of ∆ I are the left-divisorsof ∆ I . We get that w I and ∆ I have the same set of left-divisors, so are equal since M × I = { } .We finally show (vi). We take J = I ∪ { s } . The following lemma applied with S = I (resp. S = J ) gives that ∆ I is a right-lcm of I (resp. ∆ J is a right-lcm of J ).We thus get the result by associativity of the lcm. Lemma 4.6.
The Garside element ∆ = w of B + is the right-lcm of S .Proof. By [DDM, 6.27] a common multiple of S in W corresponds to an element w ∈ W such that l ( sw ) < l ( w ) for all s ∈ S . It is well known that only w satisfiesthis, so ∆ = w is the only element of W multiple of all the atoms. (cid:3)(cid:3) The category
Conj( M, I ) . Until the end of Section 4, we fix a monoid M anda transversal S of its set of atoms; we assume that M has a Garside family S associated with a Garside element ∆ so that these data satisfy properties (i) to (vi)of Proposition 4.4.The reader only interested in internal applications to this paper can assume thatwe are in the case M = B + ⋊ h φ i , the semi-direct product of a spherical Artinmonoid with a diagram automorphism (with S the usual atoms and Garside family S = W ). Our results apply also to the case of dual Artin monoids, but this willnot be used in this paper.We fix also the conjugacy class I under M of a subset of S . By property 4.4(iii)any element of I is a subset of S . We assume all elements of this class are parabolicsubsets (which is automatic in the ordinary Artin monoid case where all subsetsare parabolic).Let Conj( M, I ) be the connected component of the simultaneous conjugacy cat-egory of M whose objects are the elements of I . A morphism in Conj( M, I ) withsource I ∈ I is given by b ∈ M such that for each s ∈ I we have s b ∈ M , which byproperty 4.4(iii) implies s b ∈ S . We denote such a morphism in Conj( M, I )( I , -)by I b −→ -, or if we want to specify the target we denote it by I b −→ J where J = { s b | s ∈ I } , and in this situation we write J = I b .By Proposition 3.3 the set { I b −→ - | b ∈ S} ∩ Conj( M, I ) is a Garside family inConj( M, I ). The ribbon category.
For b ∈ M we denote by α I ( b ) the maximal left-divisorof b in M I given by Lemma 2.13, which is unique since M × I = { } . We denote by ω I ( b ) the element defined by b = α I ( b ) ω I ( b ). We say that b ∈ M is I -reduced if itis left-divisible by no element of I , or equivalently if α I ( b ) = 1. Definition 4.7.
We define the ribbon category M ( I ) as the subcategory of Conj( M, I ) obtained by restricting the morphisms to the I b −→ - such that b is I -reduced. This makes sense since the above class of morphisms is stable by composition by(ii) in the next proposition; assertion (i) of the next proposition is a motivation forrestricting to such morphisms by showing that we “lose nothing” in doing so.
Proposition 4.8. (i)
Given I ∈ I and b ∈ M then I b −→ - ∈ Conj( M, I ) ifand only if I α I ( b ) = I and I ω I ( b ) −−−→ - ∈ M ( I ) . (ii) If I b −→ J ∈ M ( I ) then for any b ′ ∈ M we have α J ( b ′ ) = α I ( bb ′ ) b . Inparticular if ( I b −→ J ) ∈ M ( I ) and ( J b ′ −→ K ) ∈ Conj( M, I ) then ( I bb ′ −−→ K ) ∈ M ( I ) if and only if ( J b ′ −→ K ) ∈ M ( I ) . (iii) If two morphisms in M ( I ) admit a right-lcm in Conj( M, I ) , then this lcmis in M ( I ) . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 17
Note that if I c −→ - is the right-lcm of two morphisms I b −→ - and I b ′ −→ - as in (iii)then by Lemma 3.2 c is the right-lcm in M of b and b ′ . Proof.
Let us prove (i). We prove that if s ∈ I and s b ∈ M then s α I ( b ) ∈ I . Thiswill prove (i) in one direction —we use that I is finite, see 4.4(ii), so that I α I ( b ) ⊂ I implies I α I ( b ) = I . The converse is obvious.By property 4.4(iii) we have sb = bt for some t ∈ S . If s b we write b = s k b ′ for some k and b ′ such that s does not left-divide b ′ . We have sb ′ = b ′ t and α I ( b ) = s k α I ( b ′ ) and we are reduced to the case where s does not left-divide b .Then any right-lcm of s and α I ( b ) left-divides sb = bt and there is such a right-lcm in M I since M I is weakly closed under right-lcm (4.4(v)). We write this lcm sv = α I ( b ) u , with v and u in M I since M I is closed under right-quotient (4.4(v))and v , u = 1 since s b . Since sv sb we get that v left-divides b , so left-divides α I ( b ), thus α I ( b ) = va for some a ∈ M I . We get sv = α I ( b ) u = vau . By property4.4(iii) we have au ∈ S , thus u is an atom which is in M I , hence u ∈ I and a = 1since S is a transversal for = × . We get s α I ( b ) = s v = u ∈ I , which gives the result.Let us prove (ii). For s ∈ I let s ′ = s b ∈ J . Since I b −→ J ∈ M ( I ) we have s b .Then bs ′ = sb is a common multiple of s and b which has to be an lcm since s ′ isan atom. So for s ∈ I we have s bb ′ if and only if bs ′ bb ′ , that is, s b b ′ whence the result.To prove (iii) we show first the statement that if for b , c ∈ M we have b c and I b −→ - ∈ M ( I ), then b ω I ( c ). We write c = bb ′ and J = I b . By (ii)we have α I ( c ) b = α J ( b ′ ), whence α I ( c ) b = b α J ( b ′ ) bb ′ = c = α I ( c ) ω I ( c ).Left-canceling α I ( c ) we get b ω I ( c ).Now (iii) is a particular case of the above statement since if c is the right-lcmof b and b ′ where I b −→ - and I b ′ −→ - are in M ( I ), we get that ω I ( c ) is a commonright-multiple of b and b ′ , thus c ω I ( c ), which implies α I ( c ) = 1. (cid:3) Note that by Proposition 4.8(i) a morphism in M ( I ) with source I correspondsby the forgetful functor to an element b ∈ M such that α I ( b ) = 1 and such thatfor each s ∈ I we have s b ∈ M . We will thus sometimes just denote by b such amorphism in M ( I ) when the context makes its source clear.The next proposition shows that ( S ∩ M ( I )) ∪ M × generates M ( I ). Note anyelement of M × gives rise to an element of M ( I ). Proposition 4.9.
All the terms of a normal decomposition in
Conj( M, I ) of amorphism of M ( I ) are in M ( I ) .Proof. Let I b −→ - ∈ M ( I ) and let b = b · · · b k be a normal decomposition in M ,which gives a normal decomposition of I b −→ - in Conj( M, I ) by Proposition 3.3.We proceed by induction on k . We have α I ( b ) α I ( b ) = 1 thus α I ( b ) = 1 and I b −→ I b ∈ M ( I ). This is the first step of the induction. Now, by 4.8(ii) we get I b b ··· b k −−−−→ - ∈ M ( I ) which concludes by induction. (cid:3) Corollary 4.10.
The set
S ∩ M ( I ) = { I w −→ - ∈ Conj( M, I ) | w ∈ S and α I ( w ) =1 } is a Garside family in M ( I ) . Proof.
By 4.8(ii) and 4.8(iii) the subcategory M ( I ) of Conj( M, I ) is closed underright-quotient and right-lcm, hence the subfamily S ∩ M ( I ) is closed under right-quotient and right-lcm in S ∩
Conj( M, I ). Thus Lemma 2.12 gives the result since( S ∩ M ( I )) ∪ M × generates M ( I ) by Proposition 4.9. (cid:3) Our aim now is Proposition 4.15 which gives a description of the atoms of M ( I ),and a convenient criterion to decide whether b ∈ M gives rise to an element of M ( I ).For I ⊂ S let Φ I be the Garside automorphism of M I associated with the Garsideelement ∆ I (see 4.4(iv)). Since I is finite (see 4.4(ii)) and is the whole set of atomsof M I , we have Φ I ( I ) = I .We denote by Φ the Garside automorphism of M associated to ∆. Since Φ is anautomorphism which preserves S , for I ⊂ S , it sends the Garside family S ∩ M I tothe Garside family S ∩ M Φ( I ) thus Φ(∆ I ) = ∆ Φ( I ) . Proposition 4.11. M ( I ) has a Garside map defined by the collection of morphisms I ∆ − I ∆ −−−−→ Φ( I ) for I ∈ I .Proof. We have Φ I ( I ) = I and ω I (∆) = ∆ − I ∆. Thus by Proposition 4.8(i) I ∆ − I ∆ −−−−→ Φ( I ) ∈ S ∩ M ( I ). We need two lemmas. Lemma 4.12.
Any morphism I b −→ - ∈ M ( I ) ∩ S left-divides I ∆ − I ∆ −−−−→ Φ( I ) .Proof. The divisibility we seek is equivalent to ∆ I b left-dividing ∆. Since ∆ I and b left-divide ∆, a right-lcm δ of these elements divides ∆. We claim that δ = × ∆ I b which will show the lemma. Since I b ⊂ S we have ∆ bI ∈ M thus δ b ∆ bI = ∆ I b .Notice that α I ( δ ) = ∆ I since ∆ I δ and α I ( δ ) α I (∆) = ∆ I . Now write δ = bx ; by Proposition 4.8(ii) we have α I b ( x ) = α I ( δ ) b = ∆ bI . Thus ∆ bI x thus b ∆ bI bx = δ , whence our claim. (cid:3) Lemma 4.13. If I b −→ J is in M ( I ) we have ∆ J = ∆ bI ; conjugation by b inducesan isomorphism of Garside monoids M I ∼ −→ M J which preserves normal forms.Proof. It is sufficient to prove the lemma for elements of the generating set (
S ∩ M ( I )) ∪ M × . So we assume b ∈ S ∪ M × . If b ∈ S , in the proof of Lemma 4.12we have ∆ bI x where x is a right-divisor hence a left-divisor of ∆, thus ∆ bI ∆.This is also clearly true if b ∈ M × . Since ∆ bI ∈ M J we get ∆ bI ∆ J . We show bycontradiction that this divisibility cannot be strict. By Lemma 4.12 we can write∆ − I ∆ = bb ′ ; then by 4.8(ii) we have J b ′ −→ Φ( I ) ∈ M ( I ) and by the same argumentas above ∆ b ′ J ∆ Φ( I ) . Now b ′ induces by conjugation a morphism M J → M Φ( I ) so we can transport the strict divisibility ∆ bI ≺ ∆ J to ∆ bb ′ I ≺ ∆ b ′ J . Composing weget Φ(∆ I ) = ∆ bb ′ I ≺ ∆ b ′ J ∆ Φ( I ) = Φ(∆ I ), a contradiction.The second part of the statement follows from the first since the first term of anormal form of an element x in a monoid with a Garside element ∆ is a left-gcd of x and ∆ (see Proposition 2.14(iii)), and the conjugation by b preserves gcds sinceit is an isomorphism. (cid:3) We now show the proposition. We know by Lemma 4.12 that any I b −→ J in S ∩ M ( I ) left-divides I ∆ − I ∆ −−−−→ Φ( I ). It remains to show that such a morphism right-divides ∆ − − ( J ) ∆, which is equivalent to b ∆ J right-dividing ∆ since Φ(∆ Φ − ( J ) ) = ARABOLIC DELIGNE-LUSZTIG VARIETIES. 19 ∆ J . This in turn is equivalent to b ∆ J left-dividing ∆ since ∆ is a Garside element.The result is then a consequence of the fact that ∆ I b divides ∆ as we have seen inLemma 4.12 and of the equality b ∆ J = ∆ I b which is given by Lemma 4.13. (cid:3) Proposition 4.14.
Let I ∈ I and let J be a parabolic subset of S such that M I ( M J . Then ∆ I ∆ J (see 4.4(vi)) and I v ( J , I ) −−−−→ Φ J ( I ) , where v ( J , I ) = ∆ − I ∆ J , is amorphism in M ( I ) .Proof. As noted after Proposition 4.8 we have to show that α I ( v ( J , I )) = 1 and thatany t ∈ I is conjugate by v ( J , I ) to an element of M . Since ∆ − I ∆ J left-divides∆ − I ∆, and α I (∆ − I ∆) = 1, by definition of ∆ I , we get the first property. Thesecond is clear since by definition v ( J , I ) conjugates t to Φ J (Φ − I ( t )). (cid:3) (i) of the next proposition is due to Paris [Pa, 5.6] in the case of Artin monoids. Proposition 4.15. (i)
Let I ∈ I and b ∈ M such that α I ( b ) = 1 and suchthat there exists p > such that (∆ p I ) b ∈ M . Then I b −→ - ∈ M ( I ) . (ii) The atoms of M ( I ) are the v ( J , I ) not strictly divisible by another v ( J ′ , I ) for I ∈ I .Proof. Since M is right-Noetherian, for (i) it suffices to prove that under our as-sumption b is either invertible or left-divisible by some non-invertible v ∈ M givingrise to an element of M ( I ); indeed if b = vb ′ where I v −→ I ′ ∈ M ( I ) then by 4.8(ii)we have α I ′ ( b ′ ) = 1 and since I v = I ′ we have (∆ p I ′ ) b ′ ∈ M by Lemma 4.13, so byNoetherian induction we have I ′ b ′ −→ - ∈ M ( I ), whence I b −→ - ∈ M ( I ). We willprove that b is left-divisible by v ( J , I ) for some parabolic J ) I which will imply(i). We proceed by decreasing induction on p . We show that if for i > s ∆ i I b for some atom s not in M I , v ( J , I ) ∆ i − I b where J is as prescribed in4.4(vi) from I and s . Indeed, the right-lcm of s and ∆ I is ∆ J by property 4.4(vi)thus from s ∆ i I b and ∆ I ∆ i I b we deduce ∆ J ∆ i I b . Since ∆ J = ∆ I v ( J , I )we get as claimed v ( J , I ) ∆ i − I b . The induction starts at i = p by taking for s any atom left-dividing b , thus not in M I since α I ( b ) = 1. Such an atom satisfies s b ∆ p I b since the assumption on b can be written b ∆ p I b . Since any atom t such that t v ( J , I ) is not in M I the induction can go on while i − > b ∈ M ( I ) satisfies the assumptionof (i) for p = 1 and I equal to the source of b ; whence the result since in the proofof (i) we have seen that b is a product of elements of the form v ( J , K ). (cid:3) Though in the current paper we need only finite Coxeter groups, we note that theabove description of the atoms also extends to the case of Artin monoids which areassociated with infinite Coxeter groups —and thus do not have a Garside element.Proposition 4.16 below can be extracted from the proof of Theorem 0.5 in [G].In the case of an Artin monoid ( B + , S ) the Garside family of Corollary 4.10 in B + ( I ) is W ∩ B + ( I ) = { I w −→ J ∈ Conj B + ( I ) | w ∈ W and α I ( w ) = 1 } . For I ⊂ S and s ∈ S we denote by I ( s ) the connected component of s in the Coxeterdiagram of I ∪ { s } , that is the vertices of the connected component of s in thegraph with vertices I ∪ { s } and an edge between s ′ and s ′′ whenever s ′ and s ′′ donot commute.When I is spherical, the subgroup W I generated by the image I of I in W isfinite even though W is not, in which case we denote by w I the lift in W of thelongest element of W I . With these notations, we have Proposition 4.16.
The atoms of B + ( I ) are the morphisms I v ( s , I ) −−−→ v ( s , I ) I where I is in I and s ∈ S − I is such that I ( s ) is spherical, and where v ( s , I ) = w I ( s ) w I ( s ) −{ s } . Application to Artin groups
We will spell out how the above results can be stated in two particular cases.We try to recall enough notation so this section can be read independently of theprevious ones.
Artin monoids with automorphism.
We first look at the case of a sphericalArtin monoid B + attached to a Coxeter system ( W, S ) with a diagram automor-phism φ , see 4.1. The category C we will take is the monoid B + ⋊ h φ i ; it has aGarside element w and an attached Garside family W . The Garside automorphismΦ is given by b b w ; it is trivial if W is central and has order 2 otherwise. Weset π = w , a central element in B + . An element b φ ∈ B + ⋊ h φ i is ( d, p )-periodicif ( b φ ) d = w p φ d , which can be written b φ b φ b · · · = w p . Theorem 5.1. If φ = Id , two periodic elements of B + of same period are cyclicallyconjugate.Proof. This results from the work of David Bessis on the dual braid monoid. Twoperiodic elements of same period in B + are also periodic and have equal periodsin the dual monoid, since the Garside element w of B + is a power of the Garsideelement of the dual monoid. By [B1, 11.21], such elements are conjugate in the dualmonoid, so are conjugate in B , hence are conjugate in B + ; indeed if b ′ = h − bh with b , b ′ ∈ B + and h ∈ B , then there exists i > h π i ∈ B + and since π is central h π i still conjugates b to b ′ . By Proposition 3.9 conjugate periodicelements are cyclically conjugate. (cid:3) We conjecture that the same result holds in the case φ = Id.Taking in account that Φ = Id, statement 3.13 gives: Proposition 5.2.
Let b ′ φ ∈ B + φ be ( d, -periodic, that is ( b ′ φ ) d = π φ d , andlet e = ⌊ d ⌋ . Then there exists b φ ∈ B + φ cyclically conjugate to b ′ φ such that b e ∈ W , and • If d is even then ( b φ ) e = w φ e . The centralizer C B + ( b φ ) identifies with cyc B + ( b φ ) , and even more specifically to the endomorphisms of b φ in thecategory of conjugacy by w φ e -stable divisors. • If d is odd there exists v ∈ W such that ( b φ ) e v = w φ e and b = v φ − e ( v w ) . The centralizer C B + ( b φ ) identifies with the endomorphismsof vw φ − e in the category of conjugacy by φ d -stable divisors. Part of the above proposition is already in [BM, 6.8]. The equation ( b φ ) d = π φ d for ( d, d -th φ -rootsof π . Ribbons in Artin monoids.
We keep in this subsection a spherical Artin monoid B + attached to ( W, S ) with a diagram automorphism φ and consider the ribboncategory B + ⋊ h φ i ( I ) defined by a conjugacy class I of subsets of S .A subset I ⊂ S and the corresponding subset I ⊂ S determine: ARABOLIC DELIGNE-LUSZTIG VARIETIES. 21 • A standard parabolic subgroup W I generated by I ; we denote by w I itslongest element (with this notation w = w S ). In every coset W I w thereis a unique shortest element called I -reduced. • A parabolic submonoid B + I generated by I ; it has the Garside family W I := W ∩ B + I and the associated Garside element is the lift w I of w I ; we set π I = w I . By Lemma 2.13 every element b ∈ B + has a unique longestdivisor α I ( b ) in B + I ; an element such that α I ( b ) = 1 is called I -reduced.The ribbon category B + ( I ) is the category whose objects are the elements of I and a morphism I b −→ J is given by an I -reduced element b ∈ B + such that I b = J ;since J is determined by I and b we denote also by I b −→ - this morphism. Propo-sition 4.8 shows that this definition makes sense, that is if we have a composition I b −→ J c −→ K in B + ( I ), then α I ( bc ) = 1.By Corollary 4.10 and Proposition 4.11 B + ( I ) has a Garside family S consistingof the morphisms I w −→ - where w ∈ W and a Garside map ∆ I ( I ) = I w − I w −−−−−→ I w .These properties include the following: Lemma 5.3. (i) S generates B + ( I ) ; specifically, if I b −→ J ∈ B + ( I ) and ( w , . . . , w k ) is the W -strict normal decomposition of b , there exist subsets I i with I = I , I k +1 = J such that for all i we have I i +1 = I w i i ; thus I w −−→ I → · · · → I k w k −−→ J is a decomposition of I b −→ J in B + ( I ) as aproduct of elements of S . (ii) The relations ( I w −−→ J w −−→ K ) = ( I w −→ K ) when w = w w ∈ W form apresentation of B + ( I ) . In our case strict normal decompositions are unique. They can be defined asfollows: for b ∈ B + , let α ( b ) be the left-gcd of b and w ; the restriction of α to B + − { } is a W -head function, thus w := α ( b ) is the first term of the normaldecomposition of b , and the other terms are defined similarly by induction, setting w = α ( w − b ), etc . . . For generating the category B + ⋊ h φ i ( I ) we need additionally the invertiblemorphisms I φ −→ I φ . The family S is still a Garside family for this category, with thesame Garside map ∆ I . When I = {∅} , B + ( I ) reduces to the Artin-Tits monoid B + and B + ⋊ h φ i ( I ) reduces to B + ⋊ h φ i , thus the results in this subsection generalizethose of the previous subsection.We will be interested in ( d, B + φ ( I ). Such an element isan endomorphism of the form I b φ −−→ I or via the correspondence between conjugacyin the semi-direct category and φ -conjugacy, a morphism I b −→ φ I in B + ( I ) where b φ I = I . Since ∆ I ( I )∆ I ( I w ) = I π / π I −−−→ I the condition for this morphism to be( d, b φ ) d = π / π I φ d .By the forgetful functor ( I b φ −−→ -) b φ the morphisms in B + φ ( I )( I , -) identifywith the elements b φ ∈ B + φ such that b φ I ⊂ S and α I ( b ) = 1. We will thussometimes write b φ ∈ B + φ ( I )( I , -) to mean I b φ −−→ - ∈ B + φ ( I )( I , -).Taking into account the above, and that the Garside automorphism associatedto ∆ I is Φ( I v −→ I v ) = I w v w −−→ I vw , the generalization of Proposition 5.2 is Proposition 5.4.
Let b ′ φ ∈ B + φ be such that ( b ′ φ ) d = π / π J φ d for some φ d -stable J ∈ I , and let e = ⌊ d ⌋ . Then b ′ φ defines an endomorphism of J in B + φ ( I ) ,that is b ′ φ J = J and α J ( b ′ ) = 1 . This endomorphism is ( d, -periodic and thereexists a φ d -stable I ∈ I and I b φ −−→ I ∈ B + φ ( I )( I ) cyclically conjugate to J b ′ φ −−→ J ∈ B + φ ( I )( J ) such that ( b φ ) d = π / π I φ d , ( b φ ) e ∈ W φ e , and • If d is even then ( b φ ) e = w − I w φ e . The centralizer Conj B + ( I )( I b φ −−→ I ) identifies with (cyc B + ( I )( I b φ −−→ I )) w φ e . • If d is odd there exists I v −→ I w φ e ∈ W ∩ B + ( I ) such that ( b φ ) e v = w − I w φ e and b = v φ − e ( v w ) . The centralizer Conj B + ( I )( I b φ −−→ I ) iden-tifies with (cyc B + ( I )( I v Φ φ − e −−−−→ I )) φ d .Proof. We need to prove that ( b ′ φ ) d = ( π J ) − π φ d implies α J ( b ′ ) = 1 and that b ′ φ J = J . The condition α J ( b ′ ) = 1 follows from α J ( b ′ ) α J (( b ′ φ ) d ) and fromthe fact that ( π J ) − π defines a morphism in B + ( I ) as we have seen above. ByProposition 4.15(i) b ′ φ defines a morphism J b ′ φ −−→ K in B + ( I ). Hence b ′ φ conju-gates π J to π K by Lemma 4.13. Since b ′ φ centralizes π / π J φ d and π is central, itthus centralizes π J φ d , hence it centralizes π δ J , where δ is the order of φ and we get π δ J = π δ K . However the support (see the proof of Lemma 4.5) of π δ J is J and thatof π δ K is K , thus J = K and b ′ φ stabilizes J .The other assertions of the proposition are straightforward translations of Corol-lary 3.13. (cid:3) We note that any element which conjugates a ( d, B + φ to another is φ d -stable. Indeed such an element conjugates some π / π J φ d to some π / π I φ d ; if δ is the order of φ since π is central it thus conjugates π δ J to π δ I thus bythe same reasoning as the end of the proof above it conjugates I to J , which finallyimplies that it commutes with φ d .We now state 3.14 in the case of ribbons. Corollary 5.5.
