Geometric Structure of Affine Deligne-Lusztig Varieties for GL_3
aa r X i v : . [ m a t h . AG ] F e b Geometric Structure of Affine Deligne-LusztigVarieties for GL
Abstract
In this paper we study the geometric structure of affine Deligne-Lusztigvarieties X λ ( b ) for GL and b basic. We completely determine the irre-ducible components of the affine Deligne-Lusztig variety. In particular,we classify the cases where all of the irreducible components are classicalDeligne-Lusztig varieties times finite-dimensional affine spaces. If this isthe case, then the irreducible components are pairwise disjoint. Let k be a field with q elements, and let ¯ k be an algebraic closure of k . Let F = k (( t )) and let L = ¯ k (( t )). We write O = ¯ k [[ t ]] , O F = k [[ t ]] for the valuationrings of L and F . Let σ denote the Frobenius morphism of ¯ k/k and also of L/F .Let G be a split connected reductive group over k and let T be a split max-imal torus of it. Let B be a Borel subgroup of G containing T . For a cochar-acter λ ∈ X ∗ ( T ), let t λ be the image of t ∈ G m ( F ) under the homomorphism λ : G m → T .We fix a dominant cocharacter λ ∈ X ∗ ( T ). Then the affine Deligne-Lusztigvariety X λ ( b ) is the locally closed reduced ¯ k -subscheme of the affine Grassman-nian defined as X λ ( b )(¯ k ) = { xG ( O ) ∈ G ( L ) /G ( O ) | x − bσ ( x ) ∈ G ( O ) t λ G ( O ) } . Analogously, we can also define the affine Deligne-Lusztig varieties associatedto arbitrary parahoric subgroups (especially Iwahori subgroups).The affine Deligne-Lusztig variety X λ ( b ) carries a natural action (by leftmultiplication) by the group J b = J b ( F ) = { g ∈ G ( L ) | g − bσ ( g ) = b } . This action induces an action of J b on the set of irreducible components.The geometric properties of affine Deligne-Lusztig varieties have been stud-ied by many people. For example, there is a simple criterion by Kottwitz andRapoport to decide whether an affine Deligne-Lusztig variety is non-empty (see[8]). Moreover, for X λ ( b ) = ∅ , we have an explicit dimension formula:dim X λ ( b ) = h ρ, λ − ν b i −
12 def( b ) , ν b is the Newton vector of b , ρ is half the sum of the positive roots, anddef( b ) is the defect of b . For split groups, the formula was obtained in [10]and [21]. Recently, the parametrization problem of top-dimensional irreduciblecomponents of X λ ( b ) was also solved. See [18] and [22].Besides the geometric properties as above, it is known that in certain cases,the (closed) affine Deligne-Lusztig variety admits a simple description. Let P bea standard rational maximal parahoric subgroup of G ( L ), where G is assumedto be simple. Then for any minuscule cocharacter µ and for b ∈ G ( L ), G¨ortz andHe in [11] (see also [12]) studied the following union of affine Deligne-Lusztigvarieties: X ( µ, b ) P = { g ∈ G ( L ) /P | g − bσ ( g ) ∈ [ w ∈ Adm( µ ) P wP } . They proved that if (
G, µ, P ) is of Coxeter type and if b lies inside the basic σ -conjugacy class, then X ( µ, b ) P is naturally a union of classical Deligne-Lusztigvarieties. Furthermore, the work [13] studied the generalized affine Deligne-Lusztig variety X ( µ, b ) K ′ associated to a standard parahoric subgroup K ′ . Themain result in [13] determines when X ( µ, b ) K ′ is naturally a union of classicalDeligne-Lusztig varieties. In particular, the existence of such a simple descrip-tion is independent of K ′ .In the Iwahori case, Chan and Ivanov [3] gave an explicit description of cer-tain Iwahori-level affine Deligne-Lusztig varieties for GL n . Each component ofthe disjoint decomposition described there is a classical Deligne-Lusztig varietytimes finite-dimensional affine space, and they point out the similarity betweentheir description and the results in [11].In this paper, we study the geometry of affine Deligne-Lusztig varieties X λ ( b )in the affine Grassmannian for G = GL and b basic. For this, it is enough toconsider the case for b whose newton vector is of the form ( i , i , i ) ( i = 0 , , λ ∈ X ∗ ( T ) with X λ ( b ) = ∅ , we completely determine theirreducible components of X λ ( b ) and the index set of them. Theorem 1.1.
The irreducible components of X λ ( b ) are parameterized by theset F µ ∈ M J b /K µ , where M is a finite set consisting of certain dominant cochar-acters µ determined by λ , and K µ is the stabilizer of a lattice depending on µ under the action of J b . Each irreducible component is an affine bundle over asimple variety.The group J b acts on the index set by left multiplication. In most cases,the irreducible component is an open subscheme of a finite-dimensional affinespace, which is also affine. For more details, see Theorem 6.1 and Theorem 6.3.As an immediate corollary of this result, we classify the cases where all ofthe irreducible components of X λ ( b ) are classical Deligne-Lusztig varieties timesfinite-dimensional affine spaces (Corollary 6.5). In such cases, X λ ( b ) is a disjointunion of the irreducible components. Theorem 1.2.
Let λ = 0 be as in Corollary 6.5, and let Ω be the Drinfeldupper half space over k of dimension 2.2i) If b = 1, then we have X λ (1) ∼ = G J /K Ω × A or X λ (1) ∼ = ( G J /K Ω × A ) ⊔ ( G J /K Ω × A )as ¯ k -varieties, where A is a finite-dimensional affine space and K i is thestabilizer of a lattice under the action of J .(ii) If b has the newton vector of the form ( i , i , i ) ( i = 1 , X λ ( b ) ∼ = G J b /H b A or X λ ( b ) ∼ = ( G J b /H b A ) ⊔ ( G J b /H b A )as ¯ k -varieties, where A is a finite-dimensional affine space and H b is thestabilizer of a lattice under the action of J b .The strategy of the proof is as follows: Using the Bruhat-Tits building ofSL , we decompose X λ ( b )(¯ k ) into closed subsets, which are actually irreduciblecomponents. This can be checked using the dimension formula above. Thenwe determine their geometric structure by embedding them into the Schubertcells. The crucial ingredient of these processes is the method of Kottwitz in [15]. Acknowledgments:
This paper is the author’s master’s thesis, written atthe University of Tokyo. The author is grateful to his advisor Yoichi Mieda forhis encouragement and helpful comments.
Let k be a field with q elements, and let ¯ k be an algebraic closure of k . We set F = k (( t )) and L = ¯ k (( t )). Further, we write O = ¯ k [[ t ]] , O F = k [[ t ]] for thevaluation rings of L and F . Let σ denote the Frobenius morphism of ¯ k/k . Weextend σ to the Frobenius morphism of L/F , i.e., σ ( P a n t n ) = P σ ( a n ) t n . In this subsection, we define the affine Grassmannian of SL and GL , cf. [9].Set G = SL or GL . The loop group LG of G is the k -space given by thefollowing functor: LG ( R ) = G ( R (( t ))) , where R is a k -algebra. Similarly, we have the positive loop group L + G , definedas L + G ( R ) = G ( R [[ t ]]) , R is a k -algebra. The positive loop group is actually an (infinite-dimensional)scheme. Definition 2.1.