Let b ′ φ ∈ B + φ be such that ( b ′ φ ) d = ( π / π J ) k φ d for some φ d -stable J ∈ I . Then b ′ φ defines a ( d, k ) -periodic endomorphism of J in B + φ ( I ) ,and up to cyclic conjugacy in B + φ ( I ) , we may assume k prime to d . Then, forany choice of integers d ′ , k ′ with dk ′ = 1 + kd ′ there exists a φ d -stable I ∈ I and I b φ −−→ I ∈ B + φ ( I )( I ) cyclically conjugate to J b ′ φ −−→ J such that ( b φ ) d = ( π / π I ) k φ d and ( b φ ) d ′ ( π / π I ) k ′ , and if we define b ∈ B + ( I ) by ( b φ ) d ′ b φ − d ′ = ( π / π I ) k ′ then ( b φ − d ′ ) d = π / π I φ − dd ′ and ( b φ − d ′ ) k = ( b φ ) φ − k ′ d .Proof. As in the beginning of the proof 5.4 we deduce from the equality ( b ′ φ ) d =( π / π J ) k φ d that b ′ φ defines an element of B + φ ( I )( J ). The only other observationneeded is that we apply 3.14 for the Garside structure corresponding to the Garsidemap ∆( J ) = J π / π J −−−→ J , the square of the previously introduced Garside map ∆ I —this is allowed by 2.15. For this Garside map the corresponding functor Φ is theidentity, as required by 3.14. (cid:3) Corollary 5.6.
As in corollary 5.5 let b ′ φ ∈ B + φ be such that ( b ′ φ ) d = ( π / π J ) k φ d for some φ d -stable J ∈ I . Then I b ′ φ −−→ J is cyclically conjugate in B + φ ( I ) to a ( d, k ) -periodic endomorphism I b φ −−→ I such that ( b φ ) ⌊ d k ⌋ ∈ W φ ⌊ d k ⌋ . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 23
Proof.
By 5.5 we may first assume that k is prime to d . We then use 5.5 toget b φ − d ′ ∈ B + φ − d ′ satisfying the assumption of 5.4 with φ replaced by φ − d ′ .By 5.4 we may find a cyclic conjugate b ′ φ − d ′ of b φ − d ′ such that ( b ′ φ − d ′ ) ⌊ d ⌋ ∈ W φ − d ′ ⌊ d ⌋ . If this cyclic conjugation conjugates b φ = ( b φ − d ′ ) k φ k ′ d to ( b ′ φ − d ′ ) k φ k ′ d we are done since k ⌊ d k ⌋ ≤ ⌊ d ⌋ . Note that the cyclic conjugacy in 5.4 conjugates J to I and π / π J φ d to π / π I φ d , so is φ d -stable ( φ − dd ′ -stable in our application). Ifwe had that any φ dd ′ -stable element is φ d -stable we would be done since the con-jugation would then commute with φ k ′ d . Thus we finish using Lemma 3.16 whichshows that we may choose d ′ prime to the order of φ . (cid:3) For b ∈ B + , let α ( b ) = gcd( b , w ). It is a W -head function in B + thus byProposition 4.9 and Corollary 4.10 ( I b −→ -) ( I α ( b ) −−−→ -) is a S -head function. Lemma 5.7.
For I b −→ - ∈ B + ( I ) and v ∈ B + I we have α ( vb ) = α ( v ) α ( b ) .Proof. Lemma 5.3 implies that α ( b ) defines an element of B + ( I )( I , -) so that v α ( b ) ∈ B + . We have α ( vb ) = α ( v α ( b )) = α ( α ( b ) v α ( b ) ) = α ( α ( b ) α ( v α ( b ) )),the first and last equalities by property ( H ) of Proposition 2.4. By Lemma 4.13 wehave α ( v α ( b ) ) = α ( v ) α ( b ) , so that α ( vb ) = α ( α ( b ) α ( v ) α ( b ) ) = α ( α ( v ) α ( b )). Since α ( b ) is I -reduced we have α ( v ) α ( b ) ∈ W , hence α ( α ( v ) α ( b )) = α ( v ) α ( b ). (cid:3) The following proposition shows a compatibility of morphisms in B + ( I ) with a“parabolic” situation. Proposition 5.8.
Fix I ∈ I , and for J ⊂ I , let J be the set of B + I -conjugates of J . Let ( I b −→ I ′ ) ∈ B + ( I ) and let ( J v −→ J ′ ) ∈ B + I ( J ) . Let ( u , . . . , u k ) be the strictnormal decomposition of vb and let ( w , w , . . . , w k ) be a normal decompositionof b (we have added some 1’s at the end of the strict normal decomposition so thedecompositions have same length); then for each i there exists v i ∈ B + such that u i = v i w i and ( v , w v , w w v , . . . ) is a normal decomposition of v .Proof. We proceed by induction on k . By Lemma 5.7, we have u = α ( v ) α ( b ) = α ( v ) w . Hence v = α ( v ) is a solution. Cancelling v we get u · · · u k = ω ( v ) α ( b ) ω ( b ).The induction hypothesis applied to ω ( v ) α ( b ) , which defines an element of B + I α ( b ) ( J α ( b ) ),and to ω ( b ) which defines an element of B + ( I ) gives the result. (cid:3) The category D I . The category cyc B + φ ( I ) will play an important role in our work:it will be interpreted as a category of morphisms between Deligne-Lusztig varieties.For this reason we will abbreviate its name to D I ; when I = {∅} it reduces to thecategory D + of [DMR, 5.1].The objects of D I are endomorphisms I w φ −−→ I in B + φ ( I ) and the morphismsare generated by the “simple” morphisms that we will denote by ad v , defined for v w such that I v ⊂ S ; such a morphism goes from I w φ −−→ I to J v − w φ v −−−−−→ J where J = I v .By Proposition 3.6 the category D I has a Garside family consisting of the simplemorphisms. In particular defining relations for D I are given by the equalitiesad v · · · ad v k = ad v ′ · · · ad v ′ k ′ whenever ad v i are simple and v · · · v k = v ′ · · · v ′ k ′ in B + . If v = v · · · v k ∈ B + where the ad v i are simple morphisms of D I , we stilldenote by ad v the composed morphism in D I . Note that for w φ ∈ B + φ ( I )( I ), the centralizer Conj B + ( I )( I w φ −−→ I ) identifiesvia the forgetful functor with the monoid B + w := { b ∈ C B + ( w φ ) | I b = I and α I ( b ) = 1 } . The following theorem gives a general case where we can describe D I ( I w φ −−→ I ): Theorem 5.9.
Assume that some power of w φ is divisible on the left by w − I w .Then D I ( I w φ −−→ I ) = Conj B + ( I )( I w φ −−→ I ) , thus consists of the morphisms ad b where b ∈ B + w .Proof. This is a special case of Proposition 3.9. (cid:3)
Note that if k is the smallest power such that φ k I = I and φ k w = w , then w ( k ) := w φ w · · · φ k − w is in B + w . Since ad w is equal up to an invertible to theGarside map of D I described in Proposition 3.7 and ad w ( k ) is equal up to aninvertible to the k -th power of that map, every element of D I ( I w φ −−→ I ) dividesa power of ad w ( k ) ; it follows that under the assumptions of Theorem 5.9 everyelement of B + w divides a power of w ( k ) . In particular, in the case I = ∅ , Theorem5.9 says that B + ∩ C B ( w φ ) = End D + ( w ), with the notations of [DM2, 2.1]. Since w divides a power of w φ , hence a power of w ( k ) , any element of the group C B ( w φ )multiplied by some power of w ( k ) lies in B + , hence the group C B ( w φ ) is generatedas a monoid by End D + ( w ) and ( w ( k ) ) − . Thus Theorem 5.9 in this particular casegives a positive answer to conjecture [DM2, 2.1].As an example of Theorem 5.9 we get that D I ( I π / π I φ −−−−→ I ) identifies with { b ∈ C B + ( I ) φ | α I ( b ) = 1 } which itself identifies with B + ( I )( I ) φ . Two examples.
In two cases we show a picture of the category associated withthe centralizer of a periodic element.We first look at the case of a (4 , w ∈ B + ( W ( D )); by Propo-sition 5.2(i) we may assume w = w ; following Proposition 5.2(i) we describe themonoid (cyc B + ( w )) w , in our case equal to cyc B + ( w ) since w is central. As inTheorem 10.11, we choose w given by the word in the generators 123423 where thelabeling of the Coxeter diagram is (cid:13) (cid:13) (cid:13) (cid:13) By Proposition 5.2(i) the monoid cyc B + ( w ) generates C B ( w ); by [B1, 12.5(ii)], C B ( w ) is the braid group of C W ( w ) ≃ G (4 , , h x , y , z | xyz = yzx = zxy i . The automorphism x y z corresponds to thetriality in D . One of the generators x corresponds to the morphism 24 in thediagram below. The other generators are the conjugates of the similar morphisms41 and 21 in the other squares. ARABOLIC DELIGNE-LUSZTIG VARIETIES. 25 / / (cid:11) (cid:11) (cid:11) (cid:11) / / / / (cid:11) (cid:11) (cid:11) (cid:11) { { / / J J J J (cid:28) (cid:28) J J / / J J l l (cid:11) (cid:11) B B (cid:11) (cid:11) o o J J H H J J o o We now look at a (3 , w ∈ B + ( W ( A )), that is w = π , and followingProposition 5.2(ii) we describe cyc B + ( v Φ) where Φ is the Garside automorphism b b w and where w = v Φ( v ) = v · v w . By Proposition 5.2(ii) the monoidcyc B + ( s Φ) generates C B ( w ) and, again by the results of Bessis, C B ( w ) is thebraid group of C W ( w ) ≃ G (3 , ,
2) (see Theorem 10.4). We choose w such that v is given by the word 21325 in the generators. The generator of C B ( w ) lifting thegenerator of order 3 of G (3 , ,
2) is given by the word 531. The other one is the conjugate of any of the length 2 cycles 23 in the diagram.21435 ) ) ! ! ❉❉❉❉❉❉❉❉❉❉❉ O O H H I I H H (cid:9) (cid:9) a a ❉❉❉❉❉❉❉❉❉❉❉ (cid:5) (cid:5) (cid:15) (cid:15) o o } } ④④④④④④④④④④④ O O Y Y (cid:22) (cid:22) I I (cid:22) (cid:22) (cid:9) (cid:9) (cid:15) (cid:15) = = ④④④④④④④④④④④ = = ④④④④④④④④④④④ U U (cid:9) (cid:9) ! ! ❉❉❉❉❉❉❉❉❉❉❉ o o E E (cid:15) (cid:15) I I (cid:9) (cid:9) O O o o h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ o o v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ I I (cid:15) (cid:15) (cid:9) (cid:9) O O o o (cid:25) (cid:25) } } ④④④④④④④④④④④ o o a a ❉❉❉❉❉❉❉❉❉❉❉ ARABOLIC DELIGNE-LUSZTIG VARIETIES. 27 Representations into bicategories
We give here a theorem on representations of categories with Garside familieswhich generalizes a result of Deligne [D, 1.11] about representations of sphericalbraid monoids into a category; just as this theorem of Deligne was used to attacha Deligne-Lusztig variety to an element of an Artin monoid, our theorem will beused to attach a Deligne-Lusztig variety to a morphism of a ribbon category. Notethat Theorem 6.2 covers the case of non-spherical Artin monoids.We follow the terminology of [McL, XII.6] for bicategories. By representation ofcategory C into bicategory X we mean a morphism of bicategories between C viewedas a trivial bicategory into the given bicategory X . This amounts to give a map T from Obj( C ) to the 0-cells of X , and for f ∈ C of source x and target y , an element T ( f ) ∈ V ( T ( x ) , T ( y )) where V ( T ( x ) , T ( y )) is the category whose objects (resp.morphisms) are the 1-cells of X with domain T ( x ) and codomain T ( y ) (resp. the 2-cells between them), together with for each composable pair ( f, g ) an isomorphism T ( f ) T ( g ) ∼ −→ T ( f g ) such that the resulting square(6.1) T ( f ) T ( f ′ ) T ( f ′′ ) ∼ / / ∼ (cid:15) (cid:15) T ( f f ′ ) T ( f ′′ ) ∼ (cid:15) (cid:15) T ( f ) T ( f ′ f ′′ ) ∼ / / T ( f f ′ f ′′ )commutes.We define a representation of the Garside family S as the same, except that theabove square is restricted to the case where f , f f ′ and f f ′ f ′′ are in S , (whichimplies f ′ , f ′′ , f ′ f ′′ ∈ S since S is closed under right-divisors). We then have Theorem 6.2.
Let C be a right-Noetherian category which admits conditional right-lcms and has a Garside family S . Then any representation of S into a bicategoryextends uniquely to a representation of C into the same bicategory.Proof. The proof goes exactly as in [D], in that what must been proven is a simpleconnectedness property for the set E ( g ) of decompositions as a product of elementsof S of an arbitrary morphism g ∈ C — this generalizes [D, 1.7] and is used in thesame way. In his context, Deligne shows more, the contractibility of the set ofdecompositions; on the other hand our proof, which follows a suggestion by SergeBouc to use a version of [Bouc, Lemma 6], is simpler and holds in our more generalcontext.Fix g ∈ C with g / ∈ C × . We denote by E ( g ) the set of decompositions of g intoa product of elements of S − C × .Then E ( g ) is a poset, the order being defined by( g , . . . , g i − , g i , g i +1 , . . . , g n ) > ( g , . . . , g i − , a, b, g i +1 , . . . , g n )if ab = g i ∈ S .We recall the definition of homotopy in a poset E (a translation of the corre-sponding notion in a simplicial complex isomorphic as a poset to E ). A path from x to x k in E is a sequence x · · · x k where each x i is comparable to x i +1 . Thecomposition of paths is defined by concatenation. Homotopy, denoted by ∼ , is thefinest equivalence relation on paths compatible with concatenation and generatedby the two following elementary relations: xyz ∼ xz if x ≤ y ≤ z and both xyx ∼ x and yxy ∼ y when x ≤ y . Homotopy classes form a groupoid, as the composition of a path with source x and of the inverse path is homotopic to the constant path at x . For x ∈ E we denote by Π ( E, x ) the fundamental group of E with base point x , which is the group of homotopy classes of loops starting from x .A poset E is said to be simply connected if it is connected (there is a path linkingany two elements of E ) and if the fundamental group with some (or any) base pointis trivial.Note that a poset with a smallest or largest element x is simply connected sinceany path xyzt · · · x is homotopic to xyxzxtx · · · x which is homotopic to the trivialloop. Proposition 6.3.
The set E ( g ) is simply connected.Proof. First we prove a version of a lemma from [Bouc] on order preserving mapsbetween posets. For a poset E we put E ≥ x = { x ′ ∈ E | x ′ ≥ x } , which is asimply connected subposet of E since it has a smallest element. If f : X → Y is an order preserving map it is compatible with homotopy (it corresponds to acontinuous map between simplicial complexes), so it induces a homomorphism f ∗ :Π ( X, x ) → Π ( Y, f ( x )). Lemma 6.4 (Bouc) . Let f : X → Y an order preserving map between two posets.We assume that Y is connected and that for any y ∈ Y the poset f − ( Y ≥ y ) isconnected and non empty. Then f ∗ is surjective. If moreover f − ( Y ≥ y ) is simplyconnected for all y then f ∗ is an isomorphism.Proof. Let us first show that X is connected. Let x, x ′ ∈ X ; we choose a path y · · · y n in Y from y = f ( x ) to y n = f ( x ′ ). For i = 0 , . . . , n , we choose x i ∈ f − ( Y ≥ y i ) with x = x and x n = x ′ . Then if y i ≥ y i +1 we have f − ( Y ≥ y i ) ⊂ f − ( Y ≥ y i +1 ) so that there exists a path in f − ( Y ≥ y i +1 ) from x i to x i +1 ; otherwise y i < y i +1 , which implies f − ( Y ≥ y i ) ⊃ f − ( Y ≥ y i +1 ) and there exists a path in f − ( Y ≥ y i ) from x i to x i +1 . Concatenating these paths gives a path connecting x and x ′ .We fix now x ∈ X . Let y = f ( x ). We prove that f ∗ : Π ( X, x ) → Π ( Y, y )is surjective. Let y y · · · y n with y n = y be a loop in Y . We lift arbitrarilythis loop into a loop x — · · · — x n in X as above, (where x i — x i +1 stands for apath from x i to x i +1 which is either in f − ( Y ≥ y i ) or in f − ( Y ≥ y i +1 )). Then thepath f ( x — x — · · · — x n ) is homotopic to y · · · y n ; this can be seen by induc-tion: let us assume that f ( x — x · · · — x i ) is homotopic to y · · · y i f ( x i ); thenthe same property holds for i + 1: indeed y i y i +1 ∼ y i f ( x i ) y i +1 as they are twopaths in a simply connected set which is either Y ≥ y i or Y ≥ y i +1 ; similarly we have f ( x i ) y i +1 f ( x i +1 ) ∼ f ( x i — x i +1 ). Putting things together gives y · · · y i y i +1 f ( x i +1 ) ∼ y y · · · y i f ( x i ) y i +1 f ( x i +1 ) ∼ f ( x — · · · — x i ) y i +1 f ( x i +1 ) ∼ f ( x — · · · — x i — x i +1 ) . We now prove injectivity of f ∗ when all f − ( Y ≥ y ) are simply connected.We first prove that if x — · · · — x n and x ′ — · · · — x ′ n are two loops lifting thesame loop y · · · y n , then they are homotopic. Indeed, we get by induction on i that x — · · · — x i — x ′ i and x ′ — · · · — x ′ i are homotopic paths, using the fact that x i − , x i , x ′ i − and x ′ i are all in the same simply connected sub-poset, namely either f − ( Y ≥ y i − ) or f − ( Y ≥ y i ). ARABOLIC DELIGNE-LUSZTIG VARIETIES. 29
It remains to prove that we can lift homotopies, which amounts to show thatif we lift as above two loops which differ by an elementary homotopy, the liftingsare homotopic. If yy ′ y ∼ y is an elementary homotopy with y < y ′ (resp. y > y ′ ),then f − ( Y ≥ y ′ ) ⊂ f − ( Y ≥ y ) (resp. f − ( Y ≥ y ) ⊂ f − ( Y ≥ y ′ )) and the lifting of yy ′ y constructed as above is in f − ( Y ≥ y ) (resp. f − ( Y ≥ y ′ )) so is homotopic to the trivialpath. If y < y ′ < y ′′ , a lifting of yy ′ y ′′ constructed as above is in f − ( Y ≥ y ) so ishomotopic to any path in f − ( Y ≥ y ) with the same endpoints. (cid:3) We now prove Proposition 6.3 by contradiction. If it fails we choose g ∈ C minimal for proper right-divisibility such that E ( g ) is not simply connected.Let L be the set of elements of S − C × which are left-divisors of g . For any I ⊂ L , since the category admits conditional right-lcms and is right-Noetherian,the elements of I have an lcm. We fix such an lcm ∆ I . Let E I ( g ) = { ( g , . . . , g n ) ∈ E ( g ) | ∆ I g } . We claim that E I ( g ) is simply connected for I = ∅ . Thisis clear if g ∈ ∆ I C × , in which case E I ( g ) = { ( g ) } . Let us assume this is notthe case. In the following, if ∆ I a , we denote by a I the element such that a = ∆ I a I . The set E ( g I ) is defined since g ∆ I C × . We apply Lemma 6.4 to themap f : E I ( g ) → E ( g I ) defined by( g , . . . , g n ) ( ( g , . . . , g n ) if g = ∆ I ( g I , g , . . . g n ) otherwise . This map preserves the order and any set f − ( Y ≥ ( g ,...,g n ) ) has a least element,namely (∆ I , g , . . . , g n ), so is simply connected. As by minimality of g the set E ( g I ) is simply connected Lemma 6.4 implies that E I ( g ) is simply connected.Let Y be the set of non-empty subsets of L . We now apply Lemma 6.4 to themap f : E ( g ) → Y defined by ( g , . . . , g n )
7→ { s ∈ L | s g } , where Y is orderedby inclusion. This map is order preserving since ( g , . . . , g n ) < ( g ′ , . . . , g ′ n ) implies g g ′ . We have f − ( Y ≥ I ) = E I ( g ), so this set is simply connected. Since Y ,having a greatest element, is simply connected, Lemma 6.4 gives that E ( g ) is simplyconnected, whence the proposition. (cid:3)(cid:3) II. Deligne-Lusztig varieties and eigenspaces
In this part, we study the Deligne-Lusztig varieties giving rise to a Lusztig induc-tion functor R GL and generalize them to varieties attached to elements of a ribboncategory.In Section 8 we consider the particular ribbons describing varieties which playa role in the Brou´e conjectures; they are associated with maximal eigenspaces ofelements of the Weyl group.Finally in Section 9 we spell out the geometric form of the Brou´e conjectures,describing how the action on the ℓ -adic cohomology of the endomorphisms of ourvarieties coming from the conjugacy category of the ribbon category should factorizethrough a cyclotomic Hecke algebra. Parabolic Deligne-Lusztig varieties
Let G be a connected reductive algebraic group over F p , and let F be an isogenyon G such that some power F δ is a Frobenius for a split F q δ -structure (this definesa positive real number q such that q δ is an integral power of p ).Let L be an F -stable Levi subgroup of a (non-necessarily F -stable) parabolicsubgroup P of G and let P = LV be the corresponding Levi decomposition of P .Let X V = { g V ∈ G / V | g V ∩ F ( g V ) = ∅} = { g V ∈ G / V | g − F g ∈ V F V }≃ { g ∈ G | g − F g ∈ F V } / ( V ∩ F V ) . On this variety G F acts by left-multiplication and L F acts by right-multiplication.We choose a prime number ℓ = p . Then the virtual G F -module- L F given by M = P i ( − i H ic ( X V , Q ℓ ) defines the Lusztig induction R GL which by definitionmaps an L F -module λ to M ⊗ Q ℓ L F λ .The map g V g P makes X V an L F -torsor over X P = { g P ∈ G / P | g P ∩ F ( g P ) = ∅} = { g P ∈ G / P | g − F g ∈ P F P }≃ { g ∈ G | g − F g ∈ F P } / ( P ∩ F P ) , a G F -variety such that R GL (Id) = P i ( − i H ic ( X P , Q ℓ ). The variety X P is theprototype of the varieties we want to study.Let T ⊂ B be a pair of an F -stable maximal torus and an F -stable Borelsubgroup of G . With this choice is associated a basis Π of the root system Φ of G with respect to T , and a Coxeter system ( W, S ) for the Weyl group W = N G ( T ) / T .Let X R = X ( T ) ⊗ R where X ( T ) is the group of rational characters of the torus T . On the vector space X R , the isogeny F acts as qφ where φ is of order δ andstabilizes the positive cone R + Π; we will still denote by φ the induced automorphismof ( W, S ).To a subset I ⊂ Π corresponds a subgroup W I ⊂ W , a parabolic subgroup P I = ` w ∈ W I B w B , and the Levi subgroup L I of P I which contains T .Given any P = LV as in the beginning of this section, where L is F -stable,there exists I ⊂ Π such that ( L , P ) is G -conjugate to ( L I , P I ); if we choose theconjugating element such that it conjugates a maximally split torus of L to T anda rational Borel subgroup of L containing this torus to B ∩ L I , then this elementconjugates ( L , P , F ) to ( L I , P I , ˙ wF ) where ˙ w ∈ N G ( T ) is such that wφ I = I , where w is the image of ˙ w in W .It will be convenient to consider I as a subset of S instead of a subset of Π; thecondition on w must then be stated as “ I w = φ I and w is I -reduced”. Note that w is then also reduced- φ I . Via the above conjugation, the variety X P is isomorphicto the variety X ( I, wφ ) = { g P I ∈ G / P I | g − F g ∈ P I w F P I } . We will denote by X G ( I, wφ ) this variety when there is a possible ambiguity on thegroup. If we denote by U I the unipotent radical of P I , we have dim X ( I, wφ ) =dim U I − dim( U I ∩ wF U I ) = l ( w ), the last equality since w is reduced- φ I . The ℓ -adiccohomology of the variety X ( I, wφ ) gives rise to the Lusztig induction from L ˙ wFI to G F of the trivial representation; to avoid ambiguity on the isogenies involved,we will sometimes denote this Lusztig induction by R G ,F L I , ˙ wF (Id). ARABOLIC DELIGNE-LUSZTIG VARIETIES. 31
Definition 7.1.
We say that a pair ( P , Q ) of parabolic subgroups is in relative po-sition ( I, w, J ) , where I, J ⊂ S and w ∈ W , if ( P , Q ) is G -conjugate to ( P I , w P J ) .We denote this as P I,w,J −−−→ Q . Since any pair ( P , Q ) of parabolic subgroups share a common maximal torus,it has a relative position ( I, w, J ) where
I, J is uniquely determined as well as thedouble coset W I wW J .Let P I be the variety of parabolic subgroups conjugate to P I ; this variety isisomorphic to G / P I . Via the map g P I g P I we have an isomorphism X ( I, wφ ) ≃ { P ∈ P I | P I,w, φ I −−−−→ F P } ;it is a variety over P I × P φ I by the first and second projection. The varieties O attached to B + ( I ) . In order to have a rich enough monoid ofendomorphisms (see Definition 7.21), we need to generalize the pairs (
I, wφ ) whichlabel our varieties to the larger set of morphisms of the category B + ( I ) of Section5, where I is the conjugacy class in B + of the lift I of I .In order to do this, we define in this subsection a representation of B + ( I ) into thebicategory X of varieties over P I ×P J , where I, J vary over I . The bicategory X has0-cells which are the elements of I , has 1-cells with domain I and codomain J whichare the P I ×P J -varieties and has 2-cells which are isomorphisms of P I ×P J -varieties.For I , J ∈ I we denote by V ( I , J ) the category whose objects (resp. morphisms) arethe 1-cells with domain I and codomain J (resp. the 2-cells between them); in otherwords, V ( I , J ) is the category of P I × P J -varieties endowed with the isomorphismsof P I × P J -varieties. The horizontal composition bifunctor V ( I , J ) × V ( J , K ) → V ( I , K ) is given by the fibered product over P J . The vertical composition is givenby the composition of isomorphisms.The representation of B + ( I ) in X we construct will be denoted by T , followingthe notations of Section 6. For I b −→ J ∈ B + ( I ), we will also write O ( I , b ) for T ( I b −→ J ), to lighten the notation. We first define T on the Garside family S of B + ( I ). Definition 7.2.
For ( I w −→ J ) ∈ S we define O ( I , w ) to be the variety { ( P , P ′ ) ∈P I × P J | P I,w,J −−−→ P ′ } , where I , w , J are the images in W of I , w , J . The following lemma constructs the isomorphism T ( f ) T ( g ) ∼ −→ T ( f g ) when f, g, f g ∈ S : Lemma 7.3.
Let ( I w −−→ I w −−→ J ) = ( I w −→ J ) where w = w w ∈ W be a definingrelation of B + ( I ) . Then ( p ′ , p ′′ ) : O ( I, w ) × P I O ( I , w ) ∼ −→ O ( I, w w ) is anisomorphism, where p ′ and p ′′ are respectively the first and last projections.Proof. First notice that for two parabolic subgroups ( P ′ , P ′′ ) ∈ P I × P J we have P ′ I,w,J −−−→ P ′′ if and only if the pair ( P ′ , P ′′ ) is conjugate to a pair containingtermwise the pair ( B , w B ). This shows that if P ′ I,w ,I −−−−−→ P and P I ,w ,J −−−−−→ P ′′ then P ′ I,w w ,J −−−−−−→ P ′′ , so ( p ′ , p ′′ ) goes to the claimed variety.Conversely, we have to show that given P ′ I,w,J −−−→ P ′′ there is a unique P suchthat P ′ I,w ,I −−−−−→ P I ,w ,J −−−−−→ P ′′ . The image of ( B , w B ) by the conjugation whichsends ( P I , w P J ) to ( P ′ , P ′′ ) is a pair of Borel subgroups ( B ′ ⊂ P ′ , B ′′ ⊂ P ′′ ) in position w . Since l ( w ) + l ( w ) = l ( w ), there is a unique Borel subgroup B suchthat B ′ w −−→ B w −−→ B ′′ . The unique parabolic subgroup of type I containing B has the desired relative positions, so P exists. And any other parabolic subgroup P ′ which has the desired relative positions contains a Borel subgroup B ′ such that B ′ w −−→ B ′ w −−→ B ′′ (take for B ′ the image of w B by the conjugation which maps( P I , w P I ) to ( P ′ , P ′ )), which implies that B ′ = B and thus P ′ = P . Thus ourmap is bijective on points. To show it is an isomorphism, it is sufficient to checkthat its target is a normal variety, which is given by Lemma 7.4.
For ( I w −→ J ) ∈ S the variety O ( I , w ) is smooth.Proof. Consider the locally trivial fibrations with smooth fibers given by G × G p −→P I × P J : ( g , g ) ( g P I , g w P J ) and G × G q −→ G : ( g , g ) g − g . It iseasy to check that O ( I , w ) = p ( q − ( w P J )) thus by for example [DMR, 2.2.3] it issmooth. (cid:3)(cid:3) From the above lemma we see also that the square 6.1 commutes for elementsof S , since the isomorphism “forgetting the middle parabolic” has clearly the cor-responding property. We have thus defined a representation T of S in X .The extension of T to the whole of B + ( I ) associates with a composition I w −−→ I → · · · → I k w k −−→ J with w i ∈ W the variety O ( I , w ) × P I · · · × P Ik O ( I k , w k ) = { ( P , . . . , P k +1 ) | P i I i ,w i ,I i +1 −−−−−−→ P i +1 } , where I = I and I k +1 = J . It is a P I × P J -variety via the first and last projectionsmapping ( P , . . . , P k +1 ) respectively to P and P k +1 , and Lemma 7.3 shows thatup to isomorphism it does not depend on the chosen decomposition of I w ··· w k −−−−−→ J .Theorem 6.2 shows that there is actually a unique isomorphism between the variousmodels attached to different decompositions, so T associates a well-defined varietyto any element of B + ( I ). Definition 7.5.
For I b −→ J ∈ B + ( I ) we denote by O ( I , b ) the variety defined byTheorem 6.2. For any decomposition ( I b −→ J ) = ( I w −−→ I → · · · w k −−→ J ) intoelements of S it has the model { ( P , . . . , P k +1 ) | P i I i ,w i ,I i +1 −−−−−−→ P i +1 } . The variety O ( I , b ) is endowed with a natural action of G by simultaneousconjugation of the P i . The Deligne-Lusztig varieties attached to B + ( I ) . The automorphism φ liftsnaturally to an automorphism of B + which stabilizes S , which we will still denote by φ , by abuse of notation. For ( I w −→ φ I ) ∈ S , the variety X ( I, wφ ) is the intersectionof O ( I , w ) with the graph of F , that is, points whose image under ( p ′ , p ′′ ) hasthe form ( P , F P ). Via the correspondance between φ -conjugacy and conjugacy inthe coset, we interpret I w −→ φ I as the endomorphism I w φ −−→ I in B + φ ( I ). Moregenerally, Definition 7.6.
Let I b φ −−→ I be an endomorphism of B + φ ( I ) ; we define the variety X ( I , b φ ) as the intersection of O ( I , b ) with the graph of F . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 33
The action of G on O ( I , b ) restricts to an action of G F on X ( I , b φ ). This lastvariety may be interpreted as an “ordinary” parabolic Deligne-Lusztig variety in agroup which is a restriction of scalars: Proposition 7.7.
For any decomposition ( I b −→ φ I ) = ( I w −−→ I → · · · w k −−→ φ I ) inelements of S the variety X ( I , b φ ) has the model { ( P , . . . , P k +1 ) | P i I i ,w i ,I i +1 −−−−−−→ P i +1 and P k +1 = F ( P ) } . Let F be the isogeny of G k defined by F ( g , . . . , g k ) =( g , . . . , g k , F ( g )) and let φ be the corresponding automorphism of W k . Then theabove model is isomorphic to X G k ( I × · · · × I k , ( w , . . . , w k ) φ ) . By this isomor-phism the action of F δ corresponds to that of F kδ and the action of G F correspondsto that of ( G k ) F —the isomorphism G F ∼ −→ ( G k ) F is via the diagonal embedding.Proof. That X ( I , b φ ) has the model given above is a consequence of the analogousstatement for O ( I , b ).An element P × · · · × P k ∈ X G k ( I × · · · × I k , ( w , . . . , w k ) φ ) by definitionsatisfies P × · · · × P k I ×··· I k , ( w ,...,w k ) ,I ×··· I k × φ I −−−−−−−−−−−−−−−−−−−−−→ P × · · · × P k × F P thus is equivalently given by a sequence ( P , . . . , P k +1 ) such that P i I i ,w i ,I i +1 −−−−−−→ P i +1 with P k +1 = F P and I k +1 = φ I , which is the same as an element( P , . . . , P k +1 ) ∈ O ( I , w ) × P I O ( I , w ) · · · × P Ik O ( I k , w k )such that P k +1 = F P . But this is a model of X G ( I , b φ ) as explained above.One checks easily that this sequence of identifications is compatible with theactions of F δ and G F as described by the proposition. (cid:3) Proposition 7.8.
The variety X ( I , b φ ) is irreducible if and only if I ∪ supp( b ) meets all the orbits of φ on S , where supp( b ) is the support of b (see the proof ofLemma 4.5).Proof. This is, using Proposition 7.7, an immediate translation in our setting of theresult [BR, Theorem 2] of Bonnaf´e-Rouquier. (cid:3)
The varieties ˜ X ( I , w φ ) . The conjugation which transforms X P into X ( I, wφ )maps X V to the G F -variety- L ˙ wFI given by(7.9) ˜ X ( I, ˙ wF ) = { g U I ∈ G / U I | g − F g ∈ U I ˙ w F U I } , where ˙ w is a representative of w (any representative can be obtained by choosingan appropriate conjugation). The map g U I g P I makes ˜ X ( I, ˙ wF ) a L ˙ wFI -torsorover X ( I, wφ ). We have written ˙ w and F together since the variety depends only onthe product ˙ wF ∈ N G ( T ) ⋊ h F i ; we will write ˜ X ( I, ˙ w · F ) to separate the Frobeniusendomorphism from the representative of the Weyl group element when needed, inthe case where the ambient group is a Levi subgroup with Frobenius endomorphismof the form ˙ xF .In this section, we define a variety ˜ X ( I , w φ ) which generalizes ˜ X ( I, ˙ wF ) byreplacing ˙ w by elements of the braid group. Since ˙ w represents a choice of a lift of w to N G ( T ), we have to make uniformly such choices for all elements of the braidgroup, which we do by using a “Tits homomorphism”.First, when w ∈ W , we define a variety ˜ O ( I, ˙ w ) “above” O ( I , w ) such that˜ X ( I, ˙ wF ) is the intersection of ˜ O ( I, w ) with the graph of F , and then we extendthis construction to B + ( I ). Definition 7.10.
Let ( I w −→ J ) ∈ S , and let ˙ w ∈ N G ( T ) be a representative of w .We define ˜ O ( I, ˙ w ) = { ( g U I , g ′ U J ) ∈ G / U I × G / U J | g − g ′ ∈ U I ˙ w U J } . The variety ˜ O ( I, ˙ w ) has a left action of G by simultaneous translation and aright action of L I by ( g U I , g ′ U J ) ( gl U I , g ′ l ˙ w U J ).We can prove an analogue of Lemma 7.3. Lemma 7.11.
Let ( I w −−→ I w −−→ J ) = ( I w w −−−−→ J ) where w w ∈ W be a definingrelation of B + ( I ) , and let ˙ w , ˙ w be representatives of the images of w and w in W . Then ( p ′ , p ′′ ) : ˜ O ( I, ˙ w ) × G / U I ˜ O ( I , ˙ w ) ∼ −→ ˜ O ( I, ˙ w ˙ w ) is an isomorphismwhere p ′ and p ′′ are the first and last projections.Proof. We first note that if I w −→ J ∈ B + ( I ) and ˙ w is a representative in N G ( T ) ofthe image of w in W , then U I ˙ w U J is isomorphic by the product morphism to thedirect product of varieties ( U I ∩ w U − J ) ˙ w × U J , where U − J is the unipotent radicalof the parabolic subgroup opposed to P J containing T . We now use the lemma: Lemma 7.12.
Under the assumptions of Lemma 7.11, the product gives an iso-morphism ( U I ∩ ˙ w U − I ) ˙ w × ( U I ∩ ˙ w U − J ) ˙ w ∼ −→ ( U I ∩ ˙ w ˙ w U − J ) ˙ w ˙ w .Proof. Since w is I -reduced and I w = J , we have U I ∩ w U − J = Q − α ∈ w N ( w ) U α as a product of root subgroups, where N ( w ) = { α ∈ Φ + | w α ∈ Φ − } . Thelemma is then a consequence of the equality N ( w ) w ` N ( w ) = N ( w w ) when l ( w ) + l ( w ) = l ( w w ). (cid:3) The lemma proves in particular that if g − g ∈ U I ˙ w U I and g − g ∈ U I ˙ w U J then g − g ∈ U I ˙ w U I ˙ w U J = ( U I ∩ ˙ w U − I ) ˙ w ( U I ∩ ˙ w U − J ) ˙ w U J = ( U I ∩ ˙ w ˙ w U − J ) ˙ w ˙ w U J = U I ˙ w ˙ w U J , so the image of the morphism ( p ′ , p ′′ ) in Lemma7.11 is indeed in the variety ˜ O ( I, ˙ w ˙ w ).Conversely, we have to show that given ( g U I , g U J ) ∈ ˜ O ( I, ˙ w ˙ w ), there existsa unique g U I such that ( g U I , g U I ) ∈ ˜ O ( I, ˙ w ) and ( g U I , g U I ) ∈ ˜ O ( I , ˙ w ).The varieties involved being invariant by left-translation by G , it is enough to solvethe problem when g = 1. Then we have g ∈ U I ˙ w ˙ w U J , and the conditions for g U I is that g U I ⊂ U I ˙ w U I . Any such coset has then a unique representativein ( U I ∩ ˙ w U − I ) ˙ w and we will look for such a representative g . But we must have g − g ∈ U I ˙ w U J = ( U I ∩ ˙ w U − J ) ˙ w U J and since by the lemma the product givesan isomorphism between ( U I ∩ ˙ w U − I ) ˙ w × ( U I ∩ ˙ w U − J ) ˙ w U J and U I ˙ w ˙ w U J ,the element g can be decomposed in one and only one way in a product g ( g − g )satisfying the conditions. To conclude as in Lemma 7.3 we show that the variety˜ O ( I , ˙ w ˙ w ) is smooth. An argument similar to the proof of Lemma 7.4, replacing P I and P J by G / U I and G / U J respectively gives the result. (cid:3) The isomorphism of Lemma 7.11 is compatible with the action of G and of L I , L I respectively.We will now use a Tits homomorphism, which is a homomorphism B t −→ N G ( T )which factors the projection B → W —the existence of such a homomorphism isproved in [T]. Theorem 6.2 implies that, setting T ( I w −→ J ) = ˜ O ( I, t ( w )) for ( I w −→ J ) ∈ S and replacing Lemma 7.3 by Lemma 7.11, we can define a representation of B + ( I ) in the bicategory ˜ X of varieties above G / U I × G / U J for I, J ∈ I . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 35
Definition 7.13.
The above representation defines for any I b −→ J ∈ B + ( I ) avariety ˜ O ( I , b ) which for any decomposition ( I b −→ J ) = ( I w −−→ I → · · · → I k w k −−→ J ) into elements of S has the model ˜ O ( I, t ( w )) × G / U I · · · × G / U Ik ˜ O ( I k , t ( w k )) . By the remarks after Lemma 7.11 the variety ˜ O ( I , b ) affords a natural left actionof G and right action of L I . Proposition 7.14.
There exists a Tits homomorphism t which is F -equivariant,that is such that t ( φ ( b )) = F ( t ( b )) .Proof. With any simple reflection s ∈ S is associated a quasi-simple subgroup G s of rank 1 of G , generated by the root subgroups U α s and U − α s ; the 1-parametersubgroup of T given by T ∩ G s is a maximal torus of G s . By [T, Theorem 4.4] iffor any s ∈ S we choose a representative ˙ s of s in G s , then these representativessatisfy the braid relations, which implies that s ˙ s induces a well defined Titshomomorphism. We claim that if s is fixed by some power φ d of φ then there exists˙ s ∈ G s fixed by F d ; we then get an F -equivariant Tits homomorphism by choosingarbitrarily ˙ s for one s in each orbit of φ . If s is fixed by φ d then G s is stableby F d ; the group G s is isomorphic to either SL or P SL and F d is a Frobeniusendomorphism of this group. In either case the simple reflection s of G s has an F d -stable representative in N G s ( T ∩ G s ), whence our claim. (cid:3) Notation 7.15.
We assume now that we have chosen, once and for all, an F -equivariant Tits homomorphism t which is used to define the varieties ˜ O ( I , b ) . The equivariance of t allows to extend it to a morphism B + ⋊ h φ i → N G ( T ) ⋊ h F i —note that here our convention that h φ i is infinite order is useful, since F is ofinfinite order. This allows to extend t by t ( φ ) = F thus we can write indifferently t ( b ) F or t ( b φ ). Definition 7.16.
For any endomorphism ( I b φ −−→ I ) ∈ B + φ ( I ) we define ˜ X ( I , b φ ) = { x ∈ ˜ O ( I , b ) | p ′′ ( x ) = F ( p ′ ( x )) } . The action of L I on ˜ O ( I , b ) restricts to an action of L t ( b φ ) I on ˜ X ( I , b φ ), com-patible with the first projection ˜ X ( I , b φ ) → G / U I .When w ∈ W we have ˜ X ( I , w φ ) = ˜ X ( I, t ( w φ )), the variety defined in 7.9 for˙ wF = t ( w φ ). We have the following analogue of Proposition 7.7 for ˜ X ( I , b φ ). Proposition 7.17.
Let I = I w −−→ I → · · · → I k w k −−→ φ I be a decomposition intoelements of S of I b −→ φ I ∈ B + ( I ) , let F be the isogeny of G k as in Proposition7.7.Then ˜ X G ( I , b φ ) ≃ ˜ X G k ( I × · · · × I k , ( t ( w ) , . . . , t ( w k )) F ) . By this isomor-phism the action of F δ corresponds to that of F kδ , the action of G F correspondsto that of ( G k ) F , and the action of L t ( b φ ) I corresponds to that of ( L I × · · · × L I k ) ( t ( w ) ,...,t ( w k )) F .Proof. An element x U I × · · · × x k U I k ∈ ˜ X G k ( I × · · · × I k , ( t ( w ) , . . . , t ( w k )) F )by definition satisfies ( x i U I i , x i +1 U I i +1 ) ∈ ˜ O ( I i , t ( w i )) for i = 1 , . . . , k , where wehave put I k +1 = F I and x k +1 U I k +1 = F ( x U I ). This is the same as an elementin the intersection of ˜ O ( I , w ) × G / U I ˜ O ( I , w ) · · · × G / U Ik ˜ O ( I k , w k ) with thegraph of F . Since, by definition, we have˜ O ( I , b ) ≃ ˜ O ( I , w ) × G / U I ˜ O ( I , w ) · · · × G / U Ik ˜ O ( I k , w k ) , via this last isomorphism we get an element of ˜ O ( I , b ) which is in ˜ X G ( I , b φ ).One checks easily that this sequence of identifications is compatible with theactions of F δ , of G F and of L t ( b φ ) I as described by the proposition. (cid:3) Lemma 7.18.
For any endomorphism ( I b φ −−→ I ) ∈ B + φ ( I ) , there is a natural pro-jection ˜ X ( I , b φ ) π −→ X ( I , b φ ) which makes ˜ X ( I , b φ ) a L t ( b φ ) I -torsor over X ( I , b φ ) .Proof. Let I w −−→ I → · · · → I r w r −−→ φ I be a decomposition into elements of S of I b −→ φ I , so that ˜ X ( I , b φ ) identifies with the set of sequences ( g U I , g U I , . . . , g r U I r )such that g − j g j +1 ∈ U I j t ( w j ) U I j +1 for j < r and g − r F g ∈ U I r t ( w r ) U φ I . Wedefine π by g j U I j g j P I j . It is easy to check that the morphism π thus de-fined commutes with an “elementary morphism” in the bicategories of varieties ˜ X or X consisting of passing from the decomposition ( w , . . . , w i , w i +1 , . . . , w r ) to( w , . . . , w i w i +1 , . . . , w r ) when ( I i w i w i +1 −−−−−→ I i +2 ) ∈ S . Thus by 6.1 the morphism π is well-defined independently of the chosen decomposition of b .The fact that π makes ˜ X ( I , b φ ) a L t ( b φ ) -torsor over X ( I , b φ ) results then viaProposition 7.17 from the same statement on the varieties of 7.9. (cid:3) We give an isomorphism which reflects the transitivity of Lusztig’s induction.
Proposition 7.19.
Let I w φ −−→ I ∈ B + φ ( I ) , and let w be the image of w in W ;the automorphism wφ of W I lifts to an automorphism that we will still denote by wφ of B + I . For J ⊂ I , let J be the set of B + I -conjugates of J and let J v wφ −−−→ J ∈ B + I wφ ( J ) . Then (i) We have an isomorphism ˜ X ( I , w φ ) × L t ( w φ ) I ˜ X L I ( J , v wφ ) ∼ −→ ˜ X ( J , vw φ ) of G F -varieties- L t ( vw φ ) J , where the variety ˜ X L I ( J , v wφ ) is defined via the(obvious) Tits homomorphism B + I ⋊ h wφ i → N L I ( T ) ⋊ h t ( w φ ) i . Thisisomorphism is compatible with the action of F n for any n such that I , J , v and w are φ n -stable. (ii) Through the quotient by L t ( v wφ ) J (see Lemma 7.18) we get an isomorphismof G F -varieties ˜ X ( I , w φ ) × L t ( w φ ) I X L I ( J , v wφ ) ∼ −→ X ( J , vw φ ) . Proof.