The affine Grassmanian G rass G for G is the quotient k -space LG/L + G .The quotient in the category of k -spaces is the sheafification of the presheafquotient R LG ( R ) /L + G ( R ).The affine Grassmannian is an ind-scheme. To check this, we define thenotion of lattices. Let R be a k -algebra, and let r ∈ Z > . The R [[ t ]]-submodule R [[ t ]] ⊂ R (( t )) is called the standard lattice and denoted by Λ R . Definition 2.2. A lattice L ⊂ R (( t )) is a R [[ t ]]-module such that(i) there exists N ∈ Z ≥ with t N Λ R ⊆ L ⊆ t − N Λ R , and(ii) the quotient t − N Λ R / L is locally free of finite rank over R .For r ∈ Z , a lattice L is said to be r - special if V L = t r Λ R .We denote the set of all lattices in R (( t )) by L att ( R ), and the set of all0-special lattices by L att ( R ). We also define, for N ≥
1, subsets L att ( N ) ( R ) ⊂ L att ( R ) , L att , ( N ) ( R ) ⊂ L att ( R ) , where the number N in part (i) of the definition of a lattice is fixed. For L att , ( N ) , the morphism of functors L att , ( N ) ( R ) → Grass N ( t − N Λ k /t N Λ k )( R ) , L L /t N Λ R , defines a closed embedding of L att , ( N ) into the Grassmann variety of 3 N -dimensional subspaces of the 6 N -dimensional k -vector space t − N Λ k /t N Λ k . Thuswe have a filtration of L att by closed subschemes: L att = [ L att , ( N ) . This gives an ind-projective scheme structure of L att because each L att , ( N ) isa closed subscheme of the projective scheme. Similarly, we can show that eachsubfunctor L att ( N ) has a projective scheme structure. So the filtration L att = [ L att ( N ) gives an ind-projective structure of L att . The ind-scheme L att is integral. See[1, Propostion 6.4] and [19, Theorem 6.1], which include the case of positivecharacteristic and also deal with other groups. On the other hand, L att is notreduced in general (cf. [9, Example 2.9] for the case G m ). However, we have( L att ) red = G r ∈ Z L att r , L att r is the k -subspace of L att consisting of r -special lattices.Finally, note that the affine Grassmannian for GL (resp. SL ) is naturallyisomorphic, as a k -space, to L att (resp. L att ). So the affine Grassmannianhas an ind-scheme structure. From now on, we write X = G rass GL , X S = G rass SL . Since L att is isomorphic to L att r through left multiplication by a r ∈ GL n ( k (( t ))) with v (det( a r )) = r , we have an isomorphism of ind-schemes X red ∼ = G r ∈ Z X S depending on the choice of ( a r ) r . We keep the notation above. Moreover, we fix a maximal torus T and a Borelsubgroup B of G . In the case G = GL , we always let T be the torus of diagonalmatrices, and we choose the subgroup of upper triangular matrices B as a Borelsubgroup. For G = SL , we make analogous choices of a maximal torus anda Borel subgroup. We often denote by T S and B S the maximal torus and theBorel subgroup of SL to avoid confusion with those of GL . Finally we set K = GL ( O ) , K S = SL ( O ).We denote by Φ the set of roots given by the choice of T , and by Φ + the setof positive roots distinguished by B . We let X ∗ ( T ) + = { λ ∈ X ∗ ( T ) | h α, λ i ≥ α ∈ Φ + } denote the set of dominant cocharacters. Then the Cartan decomposition of G ( L ) is given by G ( L ) = [ λ ∈ X ∗ ( T ) + G ( O ) t λ G ( O ) . We illustrate this with G = GL , SL . For G = GL , let χ ij be the character T → G m defined by diag( t , t , t ) t i t j − . Then we have Φ = { χ ij | i = j } , Φ + = { χ ij | i < j } with respect to our choice of T and B . So the subset X ∗ ( T ) + ⊂ X ∗ ( T ) ∼ = Z is equal to the set { ( m , m , m ) ∈ Z | m ≥ m ≥ m } and thus we have { t λ | λ ∈ X ∗ ( T ) + } = { diag( t m , t m , t m ) ∈ T | m ≥ m ≥ m , m i ∈ Z } . Similarly, for G = SL , the set { t λ | λ ∈ X ∗ ( T ) + } is equal to the set { diag( t m , t m , t m ) ∈ T | m ≥ m ≥ m , X i m i = 0 , m i ∈ Z } . For λ = ( m , m , m ) , µ = ( m ′ , m ′ , m ′ ) ∈ X ∗ ( T ) + , we write λ (cid:22) µ if m ≤ m ′ , m + m ≤ m ′ + m ′ , m + m + m = m ′ + m ′ + m ′ . This is called thedominance order. 5iven b ∈ G ( L ), its σ - stabilizer is the group { g ∈ G ( L ) | g − bσ ( g ) = b } . We denote by J b (resp. J Sb ) the σ -stabilizer for GL (resp. SL ). Lemma 2.3.
Let b ∈ GL ( L ). The restriction of η = v L ◦ det : GL ( L ) → Z to J b is surjective. Proof.
Since conjugation does not change the determinant, it is enough to provethe statement for some representatives of the σ -conjugacy classes of GL ( L ).As described in [9, Example 4.6], every σ -conjugacy class in GL ( L ) contains arepresentative b of the following form: b is a block diagonal matrix, and eachblock has the form (cid:18) t a i +1 E k i t a i E n i − k i (cid:19) ∈ GL n i ( L )Here 3 = P i n i , a i , k i ∈ Z , ≤ k i < n i . Using this description, we can easilyfind g ∈ J b such that η ( g ) = r for any r ∈ Z and b .For b ∈ GL ( L ) set H b = Ker( v L ◦ det : J b → Z ). Then v L ◦ det induces anisomorphism J b /H b ∼ = Z and if b ∈ SL ( L ), we clearly have J Sb ⊆ H b . Definition 2.4.
The relative position map isinv : G ( L ) /G ( O ) × G ( L ) /G ( O ) → X ∗ ( T ) + , which maps a pair of cosets ( xG ( O ) , yG ( O )) to the unique element λ ∈ X ∗ ( T ) + such that x − y ∈ G ( O ) t λ G ( O ).We now come to the definition of the affine Deligne-Lusztig variety. Definition 2.5.
The affine Deligne-Lusztig variety X λ ( b ) in the affine Grass-mannian associated with b ∈ G ( L ) and λ ∈ X ∗ ( T ) + is the locally closed subsetof G rass G given by X λ ( b )(¯ k ) = { xG ( O ) ∈ G ( L ) /G ( O ) | x − bσ ( x ) ∈ G ( O ) t λ G ( O ) } , provided with the reduced sub-ind-scheme structure.In fact, X λ ( b ) is a scheme locally of finite type over ¯ k ; see [14, Corollary 6.5].From now on, X λ ( b ) (resp. X Sλ ( b )) always denotes the affine Deligne-Lusztigvariety for GL (resp. SL ). Then J b (resp. J Sb ) acts by left multiplication on X λ ( b ) (resp. X Sλ ( b )). For affine Deligne-Lusztig varieties, we have the followingbasic lemma. Lemma 2.6.
Let λ = ( m , m , m ) ∈ X ∗ ( T ) + and b, g ∈ GL ( L ).(i) If X λ ( b ) is non-empty, then v L (det( b )) = m + m + m .6ii) The varieties X λ ( b ) and X λ ( g − bσ ( g )) are isomorphic.(iii) Let c = diag( t m , t m , t m ) , m ∈ Z . Then X λ ( b ) and X λ + M ( cb ) are equal assubvarieties of X , where M = ( m, m, m ). Proof. (i) follows from the equality v L (det( x − bσ ( x ))) = v L (det( b )).(ii) The map x g − x gives an isomorphism.(iii) We have xK ∈ X λ ( b ) ⇔ x − bσ ( x ) ∈ Kt λ K ⇔ x − cbσ ( x ) ∈ Kt λ + M K ⇔ xK ∈ X λ + M ( cb ).Next proposition gives a decomposition of the affine Deligne-Lusztig varietycorresponding to the decomposition of the affine Grassmannian X red ∼ = F r X S . Proposition 2.7.
Set η = v L ◦ det : GL ( L ) → Z . Then we have X λ ( b ) ∼ = G J b /H b ( X λ ( b ) ∩ η − (0))as ¯ k -varieties, and J b acts on the set of these components by left multiplication. Proof.
The scheme structure on X λ ( b ) is the reduced one, thus the inclusion X λ ( b ) ⊂ X factors through X red → X . By Lemma 2.3, we can choose ( a r ) r ∈ Z with a = 1 , a r ∈ J b and v L (det( a r )) = r . Then the isomorphism X red ∼ = F r X S determined by ( a r ) r restricts to the isomorphism X λ ( b ) ∼ = G J b /H b ( X λ ( b ) ∩ η − (0))as ¯ k -varieties. Remark 2.8.
Let b = 1 and let λ ∈ X ∗ ( T ) + with X λ (1) = ∅ . Then X λ (1) ∩ η − (0) is equal to X Sλ (1).In this paper, we will treat the affine Deligne-Lusztig variety X λ ( b ) with b basic (i.e. its newton vector ν b is central). Then by Lemma 2.6, it is enough toconsider the following three cases:(i) b = 1;(ii) b = b := t ;(iii) b = b := t
00 0 t . 7 The Bruhat-Tits Building SL In this subsection, we recall the Bruhat-Tits buildings of the groups SL ( L ) andSL ( F ). Here we will discuss mainly over L , and the same construction anddefinitions also work over F . Definition 3.1.