We first look at the case w , v ∈ W (which implies vw ∈ W ), in which casewe seek an isomorphism˜ X ( I, t ( w φ )) × L t ( w φ ) I ˜ X L I ( J, t ( v wφ )) ∼ −→ ˜ X ( J, t ( vw φ ))where ˜ X ( I, t ( w φ )) = { g U I ∈ G / U I | g − F g ∈ U I t ( w ) F U I } , ˜ X ( J, t ( vw φ )) = { g U J ∈ G / U J | g − F g ∈ U J t ( vw ) F U J } and ˜ X L I ( J, t ( v wφ )) = { l V J ∈ L I / V J | l − t ( wφ ) l ∈ V I t ( v ) t ( wφ ) V J } , where V J = L I ∩ U J .This is the content of Lusztig’s proof of the transitivity of his induction (see [Lu,Lemma 3]), that we recall and detail in our context. We claim that ( g U I , l V J ) ARABOLIC DELIGNE-LUSZTIG VARIETIES. 37 g U I l V J = gl U J induces the isomorphism we want. Using that U J = U I V J andthat V J t ( v ) t ( wφ ) V J is in L I , thus normalizes U I , we get U J t ( vw ) F U J = U I V J t ( v ) t ( wφ ) V J t ( w ) F U I = V J t ( v ) t ( wφ ) V J U I t ( w ) F U I . Hence if ( g U I , l V J ) ∈ ˜ X ( I, t ( w φ )) × ˜ X L I ( J, t ( v wφ )), we have( gl ) − F ( gl ) ∈ l − U I t ( w ) F U I F l = l − U I t ( wφ ) lt ( w ) F U I = l − t ( wφ ) l U I t ( w ) F U I ⊂ V J t ( v ) t ( wφ ) V J U I t ( w ) F U I = U J t ( vw ) F U J . Hence we have defined a morphism ˜ X ( I, t ( w φ )) × ˜ X L I ( J, t ( v wφ )) → ˜ X ( J, t ( vw φ ))of G F -varieties- L t ( v wφ ) J . We show now that it is surjective. The unicity in the de-composition P I ∩ t ( w φ ) U I = L I · ( U I ∩ t ( w φ ) U I ) implies that the product L I . ( U I t ( w ) F U I )is direct. Hence an element x − F x ∈ U J t ( vw ) F U J defines unique elements l ∈ V J t ( v ) t ( wφ ) V J and u ∈ U I t ( w ) F U I such that x − F x = lu . If, using Lang’stheorem, we write l = l ′− t ( wφ ) l ′ with l ′ ∈ L I , the element g = xl ′− satisfies g − F g = l ′ x − F x F l ′− = t ( wφ ) l ′ u F l ′− ∈ t ( wφ ) l ′ U I t ( w ) F U I F l ′− = U I t ( w ) F U I .Hence ( g U I , l ′ V J ) is a preimage of x U J in ˜ X ( I, t ( w φ )) × ˜ X L I ( J, t ( v wφ )).Let us look now at the fibers of the above morphism. If g ′ U I l ′ V J = g U I l V J then g ′− g ∈ P I so we may choose g ′ in g ′ U I such that g ′ = gλ with λ ∈ L I ; we have then λl ′ U J = l U J , so that l − λl ′ ∈ U J ∩ L I = V J ; moreover if gλ U I ∈ ˜ X ( I, t ( w φ ))with λ ∈ L I , then λ − U I t ( w ) F U I F λ = U I t ( w ) F U I which implies λ ∈ L t ( w φ ) I .Conversely, the action of λ ∈ L t ( w φ ) I given by ( g U I , l V J ) ( gλ U I , λ − l V J ) pre-serves the subvariety ˜ X ( I, t ( w φ )) × ˜ X L I ( J, t ( v wφ )), of G / U I × L I / V J . Hence thefibers are the orbits under this action of L t ( w φ ) I .Now the morphism j : ( g U I , l V J ) gl U J is an isomorphism G / U I × L I L I / V J ≃ G / U J since g U J ( g U I , V J ) is its inverse. By what we have seen abovethe restriction of j to the closed subvariety ˜ X ( I, t ( w φ )) × L t ( w φ ) I ˜ X L I ( J, t ( v wφ )) mapsthis variety surjectively on the closed subvariety ˜ X ( J, t ( vw φ )) of G / U J , hence weget the isomorphism we want.We now consider the case of generalized varieties. Let k be the number of terms ofthe strict normal decomposition of vw and let I w −−→ I w −−→ I → · · · → I k w k −−→ φ I be a normal decomposition of I w −→ φ I of same length. We have ˜ X ( I , w φ ) ≃ ˜ X ( I × I × · · · × I k , ( t ( w ) , . . . , t ( w k )) F ), where F is as in Proposition 7.7. Let uswrite ( v w , . . . , v k w k ) for the normal decomposition of vw , with same notationas in Proposition 5.8. Let J = J and J j +1 = J v j w j j ⊂ I j +1 for j = 1 , . . . , k − G k with isogeny F with I , J , w , v replaced respectively by I × · · · × I k , J × · · · × J k , ( w , . . . , w k ) ( v , . . . , v k ).Using the isomorphisms from Proposition 7.17;˜ X G k ( J × · · · × J k , ( t ( v w ) , . . . , t ( v k w k )) F ) ≃ ˜ X ( J , vw φ )and˜ X L I ×···× Ik ( J × · · · × J k , ( v , . . . , v k ) . ( t ( w ) , . . . , t ( w k )) F ) ≃ ˜ X L I ( J , v wφ ) , we get (i). Now (ii) is immediate from (i) taking the quotient on both sides by L t ( v wφ ) J . (cid:3) In the particular case where I = ∅ we write X ( w φ ) for X ( I , w φ ). Let us recallthat in [DMR, 2.3.2] we defined a monoid B + generated by B + and symbols w where w ∈ W , and attached to any u ∈ B + a Deligne-Lusztig variety X ( u φ ). Thisvariety is denoted by X ( u ) in [DMR] and roughly defined by the property thatgiven w attached to w ∈ W , we have X ( u w u φ ) = S w ′ ≤ w X ( u w ′ u φ ), where w ′ is the lift to B + of w ′ and where w ′ runs over the elements smaller than w for theBruhat order. Attached to I ⊂ S , we have an analogous monoid B + I attached to W I , which has a natural embedding B + I ⊂ B + . Corollary 7.20.
With these notations of [DMR] , for any I w φ −−→ I ∈ B + φ ( I ) andany u ∈ B + I , we have an isomorphism X ( uw φ ) ∼ −→ ˜ X ( I , w φ ) × L t ( w φ ) I X L I ( u wφ ) and a surjective morphism X ( uw φ ) → X ( I , w φ ) whose fibers are isomorphic to X L I ( u wφ ) .Proof. The variety X ( uw φ ) is the union of varieties of the form X L I ( v wφ ) with v ∈ W I . The isomorphisms given for each v by Proposition 7.19 applied with J = ∅ can be glued together to give a global morphism of varieties since theyare defined by a formula independent of v . We thus get a bijective morphism˜ X ( I , w φ ) × L t ( w φ ) I X L I ( u wφ ) → X ( uw φ ) which is an isomorphism since X ( uw φ )is normal (see [DMR, 2.3.5]). Composing this isomorphism with the projection of˜ X ( I , w φ ) × L t ( w φ ) I X L I ( u wφ ) onto X ( I , w φ ) (see 7.18), we get the second assertionof the corollary. (cid:3) Endomorphisms of parabolic Deligne-Lusztig varieties — the category D I .Definition 7.21. Given ad v ∈ D I ( I w φ −−→ I , J v − w φ v −−−−−→ J ) where J = I v , we definemorphisms of varieties: (i) D v : X ( I , w φ ) → X ( J , v − w φ v ) as the restriction of the morphism ( a, b ) ( b, F a ) : O ( I , w ) = O ( I , v ) × P J O ( J , v − w ) →O ( J , v − w ) × P φI O ( φ I , φ v ) = O ( J , v − w φ v ) . (ii) ˜ D v : ˜ X ( I , w φ ) → ˜ X ( J , v − w φ v ) as the restriction of the morphism ( a, b ) ( b, F a ) : ˜ O ( I , w ) = ˜ O ( I , v ) × G / U J ˜ O ( J , v − w ) → ˜ O ( J , v − w ) × G / U φI ˜ O ( φ I , φ v ) = ˜ O ( J , v − w φ v ) . Note that the existence of well-defined decompositions as above of O ( I , w ) andof ˜ O ( I , w ) are consequences of Theorem 6.2.Note that when v , w and v − w φ v are in W the endomorphism D v maps g P I ∈ X ( I, wφ ) to g ′ P J ∈ X ( J, v − wφv ) such that g − g ′ ∈ P I v P J and g ′− F g ∈ P J v − w F P I and similarly for ˜ D v .Note also that D v and ˜ D v are equivalences of ´etale sites; indeed, the proof of[DMR, 3.1.6] applies without change in our case.The definition of ˜ D v and D v shows the following property: ARABOLIC DELIGNE-LUSZTIG VARIETIES. 39
Lemma 7.22.
The following diagram is commutative: ˜ X ( I , w φ ) ˜ D v / / (cid:15) (cid:15) ˜ X ( J , v − w φ v ) (cid:15) (cid:15) X ( I , w φ ) D v / / X ( J , v − w φ v ) where the vertical arrows are the respective quotients by L t ( w φ ) I and L t ( v − w φ v ) J (seeLemma 7.18); for l ∈ L t ( w φ ) I we have ˜ D v ◦ l = l t ( v ) ◦ ˜ D v . As a further consequence of Theorem 6.2, the map which sends a simple mor-phism ad v to D v extends to a natural morphism from D I ( I w φ −−→ I , J v − w φ v −−−−−→ J )to Hom G F ( X ( I , w φ ) , X ( J , v − w φ v )) whose image consists of equivalences of ´etalesites. We still denote by D v the image of ad v by this morphism. Lemma 7.23.
Via the isomorphism of 7.17 and with the notations of loc. cit. themorphism D w with source ˜ X G ( I , b φ ) becomes the morphism D ( t ( w ) , ,..., withsource ˜ X G k ( I × · · · × I k , ( t ( w ) , . . . , t ( w k )) F ) .Proof. The endomorphism D w maps the element ( g U , . . . , g k U k ) of the modelof 7.17 of ˜ X G ( I , b φ ) to ( g U , . . . , g k U k , F g F U ). On the other hand the isomor-phism of Proposition 7.17 maps ( g U , . . . , g k U k ) to( g , . . . , g k )( U , . . . , U k ) ∈ ˜ X G k ( I × · · · × I k , ( t ( w ) , . . . , t ( w k )) F )which is sent by D ( t ( w ) , ,..., to ( g , . . . , g k , F g )( U , . . . , U k , F U ) which is the im-age by the isomorphism of Proposition 7.17 of ( g U , . . . , g k U k , F g F U ), whencethe lemma. (cid:3) Proposition 7.24.
For J ⊂ I let J denote the set of B + I -conjugates of J . Withsame assumptions and notation as in Proposition 7.19, let J x −→ J x ∈ B + I ( J ) be aleft-divisor of J v −→ w φ J . The following diagram is commutative: ˜ X ( I , w φ ) × L t ( w φ ) I ˜ X L I ( J , v · wφ ) ∼ / / Id × ˜ D x (cid:15) (cid:15) ˜ X ( J , vw φ ) ˜ D x (cid:15) (cid:15) ˜ X ( I , w φ ) × L t ( w φ ) I ˜ X L I ( J x , x − ( v · wφ ) x ) ∼ / / ˜ X ( J x , x − vw φ x ) Proof.
Decomposing x into a product of simples in the category analogous to D I where B + is replaced by B + I and I by J , the definitions show that it is sufficientto prove the result for x ∈ W . We use then Proposition 5.8 and Lemma 7.23 toreduce the proof to the case where v , w and x − v wφ x are in W (in which case vw and x − vw φ x are in W too): we choose compatible decompositions of v and w as in 5.8 which we refine if needed so that x is the first term of that of v and useLemma 7.23 once in G and once in in L I .Assume now v , w and x − v wφ x in W . We start with ( g U I , l V J ) ∈ ˜ X ( I, t ( w φ )) × ˜ X L I ( J, vwφ ). This element is sent by the top isomorphism of the diagram to gl U J .On the other hand, we have seen above Lemma 7.22 that it is sent by Id × ˜ D x to ( g U I , l ′ V J x ) where l − l ′ ∈ V J x V J x and l ′− t ( w φ ) l ∈ V J x x − v wF V J . This ele-ment is sent in turn to gl ′ U J x by the bottom isomorphism of the diagram. We have to check that gl ′ U J x = ˜ D x ( gl U J ). But ( gl ) − gl ′ = l − l ′ is in V J x V J x ⊂ U J x U J x and( gl ′ ) − F ( gl ) = l ′− g − F g F l ∈ l ′− U I t ( w ) F U I F l = U I l ′− t ( wφ ) lt ( w ) F U I ⊂ U I V J x x − vw F V J F U I = U J x x − vw F U J , so that ( gl ′ U J x ) = ˜ D x ( gl U J ). (cid:3) Using Proposition 7.19(ii), Proposition 7.24 and Lemma 7.22 we get
Corollary 7.25.
The following diagram is commutative: ˜ X ( I , w φ ) × L t ( w φ ) I X L I ( J , v · wφ ) ∼ / / Id × D x (cid:15) (cid:15) X ( J , vw φ ) D x (cid:15) (cid:15) ˜ X ( I , w φ ) × L t ( w φ ) I X L I ( J x , x − ( v · wφ ) x ) ∼ / / X ( J x , x − vw φ x ) Affineness.
Until the end of the text, we will be specially interested in varieties X ( I , b φ ) which satisfy the assumption of Theorem 5.9, that is some power of b φ is left-divisible by w − I w . They have many nice properties. We show in thissubsection that they are affine, by adapting the proof of Bonnaf´e and Rouquier[BR2]; we use the existence of the varieties ˜ O ( I , b ) and ˜ X ( I , b φ ) to replace doinga quotient by L I by doing a quotient by L t ( w φ ) I . Proposition 7.26.
Assume the morphism I b −→ J ∈ B + ( I ) is left-divisible by ∆ I = I w − I w −−−−−→ I w . Then the variety ˜ O ( I , b ) is affine.Proof. By assumption there exists a decomposition into elements of S of I b −→ J ofthe form I w − I w −−−−−→ I v −→ I v −→ I → · · · → I r v r −→ J . We show that the map ϕ defined by: G × i = r Y i =1 ( U I i ∩ t ( v i ) U − I i +1 ) t ( v i ) → ˜ O ( I, t ( w − I w )) × G / U I ˜ O ( I , t ( v )) · · · × G / U Ir ˜ O ( I r , t ( v r ))( g, h , . . . , h r ) ( g U I , gt ( w − I w ) U I , gt ( w − I w ) h U I , . . . , gt ( w − I w ) h · · · h r U J )is an isomorphism; since the first variety is a product of affine varieties this willprove our claim.Since U I i t ( v i ) U I i +1 is isomorphic to ( U I i ∩ t ( v i ) U − I i +1 ) t ( v i ) × U I i +1 , by com-position with the first projection we get a morphism η i : U I i t ( v i ) U I i +1 → ( U I i ∩ t ( v i ) U − I i +1 ) t ( v i ) for i = 1 , . . . , r , where I r +1 = J . Similarly we have a morphism η : U I t ( w − I w ) U I → ( U I ∩ t ( w − I w ) U − I ) t ( w − I w ). For x = ( g U I , g U I , g U I , . . . , g r U I r , g r +1 U J ) ∈ ˜ O ( I, t ( w − I w )) × G / U I ˜ O ( I , t ( v )) · · · × G / U Ir ˜ O ( I r , t ( v r )) ARABOLIC DELIGNE-LUSZTIG VARIETIES. 41 let ψ ( x ) = gη ( g − g ), ψ ( x ) = ψ ( x ) t ( w ), ψ i ( x ) = η i (( ψ ( x ) ψ ( x ) · · · ψ i − ( x )) − g i ).We claim that the map ψ (resp. ψ i ) is well defined, that is does not depend on therepresentative g (resp. g i ) chosen; the morphism x ( ψ ( x ) , ψ ( x ) , . . . , ψ r ( x )) isthen clearly inverse to ϕ . Since η i ( hu ) = η i ( h ) for all h ∈ U I i t ( v i ) U I i +1 and all u ∈ U I i +1 , we get that all ψ i are well-defined. Since moreover η ( uh ) = uη ( h ) forall h ∈ U I t ( w − I w ) U I and all u ∈ U I , we get that ψ also is well-defined, whenceour claim. (cid:3) Proposition 7.27.
Assume that we are under the assumptions of Theorem 5.9,that is ( I w φ −−→ I ) ∈ B + φ ( I ) has some power divisible by ∆ I , or equivalently somepower of w φ is left-divisible by w − I w . Then ˜ X ( I , w φ ) is affine.Proof. Let us define k as the smallest integer such that φ k I = I , φ k w = w and w − I w w ( k ) , where w ( k ) := w φ w · · · φ k − w .We will embed ˜ X ( I , w φ ) as a closed subvariety in ˜ O ( I , w ( k ) ), which will prove itto be affine.Let I w −−→ I w −−→ I → · · · → I r w r −−→ φ I be a decomposition of I w −→ φ I intoelements of S , so that ˜ O ( I , w ( k ) ) identifies with the set of sequences( g , U I , g , U I , . . . , g ,r U I r ,g , U φ I , g , U φ I , . . . , g ,r U φ I r ,. . . ,g k, U φk − I , g k, U φk − I , . . . , g k,r U φk − I r ,g k +1 , U I )such that for j < r we have g − i,j g i,j +1 ∈ U φi − I j t ( φ i − w j ) U φi − I j +1 and g − i,r g i +1 , ∈ U φi − I r t ( φ i − w r ) U φi I .Similarly ˜ X ( I , w φ ) identifies with the set of sequences ( g U I , g U I , . . . , g r U I r )such that g − j g j +1 ∈ U I j t ( w j ) U I j +1 for j < r and g − r F g ∈ U I r t ( w r ) U φ I . It isthus clear that the map( g U I , g U I , . . . , g r U I r ) ( g U I , g U I , . . . , g r U I r , F g U φ I , F g U φ I , . . . , F g r U φ I r ,. . . , F k − g U φk − I , . . . , F k − g r U φk − I r , F k g U I )identifies ˜ X ( I , w φ ) with the closed subvariety of ˜ O ( I , w ( k ) ) defined by g i +1 ,j U φi I j = F ( g i,j U φi − I j ) for all i, j . (cid:3) Corollary 7.28.
Under the assumptions of Theorem 5.9, that is ( I w φ −−→ I ) ∈ B + ( I ) has some power divisible by ∆ I , or equivalently some power of w φ is divisible onthe left by w − I w , the variety X ( I , w φ ) is affine.Proof. Indeed, by Proposition 7.27 and Lemma 7.18, X ( I , w φ ) is the quotient ofan affine variety by a finite group, so it is affine. (cid:3) Shintani descent identity.
In this subsection we give a formula for the Leftschetznumber of a variety X ( I , w F ) which we deduce from a “Shintani descent identity”.Let m be a multiple of δ ; if we identify G / B with the variety B of Borel subgroupsof G , the G F m -module Q ℓ ( G / B ) F m identifies with the permutation module of G F m on B F m . Its endomorphism algebra H q m ( W ) := End G Fm ( Q ℓ B F m ) has abasis consisting of the operators ( T w ) w ∈ W where T w : B ′ X { B ′′ ∈B Fm | B ′′ w −→ B ′ } B ′′ (see [Bou, Chapitre IV §
2, exercice 22]).Similarly, since I is F m -stable, the algebra H q m ( W, W I ) := End G Fm ( Q ℓ P F m I )has a Q ℓ -basis consisting of the operators X w : P X { P ′ ∈P FmI | P ′ I,w,I −−−→ P } P ′ , where w runs over a set of representatives of the double cosets W I \ W/W I ≃ P F m I \ G F m / P F m I . The map γ which sends P ∈ P F m I to the sum of all its F m -stableBorel subgroups makes Q ℓ P F m I into a direct summand of Q ℓ B F m . Indeed the imageof γ identifies with that of the idempotent X = | ( P I / B ) F m | − P v ∈ W I T v , and γ has a left-inverse given up to a scalar by mapping B ∈ B F m to the unique (thus F m -stable) parabolic subgroup in P I containing it. The operator X w identifies withthe restriction of X T w to the image Q ℓ P F m I of X .We may define a Q ℓ -representation of B + ( I )( I ) on Q ℓ P F m I by sending I w −→ I tothe operator X w ∈ H ( W, W I ) defined by X w ( P ) = X { x ∈O ( I , w ) Fm | p ′′ ( x )= P } p ′ ( x ) . When w ∈ W , with image w in W , the operators X w and X w coincide. In theparticular case where I = ∅ we get an operator denoted by T w , defined for any w in B + . The operator X w identifies with the restriction of X T w to the image Q ℓ P F m I of X .Similarly, to I w φ −−→ I ∈ B + φ ( I ) we associate an endomorphism X w φ of Q ℓ P F m I by the formula X w φ ( P ) = X { x ∈O ( I , w ) Fm | p ′′ ( x )= F ( P ) } p ′ ( x ) . When φ ( I ) = I we have X w φ = X w F . In general we have X w φ = X T w F on Q ℓ P F m I seen as a subspace of Q ℓ B F m : on this latter module one can separate theaction of F ; the operator F sends the submodule Q ℓ P F m I to Q ℓ P F m φ ( I ) which is sentback to Q ℓ P F m I by X T w . The endomorphism X w φ commutes with G F m like F ,hence normalizes H q m ( W, W I ); its action identifies with the conjugation action of T w φ on H q m ( W, W I ) inside H q m ( W ) ⋊ h φ i .Recall that the Shintani descent Sh F m /F is the “norm” map which maps the F -class of g ′ = h. F h − ∈ G F m to the class of g = h − . F m h ∈ G F . Proposition 7.29 (Shintani descent identity) . Let I w φ −−→ I ∈ B + φ ( I ) , and let m be a multiple of δ . We have the following equality of functions on G F : ( g
7→ | X ( I , w φ ) gF m | ) = Sh F m /F ( g ′ Trace( g ′ X w φ | Q ℓ P F m I )) . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 43
Proof.
Let g = h − . F m h and g ′ = h. F h − , so that the class of g is the image bySh F m /F of the F -class of g ′ ; we have X ( I , w φ ) gF m = { x ∈ O ( I , w ) | F m h x = h x and p ′′ ( h x ) = g ′ F p ′ ( h x ) } . Taking h x as a variable in the last formula we get | X ( I , w φ ) gF m | = |{ x ∈ O ( I , w ) F m | p ′′ ( x ) = g ′ F p ′ ( x ) }| . Putting P = p ′ ( x ) this lastnumber becomes P P ∈P FmI |{ x ∈ O ( I , w ) F m | p ′ ( x ) = P and p ′′ ( x ) = g ′ F P }| . Onthe other hand the trace of g ′ X w φ is the sum over P ∈ P F m I of the coefficient of P in P { x ∈O ( I , w ) Fm | p ′′ ( x )= F ( P ) } g ′ p ′ ( x ). This coefficient is equal to |{ x ∈ O ( I , w ) F m | g ′ p ′ ( x ) = P and p ′′ ( x ) = F P }| = |{ x ∈ O ( I , w ) F m | p ′ ( x ) = P and p ′′ ( x ) = g ′ F P }| , this last equality by changing g ′ x into x . (cid:3) The above computation can be done along different lines, without mention-ing Q ℓ P F m I ; one can use instead Corollary 7.20 for u = w I , which gives a G F -equivariant morphism X ( w I w φ ) → X ( I , w φ ) whose fibers are isomorphic to thevariety of Borel subgroups of L I ; the action of F induces that of t ( wφ ) on the fibers.One may then use directly [DMR, 3.3.7] to get | X ( w I w φ ) gF m | = Trace( g ′ T w I T w φ | Q ℓ B F m ), where T w I = P v ∈ W I T v .By, for example, [DM1, II, 3.1] the algebras H q m ( W ) and H q m ( W ) ⋊ h φ i splitover Q ℓ [ q m/ ]; corresponding to the specialization q m/ H q m ( W ) → Q ℓ W ,there is a bijection χ χ q m : Irr( W ) → Irr( H q m ( W )). Choosing an extension˜ χ to W ⋊ h φ i of each character in Irr( W ) φ , we get a corresponding extension˜ χ q m ∈ Irr( H q m ( W ) ⋊ h φ i ) which takes its values in Q ℓ [ q m/ ]. If U χ ∈ Irr( G F m ) isthe corresponding character of G F m , we get a corresponding extension U ˜ χ of U χ to G F m ⋊ h F i (see [DM1, III th´eor`eme 1.3 ]). With these notations, the Shintanidescent identity becomes Proposition 7.30. ( g
7→ | X ( I , w φ ) gF m | ) = X χ ∈ Irr( W ) φ ˜ χ q m ( X T w φ ) Sh F m /F U ˜ χ and the only characters χ in that sum which give a non-zero contribution are thosewhich are a component of Ind WW I Id .Proof. We have Trace( g ′ X w φ | Q ℓ P F m I ) = Trace( g ′ X T w φ | Q ℓ B F m ) since X is theprojector onto Q ℓ P F m I . Hence ( g
7→ | X ( I , w φ ) gF m | ) = P χ ∈ Irr( W ) φ ˜ χ q m ( X T w φ ) Sh F m /F U ˜ χ .Since X acts by 0 on the representation of character χ if χ is not a component ofInd WW I Id, we get the second assertion. (cid:3)
Finally, if λ ρ is the root of unity attached to ρ ∈ E ( G F ,
1) as in [DMR, 3.3.4],the above formula translates, using [DM1, III, 2.3(ii)] as
Corollary 7.31. | X ( I , w φ ) gF m | = X ρ ∈E ( G F , λ m/δρ ρ ( g ) X χ ∈ Irr( W ) φ ˜ χ q m ( X T w φ ) h ρ, R ˜ χ i G F , where R ˜ χ = | W | − P w ∈ W ˜ χ ( wφ ) R GT w (Id) . The only characters χ in the above sumwhich give a non-zero contribution are those which are a component of Ind WW I Id . Using the Lefschetz formula and taking the “limit for m →
0” (see for example[DMR, 3.3.8]) we get the equality of virtual characters
Corollary 7.32. X i ( − i H ic ( X ( I , w φ ) , Q ℓ ) = X { χ ∈ Irr( W ) φ |h Res
WWI χ, Id i WI =0 } ˜ χ ( x wφ ) R ˜ χ , where w is the image of w in W and x = | W I | − P v ∈ W I v . Cohomology. If π is the projection of Lemma 7.18, the sheaf π ! Q ℓ decomposesinto a direct sum of sheaves indexed by the irreducible characters of L t ( w φ ) I . Wewill denote by χ the subsheaf indexed by the character χ ∈ Irr( L t ( w φ ) I ), and inparticular by St the subsheaf indexed by the Steinberg character St ∈ Irr( L t ( w φ ) I ).We have the isomorphism of G F × L t ( w φ ) I -modules H ic ( ˜ X ( I , w φ ) , Q ℓ ) = ⊕ χ ∈ Irr( L t ( w φ ) I ) H ic ( X ( I , w φ ) , χ ) ⊗ V χ where V χ is an L t ( w φ ) I -module of character χ and H ic ( X ( I , w φ ) , χ ) is a G F -module.When χ is F δ -stable there is an action of F δ on V χ such that the inclusion of H ic ( X ( I , w φ ) , χ ) ⊗ V χ into H ic ( ˜ X ( I , w φ ) , Q ℓ ) is an inclusion of G F × L t ( w φ ) I ⋊ h F δ i -modulesThe following corollary of Proposition 7.19 relates the cohomology of a generalvariety X ( I , w φ ) to the case of the varieties X ( u φ ) considered in [DMR]; its part(ii) is a refinement of Corollary 7.32. In the following corollary, if M is a Q ℓ -vectorspace on wich F acts, we denote by M ( n ) for n ∈ Z the n -th Tate twist of M . Corollary 7.33.
Let I w −→ φ I ∈ B + ( I ) . Then (i) For any unipotent F δ -stable character χ ∈ Irr( L t ( w φ ) I ) , for any u ∈ B + I and any i, j we have the inclusion of G F × h F δ i -modules H ic ( X ( I , w φ ) , χ ) ⊗ ( H jc ( X L I ( u wφ ) , Q ℓ ) ⊗ L t ( w φ ) I V χ ) ⊂ H i + jc ( X ( uw φ ) , Q ℓ ) . (ii) For all v ∈ B + I and all i we have the following inclusions of G F × h F δ i -modules: H ic ( X ( I , w φ ) , Q ℓ ) ⊂ H i +2 l ( v ) c ( X ( vw φ ) , Q ℓ )( − l ( v )) and H ic ( X ( I , w φ ) , St ) ⊂ H i + l ( v ) c ( X ( vw φ ) , Q ℓ )(iii) For all i we have the following equality of G F × h F δ i -modules: H ic ( X ( w I w φ ) , Q ℓ ) = X j +2 k = i H jc ( X ( I , w φ ) , Q ℓ ) ⊗ Q ℓn I,k ( k ) where n I,k = |{ v ∈ W I | l ( v ) = k }| . Note that in (iii) above we have X ( w I w φ ) = S v ∈ W I X ( vw φ ). Proof.
We apply the K¨unneth formula to the isomorphism of Corollary 7.20 anddecompose the equality obtained according to the characters of L t ( w φ ) I ; we get thatfor any u ∈ B + I , we have M ≤ j ≤ l ( u ) χ ∈ Irr( L t ( w φ ) I ) H i − jc ( ˜ X ( I , w φ ) , Q ℓ ) χ ⊗ L t ( w φ ) I H jc ( X L I ( u wφ ) , Q ℓ ) χ ≃ H ic ( X ( uw φ ) , Q ℓ ) , ARABOLIC DELIGNE-LUSZTIG VARIETIES. 45 which can be written(7.34) M ≤ j ≤ l ( u ) χ ∈ Irr( L t ( w φ ) I ) H i − jc ( X ( I , w φ ) , χ ) ⊗ ( H jc ( X L I ( u wφ ) , Q ℓ ) ⊗ L t ( w φ ) I V χ ) ≃ H ic ( X ( uw φ ) , Q ℓ ) . This gives (i). We get also (ii) from equation 7.34 and the facts that for v ∈ B + I • the only j such that H jc ( X L I ( v wφ ) , Q ℓ ) Id is non-trivial is j = 2 l ( v ) andthat isotypic component is irreducible and t ( w φ ) acts by q l ( v ) on it (see[DMR, 3.3.14]) and t ( w φ ) kδ is equal to F kδ for some k . • the only j such that H jc ( X L I ( v wφ ) , Q ℓ ) St is non-trivial is j = l ( v ) andthat isotypic component is irreducible with trivial action of t ( w φ ) (see[DMR, 3.3.15]).Hence the term χ = Q ℓ in the LHS of 7.34 for u = v and j = 2 l ( v ) is H i − l ( v ) c ( X ( I , w φ ) , Q ℓ ) ⊗ Q ℓ ( − l ( v )) and is a submodule of H ic ( X ( I , wv φ ) , Q ℓ ). Similarly the term χ = St in the LHS for u = v and j = l ( v ) is H i − l ( v ) c ( X ( I , w φ ) , St ) and is a submodule of H ic ( X ( I , w φ ) , Q ℓ ).We now prove (iii). By Corollary 7.20 applied with u = w I we have an isomor-phism ˜ X ( I , w φ ) × L t ( w φ ) I B I ∼ −→ X ( w I w φ ) where B I is the variety of Borel subgroupsof L I . We get (iii) from the fact that H kc ( B I , Q ℓ ) is 0 if k is odd and if k = 2 k ′ is a trivial L t ( w φ ) I -module of dimension n I,k ′ , where F δ acts by the scalar q δk ′ ;this results for example from the cellular decomposition into affine spaces given bythe Bruhat decomposition and the fact that the action of L t ( w φ ) I extends to theconnected group L I so that it acts trivially on the cohomology. (cid:3) Corollary 7.35.
Let I w φ −−→ I ∈ B + φ ( I ) , let χ ∈ Irr( L t ( w φ ) I ) be unipotent and F δ -stable, and let i ∈ N . Then (i) The G F -module H ic ( X ( I , w φ ) , χ ) is unipotent. Given ρ ∈ Irr( G F ) unipo-tent, the eigenvalues of F δ on H ic ( X ( I , w φ ) , χ ) ρ are in q δ N λ ρ ω ρ , where λ ρ is as in Corollary 7.31 and ω ρ is the element of { , q δ/ } attached to ρ asin [DMR, 3.3.4] ; λ ρ and ω ρ are independent of i and w . (ii) The eigenvalues of F δ on H ic ( X ( I , w φ ) , χ ) have absolute value at most q δi/ . (iii) We have H ic ( X ( I , w φ ) , χ ) = 0 unless l ( w ) ≤ i ≤ l ( w ) . (iv) The Steinberg representation does not occur in H ic ( X ( I , w φ ) , χ ) unless χ = St and i = l ( w ) , in which case it occurs with multiplicity , associated withthe eigenvalue of F δ . (v) The trivial representation does not occur in H ic ( X ( I , w φ ) , χ ) unless χ = Q ℓ and i = 2 l ( w ) , in which case it occurs with multiplicity , associated withthe eigenvalue q δl ( w ) of F δ .Proof. (i) is a straightforward consequence of equation 7.34 applied for any u suchthat some term H jc ( X L I ( u wφ ) , Q ℓ ) χ is not 0 for some j , since the result is knownfor H ic ( X ( uw φ ) , Q ℓ ) (see [DMR, 3.3.4] and [DMR, 3.3.10 (i)]).(ii) and (iii) are a consequence of 7.34 applied for u ∈ B + I of minimal length suchthat χ appears in some H jc ( X L I ( u wφ ) , Q ℓ ). Then by [DMR, 3.3.21] χ appears in H l ( u ) c ( X L I ( u wφ ) , Q ℓ ) and the corresponding eigenvalue of F δ has module q δl ( u ) / . It follows then from 7.33(i) applied with j = l ( u ) that H ic ( X ( I , w φ ) , χ ) ⊗ V ⊂ H i + l ( u ) ( X ( uw φ ) , Q ℓ ) where V is an F δ -module where the eigenvalues of F δ are ofmodule q δl ( u ) / . The result follows from the facts that H i + l ( u ) ( X ( uw φ ) , Q ℓ ) = 0for i < l ( w ) and that the eigenvalues of F δ on it have a module at most q δ ( i + l ( u )) / .For (iv), we use Lemma 7.36. If χ ∈ Irr( L t ( w φ ) I ) is unipotent and χ = St there exists u ∈ B + I − B + I and j such that H jc ( X L I ( u wφ ) , Q ℓ ) χ = 0 .Proof. First, assume that χ is not in the principal series, and let v ∈ B + I be ofminimal length such that χ appears in some H jc ( X L I ( v wφ ) , Q ℓ ). Since χ is notin the principal series we have l ( v ) > s ∈ I and v ′ ∈ B + I suchthat v = sv ′ . Then H jc ( X L I ( s v ′ wφ ) , Q ℓ ) χ = H jc ( X L I ( v wφ ) , Q ℓ ) χ = 0 becauseof the minimality of v and the long exact sequence resulting from X L I ( s v ′ wφ ) = X L I ( v wφ ) ` X L I ( v ′ wφ ) where the first (resp. second) term of the RHS is an open(resp. closed) subvariety of the LHS.When χ is in the principal series, we use that if J is a wφ -stable subset of I and u ∈ B + J , then H jc ( X L I ( u wφ ) , Q ℓ ) = R L I L J H jc ( X L J ( u wφ ) , Q ℓ ). It follows thatif χ is of the form ρ ψ for ψ ∈ Irr( W wφI ) (see [DMR, 5.3.1]), and ψ is a component ofRes W wφI W wφJ ψ such that h H jc ( X L J ( u wφ ) , Q ℓ ) , ρ ψ i L FJ = 0, then h H jc ( X L I ( u wφ ) , Q ℓ ) , ρ ψ i L FI =0. If J is a wφ -orbit in I , the group W wφJ is a Coxeter group of type A and therestriction to W wφJ of a character ψ other than the sign character cannot be iso-typic of type sign for all orbits J ( ψ would then be itself isotypic of type sign). Weare thus reduced to the case where I is a single wφ -orbit, so that L t ( wφ ) I has onlytwo unipotent characters, Id and St. For such a group the identity character is acomponent of H c ( X L I ( swφ ) Q ℓ ) where W wφI = h s i , so that the lemma is true. (cid:3) Since for u as in the lemma we have H ∗ c ( X ( uw φ ) , Q ℓ ) St = 0 (see [DMR, 3.3.15]),by 7.34 we deduce that for χ = St we have H ic ( X ( I , w φ ) , χ ) St = 0 for all i . Thus,for any u ∈ B + I , using that H jc ( X L I ( u wφ ) , Q ℓ ) ⊗ L t ( w φ ) I V St = 0 when j = l ( u ), theSt-part of 7.34 reduces to H i − l ( u ) c ( X ( I , w φ ) , St ) St ⊗ ( H l ( u ) c ( X L I ( u wφ ) , Q ℓ ) ⊗ L t ( w φ ) I V St ) ≃ H ic ( X ( uw φ ) , Q ℓ ) St . We apply this for u = v ∈ B + I in which case H l ( v ) c ( X L I ( v wφ ) , Q ℓ ) ⊗ L t ( w φ ) I V St = Q ℓ with trivial action of F δ , which gives the isomorphism of G F × h F i δ -modules H ic ( X ( I , w φ ) , St ) St ≃ H i + l ( v ) c ( X ( vw φ ) , Q ℓ ) St . using the values of the RHS (known by [DMR, 3.3.15]) H i + l ( v ) c ( X ( vw φ ) , Q ℓ ) St = ( i = l ( w ) Q ℓ with trivial action of F δ otherwise , we get (iv).For (v), we use Lemma 7.37. If χ ∈ Irr( L t ( w φ ) I ) is unipotent and χ = Id there exists u ∈ B + I and j = 2 l ( u ) such that H jc ( X L I ( u wφ ) , Q ℓ ) χ = 0 .Proof. First, assume that χ is not in the principal series, and let u ∈ B + I be ofminimal length such that χ appears in some H jc ( X L I ( u wφ ) , Q ℓ ). Then by [DMR, ARABOLIC DELIGNE-LUSZTIG VARIETIES. 47 H l ( u ) c ( X L I ( u wφ ) , Q ℓ ) χ = 0. Since χ is not in the principal serieswe have l ( u ) = 2 l ( u ), whence the lemma in this case.Now assume χ in the principal series and take u = π I . It results for examplefrom [DMR, 3.3.8 (i)] that there exists j such that H jc ( X L I ( π I wφ ) , Q ℓ ) = 0. Onthe other hand, it results for example from Proposition 7.8 that X L I ( π I wφ ) isirreducible, thus H l ( π I ) c ( X L I ( π I wφ ) , Q ℓ ) is a 1-dimensional module affording onlythe trivial representation of G F . It follows that j = 2 l ( π I ), whence the lemma. (cid:3) Applying 7.34 for an u as in Lemma 7.37 and using that H ic ( X ( uw φ ) , Q ℓ ) Id = 0for i = 2( l ( w ) + l ( u )), we deduce that for χ = Id we have H ic ( X ( I , w φ ) , χ ) Id = 0for all i . Taking now u = 1 and using that H c ( X L I ( wφ ) , Q ℓ ) ⊗ L t ( w φ ) I Id = Q ℓ , theId-part of 7.34 reduces to H ic ( X ( I , w φ ) , Id) Id ≃ H ic ( X ( w φ ) , Q ℓ ) Id . whence the result using the value of the RHS given by [DMR, 3.3.14]. (cid:3) Eigenspaces and roots of π / π I Let ℓ = p be a prime such that a Sylow ℓ -subgroup S of G F is abelian.Then “generic block theory” (see [BMM]) associates with ℓ a root of unity ζ and some wφ ∈ W φ such that its ζ -eigenspace V in X := X R ⊗ C is non-zero andmaximal among ζ -eigenspaces of elements of W φ ; for any such ζ , there exists aunique minimal subtorus S of T such that V ⊂ X ( S ) ⊗ C . The space X ( S ) ⊗ C isthe kernel of Φ( wφ ), where, if the coset W φ is rational (that is, φ preserves X ( T ))then Φ is the d -th cyclotomic polynomial, where d is the order of ζ . Otherwise, inthe “very twisted” cases B , F (resp. G ) we have to take for Φ the irreduciblecyclotomic polynomial over Q ( √
2) (resp. Q ( √ ζ is a root. The torus S is wF -stable thus has an F -stable G -conjugate S ′ in a maximal torus of type w ;the torus S ′ is called a Φ-Sylow; we have | S ′ F | = Φ( q ) dim V .The relationship with ℓ is that S is a subgroup of S ′ F , and thus that | G F | / | S ′ F | is prime to ℓ ; we have N G F ( S ) = N G F ( S ′ ) = N G F ( L ) where L := C G ( S ′ ) is a Levisubgroup of G whose Weyl group is C W ( V ). Conversely, any non-zero maximal ζ -eigenspace determines some primes ℓ giving an abelian Sylow, those which divideΦ( q ) and no other cyclotomic factor of | G F | .The classes C W ( V ) wφ , where V = Ker( wφ − ζ ) is maximal, form a single orbitunder W -conjugacy [see eg. [Br2, 5.6(i)]]; the maximality implies that all elementsof C W ( V ) wφ have same ζ -eigenspace.We will see in Theorem 8.1(i) that up to conjugacy we may assume that C W ( V )is a standard parabolic group W I ; then the Brou´e conjectures predict that for anappropriate choice of coset C W ( V ) wφ in its N W ( W I )-conjugacy class the cohomol-ogy complex of the variety X ( I , w φ ) should be a tilting complex realizing a derivedequivalence between the unipotent parts of the principal ℓ -blocks of G F and of N G F ( S ′ ). We want to describe explicitly what should be a “good” choice of w (seeConjectures 9.1).Since it is no more effort to have a result in the context of any finite real reflectiongroup than for a context which includes the Ree and Suzuki groups, we give amore general statement. Our situation generalizes that studied in [BM], whichcorresponds to the case I = ∅ , or ζ -regular elements, that is elements of W φ whichhave an eigenvector for the eigenvalue ζ outside the reflecting hyperplanes (see [S, above 6.5]); in particular Theorem 8.1 generalizes [BM, 3.11, 6.5] and Theorem8.3 generalizes [BM, 3.12, 6.6]; in the [BM] case, the “ d -good periodic maximal”elements we consider here reduce to “good d -th φ -roots of π ”. Note that we focusour study on the ℓ -principal block (or Φ( q )-principal block), which correspondsto the maximality condition on eigenspaces and to what we call “non-extendable”periodic elements. Extendable periodic elements would be needed in consideringmore general blocks.In what follows we look at real reflection cosets W φ of finite order, that is W isa finite reflection group acting on the real vector space X R and φ is an element of N GL( X R ) ( W ), such that W φ is of finite order δ , that is δ is the smallest integer suchthat ( W φ ) δ = W (equivalently φ is of finite order). Since W is transitive on thechambers of the real hyperplane arrangement it determines, one can always choose φ in its coset so that it preserves a chamber of this arrangement. We will do this;thus φ is 1-regular, since it has a fixed point outside the reflecting hyperplanes,thus is of order δ since 1 is the only 1-regular element of W . Theorem 8.1.
Let
W φ ⊂ GL( X R ) be a finite order real reflection coset, such that φ preserves a chamber of the hyperplane arrangement on X R determined by W , thusinduces an automorphism of the Coxeter system ( W, S ) determined by this chamber.We call again φ the induced automorphism of the braid group B of W , and denoteby S , W the lifts of S, W to B (see Example 4.1).Let ζ = e iπk/d , and let V be a subspace of X := X R ⊗ C on which some elementof W φ acts by ζ . Then we may choose V in its W -orbit such that: (i) C W ( V ) = W I for some I ⊂ S . (ii) If W I wφ is the W I -coset of elements which act by ζ on V , where w is I -reduced, then l (( wφ ) i ) = (2 ik/d ) l ( w w − I ) for ik ≤ d , where we haveextended the length function to W ⋊ h φ i by l ( wφ i ) = l ( w ) .Further, we may lift w as in (ii) to w ∈ B + such that w φ I = I and ( w φ ) d = φ d ( π / π I ) k , where I ⊂ S lifts I . Thus I w φ −−→ I is a ( d, k ) -periodic element in B + φ ( I ) , where I is the set of subsets of S conjugate to I . Note that the last part implies that for w as in (ii) we have ( wφ ) d = φ d . Notealso that if 2 k ≤ d , then (ii) is applicable for i = 1 and we get l ( w ) = l ( w ) =(2 k/d ) l ( w w − I ) thus w is the unique lift of w to W .Since we assume W φ real, if e iπk/d is an eigenvalue of wφ , then the complexconjugate e iπ ( d − k ) /d is also an eigenvalue, for the complex conjugate eigenspace;thus we may always assume that 2 k ≤ d , so that w ∈ W .If the coset W φ preserves a Q -structure on X R (which is the case for cosetsassociated with finite reductive groups, except for the “very twisted” cases B , G and F ), we have more generally that if e iπk/d is an eigenvalue of wφ , with k prime d , the Galois conjugate e iπ/d is also an eigenvalue, for a Galois conjugateeigenspace; in these cases we may assume k = 1.Recall that by our conventions, even though φ is a finite order automorphism of B + , in the semi-direct product B + ⋊ h φ i we take h φ i of infinite order. Proof of Theorem 8.1.
Since W h φ i is finite, we may find a scalar product on X R (extending to an Hermitian product on X ) invariant by W and φ . The subspace X ′ R of X R orthogonal to the fixed points of W (the subspace spanned by the root linesof W ) identifies with the reflection representation of the Coxeter system ( W, S ) (see
ARABOLIC DELIGNE-LUSZTIG VARIETIES. 49 for example [Bou, Chapitre V § X ′ R consistingof the vectors of norm 1 (for the scalar product) along the root lines of W , whichis thus preserved by W h φ i . By [Bou, Chapitre V § X is a parabolic subgroup of W , hence conjugate to a standardparabolic subgroup, whence (i).To prove (ii) we reprove (i) by changing the order on Φ, which is equivalent to do aconjugation by some element of W . Let v be a regular vector in V , that is v ∈ V suchthat C W ( v ) = C W ( V ). Multiplying v if needed by a complex number of absolutevalue 1, we may assume that for any α ∈ Φ we have ℜh v, α i = 0 if and only if h v, α i = 0. Then there exists an order on Φ such that Φ + ⊂ { α ∈ Φ | ℜ ( h v, α i ) ≥ } .Let Π be the corresponding basis; the subset I = { α ∈ Π |ℜ ( h v, α i ) = 0 } is suchthat C W ( V ) = C W ( v ) = W I , and Φ I = { α ∈ Φ | h v, α i = 0 } is a root system for W I .Note that ( wφ ) d = φ d . Indeed ( wφ ) d fixes v , thus preserves the sign of any rootnot in Φ I ; since w is chosen I -reduced we have wφ I = I , so that wφ also preservesthe sign of roots in Φ I . It is thus equal to the only element φ d of W φ d whichpreserves the signs of all roots. We get also that φ d I = I . If we notice that we maylift φ to φ π / π I , this completes the proof in the case d = 1.We now assume that d = 1 and we first prove the theorem in the case k = 1.Since h v, ( wφ ) m α i = h ( wφ ) − m v, α i = ζ − m h v, α i , we get that all orbits of wφ on Φ − Φ I have cardinality a multiple of d ; it is thus possible by partitioning suitably thoseorbits, to get a partition of Φ − Φ I in subsets O of the form { α, wφ α, . . . , ( wφ ) d − α } ;and the numbers {h v, β i | β ∈ O} for a given O form the vertices of a regular d -goncentered at 0 ∈ C ; the action of wφ is the rotation by − π/d of this d -gon. Lookingat the real parts of the vertices of this d -gon, we see that for m ≤ d/
2, exactly m positive roots in O are sent to negative roots by ( wφ ) m . Since this holds for all O ,we get that for m ≤ d/ l (( wφ ) m ) = m | Φ − Φ I | d ; thus if w is the lift of w to W we have ( w φ ) i ∈ W φ i if 2 i ≤ d .Now we finish the case k = 1 , d = 1 with the following Lemma 8.2.
Assume that wφ W I = W I , that w is I -reduced, and that for some d > we have ( wφ ) d = φ d and l (( wφ ) i ) = (2 i/d ) l ( w w − I ) if i ≤ d . Then if w isthe lift of w to W we have w φ I = I and ( w φ ) d = φ d π / π I .Proof. Since w is I -reduced and wφ normalizes W I we get that wφ stabilizes I ;these properties imply in the braid monoid the equality w φ I = I .Assume first d even and let d = 2 d ′ and x = φ − d ′ ( wφ ) d ′ . Then l ( x ) = (1 / l ( π / π I ) = l ( w ) − l ( w I ) and since x is reduced- I it is equal to the only reduced- I elementof that length which is w w − I . Since the lengths add we can lift the equality( wφ ) d ′ = φ d ′ w w − I to the braid monoid as ( w φ ) d ′ = φ d ′ w w − I . By a similar rea-soning using that ( wφ ) d ′ φ − d ′ is the unique I -reduced element of its length, we getalso ( w φ ) d ′ = w − I w φ d ′ . Thus ( w φ ) d = w − I w φ d ′ φ d ′ w w − I = φ d π / π I , wherethe last equality uses that φ d = ( wφ ) d preserves I .Assume now that d = 2 d ′ + 1; then ( wφ ) d ′ φ − d ′ is I -reduced and φ − d ′ ( wφ ) d ′ isreduced- I . Using that any reduced- I element of W is a right-divisor of w w − I (resp. any I -reduced element of W is a left-divisor of w − I w ), we get that thereexists t , u ∈ W such that φ d ′ w − I w = t ( w φ ) d ′ and w w − I φ d ′ = ( w φ ) d ′ u . Thus φ d π / π I = w w − I φ d w − I w = ( w φ ) d ′ u φ t ( w φ ) d ′ , the first equality since φ d I = I . The image in
W φ d of the left-hand side is φ d , and ( wφ ) d = φ d . We deduce that theimage in W φ of u φ t is wφ . If d = 1 then d ′ = 0 and we have l ( u ) = l ( t ) = l ( w ) / u φ t = w φ and ( w φ ) d = φ d π / π I . (cid:3) We now consider the case k = 1, d = 1. We have seen (before assuming k = 1)that (i) holds and that the I -reduced element w of the coset W I wφ acting by ζ on V satisfies ( wφ ) d = φ d .We first consider the case when k is prime to d . Let d ′ , k ′ be positive integerssuch that kd ′ = 1 + dk ′ , and let w φ = ( wφ ) d ′ , where φ = φ d ′ . Then w φ acts on V by e iπ/d , so by the case k = 1 we have l (( w φ ) i ) = (2 i/d ) l ( w w − I ) for 2 i ≤ d .Since ( w φ ) ik = ( wφ ) ikd ′ = ( wφ ) i (1+ dk ′ ) = ( wφ ) i φ idk ′ , we get (ii).By Lemma 8.2 the lift w of w to B satisfies w φ I = I and ( w φ ) d = φ d π / π I ,thus if we define w by ( w φ ) k = w φ dk ′ , then w lifts w and satisfies ( w φ ) d = φ d ( π / π I ) k , using φ d I = I .We finally consider the general case d = λd , k = λk where d is prime to k .The theorem holds for d , k ; statement (ii) depends only on k/d thus holds, andwe just have to raise the equation ( w φ ) d = ( π / π I ) k φ d to the λ -th power to getthe desired equation ( w φ ) d = ( π / π I ) k φ d . (cid:3) We give now a kind of converse of Theorem 8.1.