The
Bruhat-Tits Building of SL ( L ) is the simplicial complex B ∞ such that(i) The set of vertices of B ∞ is the set of equivalence classes of O -lattices L ⊆ L , where the equivalence relation is given by homothety, i.e., L ∼ L ′ ifand only if there exists c ∈ L × such that L ′ = c L .(ii) A set { L , . . . , L m } of m vertices is a simplex if and only if there existrepresentatives L i of L i such that L ⊃ · · · ⊃ L m ⊃ t L . We denote the simplicial complex arising from this construction over F by B . We see B as a subset of B ∞ by sending an O F -lattice L ⊆ F to L ⊗ O F O ⊆ L . Maximal simplices are called alcoves or chambers . For a vertex of B ∞ represented by L = g Λ ¯ k with g ∈ GL ( L ), we call the residue class of v L (det( g )) in Z / type of the vertex. This number is independent of thechoice of a representative.We say that an O -lattice L ⊂ L is adapted to a basis f , f , f of L , if L has an O -basis of the form t i f , t i f , t i f . The apartment corresponding tothe basis f i is the subcomplex of B ∞ whose simplices consist of vertices givenby lattices adapted to this basis. The apartment corresponding to the standardbasis is called the main apartment and denoted by A M .The action of SL ( L ) (resp. SL ( F )) on all vertices with the same type in B ∞ (resp. B ) is transitive. One has a base vertex of type 0 represented by Λ ¯ k (resp. Λ k ) with stabilizer SL ( O ) (resp. SL ( O F )). Thus we have X S ( k ) = the set of all vertices of type 0 in B X S (¯ k ) = the set of all vertices of type 0 in B ∞ . The Frobenius morphism σ acts on B ∞ in the obvious way. Then the fullsubcomplex consisting of all vertices fixed by σ is B . In this subsection, our goal is to describe the formula on the relative position oftwo vertices in B ∞ . For this, we mainly refer to [15] (and especially to the firstthree sections). Moreover, see [7] for the general notion on buildings.We begin by introducing an equivalence relation ∼ on X ∗ ( T ) + ⊂ Z . We willsay that ( m , m , m ) and ( m ′ , m ′ , m ′ ) are equivalent (and write ( m , m , m ) ∼ m ′ , m ′ , m ′ )) if there exists an integer m such that ( m , m , m ) = ( m ′ + m, m ′ + m, m ′ + m ). Let X ∗ ( T ) ′ + be X ∗ ( T ) + modulo ∼ , and [ λ ] = [ m , m , m ]denotes the class of a dominant cocharacter λ = ( m , m , m ). For any [ λ ] , [ µ ] ∈ X ∗ ( T ) ′ + , we write [ λ ] (cid:22) [ µ ] if there exists M = ( m, m, m ) such that λ + M (cid:22) µ .For any vertex P in the Bruhat-Tits building of SL ( L ), we can choose amatrix x ∈ GL ( L ) such that P = [ x Λ ¯ k ], where [ L ] denotes the class of alattice L . Then, for any two vertices P = [ x Λ ¯ k ] , Q = [ y Λ ¯ k ] ∈ B ∞ , one candefine their relative position by (the class of) inv( xK, yK ) ∈ X ∗ ( T ) ′ + , which isclearly independent of the choice of x and y . We denote it by inv ′ ( P, Q ).Let S be a subset of B ∞ contained in some apartment, and define cl ( S ) tobe the intersection of all the apartments which contain S . If S is containedin an apartment A of B ∞ , then cl ( S ) is equal to the intersection of all half-apartments in A containing S . Let S and S be two faces in B ∞ . We define cl ( S , S ) to be cl ( S ∪ S ). If each of S and S consists of a single vertex of B ∞ , then we will write cl ( P , P ) for it, where S = { P } and S = { P } . Let[ e , e , e ] = inv ′ ( P , P ), and let A be an apartment containing P and P . If e = e or e = e , then there is a wall in A which contains P and P , and cl ( P , P ) consists of the vertices of the wall which lie between P and P . P P • • • • In this case we will say that cl ( P , P ) is a line segment. Note that each minimalgallery ( C , . . . , C r ) connecting P and P is of the form P P C C r where the points in the bottom row are the points of cl ( P , P ). Such a galleryis called a special gallery connecting P and P . We call C a first chamber of cl ( P , P ) ( C is a chamber containing P ). If e > e > e , then cl ( P , P )forms a parallelogram. P P P C C C C r − C r In this case, the gallery ( C , . . . , C r ) right above is called the special galleryconnecting P and P , where inv ′ ( P , P ) = [ e − e , e − e , C thefirst chamber of cl ( P , P ) ( C is the chamber containing P ).9et P be a vertex of B ∞ . Let C , C be two chambers containing P , andlet P be a vertex of C such that inv ′ ( P , P ) = [1 , , C and C . We saythat C and C have relative position I (resp. relative position II , resp. relativeposition III ) if cl ( C , C ) looks like Figure 1 (resp. Figure 2, resp. Figure 3). P C C Figure 1: Relative position I P P C C Figure 2: Relative position II P P C C Figure 3: Relative position IIILet A be an apartment of B ∞ and C a chamber contained in A . Thenwe denote by ρ A ,C the retraction of B ∞ onto A with center C . From nowon, we will consider the following situation: Let P , P , P belong to B ∞ with P = P and P = P . Let ( C , . . . , C r ) (resp. ( D , . . . , D s )) be a special galleryconnecting P to P (resp. P to P ). Moreover, we assume that C and D haverelative position I, II or III. We want to compute inv ′ ( P , P ) from inv ′ ( P , P )and inv ′ ( P , P ). For this, the main machineries are the following two lemmas. Lemma 3.2.
Let C be a chamber of B ∞ , and let A be an apartment containing C . If a vertex P belongs to C and a vertex Q belongs to B ∞ , then inv ′ ( P, Q ) =inv ′ ( P, ρ ( Q )), where ρ = ρ A ,C . Proof.
This is [15, LEMMA 1.1].
Lemma 3.3.
Let A be an apartment of B ∞ , C a chamber of A and ρ = ρ A ,C .Let E be an edge of B ∞ , and let C (resp. C ) be the chamber of A containing ρ ( E ) which is on the same (resp. opposite) side as C of the wall of A containing ρ ( E ). Then there is a unique chamber C ′ containing E such that ρ ( C ′ ) = C ,and for every chamber C ′′ containing E which is distinct from C ′ , we have ρ ( C ′′ ) = C . 10 roof. This is [15, LEMMA 1.2].Let A be an apartment containing C , . . . , C r , and let ρ = ρ A ,C r be theretraction of B ∞ onto A with center C r . Our method will be to retract thegallery ( D , . . . , D s ) onto the apartment A , using Lemma 3.3 each step of theway. Lemma 3.2 guarantees that such retraction does not change inv ′ ( P , P ).Let E i be the edge between D i and D i +1 . If ρ ( D i ) is on the same side of ρ ( E i )as C r , then ρ ( D i +1 ) is the other chamber in A besides ρ ( D i ) containing ρ ( E i ).However, if ρ ( D i ) is on the opposite side of ρ ( E i ) as C r , then it may happen that E i +1 retracts to ρ ( E i ). In this case, we will say that “a bend has occurred” atthe edge between D i and D i +1 . There is at most one edge at which a bend occurs([15, p. 342]). If C and D have relative position I, then a bend never occurs( ρ ( D i ) is always on the same side of ρ ( E i ) as C r ). If C and D have relativeposition II, then ρ ( D i ) is on the same side of ρ ( E i ) as C r for i = − , , . . . , a ,as shown in Figure 4. However ρ ( D a +2 j − ) (1 ≤ j, a + 2 j − ≤ s ) is on the sideof ρ ( E a +2 j − ) opposite from C r , so a bend can occur at the edge E a +2 j − . Onecan show that the condition that a bend occur at the edge between D a +2 j and D a +2 j − is D a +2 j ⊂ cl ( D a +2 j − , C j ) for 1 ≤ j < j , and D a +2 j * cl ( D a +2 j − , C j )or, equivalently, C j ⊂ cl ( D a +2 j , C j − ) for 1 ≤ j < j , and C j * cl ( D a +2 j , C j − ) . If C and D have relative position III, then the condition that a bend occur atthe edge between D j and D j − is D j ⊂ cl ( D j − , C a +2 j ) for 1 ≤ j < j , and D j * cl ( D j − , C a +2 j )or, equivalently, C a +2 j ⊂ cl ( D j , C a +2 j − ) for 1 ≤ j < j , and C a +2 j * cl ( D j , C a +2 j − ) . For more details on these conditions, see Case II and Case III in [15, Section 2]. C r C D − D D a D a +1 D s Figure 4: Computation for relative position II11 C a C r D − D D s Figure 5: Computation for relative position IIIThe retraction of ( D , . . . , D s ) onto A allows us to compute inv ′ ( P , P ).The next proposition is the summary of this computation (cf. [15, LEMMA2.1]). Proposition 3.4.
Let P , P , P belong to B ∞ with P = P and P = P .Let [ e , e , e ] = inv ′ ( P , P ) and let [ f , f , f ] = inv ′ ( P , P ). Let ( C , . . . , C r )(resp. ( D , . . . , D s )) be a special gallery connecting P to P (resp. P to P ).Assume that C and D have relative position I, II or III.(i) If C and D have relative position I, theninv ′ ( P , P ) = [ f − e , f − e , f − e ] . (ii) Let m = min { e − e , f − f } . If C and D have relative position II,then inv ′ ( P , P ) = [ f − e , f − e − j, f − e + j ], where j = m if nobend occurs, and j = j if a bend occurs at the edge between D a +2 j − and D a +2 j (Figure 4). The possible values for j are 1 , , . . . , m − m = min { f − f , e − e } . If C and D have relative position III,then inv ′ ( P , P ) = [ f − e − j, f − e + j, f − e ], where j = m if nobend occurs, and j = j if a bend occurs at the edge between D j − and D j (Figure 5). The possible values for j are 1 , , . . . , m − Remark 3.5.
In the cases (ii) and (iii) of Proposition 3.4, it may happen that m = 0. If it occurs, then cl ( P , P ) or cl ( P , P ) is a line segment and we cantake a pair ( C , D ) such that C and D have relative position I. However, thisdoes not contradict the results in Proposition 3.4 because m = 0 implies that j is always equal to 0. In the following, we will consider the set X λ ( b )(¯ k ) ∩ η − (0). Note that we have X λ ( b )(¯ k ) ∩ η − (0) = { xK S ∈ SL ( L ) /K S | x − bσ ( x ) ∈ Kt λ K } .
12n terms of buildings, this is equal to the set { P | P is a vertex of type 0 in B ∞ with inv ′ ( P, bσP ) = [ λ ] } . SL In this subsection, We will study the affine Deligne-Lusztig varieties for SL and b = 1.Let Q be a vertex in B , and let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + . We denote by G r [ µ ] ( Q ) the non-empty subset { P | P is a vertex in B ∞ with inv ′ ( Q, P ) = [ µ ] } . Clearly, all vertices in G r [ µ ] ( Q ) have the same type. Let i ∈ { , , } be the typeof vertices in this set. Then, by choosing the i -special representative, we alwayssee G r [ µ ] ( Q ) as a subset of L att i (¯ k ). Define G B [ µ ] ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) there exists a minimal gallery from Q to P containing no vertices in B except Q (cid:27) . Let λ = ( m , m , m ) ∈ X ∗ ( T S ) + such that X Sλ (1) is non-empty (for the explicitcriterion, see Remark 6.4). Set M λ (1) = { [ µ ] ∈ X ∗ ( T ) ′ + | X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = ∅ for some vertex Q in B } . Lemma 4.1.
Let notation be as above. We have M λ (1) = { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m } ( m = 0) { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m , m µ ≥ − m } ( m < { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m , m µ ≥ m } ( m > , where m µ = min { e − e , e − e } . In particular, M λ (1) is a finite set. Proof.
Let P ∈ G B [ µ ] ( Q ) and let ( C , . . . , C r ) be a special gallery connecting Q to P , where [ µ ] = [ e , e , e ] ∈ M λ (1). Then it follows that λ is one of the forms(i) ( e − e , , e − e )(ii) ( e − e , − j, e − e + j ) , min { , m µ } ≤ j ≤ m µ (iii) ( e − e − j, j, e − e ) , min { , m µ } ≤ j ≤ m µ ,where m µ = min { e − e , e − e } (and note that m µ = 0 implies j = 0). Indeed,using Proposition 3.4, we can compute inv ′ ( P, σP ) by connecting ( C , . . . , C r )and ( σC , . . . , σC r ). Then, by Lemma 2.6, λ = ( m , m , m ) is the representa-tive of inv ′ ( P, σP ) with m + m + m = 0.Conversely, for any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + with e > e > e and λ = ( e − e , − j , e − e + j ) ∈ X ∗ ( T S ) + with 1 ≤ j < m µ (resp. λ =( e − e , − m µ , e − e + m µ ) ∈ X ∗ ( T S ) + ), there exists P ∈ G B [ µ ] ( Q ) belonging13o X Sλ (1)(¯ k ). If m µ = 1, then this is obvious. So we may assume m µ ≥
2. Tocheck the assertion, we use the condition that a bend occur (Section 3). Let a = 2( e − e − λ is of the form ( e − e , − j , e − e + j )(resp. λ = ( e − e , − m µ , e − e + m µ )) if and only if σC a +2 j ⊂ cl ( σC a +2 j − , C j ) for 1 ≤ j < j , and σC a +2 j * cl ( σC a +2 j − , C j )(resp. σC a +2 i ⊂ cl ( σC a +2 i − , C i ) for 1 ≤ i < m µ ).So, for any j with 1 ≤ j < j , σC a +2 j (hence C a +2 j ) is the unique cham-ber determined by the gallery ( C , . . . , C j , . . . , C a +2 j − ), and σC a +2 j (hence C a +2 j ) is a chamber distinct from the one uniquely determined by the gallery( C , . . . , C j , . . . , C a +2 j − ) (such C a +2 j exists because the Bruhat-Tits build-ing of SL ( L ) is actually a thick building). Similarly, for any i with 1 ≤ i Let Q be a vertex in B . If X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) is non-empty,then we have X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = X Sλ (1)(¯ k ) ∩ G r [ µ ] ( Q ) = X Sλ (1)(¯ k ) ∩ G r [ µ ] ( Q ) , where G r [ µ ] ( Q ) denotes the closure of G r [ µ ] ( Q ) in X S . In particular, X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) is closed in X λ (1)(¯ k ). Proof. It is enough to show X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = X Sλ (1)(¯ k ) ∩ G r [ µ ] ( Q ). Notethat we have G r [ µ ] ( Q ) = [ [ µ ′ ] (cid:22) [ µ ] G r [ µ ′ ] ( Q ) . Set µ = ( e , e , e ) , µ ′ = ( e ′ , e ′ , e ′ ) with [ µ ′ ] (cid:22) [ µ ]. Let µ ′′ = ( e ′′ , e ′′ , e ′′ ) be adominant cocharacter satisfying µ ′ − µ ′′ ∈ X ∗ ( T ) + . Then it suffices to showthat X Sλ (1)(¯ k ) ∩ G B [ µ ′′ ] ( Q ) is the empty set unless [ µ ′′ ] = [ µ ]. Indeed, for any P ∈ G r [ µ ] ( Q ), there exists such µ ′′ with P ∈ G B [ µ ′′ ] ( Q ).We may assume that µ ′ (cid:22) µ . We have e ′′ − e ′′ ≤ e ′ − e ′ ≤ e − e . Theformula in the proof of Lemma 4.1 shows that if P ∈ G B [ µ ′′ ] ( Q ) is contained in X Sλ (1)(¯ k ), then e ′′ − e ′′ = e − e , and this equation implies [ µ ′′ ] = [ µ ′ ] = [ µ ].So X Sλ (1)(¯ k ) ∩ G B [ µ ′′ ] ( Q ) = ∅ if and only if [ µ ′′ ] = [ µ ]. The last assertion followsfrom Proposition 2.7. 14or any λ ∈ X ∗ ( T S ) + such that X Sλ (1) is non-empty, set P λ (1) = { ( Q, [ µ ]) | Q is a vertex in B with X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = ∅} . Then for any [ µ ] = [ e , e , e ] ∈ M λ (1), a tuple ( Q, [ µ ]) is contained in P λ (1) ifand only if Q is a vertex of type − ( e + e + e ) ∈ Z / B . Indeed, the actionof SL ( F ) on all vertices with the same type in B is transitive. Proposition 4.3. For any λ ∈ X ∗ ( T S ) + such that X Sλ (1) is non-empty, wehave a decomposition X Sλ (1)(¯ k ) = [ ( Q, [ µ ]) ∈P λ (1) ( X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q )) . Proof. For any P ∈ X Sλ (1)(¯ k ), we have a minimal gallery to B . Let Q be avertex in B such that the distance between P and Q is minimal. Then P iscontained in X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ), where [ µ ] = inv ′ ( Q, P ). So we obtain thedecomposition in the proposition. We keep the notation above. Set b = t , b = t 00 0 t . Then the newton vector of b (resp. b ) is ( , , ) (resp. ( , , )).Let C M be the main chamber consisting of three vertices [ O ⊕ O ⊕ O ] , [ t O ⊕O ⊕ O ] , [ t O ⊕ t O ⊕ O ], and let Q be a vertex in C M . Let λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b i ) = ∅ ( i = 1 , 2) (for the explicit criterion, see Remark 6.4).Set M λ ( b i ) = (cid:26) [ µ ] ∈ X ∗ ( T ) ′ + (cid:12)(cid:12)(cid:12)(cid:12) ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) = ∅ for some vertex Q in C M (cid:27) where G C M [ µ ] ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) there exists a minimal gallery from Q to P containing no vertices in C M except Q (cid:27) . Lemma 4.4. Let notation be as above. We have M λ ( b ) = { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m − } ( m = 0) { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m − , m µ, I ≥ − m } ( m < { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m , max { m µ, II , m µ, III } ≥ m } ( m > , λ ( b ) = { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m − } ( m = 1) { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m − , max { m µ, II , m µ, III } ≥ − m + 1 } ( m < { [ e , e , e ] ∈ X ∗ ( T ) ′ + | e − e = m , m µ, I ≥ m − } ( m > , where m µ, I = min { e − e , e − e } , m µ, II = min { e − e + 1 , e − e } , m µ, III =min { e − e , e − e + 1 } . In particular, M λ ( b ) and M λ ( b ) are finite sets. Proof. Set [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + . Let P ∈ G C M [ µ ] ( Q ) and let ( C , . . . , C r ) bea special gallery connecting Q to P . Let us first consider the case for b . Giventhe relative position of C M and C , we can compute inv ′ ( b Q, P ) , inv ′ ( Q, b σP )and inv ′ ( P, b σP ) using Proposition 3.4.If C M and C have relative position III, then inv ′ ( b Q, P ) = [ e , e , e − b Q to P whose first chamberis C M . To check this, note that one can take an apartment containing C M and cl ( Q, P ) (e.g., an apartment containing C M and P ). Since C M and C haverelative position III, C M = b σC M and b σC also have relative position III. So,by connecting ( C M , . . . , C , . . . , C r ) and ( b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 − j, j, e − e ] , min { , m µ, III } ≤ j ≤ m µ, III , where m µ, III = min { e − e , e − e + 1 } .If C M and C have relative position II, then inv ′ ( Q, b σP ) = [ e + 1 , e , e ]because C M and b σC also have relative position II. In this case, we can takea special gallery from Q to b σP whose first chamber is C M . Since C M and C have relative position II, C and C M have relative position III. So, by connecting( C , . . . , C r ) and ( C M , . . . , b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 − j, j, e − e ] , min { , m µ, II } ≤ j ≤ m µ, II , where m µ, II = min { e − e + 1 , e − e } .If C M and C have relative position I, then inv ′ ( Q, b σP ) = [ e + 1 , e , e ]because C M and b σC also have relative position I. Let D = C M be the uniquechamber in cl ( C M , C ) containing Q and b Q . Then we can take a special galleryfrom Q to b σP whose first chamber is b σD . b Q QC M DC Qb QC M b σDb σC Using Lemma 3.2 and Lemma 3.3, one can show that C and b σD have relativeposition I or II (and both of the two cases actually occur). If C and b σD have16elative position I, then by connecting ( C , . . . , C r ) and ( b σD, . . . , b σC , . . . , b σC r ),we have inv ′ ( P, b σP ) = [ e − e + 1 , , e − e ] . If C and b σD have relative position II, then by connecting ( C , . . . , C r ) and( b σD, . . . , b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 , − j, e − e + j ] , min { , m µ, I } ≤ j ≤ m µ, I , where m µ, I = min { e − e , e − e } .Next, we will consider the case for b . In the same way as the case for b ,we can compute inv ′ ( P, b σP ). We will state only the results but will not givedetails of the proofs. If C M and C have relative position III, then by connecting( C , . . . , C r ) and ( C M , . . . , b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 , − j, e − e + j ] , min { , m µ, III } ≤ j ≤ m µ, III . If C M and C have relative position II, then by connecting ( C M , . . . , C , . . . , C r )and ( b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 , − j, e − e + j ] , min { , m µ, II } ≤ j ≤ m µ, II . If C M and C have relative position I, then let D ′ = C M be the unique chamberin cl ( C M , C ) containing Q and b Q . We can always take a special gallery from Q to b σP whose first chamber is b σD ′ . Moreover, C and b σD ′ have relativeposition I or III. So if C and b σD ′ have relative position I, then by connecting( C , . . . , C r ) and ( b σD ′ , . . . , b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 , , e − e ] , and if C and b σD ′ have relative position III, then by connecting ( C , . . . , C r )and ( b σD ′ , . . . , b σC , . . . , b σC r ), we haveinv ′ ( P, b σP ) = [ e − e + 1 − j, j, e − e ] , min { , m µ, I } ≤ j ≤ m µ, I . Finally, the result follows from these formulas (compare the proof of Lemma4.1). Lemma 4.5. Let Q be a vertex in C M . If ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q )( i = 1 , 2) is non-empty, then we have( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) = ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G r [ µ ] ( Q )= ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G r [ µ ] ( Q ) , where G r [ µ ] ( Q ) denotes the closure of G r [ µ ] ( Q ) in X S . In particular, ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) is closed in X λ ( b i )(¯ k ). Proof. This lemma follows from the computation in Lemma 4.4 (see the proofof Lemma 4.2). 17or any λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b i ) = ∅ ( i = 1 , P λ ( b i ) = (cid:26) ( Q, [ µ ]) (cid:12)(cid:12)(cid:12)(cid:12) Q is a vertex in C M with( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) = ∅ (cid:27) . Then for any [ µ ] = [ e , e , e ] ∈ M λ ( b i ), a tuple ( Q, [ µ ]) is contained in P λ ( b i )if and only if Q is a vertex of type − ( e + e + e ) ∈ Z / C M . Indeed, C M has the only one vertex of type i ∈ Z / Proposition 4.6. For any λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b i ) = ∅ ( i =1 , X λ ( b i )(¯ k ) ∩ η − (0) = [ ( Q, [ µ ]) ∈P λ ( b i ) (( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q )) . Proof. For any P ∈ X λ ( b i )(¯ k ) ∩ η − (0), we have a minimal gallery to C M . Let Q be a vertex in C M such that the distance between P and Q is minimal. Then P is contained in ( X λ ( b i )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ), where [ µ ] = inv ′ ( Q, P ). Sowe obtain the decomposition in the proposition. Remark 4.7. Although the set P λ (1) is always infinite, the sets P λ ( b ) and P λ ( b ) are finite. The Schubert cell Kt λ K/K is locally closed in X for any λ ∈ X ∗ ( T ) + , so itinherits the structure of a reduced sub-ind-scheme of X . We denote it by G r λ . Proposition 5.1. Let λ = ( m , m , m ) , µ = ( m ′ , m ′ , m ′ ) ∈ X ∗ ( T ) + .(i) The Schubert cell G r λ forms an L + SL -orbit and is a smooth quasi-projectivevariety.(ii) We have a canonical projection G r λ → G r µ if λ − µ ∈ X ∗ ( T ) + . Proof. Let r = m + m + m . First note that G r λ is actually a locally closedsubscheme of L att r, ( N ) for some sufficiently large N , and the left action on theprojective variety L att r, ( N ) of the group scheme L + SL actually acts throughits finite dimensional quotient SL (¯ k [ t ] / ( t N )), which is (formally) smooth.(i) The stabilizer of t λ for the action of L + SL is L + SL ∩ t λ L + SL t − λ . Theinduced map L + SL / ( L + SL ∩ t λ L + SL t − λ ) → X, g gt λ then is a locally closed embedding. Since G r λ (¯ k ) = Kt λ K/K = K S t λ K/K , theimage is exactly G r λ . 18ii) Let us show that L + SL ∩ t λ L + SL t − λ is contained in L + SL ∩ t µ L + SL t − µ if and only if λ − µ ∈ X ∗ ( T ) + . Then G r λ → G r µ is the canonical quotient map L + SL / ( L + SL ∩ t λ L + SL t − λ ) → L + SL / ( L + SL ∩ t µ L + SL t − µ ) . Let R be a k -algebra. Then we haveSL ( R [[ t ]]) ∩ t λ SL ( R [[ t ]]) t − λ = { ( a ij ) ∈ SL ( R [[ t ]]) | ∀ i < j, a ij ∈ ( t m i − m j ) } . Thus SL ( R [[ t ]]) ∩ t λ SL ( R [[ t ]]) t − λ ⊆ SL ( R [[ t ]]) ∩ t µ SL ( R [[ t ]]) t − µ is equivalentto saying that m i − m j ≥ m ′ i − m ′ j for all i < j , i.e., λ − µ = ( m − m ′ , m − m ′ , m − m ′ ) ∈ X ∗ ( T ) + .Let us denote by U the unipotent radical of B . Let λ = ( m , m , m ) ∈ X ∗ ( T ) + and let J λ be a k -space defined as J λ ( R ) = { ( a ij ) ∈ U ( R [ t ]) | ∀ i < j, deg a ij ≤ m i − m j − } . Then by definition, we have J λ + M = J λ , where M = ( m, m, m ) ∈ X ∗ ( T ) + . Forany α ∈ Φ and k ∈ Z , we denote by U α,k the image of the homomorphism G a → L GL defined by x U α ( t k x ), where U α is the root subgroup. Multiplicationdefines an isomorphism Y α ∈ Φ + , h α,λ i > h α,λ i− Y k =0 U α,k → J λ . In particular, J λ is isomorphic to the affine space of dimension 2 h ρ, λ i , where ρ is half the sum of the positive roots. We will often write U ij,k instead of U χ ij ,k . Lemma 5.2. The morphism J λ → G r λ defined by g gt λ is an open immer-sion. Moreover, G r λ is irreducible and of dimension 2 h ρ, λ i . Proof. See [17, Lemme 2.2].From now on, we see J λ as an open subscheme of G r λ by this open immersion.By Proposition 5.1, there exists a canonical projection G r λ → G r (1 , , (resp. G r λ → G r (0 , , − ) if m > m (resp. m > m ). To shorten notation we set G r = G r (1 , , , G r − = G r (0 , , − , J = J (1 , , , J − = J (0 , , − . Let Flag bethe reduced closed subscheme of G r × G r − defined as Flag(¯ k ) = { ( L , L ′ ) ∈G r (¯ k ) × G r − (¯ k ) | L ⊃ t L ′ } . Then Flag can be covered by open subsetsisomorphic to the 3-dimensional affine space. In particular, we have an isomor-phism U , × U , × U , ∼ = ( J × J − ) ∩ Flag . Let E, S be ¯ k -schemes. Then a morphism p : E → S is an affine bundle ofrank n over S if S has an open covering by U i , and there are isomorphisms p − ( U i ) ∼ = U i × A n such that p restricted to p − ( U i ) corresponds to the projection from U i × A n to U i . 