Theorem 8.3.
Let ( W, S ) , φ , X R , X , S , B, B + be as in Theorem 8.1. For d ∈ N ,let w ∈ B + be such that ( w φ ) d = φ d ( π / π I ) k for some φ d -stable I ⊂ S . Then (i) w φ I = I , and I w φ −−→ I is a ( d, k ) -periodic element in B + φ ( I ) , where I isthe set of subsets of S conjugate to I .Denote by w and I the images in W of w and I , let ζ = e iπk/d , let V ⊂ X be the ζ -eigenspace of wφ , and let X W I be the fixed point space of W I ; then (ii) W I = C W ( X W I ∩ V ) , in particular C W ( V ) ⊂ W I .Further, the following two assertions are equivalent: (iii) No element of the coset W I wφ has a non-zero ζ -eigenvector on the subspacespanned by the root lines of W I . (iv) w φ is “non-extendable”, that is, there do not exist a φ d -stable J ( I and v ∈ B + I such that ( vw φ ) d = φ d ( π / π J ) k .Proof. We will deduce the general case from the case k = 1.So we first assume k = 1. Then (i) is already in Proposition 5.4 which alsostates that there exists I v −→ J ∈ B + ( I ) such that if w ′ φ = ( w φ ) v then w ′ φ ∈ B + φ ,( w ′ φ ) d = φ d π / π J and ( w ′ φ ) ⌊ d ⌋ ∈ W φ ⌊ d ⌋ .As (ii) and the equivalence of (iii) and (iv) are invariant by a conjugacy in B which sends w φ to B + φ and I to another subset of S , we may replace ( w φ, I ) bythe conjugate ( w ′ φ, J ), thus assume that w and I satisfy the assumptions of thenext lemma. Lemma 8.4.
Let w ∈ W, I ⊂ S be such that ( wφ ) d = φ d , wφ I = I and such that l (( wφ ) i ) = id l ( w − I w ) for any i ≤ d/ . We have (i) If Φ is a root system for W and Φ + is chosen such that φ (Φ + ) = Φ + (asin the proof of Theorem 8.1), then Φ − Φ I is the disjoint union of sets ofthe form { α, wφ α, . . . , ( wφ ) d − α } with α, wφ α, . . . , ( wφ ) ⌊ d/ ⌋− α of same signand ( wφ ) ⌊ d/ ⌋ α, . . . , ( wφ ) d − α of the opposite sign. ARABOLIC DELIGNE-LUSZTIG VARIETIES. 51 (ii)
The order of wφ is lcm( d, δ ) . (iii) If d > , then W I = C W ( X W I ∩ ker( wφ − ζ )) .Proof. The statement is empty for d = 1 so in the following proof we assume d > x ∈ W ⋊ h φ i let N ( x ) = { α ∈ Φ + | x α ∈ Φ − } ; for x ∈ W we have l ( x ) = | N ( x ) | (see [Bou, Chapitre VI §
1, Corollaire 2]). This still holds for x = wφ i ∈ W ⋊ h φ i since N ( wφ i ) = φ − i N ( w ). It follows that for x, y ∈ W ⋊ h φ i wehave l ( xy ) = l ( x ) + l ( y ) if and only if N ( xy ) = N ( y ) ` y − N ( x ). In particular l (( wφ ) i ) = il ( wφ ) for i ≤ d/ ( wφ ) − i N ( wφ ) ⊂ Φ + for i ≤ d/ − wφ -orbit in Φ − Φ I into “pseudo-orbits” of the form { α, wφ α, . . . , ( wφ ) k − α } , where k is minimal such that ( wφ ) k α = φ k α (then k di-vides d ); a pseudo-orbit is an orbit if φ = 1. The action of wφ defines a cyclicorder on each pseudo-orbit. The previous paragraph shows that when there is asign change in a pseudo-orbit, at least the next ⌊ d/ ⌋ roots for the cyclic order havethe same sign. On the other hand, as φ k preserves Φ + , each pseudo-orbit containsan even number of sign changes. Thus if there is at least one sign change we have k ≥ ⌊ d/ ⌋ . Since k divides d , we must have k = d for pseudo-orbits which have asign change, and then they have exactly two sign changes. As the total number ofsign changes is 2 l ( w ) = 2 | Φ − Φ I | /d , there are | Φ − Φ I | /d pseudo-orbits with signchanges; their total cardinality is | Φ − Φ I | , thus there are no other pseudo-orbitsand up to a cyclic permutation we may assume that each pseudo-orbit consists of ⌊ d/ ⌋ roots of the same sign followed by d − ⌊ d/ ⌋ of the opposite sign. We haveproved (i).Let d ′ = lcm( d, δ ). The proof of (i) shows that the order of wφ is a multiple of d . Since the order of ( wφ ) d = φ d is d ′ /d , we get (ii).We now prove (iii). Let V = ker( wφ − ζ ). Since W h φ i is finite, we may find ascalar product on X invariant by W and φ . We have then X W I = Φ ⊥ I . The map p = d ′ P d ′ − i =0 ζ − i ( wφ ) i is a wφ -invariant projector on V , thus is the orthogonalprojector on V .We claim that p ( α ) < Φ I > for any α ∈ Φ − Φ I . As p (( wφ ) i α ) = ζ i p ( α )it is enough to assume that α is the first element of a pseudo-orbit; replacing ifneeded α by − α we may even assume α ∈ Φ + . Looking at imaginary parts, wehave ℑ ( ζ i ) ≥ ≤ i < ⌊ d/ ⌋ , and ℑ ( ζ i ) < ⌊ d/ ⌋ ≤ i < d . Let λ be alinear form such that λ is 0 on Φ I and is real strictly positive on Φ + − Φ I ; we have λ ( ( wφ ) i α ) > ≤ i < ⌊ d/ ⌋ , and λ ( ( wφ ) i α ) < ⌊ d/ ⌋ ≤ i < d ; it followsthat ℑ ( λ ( ζ i ( wφ ) i α )) = ℑ ( ζ i λ ( ( wφ ) i α )) > d ′ = d we have thus ℑ ( λ ( p ( α ))) >
0, in particular p ( α ) < Φ I > . If d ′ > d , since φ d α is also a positive root and the first term of the next pseudo-orbit the samecomputation applies to the other pseudo-orbits and we conclude the same way.Now C W ( X W I ∩ V ) is generated by the reflections whose root is orthogonal to X W I ∩ V , that is whose root is in < Φ I > + V ⊥ . If α is such a root we have p ( α ) ∈ < Φ I > , whence α ∈ Φ I by the above claim. This proves that C W ( X W I ∩ V ) ⊂ W I .Since the reverse inclusion is true, we get (iii). (cid:3) We return to the proof of the case k = 1 of Theorem 8.3. Assertion (iii) of Lemma8.4 gives the second assertion of the theorem. We now show ¬ (iv) ⇒ ¬ (iii). If w φ isextendable, there exists a φ d -stable J ( I and v ∈ B + I such that ( vw φ ) d = φ d π / π J ,which implies vw φ J = J . If we denote by ψ the automorphism of B I induced bythe automorphism w φ of I , we have v ψ J = J and ( v ψ ) d = ψ d π I / π J . Let X I be the subspace of X spanned by Φ I . It follows from the first part of the theoremapplied with X , φ , w , w respectively replaced with X I , ψ , v , v that vψ = vwφ has a non-zero ζ -eigenspace in X I , since if V ′ is the ζ -eigenspace of vwφ we get C W I ( V ′ ) ⊂ W J ( W I ; this contradicts (iv).We show finally that ¬ (iii) ⇒ ¬ (iv). If some element of W I ψ has a non-zero ζ -eigenvector on X I , by Theorem 8.1 applied to W I ψ acting on X I we get theexistence of J ( I and v ∈ B + I satisfying v ψ J = J and ( v ψ ) d = ψ d π I / π J . Usingthat ( w φ ) d = φ d π / π I , it follows that ( vw φ ) d = ( w φ ) d π I / π J = φ d π / π I · π I / π J = φ d π / π J so w φ is extendable.We now deal with the general case k = 1. This time we use 5.5, which givesimmediately (i). Let us first consider the case when k is prime to d . Then, by 5.5,up to conjugacy in B + ( I ), which we may as well do as observed at the beginning ofthe proof, we get that with d ′ and k ′ as in 5.5 we have ( w φ ) d ′ ( π / π I ) k ′ and theelement w defined by ( w φ ) d ′ w φ − d ′ = ( π / π I ) k ′ satisfies ( w φ − d ′ ) k = ( w φ ) φ − k ′ d and ( w φ − d ′ ) d = π / π I φ − dd ′ . Since I is φ − dd ′ -stable the last equality shows thatwe may apply the case k = 1 to w φ − d ′ . Since k is prime to d the defining relationfor w gives in W that ( wφ ) − d ′ = w φ − d ′ , where w is the image of w in W ,which (since d ′ is prime to d ) shows that that the ζ -eigenspace of wφ is the e iπ/d -eigenspace of w φ − d ′ . This gives (ii).Similarly the coset W I w φ − d ′ is the − d ′ -th power of the coset W I wφ , so condition(iii) for wφ and ζ is equivalent to (iii) for w φ − d ′ and e iπ/d .Item (ii) of the following lemma completes the proof of the case gcd( d, k ) = 1since by Lemma 3.16 we may choose d ′ prime to δ ; Lemma 8.5.
Let k, d, k ′ , d ′ be positive integers satisfying dk ′ = 1 + kd ′ with d ′ prime to the order of φ . Let w φ − d ′ be ( d, -periodic element. Define w φ by w φ = ( w φ − d ′ ) k φ k ′ d . Then (i) w φ is ( d, k ) -periodic. (ii) w φ is non-extendable if and only if w φ − d ′ is non-extendable.Proof. Assertion (i) is an immediate translation of 3.15. Assume w φ − d ′ ex-tendable, that is there exists v such that ( v w φ − d ′ ) d = φ − dd ′ π / π J for some J ( I . The k -th power of this equality gives ( v ( w φ − d ′ ) k ) d = ( vw φ · φ − k ′ d ) d = φ − kdd ′ ( π / π J ) k , where v is defined by ( v w φ − d ′ ) k = v ( w φ − d ′ ) k . Since w φ − d ′ is( d, φ dd ′ -stable, and the defining equality for v shows that v alsois φ dd ′ -stable. It follows that vw φ is also φ dd ′ -stable. Since d ′ is prime to δ anyelement commuting to φ dd ′ commutes to φ d , in particular ( vw φ · φ − k ′ d ) d φ kdd ′ =( vw φ ) d φ − k ′ d + kdd ′ = ( vw φ ) d φ − d , whence the result.For the converse, if wφ denotes the automorphism of B + I induced by w φ , us-ing that ( w φ ) d = φ d ( π / π I ) k and that π / π J = ( π / π I )( π I / π J ) we may writethe equation ( vw φ ) d = φ d ( π / π J ) k as ( v wφ ) d = φ d ( π I / π J ) k . We now use arelative version of 5.5, where we replace B + ( I ) by B + I ( J ) where J is the setof B + I -conjugates of J , replace φ by wφ and replace b by v ; we get the ex-istence of v such that ( v ( wφ ) − d ′ ) d = π I / π J ( wφ ) − dd ′ , which can be written( v ( w φ ) − d ′ ) d ( π / π I ) kd ′ = π I / π J φ − dd ′ or ( v ( w φ ) − d ′ ( π / π I ) k ′ ) d = π / π J φ − dd ′ which using that ( w φ ) d ′ w φ − d ′ = ( π / π I ) k ′ transforms into the equality we seek( v w φ − d ′ ) d = π / π J φ − dd ′ . (cid:3) ARABOLIC DELIGNE-LUSZTIG VARIETIES. 53
We now consider the case when λ = gcd( d, k ) = 1. We set d = d/λ and k = k/λ . Up to cyclic conjugacy, which we may as well do, we may assume by5.5 that ( w φ ) d = ( π / π I ) k φ d . Since e iπk /d = e iπk/d we have (i), (ii) of thetheorem as well as the equivalence of (iii) with the “ d -extendability” of w , that isthe existence of v ∈ B + I such that ( vw φ ) d = φ d ( π / π J ) k . The d -extendabilityimplies trivially the d -extendability by raising the equation to the λ -th power.Conversely, using as above that the equation ( vw φ ) d = φ d ( π / π J ) k is equivalent to( v wφ ) d = φ d ( π I / π J ) k the relative version of 5.5 as used above shows that up tocyclic conjugacy we have ( v wφ ) d = φ d ( π I / π J ) k which in turn is equivalent to( vw φ ) d = φ d ( π / π J ) k . (cid:3) The non-extendability condition (iii) or (iv) of Theorem 8.3 is equivalent tothe conjunction of two others, thanks to the following lemma which holds for anycomplex reflection coset and any ζ . For definitions and basic results on complexreflection groups we refer to [Br2]. Recall that a complex reflection group is a finitegroup generated by pseudo-reflections acting on a finite dimensional complex vectorspace and that the fixator of a subspace is called a parabolic subgroup. It is still acomplex reflection group. Lemma 8.6.
Let W be finite a reflection group on the complex vector space X andlet φ be an automorphism of X of finite order which normalizes W . Let V be the ζ -eigenspace of an element wφ ∈ W φ . Assume that W ′ is a parabolic subgroup of W which is wφ -stable and such that C W ( V ) ⊂ W ′ , and let X ′ denote the subspaceof X spanned by the root lines of W ′ . Then the condition (i) V ∩ X ′ = 0 .is equivalent to (ii) C W ( V ) = W ′ .While the stronger condition (iii) No element of the coset W ′ wφ has a non-zero ζ -eigenvector on X ′ .is equivalent to the conjunction of (ii) and (iv) The space V is maximal among the ζ -eigenspaces of elements of W φ .Proof.
Since W h φ i is finite we may endow X with a W h φ i -invariant Hermitianscalar product, which we shall do.We show (i) ⇔ (ii). Assume (i); since wφ has no non-zero ζ -eigenvector in X ′ and X ′ is wφ -stable, we have V ⊥ X ′ , so that W ′ ⊂ C W ( V ), whence (ii) since thereverse inclusion is true by assumption. Conversely, (ii) implies that V ⊂ X ′⊥ thus V ∩ X ′ = 0.We show (iii) ⇒ (iv). There exists an element of W φ whose ζ -eigenspace V ismaximal with V ⊂ V . Then C W ( V ) ⊂ C W ( V ) ⊂ W ′ and the C W ( V )-coset ofelements of W φ which act by ζ on V is a subset of the coset C W ( V ) wφ of elementswhich act by ζ on V . Thus this coset is of the form C W ( V ) vwφ for some v ∈ W ′ .By (i) ⇒ (ii) applied with wφ replaced by vwφ we get C W ( V ) = W ′ . Since v ∈ W ′ this implies that vwφ and wφ have same action on V so that wφ acts by ζ on V ,thus V ⊂ V .Conversely, assume that (ii) and (iv) are true. If there exists v ∈ W ′ such that vwφ has a non-zero ζ -eigenvector in X ′ , then since v acts trivially on V by (ii), theelement vwφ acts by ζ on V and on a non-zero vector of X ′ so has a ζ -eigenspacestrictly larger that V , contradicting (iv). (cid:3) Let us give now examples which illustrate the need for the conditions in Theorem8.3 and Lemma 8.6.We first give an example where w φ is a root of π / π I which is extendable in thesense of Theorem 8.3(iv) and ker( wφ − ζ ) is not maximal: let us take W = W ( A ), φ = 1, d = 2, ζ = − I = { s } (where the conventions for the generators of W areas in the appendix, see Subsection 10.2), w = w − I w . We have w = π / π I butker( w + 1) is not maximal: its dimension is 1 and a 2-dimensional − w = w .In the above example we still have C W ( V ) = W I but even this need not happen;at the same time we illustrate that the maximality of V = ker( wφ − ζ ) does notimply the non-extendability of w if C W ( V ) ( W I ; we take W = W ( A ), φ = 1, d = 2, ζ = −
1, but this time I = { s , s } , w = w − I w . We have w = π / π I andker( w + 1) is maximal ( w is conjugate to w , thus − w is extendable.In this case C W ( V ) = { } .The smallest example with a non-extendable w and non-trivial I is for W = W ( A ), φ = 1, d = 3, w = s s s s s s and I = { s } . Then w = π / π I ; thiscorresponds to the smallest example with a non-regular eigenvalue (we call regularan eigenvalue of a regular element for which the eigenspace has trivial centralizer): ζ is not regular in A .Finally we give an example which illustrates the necessity of the condition φ d ( I ) = I in Theorem 8.3. We take W φ of type D , thus φ is the triality au-tomorphism s s s . Let w = w s − s − s . Then, for I = { s } we have( w φ ) = π / π I φ , but I w φ = { s } . The other statements of Theorem 8.3 alsofail: if V is the − wφ the group C W ( V ) is the parabolic subgroupgenerated by s , s and s . Lemma 8.7.
Let
W φ be a complex reflection coset and let V be the ζ -eigenspaceof wφ ∈ W φ ; then (i) N W ( V ) = N W ( C W ( V ) wφ ) . (ii) If W φ is real, and C W ( V ) = W I where ( W, S ) is a Coxeter system and I ⊂ S , and w is I -reduced, then the subgroup { v ∈ C W ( wφ ) ∩ N W ( W I ) | v is I -reduced } is a section of N W ( V ) /C W ( V ) in W .Proof. Let W denote the parabolic subgroup C W ( V ). All elements of W wφ havethe same ζ -eigenspace V , so N W ( W wφ ) normalizes V ; conversely, an elementof N W ( V ) normalizes W and conjugates wφ to an element w ′ φ with same ζ -eigenspace, thus w and w ′ differ by an element of W , whence (i).For the second item, N W ( W I wφ ) /W I admits as a section the set of I -reducedelements, and such an element will conjugate wφ to the element of the coset W I wφ which is I -reduced, so will centralize wφ . (cid:3) We call essential rank of a (complex) reflection coset
W φ ⊂ GL( X ) the dimensionof the space generated by its root lines (the dimension of X minus the dimensionof the intersection of the reflection hyperplanes of W ).We call ζ -rank of an element of W φ the dimension of its ζ -eigenspace, and ζ -rankof W φ the maximal ζ -rank of its elements.Let us say that a ( d, k )-periodic element of B + φ ( I ) is non-extendable if it isnon-extendable in the sense of Theorem 8.3(iv). Another way to state the non-extendability of a periodic element I w φ −−→ I ∈ B + φ ( I ) is to require that | I | be nomore than the essential rank of the centralizer of a maximal ζ -eigenspace of W φ , ARABOLIC DELIGNE-LUSZTIG VARIETIES. 55 where ζ = e ikπ/d : indeed if I w φ −−→ I is extendable there exists J and v as inTheorem 8.3(iv) and, since condition 8.3(iv) implies Lemma 8.6(iii), the element vwφ has maximal ζ -rank, and the centralizer of its ζ -eigenspace has essential rank | J | < | I | . Note that the notion of non-extendable ( d, k )-periodic element makessense without specifying I , as ζ = e ikπ/d is determined by k/d , and I in turn is de-termined as the class of parabolic subgroups which are centralizers of ζ -eigenspacesof elements of W φ of maximal ζ -rank.The correspondence between maximal eigenspaces and non-extendable periodicelements, as described by Theorems 8.1 and 8.3, can be summarized as follows: Corollary 8.8.
Let V ′ be the ζ -eigenspace of an element of W φ of maximal ζ -rank, where ζ = e iπk/d . Then there is a W -conjugate V of V ′ and I ⊂ S such that C W ( V ) = W I and the corresponding I -reduced wφ (see Theorem 8.1(ii)) lifts to anon-extendable ( d, k ) -periodic element I w φ −−→ I . Conversely, for a ( d, k ) -periodicnon-extendable I w φ −−→ I the image wφ in W φ has maximal ζ -rank. We conjecture that Bessis’s theorem [B1, 11.21] extends to
Conjecture 8.9.
Two non-extendable ( d, k ) -periodic elements of B + φ ( I ) arecyclically conjugate. Note that because of Lemma 8.6 the non-extendability condition is necessary inthe above.By 5.6 a ( d, k )-periodic element is cyclically conjugate to an element which sat-isfies in addition ( w φ ) ⌊ d k ⌋ ∈ W φ ⌊ d k ⌋ . We will call good a ( d, k )-periodic elementwhich satisfies this additional condition.When k = 1 we can give conditions purely in terms of W for an element to liftto a good ( d, d, Lemma 8.10.
Let
W φ ⊂ GL( X R ) be a finite order real reflection coset such that φ preserves the chamber of the corresponding hyperplane arrangement determiningthe Coxeter system ( W, S ) .Let w ∈ W and I ⊂ S and let w ∈ W and I ⊂ S be their lifts; let I be theconjugacy orbit of I , then w induces a morphism I w φ −−→ I ∈ B + ( I ) if and only if: (i) wφ I = I and w is I -reduced.If w satisfies (i), for d > the element I w φ −−→ I is good ( d, -periodic if and onlyif the following two additional conditions are satisfied. (ii) l (( wφ ) i ) = id l ( w − I w ) for ≤ i ≤ d . (iii) ( wφ ) d = φ d .If, moreover, (iv) W I wφ has ζ -rank on the subspace spanned by the root lines of W I where ζ = e iπ/d ,then w is non-extendable in the sense of Theorem 8.3(iv).Proof. By definition w induces a morphism I w φ −−→ I if and only if it satisfies (i).By definition again if I w φ −−→ I is good ( d, w satisfying(i), (ii), (iii) is good ( d, Property (iv) means that no element vwφ with v ∈ W I has an eigenvalue ζ onthe subspace spanned by the root lines of W I which is exactly the characterizationof Theorem 8.3(iv) of a non-extendable element. (cid:3) Note that d and I are uniquely determined by wφ satisfying (i), (ii), (iii) abovesince d is the smallest power of wφ which is a power of φ and I is determined by( w φ ) d = π / π I φ d . Definition 8.11.
We say that wφ ∈ W φ is d -good if it satisfies (i), (ii), (iii) inLemma 8.10.We say wφ is d -good maximal if it satisfies in addition (iv) in Lemma 8.10. In particular, d -good elements lift to good ( d, d -goodmaximal elements lift to good non-extendable ( d, d, k )-periodic element for each W φ ,each d and each k . Actually, we will do this only for k = 1 (by constructing d -goodmaximal elements of W φ ), which is sufficient by
Lemma 8.12. (i) If λ = gcd( d, k ) and we set d = d/λ and k = k/λ and w φ is ( d , k ) -periodic (resp. non-extendable ( d , k ) -periodic) then w φ is ( d, k ) -periodic (resp. non-extendable ( d, k ) -periodic). (ii) If k is prime to d there exists integers k ′ and d ′ such that dk ′ = 1 + kd ′ such that if w φ − d ′ is ( d, -periodic (resp. non-extendable ( d, -periodic)then the element w φ defined by ( w φ − d ′ ) k = ( w φ ) φ − k ′ d is ( d, k ) -periodic(resp. non-extendable ( d, k ) -periodic).Proof. (i) is part of what is proved in the last paragraph of the proof of 8.3 and (ii)is Lemma 8.5. (cid:3) Any element of
W φ with a maximal ζ -eigenspace is conjugate to an element of C W ( V ) wφ since the maximal eigenspaces are conjugate, see [S, Theorem 3.4(iii) andTheorem 6.2(iii)]. If wφ is the image of a non-extendable ( d, k )-periodic element,where ζ = e ikπ/d , it is 1-regular in this coset by Theorem 8.3 (ii) which impliesthat it preserves a chamber of the corresponding real arrangement (see remarksabove Theorem 8.1). The following lemma shows that the images in W φ of non-extendable ( d, k )-periodic elements (thus in particular d -good maximal elements)belong to a single conjugacy class under W , characterized by the above property. Lemma 8.13.