19 emma 5.3. Let λ = ( m , m , m ) ∈ X ∗ ( T ) + .(i) If m > m > m , then the canonical projection G r λ → G r × G r − factors through Flag. Then ϕ λ : G r λ → Flag is an affine bundle of rank2( m − m − − 1. In particular, we have an isomorphism ϕ − λ (( J × J − ) ∩ Flag) = J λ ∼ = (( J × J − ) ∩ Flag) × A m − m − − such that ϕ λ restricted to ϕ − λ (( J × J − ) ∩ Flag) corresponds to theprojection from (( J × J − ) ∩ Flag) × A m − m − − to ( J × J − ) ∩ Flag.(ii) If m > m = m , then the canonical projection ϕ λ : G r λ → G r ∼ = P is an affine bundle of rank 2( m − m − ϕ − λ ( J ) = J λ ∼ = J × A m − m − such that ϕ λ restricted to ϕ − λ ( J ) corresponds to the projection from J × A m − m − to J .(iii) If m = m > m , then the canonical projection ϕ λ : G r λ → G r − ∼ = P is an affine bundle of rank 2( m − m − ϕ − λ ( J − ) = J λ ∼ = J − × A m − m − such that ϕ λ restricted to ϕ − λ ( J − ) corresponds to the projection from J − × A m − m − to J − . Proof. Since G r λ + M ∼ = G r λ , where M = ( m, m, m ), we may assume m = 0.Then let us write φ m ,m : G r ( m +1 , ,m ) → G r ( m , ,m ) ( m > ,ψ m ,m : G r ( m , ,m − → G r ( m , ,m ) ( m < J λ = ϕ − λ (( J × J − ) ∩ Flag) (2) J λ = ϕ − λ ( J ) (3) J λ = ϕ − λ ( J − ) corresponding to the equation in(i), (ii) and (iii) respectively. For this, it suffices to show J (1 , , − = ϕ − , , − (( J × J − ) ∩ Flag) ,J ( m +1 , ,m ) = φ − m ,m ( J λ ) , J ( m , ,m − = ψ − m ,m ( J λ )where λ = ( m , , m ). Indeed, any ϕ λ can be obtained as the composite ofsome φ m ′ ,m ′ , ψ m ′ ,m ′ , and ϕ (1 , , − .Obviously, we have φ m ,m ( J ( m +1 , ,m ) ) ⊆ J λ . To see J ( m +1 , ,m ) = φ − m ,m ( J λ ), assume that φ m ,m (¯ g ) ∈ J λ (¯ k ) for a matrix g ∈ SL ( O ). Thisis equivalent to saying that there exist matrices v ∈ J λ (¯ k ) and t λ at − λ ∈ SL ( O ) ∩ t λ SL ( O ) t − λ such that g = vt λ at − λ . g , there exists a matrix u = u t m u t m − m ∈ U ,m × U ,m − m , u , u ∈ ¯ k satisfying u − v − g = u − t λ at − λ ∈ SL ( O ) ∩ t ( m +1 , ,m ) SL ( O ) t ( − m − , , − m ) . Let a = ( a ij ) = ( a ij ( t )). Then this condition holds if and only if( u − t λ at − λ ) = t m ( a − u a − u a ) ∈ ( t m +1 ) and( u − t λ at − λ ) = t m − m ( a − u a − u a ) ∈ ( t m − m +1 ) . Since t λ at − λ ∈ SL ( O ) ∩ t λ SL ( O ) t − λ and m > a and a are not units. Sothe cofactor expansion of a along the first column implies that the determinantof the matrix A ( t ) = (cid:18) a ( t ) a ( t ) a ( t ) a ( t ) (cid:19) is a unit. Equivalently we have det A (0) = 0, and then we can find a solution( u , u ) of the conditions a − u a − u a , a − u a − u a ∈ ( t ),i.e., A (0) (cid:18) u u (cid:19) = (cid:18) a (0) a (0) (cid:19) . Thus we get J ( m +1 , ,m ) = φ − m ,m ( J λ ) by what we have just proven. Theproof for J ( m , ,m − = ψ − m ,m ( J λ )is similar ( m < a (0) = 0, and then we may find the solutions u , u ).Next we must show J (1 , , − = ϕ − , , − (( J × J − ) ∩ Flag). Assume that amatrix g ∈ SL ( O ) satisfies ϕ (1 , , − (¯ g ) ∈ J (¯ k ) × J − (¯ k ). Let λ = (1 , , , λ − =(0 , , − ϕ (1 , , − ( g ) ∈ J (¯ k ) × J − (¯ k ) is equivalent to saying that thereexist matrices v i ∈ J i (¯ k ) , t λ i a λ i t λ i ∈ SL ( O ) ∩ t λ i SL ( O ) t − λ i ( i = ± 1) suchthat g = v t λ a λ t − λ = v − t λ − a λ − t − λ − . Let us write v = z z , v − = z z , z , z , z , z ∈ ¯ k. Further, set a λ − = ( a ij ) = ( a ij ( t )) and u = z a (0) − ( a (0) − z a (0)) t . a (0) = 0 and a (0) = 0 because t λ − a λ − t λ − ∈ SL ( O ) ∩ t λ − SL ( O ) t − λ − . Then ( v ) − v − = t λ a λ t − λ ( t λ − a λ − t − λ − ) − yields z = z − z z (resp. a (0) − z a (0) = 0) by comparing the (1,3) (resp. (1,2)) entryof the matrices. In particular, we have ϕ (1 , , − ( J (1 , , − ) ⊆ ( J × J − ) ∩ Flag.Moreover, using these equations, one can check that u − ( v − ) − g ∈ SL ( O ) ∩ t (1 , , − SL ( O ) t ( − , , . Thus we get J (1 , , − = ϕ − , , − (( J × J − ) ∩ Flag).We can cover Flag (resp. G r , resp. G r − ) by suitable open subvarietiesisomorphic to U = ( J × J − ) ∩ Flag (resp. J , resp. J − ). Indeed, there existsa finite set { g i } i with g i ∈ SL (¯ k ) such that { g i U } i is an open covering (forexample, we can take the set of standard representatives of the finite Weyl groupof SL ). Since each J λ is a product of root subgroups, one easily verifies thatsuch an open covering defines the structure of an affine bundle, and its relativedimension is 2( m − m − − m − m − m − m − Remark 5.4. For λ ∈ X ∗ ( T ) + , there is a natural projection G r λ → SL n /P λ induced by L + SL n → SL n , t 0, where P λ is the stabilizer of t λ in SL n . Thisprojection is actually an affine bundle, and the lemma is the special case of thisfact (see for example [16, Section 2] that treats the affine Grassmannian over C ). Note that the set G r [ λ ] ( Q ) (Section 4.1) can be identified with the Schubertcell G r λ (¯ k ) for any vertex Q in B and λ ∈ X ∗ ( T ) + . So we can see this set as avariety, and denote it also by G r [ λ ] ( Q ) (of course, this definition is independent ofthe choice of λ ). Let µ ∈ X ∗ ( T ) + with λ − µ ∈ X ∗ ( T ) + . Via this identification,the projection (Proposition 5.1) ϕ λ,µ : G r λ (¯ k ) → G r µ (¯ k )can also be described as follows. Let P ∈ G r [ λ ] ( Q ). Then there exists a uniquevertex in G r [ µ ] ( Q ) ∩ cl ( Q, P ). So we have a map G r [ λ ] ( Q ) → G r [ µ ] ( Q ) , which sends P to the unique vertex in G r [ µ ] ( Q ) ∩ cl ( Q, P ). This map correspondsto ϕ λ,µ through the identification. For example, if Q = [Λ ¯ k ] and P = [ gt λ Λ ¯ k ]with g ∈ SL ( O ), then the unique vertex in G r [ µ ] ([Λ ¯ k ]) ∩ cl ([Λ ¯ k ] , P ) is [ gt µ Λ ¯ k ].Indeed, cl ([Λ ¯ k ] , [ t λ Λ ¯ k ]) is isomorphic to cl ([Λ ¯ k ] , [ gt λ Λ ¯ k ]) by multiplication with g . In particular, ϕ λ (Lemma 5.3) can be seen as a morphism mapping P ∈G r [ λ ] ( Q ) to the first alcove or vertex of cl ( Q, P ).22 .2 Subvarieties of the Schubert Cells We can identify Flag(¯ k ) with the set of chambers containing [Λ ¯ k ]. Then wedefine a locally closed subvariety X u of Flag by X u (¯ k ) = { C ∈ Flag(¯ k ) | C and σC have relative position u } , where u = I, II or III. Let W be the finite Weyl group of GL . Obviously, if u =I (resp. II, resp. III), then X u is the classical Deligne-Lusztig variety associatedwith the maximal length (resp. a Coxeter, resp. a Coxeter) element in W . Inparticular, if u = II or III, then X u can be identified with the Drinfeld upperhalf space (of dimension 2) Ω = P \ [ H ∈H H, where H is the set of k -rational hyperplanes in P .Next we introduce subsets of G r [ µ ] ( Q ) for each vertex Q in B (Section 4).For any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + , set G r I[ µ ] ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) there exists a first chamber C of cl ( Q, P )such that C and σC have relative position I (cid:27) . In case e = e or e = e , then P ∈ G r [ µ ] ( Q ) belongs to G r I[ µ ] ( Q ) if and only ifthe first “vertex” of cl ( Q, P ) is not contained in B . Indeed, let P be the firstvertex of cl ( Q, P ) which is not contained in B . Then chambers C = { Q, P , P } and σC have relative position II or III if and only if P or σP ∈ cl ( P , σP ).So if we take P such that P , σP / ∈ cl ( P , σP ), then C and σC have relativeposition I. Further, for any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + with e > e > e , set G r u [ µ ] ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C and σC have relative position u , where C is the unique first chamber of cl ( Q, P ) (cid:27) , where u = II or III.For any P ∈ G r u [ µ ] ( Q ), let ( C , . . . , C r ) be the special gallery connecting Q to P , and let 1 ≤ j < m µ , m µ = min { e − e , e − e } . Let a = 2( e − e − u = II, then we define the sets G r II ,j [ µ ] ( Q ) = (cid:26) P ∈ G r II[ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) σC a +2 i ⊂ cl ( C i , σC a +2 i − ) for 1 ≤ i < j , and σC a +2 j * cl ( C j , σC a +2 j − ) (cid:27) and G r II ,m µ [ µ ] ( Q ) = { P ∈ G r II[ µ ] ( Q ) | σC a +2 i ⊂ cl ( C i , σC a +2 i − ) for 1 ≤ i < m µ } . If u = III, then we define the sets G r III ,j [ µ ] ( Q ) = (cid:26) P ∈ G r III[ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C a +2 i ⊂ cl ( C a +2 i − , σC i ) for 1 ≤ i < j , and C a +2 j * cl ( C a +2 j − , σC j ) (cid:27) G r III ,m µ [ µ ] ( Q ) = { P ∈ G r III[ µ ] ( Q ) | C a +2 i ⊂ cl ( C a +2 i − , σC i ) for 1 ≤ i < m µ } . Recall that G r [ µ ] ( Q ) can be identified with the Schubert cell G r µ for anyvertex Q in B and µ ∈ X ∗ ( T ) + . Moreover, the subsets of G r [ µ ] ( Q ) definedabove can be seen as locally closed reduced subvarieties of the variety G r [ µ ] ( Q ),and are denoted by the same symbols (see the next proposition). Proposition 5.5. Let notation be as above.(i) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + . If e = e > e or e > e = e (resp. e > e > e ), then the ¯ k -variety G r I[ µ ] ( Q ) is an affine bundle of rank2( e − e ) − e − e ) − 3) over P \ P ( k ) (resp. X I ).(ii) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + with e > e > e . Then the ¯ k -variety G r II[ µ ] ( Q ) (resp. G r II ,j [ µ ] ( Q ), resp. G r II ,m µ [ µ ] ( Q )) is contained in J µ via theidentification above, and isomorphic to Ω × A e − e ) − (resp. Ω × G m × A e − e ) − j − , resp. Ω × A e − e ) − m µ − ).(iii) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + with e > e > e . Then the ¯ k -variety G r III[ µ ] ( Q ) (resp. G r III ,j [ µ ] ( Q ), resp. G r III ,m µ [ µ ] ( Q )) is contained in J µ via theidentification above, and isomorphic to Ω × A e − e ) − (resp. Ω × G m × A e − e ) − j − , resp. Ω × A e − e ) − m µ − ). Proof. It suffices to prove the case for Q = [Λ ¯ k ]. We omit [Λ ¯ k ] from the notation(for instance, G r I[ µ ] = G r I[ µ ] ([Λ ¯ k ])). Further, we set µ = ( e , e , e ) ∈ X ∗ ( T ) + .As explained in the last part of Section 5.1, for any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + with e > e > e (resp. e = e > e or e > e = e ), we have the affinebundle ϕ µ : G r [ µ ] → Flag (resp. ϕ µ : G r [ µ ] → P ) . So (i) follows immediately from the facts explained before the proposition. Letus prove (ii). The proof for (iii) is similar.First note that X II ⊂ Flag is actually contained in ( J × J − ) ∩ Flag. Indeed,for any C ∈ X II (¯ k ) and the vertex P i ( i = 1 , 2) of type i in C , the only vertex in B contained in cl ( P i , σP i ) is [Λ ¯ k ]. In particular, both of the sets { [Λ ¯ k ] , P , [ O ⊕ t O ⊕ t O ] } and { [Λ ¯ k ] , P , [ O ⊕ O ⊕ t O ] } are not chambers, so P ∈ J (¯ k ) and P ∈ J − (¯ k ). [Λ ¯ k ] P σP σP P C σC G r II[ µ ] ⊂ J µ and G r II[ µ ] = ϕ − µ ( X II ) ∼ = X II × A e − e ) − ∼ = Ω × A e − e ) − as a locally closed subvariety of G r [ µ ] .Next, we show the statement for G r II ,m µ [ µ ] by induction on m µ = min { e − e , e − e } . If m µ = 1, then G r II ,m µ [ µ ] = G r II[ µ ] and the statement follows from theisomorphism above. Let us suppose that m µ ≥ G r II ,m ν [ ν ] with m ν < m µ . If e − e = m µ , we have a canonical projection G r II[ µ ] → G r II[ µ ′ ] , where [ µ ′ ] = [2 e − e , e , e ]. Then the inverse image of G r II ,m µ ′ [ µ ′ ] by this projection is G r II ,m µ [ µ ] . To check this, let P ∈ G r II[ µ ] and let P be theunique vertex in G r II[ µ ′ ] which is also contained in cl ([Λ ¯ k ] , P ). Let ( C , . . . , C r )(resp. ( D , . . . , D s )) be the special gallery connecting [Λ ¯ k ] to P (resp. P ), andlet a = 2( e − e − 1) (resp. a = 2( e − e − cl ([Λ ¯ k ] , P ), and let ρ (resp. ρ ) be the retraction of B ∞ onto this apartmentwith center C r (resp. D s ). C C i C a D a C r D s σC σC a σC a +2 i σD a +2 i σD a σC r σD s P P σP σP Recall that C a +2 i (resp. D a +2 i ) satisfies σC a +2 i ⊂ cl ( C i , σC a +2 i − ) for 1 ≤ i < m µ ′ (resp. σD a +2 i ⊂ cl ( D i , σD a +2 i − ) for 1 ≤ i < m µ )if and only if ρ ( σC a +2 i ) (resp. ρ ( σD a +2 i )) is on the same side as C r (resp. D s ) of the wall containing ρ ( σC a +2 i ∩ σC a +2 i − ) (resp. ρ ( σD a +2 i ∩ σD a +2 i − ))for 1 ≤ i < m µ ′ (resp. 1 ≤ i < m µ ). Retracting minimal galleries ( C i = D i , . . . , σC a +2 i ) and ( σC a +2 i , . . . , σD a +2 i ) successively by ρ , we can alsocheck that σC a +2 i ⊂ cl ( C i , σC a +2 i − ) for 1 ≤ i < m µ ′ if and only if ρ ( σD a +2 i )is on the same side as D s of the wall containing ρ ( σD a +2 i ∩ σD a +2 i − ) for1 ≤ i < m µ ′ = m µ (see Lemma 3.3 and [15, p. 340]). This implies that the in-verse image of G r II ,m µ ′ [ µ ′ ] by the projection G r II[ µ ] → G r II[ µ ′ ] is G r II ,m µ [ µ ] . Thus, by theproof of Lemma 5.3, this is equivalent to saying that we have an isomorphism G r II ,m µ [ µ ] ∼ = G r II ,m µ ′ [ µ ′ ] × A e + e − e ) as a locally closed subvariety of G r [ µ ] . So it is enough to consider the case e − e = m µ . 25n the sequel, we assume that e − e = m µ . In this case, we have a morphismof ¯ k -spaces G r II ,m µ ′ [ µ ′ ] → X S , which actually factors through G r [ λ ] , where µ ′ = ( e − , e , e ) , λ = ( e − e − , − m µ ′ , e − e + 1 + m µ ′ ). Indeed, we have a morphism of ¯ k -spaces J µ ′ → X S , given on R -valued points by sending j ∈ J µ ′ ( R ) to the lattice t − µ ′ j − σ ( j ) t µ ′ Λ R ∈ X S ( R ). By Proposition 3.4, the composition G r II ,m µ ′ [ µ ′ ] ⊂ J µ ′ → X S actu-ally factors through G r [ λ ] . Moreover, the composition ϕ of the morphism G r II ,m µ ′ [ µ ′ ] → G r [ λ ] and the canonical projection G r [ λ ] → G r [0 , , − ∼ = P is amorphism of varieties factoring through a locally closed immersion A ⊂ P . Tocheck this, let P ∈ G r II ,m µ ′ [ µ ′ ] . Then there exists a unique matrix g ∈ J µ ′ (¯ k )such that P = [ g t µ ′ Λ ¯ k ]. Let ( C , . . . , C r ) be the special gallery connecting[Λ ¯ k ] to P , and let A be an apartment containing C r and σC r . Then the im-age ϕ ( P ) corresponds to a vertex in the apartment t − µ ′ g − A , which contains[ O ⊕ O ⊕ t O ] and differs from [ O ⊕ t O ⊕ t O ]. So ϕ is actually a regular functionon G r II ,m µ ′ [ µ ′ ] . C C a C r D D P ′ σC σC a σC r σDP σPP σP We have a canonical projection φ : G r II[ µ ] → G r II[ µ ′ ] , where [ µ ′ ] = [ e − , e , e ]. Using Lemma 3.3, we can check that φ ( G r II ,m µ [ µ ] ) ⊂G r II ,m µ ′ [ µ ′ ] (cf. [15, p. 340]). Moreover, by the proof of Lemma 5.3, we have φ − ( J µ ′ ) = J µ ∼ = J µ ′ × U ,e − e − × U ,e − e − . This is equivalent to saying that any matrix in J µ (¯ k ) can be written as a productof a matrix in J µ ′ (¯ k ) and a matrix of the form u t e − e − u t e − e − , u , u ∈ ¯ k . Let P = [ gt µ Λ ¯ k ] ∈ G r II[ µ ] with g ∈ J µ (¯ k ), and let D be thefirst chamber cl ( P, [Λ ¯ k ]). Further, we assume that P = [ g t µ ′ Λ ¯ k ] ∈ G r II ,m µ ′ [ µ ′ ] with g ∈ J µ ′ (¯ k ) belongs to cl ([Λ ¯ k ] , P ). Let ( C , . . . , C r ) be the special