Let
W φ be a finite order real reflection coset. The elements of
W φ which have a ζ -eigenspace V of maximal dimension and among those, have thelargest dimension of fixed points, are conjugate.Proof. As remarked above, up to W -conjugacy we may fix a ζ -eigenspace V andconsider only elements of the coset C W ( V ) wφ where wφ is some element with ζ -eigenspace V ; then W -conjugacy is reduced to C W ( V )-conjugacy. Since C W ( V )is a parabolic subgroup of the Coxeter group W and is normalized by wφ , thecoset C W ( V ) wφ is a real reflection coset; in this coset there are 1-regular elements,which are those which preserve a chamber of the corresponding real hyperplanearrangement; the 1-regular elements have maximal 1-rank, that is have the largestdimension of fixed points, and they form a single C W ( V )-orbit under conjugacy,whence the lemma. (cid:3) ARABOLIC DELIGNE-LUSZTIG VARIETIES. 57
Lemma 8.14.
Let wφ be the image in W φ of a non-extendable ( d, k ) -periodicelement I w φ −−→ I , let I be the image of I and let V be the fixed point subspace of wφ in the space spanned by the root lines of W I ; then wφ is regular in the coset C W ( V ) wφ .Proof. Let W ′ = C W ( V ); we first note that since wφ normalizes V it normalizesalso W ′ , so W ′ wφ is indeed a reflection coset. We have thus only to prove that C W ′ ( V ) is trivial, where V is the e ikπ/d -eigenspace of wφ . This last group isgenerated by the reflections with respect to roots both orthogonal to V and to V , which are the roots of W I = C W ( V ) orthogonal to V . Since wφ preserves achamber of W I , the sum v of the positive roots of W I with respect to the orderdefined by this chamber is in V and is in the chamber: this is well known for atrue root system; here we have taken all the roots to be of length 1 but the usualproof (see [Bou, Chapitre VI §
1, Proposition 29]) is still valid. Since no root isorthogonal to a vector v inside a chamber, W I has no root orthogonal to V , hence C W ′ ( V ) = { } . (cid:3) One could hope that the above lemma reduces the classification of d -good maxi-mal elements to that of regular elements; however the map C W ′ ( wφ ) = N W ′ ( V ) → N W ( V ) /C W ( V ) with the notations of the above proof is injective, but not al-ways surjective: for example, if W of type E , and φ = Id and d = 4, then N W ( V ) /C W ( V ) is the complex reflection group G , while W ′ is of type D and N W ′ ( V ) /C W ′ ( V ) is the complex reflection group G (4 , , W ; the group N W ( V ) /C W ( V ) was determinedin appendix 1 in all other cases by the equality C W ′ ( wφ ) ≃ N W ( V ) /C W ( V ), whichis proved by checking that C W ′ ( wφ ) and N W ( V ) /C W ( V ) have the same reflectiondegrees, a simple arithmetic check on the reflection degrees of W and W ′ ; indeed,recall that when V is a maximal ζ -eigenspace, the group N W ( V ) /C W ( V ) is a com-plex reflection group acting on V , with reflection degrees the reflection degrees of W satisfying the arithmetic condition given for instance in [Br2, 5.6] (when φ = Id,the reflection degrees divisible by d ).9. Conjectures
The following conjectures extend those of [DM2, § Conjectures 9.1.
Let I w φ −−→ I ∈ B + ( I ) φ be non-extendable ( d, k ) -periodic. Then (i) The group B w generated by the monoid B + w of Theorem 5.9 is isomorphic tothe braid group of the complex reflection group W ( wφ ) := N W ( W I wφ ) /W I . (ii) The natural morphism D I ( I w φ −−→ I ) → End G F ( X ( I , w φ )) (see below Lemma7.22) gives rise to a morphism B w → End G F H ∗ c ( X ( I , w φ )) which factorsthrough a special representation of a ζ -cyclotomic Hecke algebra H w for W ( wφ ) , where ζ = e ikπ/d . (iii) The odd and even H ic ( X ( I , w φ )) are disjoint G F -modules, and the abovemorphism extends to a surjective morphism Q ℓ [ B w ] → End G F ( H ∗ c ( X ( I , w φ ))) . The group W ( wφ ) above is a complex reflection group by the remarks at theend of last section and Lemma 8.7 (i).The condition that the periodic elements we consider are non-extendable is nec-essary for assertion (ii) above to hold; in the case of extendable periodic elements the endomorphism algebra should, instead of being a deformation of the groupalgebra of W ( wφ ), be a deformation of an endomorphism algebra of an inducedrepresentation from a complex reflection group to another. Whenever a periodicelement is extendable, a decomposition as in Theorem 7.19 can be applied. See[Du, 1.3] for such computations.David Craven has made (iii) above more specific by giving a conjectural formulacomputing the cohomology degree in which a given unipotent character shouldoccur (see [C]); Craven’s formula should be valid for any ( d, k )-periodic element,not only the non-extendable ones. In the current paper we focus on the study ofnon-extendable periodic elements; this should be a start for the general study of allperiodic elements. Lemma 9.2.
Let I w φ −−→ I ∈ B + φ ( I ) be non-extendable ( d, k ) -periodic and as-sume Conjectures 9.1; then for any i = j the G F -modules H ic ( X ( I , w φ )) and H jc ( X ( I , w φ )) are disjoint.Proof. Since the image of the morphism of Conjecture 9.1(ii) consists of equiva-lences of ´etale sites, it follows that the action of H w on H ∗ c ( X ( I , w φ )) preservesindividual cohomology groups. The surjectivity of the morphism of (iii) implies thatfor ρ ∈ Irr( G F ), the ρ -isotypic part of H ∗ c ( X ( I , w φ )) affords an irreducible H w -module; this would not be possible if this ρ -isotypic part was spread over severaldistinct cohomology groups. (cid:3) We will now explore the information given by the Shintani descent identity onthe above conjectures
Lemma 9.3.
Let I w φ −−→ I ∈ B + φ ( I ) be ( d, k ) -periodic With the notations ofProposition 7.30, we have ˜ χ q m ( X T w φ ) = q m kd ( l ( π / π I ) − a χ − A χ ) ˜ χ ( e I wF ) for χ ∈ Irr( W ) φ , where a χ (resp. A χ ) is the valuation (resp. the degree) of the genericdegree of χ and e I = | W I | − P v ∈ W I v .Proof. We have ( X T w φ ) d = X ( T π /T π I ) k φ d = q − kl ( π I ) X T k π φ d since X com-mutes with T w φ and since for any v ∈ W I we have X T v = q l ( v ) X . Since T π acts on the representation of character χ q m as the scalar q m ( l ( π ) − a χ − A χ ) (see [BM,Corollary 4.20]), it follows that all the non-zero eigenvalues of X T w φ on this rep-resentation are equal to q m kd ( l ( π / π I ) − a χ − A χ ) times a root of unity. To compute thesum of these roots of unity, we may use the specialization q m/
1, through which˜ χ q m ( X T w φ ) specializes to ˜ χ ( e I wφ ). (cid:3) Proposition 9.4.
Let I w φ −−→ I ∈ B + φ ( I ) be ( d, k ) -periodic. For any m multipleof δ , we have | X ( I , w φ ) gF m | = X ρ ∈E ( G F , λ m/δρ q m kd ( l ( π / π I ) − a ρ − A ρ ) h ρ, R G ,F L I ,t ( w φ ) Id i G F ρ ( g ) , where a ρ and A ρ are respectively the valuation and the degree of the generic degreeof ρ . ARABOLIC DELIGNE-LUSZTIG VARIETIES. 59
Proof.
We start with Corollary 7.31, whose statement reads, using the value of˜ χ q m ( X T w φ ) given by Lemma 9.3: | X ( I , w φ ) gF m | = X ρ ∈E ( G F , λ m/δρ ρ ( g ) X χ ∈ Irr( W ) φ q m kd ( l ( π / π I ) − a χ − A χ ) ˜ χ ( e I wφ ) h ρ, R ˜ χ i G F . Using that for any ρ such that h ρ, R ˜ χ i G F = 0 we have a ρ = a χ and A ρ = A χ (see[BM] around (2.4)) the right-hand side can be rewritten X ρ ∈E ( G F , λ m/δρ q m kd ( l ( π / π I ) − a ρ − A ρ ) ρ ( g ) h ρ, X χ ∈ Irr( W ) φ ˜ χ ( e I wφ ) R ˜ χ i G F . The proposition is now just a matter of observing that X χ ∈ Irr( W ) φ ˜ χ ( e I wφ ) R ˜ χ = | W I | − X v ∈ W I X χ ∈ Irr( W ) φ ˜ χ ( vwφ ) R ˜ χ = | W I | − X v ∈ W I R GT vw (Id) = R G ,F L I ,t ( w φ ) (Id) . Where the last equality is obtained by transitivity of R GL and the equality Id L t ( w φ ) I = | W I | − P v ∈ W I R L I ,t ( w φ ) T vw (Id), a torus T of L I of type v for the isogeny t ( w φ ) beingconjugate to T vw in G . (cid:3) Corollary 9.5.
Let I w φ −−→ I ∈ B + φ ( I ) be non-extendable ( d, k ) -periodic andassume Conjectures 9.1; then for any ρ ∈ Irr( G F ) such that h ρ, R G ,F L I ,t ( w φ ) (Id) i G F =0 the isogeny F δ has a single eigenvalue on the ρ -isotypic part of H ∗ c ( X ( I , w φ )) ,equal to λ ρ q δ kd ( l ( π / π I ) − a ρ − A ρ ) .Proof. This follows immediately, in view of Lemma 9.2, from the comparison be-tween Proposition 9.4 and the Lefschetz formula: | X ( I , w φ ) gF m | = X i ( − i Trace( gF m | H ic ( X ( I , w φ ) , Q ℓ )) . (cid:3) In view of Corollary 7.35(i) it follows that if h ρ, R GL I (Id) i G F = 0 then if ω ρ = 1then kd ( l ( π / π I ) − a ρ − A ρ ) ∈ N , and if ω ρ = p q δ then kd ( l ( π / π I ) − a ρ − A ρ ) ∈ N +1 / q /a ζ /a , where a ∈ N is large enough such that H w ⊗ Q ℓ [ q /a ] is split. This givesa bijection ϕ ϕ q : Irr( W ( wφ )) → Irr( H w ), and the conjectures give a furtherbijection ϕ ρ ϕ between Irr( W ( wφ )) and the set { ρ ∈ Irr( G F ) | h ρ, R GL I (Id) i G F =0 } , which is such that h ρ ϕ , R GL I (Id) i G F = ϕ (1). Corollary 9.6.
Under the assumptions of Corollary 9.5, if ω ϕ is the central char-acter of ϕ , then λ ρ ϕ = ω ϕ (( wφ ) δ ) ζ − δ kd ( l ( π / π I ) − a ρϕ − A ρϕ ) . Proof.
We first note that it makes sense to apply ω ϕ to ( wφ ) δ , since ( wφ ) δ isa central element of W ( wφ ). Actually ( w φ ) δ is a central element of B w andmaps by the morphism of Conjecture 9.1(iii) to F δ , thus the eigenvalue of F δ onthe ρ ϕ -isotypic part of H ∗ c ( X ( I , w φ )) is equal to ω ϕ q (( w φ ) δ ); thus ω ϕ q (( w φ ) δ ) = λ ρ ϕ q δ kd ( l ( π / π I ) − a ρϕ − A ρϕ ) . The statement follows by applying the specialization q /a ζ /a to this equality. (cid:3) Appendix: d -good maximal elements in finite Coxeter cosets. We will describe, in a finite Coxeter coset, for each d , a d -good maximal element.As explained the introduction of Section 8, when the Coxeter coset is attachedto a reductive group G , such an element defines a parabolic Deligne-Lusztig varietywhose cohomology should be a tilting complex for the Brou´e conjectures for an ℓ dividing Φ d ( q ). The properties of this variety do not depend on the isogenytype, thus it is sufficient to study the case when G is semi-simple and simplyconnected. Now, a semi-simple and simply connected group is a direct productof restrictions of scalars of simply connected quasi-simple groups. A restriction ofscalars is a group of the form G n , with an isogeny F such that F ( x , . . . , x n − ) =( x , . . . , x n − , F ( x )). Then ( G n ) F ≃ G F . If F induces φ on the Weyl group W of G then ( G n , F ) corresponds to the reflection coset W n · σ , where σ ( x , . . . , x n − ) =( x , . . . , x n − , φ ( x )).10.1. Restrictions of scalars.
Restrictions of scalars as above appear in the clas-sification of arbitrary complex reflection cosets. Arbitrary cosets
W φ are directproducts of cosets where φ is transitive on the irreducible components of W ; wecall restriction of scalars a complex reflection coset with this last property. It is ofthe form W n · σ ⊂ GL( V n ), where V is a complex vector space and W φ ⊂ GL( V ) isa complex reflection coset and where σ ( x , . . . , x n − ) = ( x , . . . , x n − , φ ( x )). Wesay that W n σ is a restriction of scalars of W φ , by analogy with the terminologyfor reductive groups.We first look at the invariant theory of a restriction of scalars. Recall (see forexample [Br2]) that, if S W is the coinvariant algebra of W (the quotient of thesymmetric algebra of V ∗ by the ideal generated by the W -invariants of positivedegree), for any W -module X the graded vector space ( S W ⊗ X ∗ ) W admits ahomogeneous basis formed of eigenvectors of φ . The degrees of the elements of thisbasis are called the X -exponents of W and the corresponding eigenvalues of φ the X -factors of W φ . For X = V , the V -exponents n i satisfy n i = d i − d i ’sare the reflection degrees of W , and the V -factors ε i are called the factors of W φ .For X = V ∗ , the n i − n i are the V ∗ -exponents are called the codegrees d ∗ i of W and the corresponding V ∗ -factors ε ∗ i are called the cofactors of W φ . BySpringer [S, 6.4], for a root of unity ζ , the ζ -rank of W φ is equal to |{ i | ζ d i = ε i }| .By analogy, we define the ζ -corank of W φ as |{ i | ζ d ∗ i = ε ∗ i }| . By for example [Br2,5.19.2] an eigenvalue is regular if it has same rank and corank. Proposition 10.1.
Let W n · σ be a restriction of scalars of the complex reflectioncoset W φ . Then the ζ -rank (resp. corank) of W n · σ is equal to the ζ n -rank (resp.corank) of W φ .In particular, ζ is regular for W n · σ if and only if ζ n is regular for W · φ .Proof. It is easy to see from the construction that the pairs of a reflection degreeand the corresponding factor of σ for the coset W n · σ are the pairs ( d i , η i,j ), where ARABOLIC DELIGNE-LUSZTIG VARIETIES. 61 i ∈ { , . . . , r } and where { η i,j } j ∈{ ...n } run over the n -th roots of ε i . Similarly, thepairs of a reflection codegree and the corresponding cofactor are ( d ∗ i , η ∗ i,j ) where { η ∗ i,j } j ∈{ ...n } run over the n -th roots of ε ∗ i .In particular the ζ -rank of W n · σ is |{ ( i, j ) | ζ d i = η i,j }| and the ζ -corank is |{ ( i, j ) | ζ d ∗ i = η ∗ i,j }| .Given d , there is at most one j such that ζ d = η i,j , and there is one if and only if ζ nd = ε i . Thus |{ ( i, j ) | ζ d i = η i,j }| = |{ i | ζ nd i = ε i }| and similarly for the corank,whence the two assertions of the statement. (cid:3) The next lemma can also be used to give a direct proof of the statement on ζ -ranks. Lemma 10.2.
Let W n · σ be a restriction of scalars of W φ . Then (i)
Any element of W n σ is conjugate to an element of the form (1 , . . . , , w ) σ . (ii) The vector ( x , . . . , x n − ) ∈ V n is a ζ -eigenvector of (1 , . . . , , w ) σ if andonly if x is a ζ n -eigenvector of wφ and x i = ζ i x for i = 1 , . . . , n − .Proof. The element (1 , w , w w , . . . , w w . . . w n − ) conjugates ( w , . . . , w n − ) σ to(1 , . . . , , w . . . w n − ) σ , whence (i). Property (ii) results from an immediate com-putation. (cid:3) In view of Lemma 10.1, the following proposition is enough to determine allpossible non-extendable ( d, k )-periodic elements of W n σ . Proposition 10.3.
Let W n · σ be a restriction of scalars of the finite Coxeter coset W φ . Let ( B + ) n σ and B + φ be the corresponding cosets of braid monoids. Then (i) Any element in ( B + ) n σ is conjugate under ( B + ) n to an element of theform (1 , . . . , , w ) σ . (ii) The element (1 , . . . , , w ) σ ∈ ( B + ) n σ is ( nd, k ) -periodic if and only if w φ is ( d, k ) periodic. Moreover the latter is non-extendable if and only if theformer is non-extendable.Proof. The element (1 , w , w w , . . . , w w . . . w n − ) conjugates ( w , . . . , w n − ) σ to (1 , . . . , , w . . . w n − ) σ , whence (i).For (ii), we have ((1 , . . . , , w ) σ ) nd = (( w φ ) d φ − d , . . . , ( w φ ) d φ − d ) σ nd , whence thefirst assertion: ( w φ ) d = ( π / π I ) k φ d is equivalent to ((1 , . . . , , w ) σ ) nd = (( π / π I ) k , . . . , ( π / π I ) k ) σ nd .If the last equalities hold and w φ is extendable, that is there exist v ∈ B + I and J ( I such that ( vw ) d = ( π / π J ) k φ d , then ((1 , . . . , , v )(1 , . . . , , w ) σ ) nd =(( π / π J ) k , . . . , ( π / π J ) k ) σ nd , so that (1 , . . . , , w ) σ is extendable.Conversely assume that (1 , . . . , , w ) σ is extendable, that is, there exist ( v . . . , v n − ) ∈ ( B + I ) n and J × · · · × J n − ( I n such that(( v , . . . , v n − , v n − w ) σ ) nd = ( π / π J , . . . , π / π J n − ) k σ nd . Then since ( v , . . . , v n − , v n − w ) σ stabilizes J × · · · × J n − , we have J i = v i J i +1 for i < n − J i ( I for all i . By the same conjugation as in thefirst line of the proof (by (1 , v , v v , . . . , v v · · · v n − )) the above equality con-jugates to ((1 , . . . , v · · · v n − w ) σ ) nd = ( π / π J , . . . , π / π J ) k σ nd , or equivalently( v · · · v n − w φ ) d = ( π / π J ) k φ d , thus w φ is extendable. (cid:3) Case of irreducible Coxeter cosets.
We are going to give, for each ir-reducible finite Coxeter group W , each possible corresponding coset W φ where φ preserves a chamber of the corresponding hyperplane arrangement, and each possi-ble d , a representative wφ of the d -good maximal elements. Since conjecturally allnon-extendable ( d, ζ d -eigenspace V where ζ d = e iπ/d , the set I and the relative complex reflection group W ( wφ ) := N W ( V ) /C W ( V ). In the caseswhere the injection C W ′ ( wφ ) → N W ( V ) /C W ( V ) = W ( wφ ) of the remark afterLemma 8.14, is surjective, where W ′ = C W ( V ) and V is the fixed point subspaceof wφ in the space spanned by the root lines of W I , we use it to deduce W ( wφ )from W ′ = C W ( V ) using the description of centralizers of regular elements in [BM,Annexe 1]. Types A n and A n (cid:13) s (cid:13) s · · ·(cid:13) s n . A n is defined by the diagram automorphism φ which exchanges s i and s n +1 − i .For any integer 1 < d ≤ n + 1, we define v d = s s · · · s n −⌊ d ⌋ s n s n − · · · s ⌊ d +12 ⌋ and J d = { s i | ⌊ d + 12 ⌋ + 1 ≤ i ≤ n − ⌊ d ⌋} . If d is odd we have v d = v ′ d φ v ′ d , where v ′ d = s s · · · s n −⌊ d ⌋ .Now, for 1 < d ≤ n + 1, let kd be the largest multiple of d less than or equal to n + 1, so that n +12 < kd ≤ n + 1 and k = ⌊ n +1 d ⌋ . We then define w d = v kkd , I d = J kd and if d is odd we define w ′ d by w ′ d φ = ( ( v ′ kd φ ) k if k is odd, v k/ kd φ if k is even, Theorem 10.4.
For W = W ( A n ) , d -good maximal elements exist for < d ≤ n + 1 ; a representative is w d , with I = I d and W ( w d ) = G ( d, , ⌊ n +1 d ⌋ ) .For W φ , d -good maximal elements exist for the following d with representativesas follows: • d ≡ , < d ≤ n + 1 ; a representative is w d φ with I = I d and W ( w d φ ) = G ( d, , ⌊ n +1 d ⌋ ) . • d ≡ , < d ≤ n + 1) ; a representative is w ′ d/ φ with I = I d/ and W ( w ′ d/ φ ) = G ( d/ , , ⌊ n +1) d ⌋ ) . • d odd, < d ≤ n +12 . If d = 1 a representative is w d φ with I = I d and W ( w d φ ) = G (2 d, , ⌊ n +12 d ⌋ ) .Proof. We identify the Weyl group of type A n as usual with S n +1 by s i ( i, i + 1);the automorphism φ maps to the exchange of i and n + 2 − i . An easy computationshows that the element v d maps to the d -cycle (1 , , . . . , ⌊ d +12 ⌋ , n + 1 , n, . . . , n + 2 −⌊ d ⌋ ) and that for d odd v ′ d maps to the cycle (1 , , . . . , n − d − ). Lemma 10.5. If d is even v d and w d are φ -stable. If d is odd we have w d = w ′ d . φ w ′ d .Proof. That d is even implies ⌊ d +12 ⌋ = ⌊ d ⌋ , thus in the above cycle φ exchangesthe two sequences 1 , , . . . , ⌊ d +12 ⌋ and n + 1 , n, . . . , n + 2 − ⌊ d ⌋ , thus v d is φ -stable.The same follows for w d , with k = ⌊ n +1 d ⌋ , since kd is even if d is even. ARABOLIC DELIGNE-LUSZTIG VARIETIES. 63
For d odd we have w ′ d . φ w ′ d = ( w ′ d φ ) = ( ( v ′ kd φ ) k if k is odd, v k/ kd . φ ( v k/ kd ) if k is even.If k is odd we have ( v ′ kd φ ) k = ( v ′ kd φ v ′ kd ) k = v kkd = w d . If k is even then v kd is φ -stable thus v k/ kd . φ ( v k/ kd ) = v kkd = w d . (cid:3) Lemma 10.6.
For < d ≤ n + 1 , • the element v d is J d -reduced and stabilizes J d . • the element w d is I d -reduced and stabilizes I d . • for d odd, the element v ′ d is J d -reduced and v ′ d φ stabilizes J d . • for d odd, the element w ′ d is I d -reduced and w ′ d φ stabilizes I d .Proof. The property for w d (resp. w ′ d ) follows from that for v d (resp. v ′ d ) and thedefinitions since being I d -reduced and stabilizing I d are properties stable by takinga power.It is clear on the expression of v d as a cycle that it fixes i and i + 1 if s i ∈ J d thus it fixes the simple roots corresponding to J d , whence the lemma for v d .For d odd, 1 < d ≤ n +1, an easy computation shows that v ′ d = (1 , , . . . , n − d − ),and that v ′ d φ preserves the simple roots corresponding to J d . (cid:3) Lemma 10.7.
For < d ≤ n + 1 and for ≤ i ≤ ⌊ d ⌋ , we have • l ( v id ) = id l ( w − J d w ) and l ( w id ) = id l ( w − I d w ) • (for d odd) l (( v ′ d φ ) i φ − i ) = id l ( w − J d w ) and l (( w ′ d φ ) i φ − i ) = id l ( w − I d w ) .Proof. It is straightforward to see that the result for w d (resp. w ′ d ) results from theresult for v d (resp. v ′ d or v d ) and the definitions.Note that the group W J d is of type A n − d , thus l ( w − J d w ) = n ( n +1)2 − ( n − d )( n − d +1)2 = (2 n − d +1) d .We first prove the result for v d and v ′ d when i = 1. For odd d we have bydefinition l ( v ′ d ) = n − d − = n − d +12 which is the formula we want for v ′ d . To findthe length of v d one can use that s n s n − · · · s ⌊ d +12 ⌋ is { s , s , . . . , s n − } -reduced,thus adds to s s · · · s n −⌊ d ⌋ , which gives l ( v d ) = 2 n − d + 1, the result for v d .We now show by direct computation that when d is even v d/ d = w − J d w . Rais-ing the cycle (1 , , . . . , d , n + 1 , n, . . . , n + 2 − d ) to the d/ , n + 1)(2 , n ) · · · ( d , n + 2 − d ) which gives the result since w J d = ( d + 1 , n +1 − d ) · · · ( ⌊ n ⌋ , ⌊ n +12 ⌋ ). The lemma follows for v d with d even since its truth for i = 1 and i = d implies its truth for all i between these values.We show now similarly that for odd d we have ( v ′ d φ ) d = w − J d w φ d . Since φ actson W by the inner automorphism given by w , this is the same as ( v ′ d w ) d = w J d .We find that (1 , , . . . , n − d − ) w = (1 , n +1 , , n, , n − , . . . , n − d − , d +12 )( d +32 , n − d − ) · · · ( ⌊ n +32 ⌋ , ⌊ n +42 ⌋ ) as a product of disjoint cycles, which gives the result since(1 , n +1 , , n, , n − , . . . , n − d − , d +12 ) is a d -cycle and ( d +32 , n − d − ) · · · ( ⌊ n +32 ⌋ , ⌊ n +42 ⌋ ) = w J d . This proves the lemma for w ′ d by interpolating the other values of i as above.It remains the case of v d for odd d . We then have v d = ( v ′ d φ ) where the lengthsadd, and we deduce the result for v d from the result for v ′ d . (cid:3) Lemma 10.8.
The following elements are d -good • For < d ≤ n + 1 , the elements v d and w d . • For d ≡ , d ≤ n + 1 the elements v d φ and w d φ . • For d ≡ , d ≤ n + 1) the elements v ′ d/ φ and w ′ d/ φ . • For d odd, d ≤ n +12 the elements v d φ and w d φ .Proof. In view of the previous lemmas, the only thing left to check is that in eachcase, the chosen element x in W (resp. W φ ) satisfies x d = 1 (resp. ( xφ ) d = φ d ).Once again, it is easy to check that the property for w d (resp. w ′ d ) results from thatfor v d (resp. v ′ d or v d ) and the definitions.It is clear that v dd = 1 since then it is a d -cycle, from which it follows that when d ≡ v ′ d/ φ ) d = v d/ d/ = 1. The other cases are obvious. (cid:3) To prove the theorem, it remains to check that: • The possible d for which the ζ d -rank of W (resp. W φ ) is non-zero are asdescribed in the theorem. In the untwisted case they are the divisors of one ofthe degrees, which are 2 , . . . , n + 1. In the twisted case the pairs of degrees andfactors are (2 , , . . . , ( i, ( − i ) , . . . , ( n + 1 , ( − n +1 ) and we get the given list bythe formula for the ζ d -rank recalled above Proposition 10.1. • The coset W I wφ has ζ d -rank 0 on the subspace spanned by the root lines of W I . For this we first have to describe the type of the coset, which is a consequenceof the analysis we did to show that wφ stabilizes I . We may assume I non-empty.Let us look first at the untwisted case. We found that w d acts trivially on I d ,so the coset is of untwisted type A n − kd where k = ⌊ n +1 d ⌋ . Since 1 + n − kd < d byconstruction, this coset has ζ d -rank 0.In the twisted case, if d ≡ W I d w d φ , which since w d actstrivially on I d and φ acts by the non-trivial diagram automorphism, is of type A n − kd where k = ⌊ n +1 d ⌋ . Since n − kd = n − ⌊ n +1 d ⌋ d < d −
1, this coset has ζ d -rank 0.If d is odd, the coset is W I d w d φ , which since w d acts trivially on I d and φ acts by the non-trivial diagram automorphism, is of type A n − kd where k = ⌊ n +12 d ⌋ .Since n − kd = n − ⌊ n +12 d ⌋ d < d , this coset has ζ d -rank 0.Finally, if d ≡ W I d/ w ′ d/ φ . Let k = ⌊ n +1) d ⌋ ; then W I d/ is of type A n − kd/ . If k is even then w ′ d/ = w k/ kd/ and the coset is of type A n − kd/ . Since n − kd/ < d/ −
1, this coset has ζ d -rank 0. Finally if k is odd w ′ d/ φ = ( w ′ kd/ φ ) k . Since kd/ w ′ kd/ φ acts trivially on I d/ so the coset is of type A n − kd/ , and has also has ζ d -rank 0. • Determine the group W ( wφ ) (resp. W ( w )) in each case, We first give V andthe coset C W ( V ) wφ or C W ( V ) w . In the untwisted case w d acts trivially on theroots of W I d , hence V is spanned by these roots and C W ( V ) is generated by thereflection with respect to the roots orthogonal to those, which gives that C W ( V ) isof type A d ⌊ n +1 d ⌋− if d n and A n otherwise. In the twisted case if d ≡ w d acts trivially on the roots of W I d the space V is spanned by the sumsof the orbits of the roots under φ which is the non-trivial automorphism of thatroot system. Hence the type of the coset C W ( V ) w d φ is A d ⌊ n +1 d ⌋− if n is oddand A d ⌊ n +1 d ⌋ if n is even. If d is odd a similar computation gives that the typeof the coset C W ( V ) w d φ is A d ⌊ n +12 d ⌋− if n is odd and A d ⌊ n +12 d ⌋ if n is even. If d ≡ w ′ d/ φ acts also by the non-trivial automorphism on W I d/ and we ARABOLIC DELIGNE-LUSZTIG VARIETIES. 65 get that the coset C W ( V ) w ′ d/ φ is of type A d ⌊ n +1) d ⌋ if n and ⌊ n +1) d ⌋ have thesame parity and A d ⌊ n +1) d ⌋− otherwise.Knowing the type of the coset in each case, we deduce the group W ( wφ ) (resp. W ( w )) as in the remark at the beginning of Subsection 10.2. (cid:3) Type B n (cid:13) s (cid:13) s (cid:13) s · · · (cid:13) s n . For d even, 2 ≤ d ≤ n we define v d = s n +1 − d/ · · · s s s · · · s n and J d = { s i | ≤ i ≤ n − d/ } . Note that v n is the Coxeter element s s · · · s n . Now for 1 ≤ d ≤ n , that werequire even if d > n , we define w d as follows: let kd be the largest even multipleof d less than or equal to 2 n so that k = ⌊ nd ⌋ if d is even and k = 2 ⌊ nd ⌋ is d is odd.We define w d = v kkd and I d = J kd . Theorem 10.9.
For W = W ( B n ) , d -good maximal elements exist for odd d lessthan or equal to n and even d less than or equal to n . A representative is w d , with I = I d ; we have W ( w d ) = G ( d, , ⌊ nd ⌋ ) if d is even and W ( w d ) = G (2 d, , ⌊ nd ⌋ ) if d is odd.Proof. We identify as usual the Weyl group of type B n with the group of signedpermutations on { , . . . , n } by s i ( i − , i ) for i ≥ s (1 , − v d maps to the d -cycle (or signed d/ n + 1 − d/ , n + 2 − d/ , . . . , n − , n, d/ − n − , d/ − n − , . . . , − n ). This element normalizes J d andacts trivially on the corresponding roots, so is J d -reduced. The same is thus truefor w d and I d , since these properties carry to powers. Lemma 10.10.
For ≤ i ≤ ⌊ d ⌋ we have l ( v id ) = id l ( w − J d w ) and l ( w id ) = id l ( w − I d w ) .Proof. As in Lemma 10.7 it is sufficient to prove the lemma for v d , which we donow. To find the length of v d we note that s s · · · s n is { s , s , . . . , s n } -reduced sothat the lengths of s n +1 − d/ · · · s and of s s · · · s n add, whence l ( v d ) = 2 n − d/ l ( w ) = n and l ( w J d ) = ( n − d/ we have l ( w − I d w ) = nd − d /
4, whichgives the result for i = 1. Written as permutations w is the product of all signchanges and w I d is the product of all sign changes on the set { , . . . , n − d/ } ; adirect computation shows that v d/ d is the product of all sign changes on { n + 1 − d/ , . . . , n } , hence v d/ d = w − I d w . The lemma follows for the other values of d . (cid:3) Since v d/ d = w − I d w we have v dd = 1, so the same property is true for w d , thusthe above lemma shows that v d and w d are d -good elements.Note also that Theorem 10.9 describes all d such that W has non-zero ζ d -ranksince the degrees of W ( B n ) are all the even integers from 2 to 2 n . We prove nowthe maximality property 8.10(iv) for w d . If k is as in the definition of w d , the group W I d is a Weyl group of type B n − kd/ and w d acts trivially on I d . Since n − kd/ < d the ζ d -rank of W I d w d is zero on the subspace spanned by the roots correspondingto I d .It remains to get the type of W ( w d ). Since w d acts trivially on I d the space V of Lemma 8.14 is spanned by the root lines of W I d and C W ( V ) is spanned by theroots orthogonal to those, so is of type B kd/ . We then deduce the group W ( w d ) asin the remark at the beginning of Subsection 10.2, as the centralizer of a ζ d -regularelement in a group of type B kd/ . (cid:3) Types D n and D n (cid:13) s (cid:13) s (cid:13) s (cid:13) s · · · (cid:13) s n . D n is defined by the diagram automorphism φ which exchanges s and s and fixes s i for i > d even, 2 ≤ d ≤ n −
1) we define v d = s n +1 − d/ · · · s s s s · · · s n and J d = ( ∅ if d = 2( n − { s i | ≤ i ≤ n − d/ } otherwise.Note that v n − is a Coxeter element. Then for 1 ≤ d ≤ n − d > n , we let kd be the largest even multiple of d less than 2 n , so that k = ⌊ n − d ⌋ if d is even and k = 2 ⌊ n − d ⌋ if d is odd, and define w d = v kkd and I d = J kd .Note that v d , and thus w d , are φ -stable. Theorem 10.11. • For W = W ( D n ) there exist d -good maximal elementsfor odd d less than or equal to n and even d less than or equal to n − .When d does not divide n a representative is w d , with I = I d ; in thiscase, if d is odd W ( w d ) = G (2 d, , ⌊ n − d ⌋ ) and if d is even W ( w d ) = G ( d, , ⌊ n − d ⌋ ) .If d | n a representative is w n/dn where w n = s s s · · · s n s s · · · s n − . Inthis case I = ∅ and W ( w n/dn ) = G (2 d, , n/d ) . • For
W φ there exist d -good maximal elements for odd d less than n , for even d less than n − and for d = 2 n . Except in the case when d divides n and n/d is odd a representative is w d φ , with I = I d and W ( w d φ ) = G (2 d, , ⌊ n − d ⌋ ) if d is odd and W ( w d φ ) = G ( d, , ⌊ n − d ⌋ ) if d is even. Inthe excluded case a representative is ( w n φ ) n/d where w n = s s s · · · s n .In this case I = ∅ and W (( w n φ ) n/d ) = G ( d, , n/d ) .Proof. The cases D n with d | n or D n with d | n and 2 n/d odd involve regularelements, so are dealt with in [BM]. We thus consider only the other cases.We identify the Weyl group of type D n with the group of signed permutationson { , . . . , n } with an even number of sign changes, by mapping s i to ( i − , i ) for i = 2 and s to (1 , − − , d even v d maps to (1 , − n + 1 − d/ , n + 2 − d/ , . . . , n − , n, d/ − n − , . . . , − n, − n ). This element normalizes J d : when J d = ∅ , it exchanges the simple roots corresponding to s and s and acts triviallyon the other simple roots indexed by J d , so it is J d -reduced. It follows that w d normalizes I d and is I d -reduced. Lemma 10.12.
For ≤ i ≤ ⌊ d ⌋ we have l ( v id ) = id l ( w − J d w ) and l ( w id ) = id l ( w − I d w ) .Proof. As in Lemma 10.7 it is sufficient to prove the lemma for v d . To find thelength of v d we note that s s s s · · · s n is { s , . . . , s n } -reduced so that the lengthsof s n +1 − d/ · · · s and of s s s · · · s n add, whence l ( v d ) = 2 n − − d/
2. Since l ( w ) = n − n and l ( w J d ) = ( n − d/ − ( n − d/ l ( w − J d w ) = d/ n − − d/ i = 1. Written as permutations w = (1 , − n (2 , − · · · ( n, − n ) and w J d = (1 , − n − d/ (2 , − · · · ( n − d/ , d/ − n );a direct computation shows that v d/ d = (1 , − d/ ( n +1 − d/ , d/ − n − · · · ( n, − n ),hence v d/ d = w − J d w . The lemma follows for smaller i . (cid:3) ARABOLIC DELIGNE-LUSZTIG VARIETIES. 67
Since v d/ d = w − J d w and J d is w stable we have v dd = 1, so the same propertyfollows for w d which shows that v d and w d are d -good elements.We also note that the theorem describes all d such that the ζ d -rank is not zero,since the degrees of W ( D n ) are all the even integers from 2 to 2 n − n , and inthe twisted case the factor associated with the degree n is -1 and the other factorsare equal to 1.Since w d is φ -stable the element w d φ is also d -good.We now check Lemma 8.10(iv), that is that the ζ d -rank of W I d w d in the untwistedcase, resp. W I d w d φ in the twisted case is 0 on the subspace spanned by the rootscorresponding to I d . This property is clear if I d = ∅ . Otherwise: • In the untwisted case the type of the coset is D n − kd/ if k is even and D n − kd/ if k is odd, where k is as in the definition of w d . In both cases the set of values i such that the ζ i -rank is not 0 consists of the even i less than 2 n − kd , the odd i less than n − kd/ k odd) i = 2 n − kd . Since if d is evenwe have 2 n − kd ≤ d and if d is odd we have n − kd/ ≤ d , the only case where d could be in this set is k odd and d = 2 n − kd , which means that k +12 d = n . But d is assumed not to divide n , so this case does not happen. • In the twisted case the type of the coset is D n − kd/ if k is odd and D n − kd/ if k is even. In both cases the set of values i such that the ζ i -rank is not 0 consistsof the even i less than 2 n − kd , the odd i less than n − kd/ k even) i = 2 n − kd . Since if d is even we have 2 n − kd ≤ d and if d is odd we have n − kd/ ≤ d , the only case where d could be in this set is k even and d = 2 n − kd ,which means that ( k + 1) d = 2 n . But this is precisely the excluded case.We now give C W ( V ), where V is as in Lemma 8.14, in each case where I is notempty. In the untwisted case, if d is odd the group C W ( V ) is of type D d ⌊ n − d ⌋ ; if d is even the group C W ( V ) is of type D d ⌊ n − d ⌋ +1 if ⌊ n − d ⌋ is odd and D d ⌊ n − d ⌋ if ⌊ n − d ⌋ is even. In the twisted case, if d is odd the coset C W ( V ) wφ is of type D d ⌊ n − d ⌋ +1 and if d is even the coset is of type D d ⌊ n − d ⌋ +1 if ⌊ n − d ⌋ is even and D d ⌊ n − d ⌋ if ⌊ n − d ⌋ is odd. In all cases except if d is even and ⌊ n − d ⌋ is even (resp.odd) in the untwisted case (resp. twisted case) we then deduce the group W ( wφ )(resp. W ( w )) as in the remarks at the beginning of Subsection 10.2 and after Lemma8.14, since in these cases the centralizer of the regular element wφ (resp. w ) in theparabolic subgroup W ′ = C W ( V ) has the (known) reflection degrees of W ( wφ )(resp. W ( w )). In the excluded cases the group C W ′ ( wφ ) or C W ′ ( w ) is isomorphicto G ( d, , ⌊ n − d ⌋ ) which does not have the reflection degrees of W ( wφ ), resp. W ( w ).This means that the morphism of the remark after Lemma 8.14 is not surjective.We can prove in this case that W ( wφ ) or W ( w ) is G ( d, , ⌊ n − d ⌋ ) since it is anirreducible complex reflection group by [Br2, 5.6.6] and it is the only one whichhas the right reflection degrees apart from the exceptions in low rank given by G , G , G , G , G ; we can exclude these since they do not have G ( d, , ⌊ n − d ⌋ )as a reflection subgroup. (cid:3) Types I ( n ) and I ( n ) . All eigenvalues ζ such that the ζ -rank is non-zero areregular, so this case can be found in [BM]. Exceptional types.
Below are tables for exceptional finite Coxeter groups givinginformation on d -good maximal elements for each d . They were obtained with the GAP package
Chevie (see [Chevie]): first, the conjugacy class of good ζ d -maximal elements as described in Lemma 8.13 was determined; then we determined I for anelement of that class, which gave l ( w I ). The next step was to determine the elementsof the right length 2( l ( w ) − l ( w I )) /d in that conjugacy class; this required care inlarge groups like E . The best algorithm is to start from an element of minimallength in the class (known by [GP]) and conjugate by Coxeter generators until allelements of the right length are reached.In the following tables, we give for each possible d and each possible I for that d arepresentative good wφ , and give the number of possible wφ . We then describe thecoset W I wφ by giving, if I = ∅ , in the column I the permutation induced by wφ ofthe nodes of the Coxeter diagram indexed by I . Then we describe the isomorphismtype of the complex reflection group N W ( W I wφ ) /W I = N W ( V ) /C W ( V ), where V is the ζ d -eigenspace of wφ . Finally, in the cases where I = ∅ , we give theisomorphism type of W ′ = C W ( V ), where V is the 1-eigenspace of wφ on thesubspace spanned by the root lines of I . We note that there are 3 cases where N W ′ ( V ) /C W ′ ( V ) (cid:12) N W ( V ) /C W ( V ): for d = 4 or 5 in E and for d = 9 in E . H : (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , d representative w w C W ( w )10 w = 123 4 Z w = 32121 6 Z w Z w Z w H · H H : (cid:13) (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , , d representative w w C W ( w )30 w = 1234 8 Z w = 432121 12 Z w Z w = 2121432123 22 Z w or w G w or w G w or w G w or w G w or w G w H · H D : (cid:13) (cid:13) (cid:13) (cid:13) φ does the permutation (1 , , , , , , ζ , ζ , d representative wφ wφ C W ( wφ )12 w φ = 13 φ Z w φ = 1243 φ G w φ G w φ G φ G F : (cid:13) (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , , d representative w w C W ( w )12 w = 1234 8 Z w = 214323 14 Z w G w or w G w G w F · F F : φ does the permutation (1 , , , − , , − d representative wφ wφ C W ( wφ )24 w φ = 12 φ Z w φ = 3231 φ Z w φ ) G w φ ) G w φ I (8)1 φ I (8) E : (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , , , , d representative w w I N W ( W I w ) /W I C W ( V )12 w = 123654 8 Z w = 12342654 24 Z w = 123436543 14 Z w G Z A w or w G w or w G w F · E
60 F. DIGNE AND J. MICHEL2 E : φ does the permutation (1 , , , − , , , − , d representative wφ wφ I N W ( W I wφ ) /W I C W ( V ) wφ w φ = 1234 φ Z w φ = 123654 φ Z
10 2431543 φ Z A φ φ w φ = 123436543 φ Z w φ ) G w φ ) G w φ G w φ E φ F E : (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , , , , , d representative w w I N W ( W I w ) /W I C W ( V )18 w = 1234567 64 Z w = 123425467 160 Z w = 1342546576 8 (2 , , Z E w a = 134254234567 8 (2 , Z D w b = 243154234567 8 (3 , w c = 124354265437 8 (4 , w Z , Z D w Z w or w G w a Z A w b w c w or w
12 (2)(5)(7) G D w or w G w E · E E : (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) The reflection degrees are 2 , , , , , , , d representative w w I N W ( W I w ) /W I C W ( V )30 w = 12345678 128 Z w = 1234254678 320 Z w = 123425465478 624 Z w a = 1342542345678 16 (2 , Z E w b = 2431542345678 16 (3 , w c = 1243542654378 16 (4 , w Z w a = 13423454234565768 128 (2) Z E w b = 24231454234565768 88 (3) w c = 12435423456543768 108 (4) w d = 12342543654276548 68 (5)12 w G w or w G w a
16 (2)(4) Z E w b
16 (3)(4) w c
16 (4)(5)8 w G w a
128 (2) Z E w b
88 (3) w c
108 (4) w d
68 (5)6 w or w G w or w G w or w G w or w G w E · E References [B1] D. Bessis, Finite complex reflection arrangements are K (Π , arXiv:math/0610777 .[BR] C. Bonnaf´e and R. Rouquier, On the irreducibility of Deligne-Lusztig varieties C.R.A.S. (2006) 37–39.[BR2] C. Bonnaf´e and R. Rouquier, Affineness of Deligne-Lusztig varieties for minimal lengthelements
J. Algebra (2008) 1200–1206.[Bouc] S. Bouc, Homologie de certains ensembles de 2-sous-groupes des groupes sym´etriques,
Journal of Algebra , 158–186 (1992).[Bou] N. Bourbaki, Groupes et alg`ebres de Lie, Chap. 4,5 et 6,
Masson (1981).[Br1] M. Brou´e, Isom´etries parfaites, types de blocs, cat´egories d´eriv´ees.
Ast´erisque (1990), 61–92.[Br2] M. Brou´e, Introduction to complex reflection groups and their braid groups,
SpringerSLN (2010).[BMM] M. Brou´e, G. Malle and J. Michel, Generic blocks of finite reductive groups,
Ast´erisque (1993) 7–92.[BM] M. Brou´e et J. Michel, Sur certains ´el´ements r´eguliers des groupes de Weyl et lesvari´et´es de Deligne-Lusztig associ´ees,
Progress in Math. , 73–139 (1997).[C] D. Craven, On the cohomology of Deligne-Lusztig varieties arXiv:1107.1871 [DDGKM] P. Dehornoy, F. Digne, E. Godelle, D. Krammer and J. Michel, Garside theory, bookin preparation, see .[DDM] P. Dehornoy, F. Digne and J. Michel, Garside families and Garside germs,
J. Algebra (2013) 109–145.[D] P. Deligne, Action du groupe des tresses sur une cat´egorie,
Invent. Math. (1997).[DM1] F. Digne et J. Michel, Fonctions L des vari´et´es de Deligne-Lusztig et descente deShintani, M´emoires de la SMF (1985).[DM2] F. Digne and J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya Math.J. 183 (2006) 35–103[DMR] F. Digne, J. Michel and R. Rouquier, Cohomologie des vari´et´es de Deligne-Lusztig, Advances in Math. (2007) 749–822.[Du] O. Dudas, Cohomology of Deligne-Lusztig varieties for unipotent blocks of GL n ( q ), Representation Theory (2013), 647–662.[G] E. Godelle, Normalisateur et groupe d’Artin de type sph´erique, J. Algebra (2003)263–274.[GP] M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras,
Advances inMath. (1993) 79–94.[HN] X. He and S. Nie, Minimal length elements of finite Coxeter groups,
Duke math. J. (2012) 2945–2967.[Lu] G. Lusztig, On the finiteness of the number of unipotent classes,
Inventiones (1976)201–213[McL] S. Mac Lane, Categories for the working mathematician, 2nd edition Springer-Verlag(1998).[Chevie] J. Michel, The development version of the CHEVIE package of
GAP3 . arXiv:1310.7905 .[Pa] L. Paris, Parabolic subgroups of Artin groups, J. Algebra (1997) 369–399.[S] T. Springer, Regular elements of finite reflection groups,
Inventiones (1974) 159–198.[T] J. Tits, Normalisateurs de tores. I. Groupes de Coxeter ´etendus, J. Algebra (1966)96–116. LAMFA, CNRS UMR 7352, Universit´e de Picardie-Jules Verne
E-mail address : [email protected] IMJ, CNRS UMR 7586, Universit´e Paris VII
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