galleryconnecting [Λ ¯ k ] to P , and let ( C r , D , D ) be the unique minimal gallery. Fixan apartment A containing both ( σC , . . . , σC r ) and σD . Then, using Lemma3.3, we can check that P is contained in G r II ,m µ [ µ ] if and only if ρ A ,σD ( D ) is onthe same side as σD of the wall containing ρ A ,σD ( C r ∩ D ). We can also checkthat this is equivalent to saying that D ⊂ cl ( C r , σC r ). Set D = { P , P, P ′ } .Then we have P ′ = [ g ′ Λ ¯ k ], where g ′ = g t µ ′ t ϕ ( P ) 00 1 00 0 t .This implies easily that P is contained in G r II ,m µ [ µ ] if and only if g can be writtenas g = g t µ ′ t ϕ ( P ) u t − µ = g ϕ ( P ) t e − e − u t e − e − , where g ∈ G r II ,m µ ′ [ µ ′ ] and ϕ ( P ) , u ∈ ¯ k . Thus G r II ,m µ [ µ ] is a closed subvariety of φ − ( G r II ,m µ ′ [ µ ′ ] ) ∼ = G r II ,m µ ′ [ µ ′ ] × U ,e − e − × U ,e − e − defined by the equation u = ϕ . By the induction hypothesis, this is isomorphic to G r II ,m µ ′ [ µ ′ ] × A ∼ = Ω × A e − e ) − m µ ′ − × A ∼ = Ω × A e − e ) − m µ − . Finally, we show the statement for G r II ,j [ µ ] . Set G II ,j [ µ ] = { P ∈ G r II[ µ ] | σC a +2 i ⊂ cl ( C i , σC a +2 i − ) for 1 ≤ i < j } . Then the similar proof as above shows that G II ,j [ µ ] is locally closed and its reducedsubscheme structure is isomorphic to Ω × A e − e ) − j − . Again, the similarargument as above shows that G r II ,j [ µ ] is an open subvariety of G II ,j [ µ ] isomorphicto Ω × G m × A e − e ) − j − , which completes the proof.We keep the notation in the proof of Lemma 4.4, and introduce the subsetsof G r [ µ ] ( Q ) for any vertex Q in C M . For any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + , set G r u [ µ ] ,C M ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) there exists a first chamber C of cl ( Q, P )such that C M and C have relative position u (cid:27) , where u = II or III. In case e = e or e = e , then P ∈ G r [ µ ] ( Q ) belongsto G r u [ µ ] ( Q ) if and only if the first “vertex” of cl ( Q, P ) is not contained in27 M . In the case where C M and C have relative position I, we will define twosubvarieties. For any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e , set G r u [ µ ] ,D ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C and b σD have relative position u , where C is the unique first chamber of cl ( Q, P ) (cid:27) , where u = I or II, and similarly, set G r u [ µ ] ,D ′ ( Q ) = (cid:26) P ∈ G r [ µ ] ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C and b σD ′ have relative position u , where C is the unique first chamber of cl ( Q, P ) (cid:27) , where u = I or III.Let a = 2( e − e − µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e ,we also define the sets G r II ,j [ µ ] ,b ( Q ) = (cid:26) P ∈ G r II[ µ ] ,C M ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C a +2 i ⊂ cl ( b σC i − , C a +2 i − ), 1 ≤ i < j and C a +2 j * cl ( b σC j − , C a +2 j − ) (cid:27) , where 1 ≤ j < m µ, II , and G r II ,j [ µ ] ,b ( Q ) = { P ∈ G r II[ µ ] ,C M ( Q ) | C a +2 i ⊂ cl ( b σC i − , C a +2 i − ), 1 ≤ i < m µ, II } , where j = m µ, II . These conditions appear when we connect the two galleries( C , . . . , C r ) and ( C M , . . . , b σC , . . . , b σC r ). In the case where C M and C have relative position I, we define G r II ,j [ µ ] ,D ( Q ) = (cid:26) P ∈ G r II[ µ ] ,D ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) C i ⊂ cl ( b σC a +2 i , C i − ), 1 ≤ i < j and C j * cl ( bσC a +2 j , C j − ) (cid:27) , where 1 ≤ j < m µ, I , and G r II ,j [ µ ] ,D ( Q ) = { P ∈ G r II[ µ ] ,D ( Q ) | C i ⊂ cl ( b σC a +2 i , C i − ), 1 ≤ i < m µ, I } , where j = m µ, I . These conditions appear when we connect the two galleries( C , . . . , C r ) and ( b σD, . . . , b σC , . . . , b σC r ). Similarly, by writing down theconditions on a bend of galleries appearing in the proof of Lemma 4.4, we define G r II ,j [ µ ] ,b ( Q ) (1 ≤ j ≤ m µ, II ) , G r III ,j [ µ ] ,b ( Q ) , G r III ,j [ µ ] ,b ( Q ) (1 ≤ j ≤ m µ, III ) , G r III ,j [ µ ] ,D ′ ( Q ) (1 ≤ j ≤ m µ, I ) . The subsets of G r [ µ ] ( Q ) defined above can be seen as the locally closed reducedsubvarieties of the variety G r [ µ ] ( Q ), and denoted by the same symbols (see thenext proposition). Proposition 5.6. Let notation be as above. Let b = b or b .(i) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e = e > e . Then the ¯ k -variety G r II[ µ ] ,C M ( Q ) (resp. G r III[ µ ] ,C M ( Q )) is isomorphic to A e − e ) − (resp. A e − e ) ).28ii) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e = e . Then the ¯ k -variety G r II[ µ ] ,C M ( Q ) (resp. G r III[ µ ] ,C M ( Q )) is isomorphic to A e − e ) (resp. A e − e ) − ).(iii) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e . Then the ¯ k -variety G r II[ µ ] ,C M ( Q ) (resp. G r II ,j [ µ ] ,b ( Q ), resp. G r II ,m µ, II [ µ ] ,b ( Q )) is isomorphic to A e − e ) − (resp. G m × A e − e ) − j − , resp. A e − e ) − m µ, II ).(iv) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e . Then the ¯ k -variety G r III[ µ ] ,C M ( Q ) (resp. G r III ,j [ µ ] ,b ( Q ), resp. G r III ,m µ, III [ µ ] ,b ( Q )) is isomorphic to A e − e ) − (resp. G m × A e − e ) − j − , resp. A e − e ) − m µ, III ).(v) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e . Then the ¯ k -variety G r I[ µ ] ,D ( Q ) (resp. G r II[ µ ] ,D ( Q ), resp. G r II ,j [ µ ] ,D ( Q ), resp. G r II ,m µ, I [ µ ] ,D ( Q )) is iso-morphic to G m × A e − e ) − (resp. A e − e ) − , resp. G m × A e − e ) − j − ,resp. A e − e ) − m µ, I ).(vi) Let [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) + with e > e > e . Then the ¯ k -variety G r I[ µ ] ,D ′ ( Q ) (resp. G r III[ µ ] ,D ′ ( Q ), resp. G r III ,j [ µ ] ,D ′ ( Q ), resp. G r III ,m µ, I [ µ ] ,D ′ ( Q )) is iso-morphic to G m × A e − e ) − (resp. A e − e ) − , resp. G m × A e − e ) − j − ,resp. A e − e ) − m µ, I ). Proof. It suffices to prove the case for Q = [Λ ¯ k ]. We omit [Λ ¯ k ] from the notation.Define G = { P ∈ G r [1 , , | { [Λ ¯ k ] , P, [ t O ⊕ t O ⊕ O ] } is not a chamber } and G ′ = (cid:26) P ∈ G r [1 , , (cid:12)(cid:12)(cid:12)(cid:12) P = [ t O ⊕ O ⊕ O ] and { [Λ ¯ k ] , P, [ t O ⊕ t O ⊕ O ] } is a chamber (cid:27) . Then both G and G ′ are locally closed subsets, and we have G ∼ = A , G ′ ∼ = A as reduced ¯ k -varieties. These isomorphisms and the proof of Lemma 5.3 imply(ii), and the proof for (i) is similar.Let C be a chamber containing [Λ ¯ k ] such that C M and C have relativeposition II. Then C is completely determined by the vertex P of type 1 in C .Indeed, the vertex P of type 2 in C belongs to cl ( C M , P ).[Λ ¯ k ] P P C M C G r (¯ k ) × G r − (¯ k )) is isomorphic to G and the proof of Lemma 5.3 implies thestatement for G r II[ µ ] ,C M . Moreover, by an argument similar to Proposition 5.5,we can show the statements for G r II ,j [ µ ] ,b and G r m µ, II [ µ ] ,b . So we obtain (iii), and theproof for (iv) is the same.Finally, we show (v), and (vi) follows similarly. Define F I = (cid:26) C ∈ Flag(¯ k ) (cid:12)(cid:12)(cid:12)(cid:12) C M and C have relative position I and C and b σD have relative position I (cid:27) and F II = (cid:26) C ∈ Flag(¯ k ) (cid:12)(cid:12)(cid:12)(cid:12) C M and C have relative position I and C and b σD have relative position II (cid:27) . Using the techniques explained above, we can show that F I ∼ = G m × A , F II ∼ = A as reduced ¯ k -varieties. These isomorphisms show the assertion in the same wayas above. Let λ ∈ X ∗ ( T ) + with X λ (1) = ∅ . Recall that P λ (1) can be seen as the set (cid:26) ( Q, [ µ ]) (cid:12)(cid:12)(cid:12)(cid:12) [ µ ] = [ e , e , e ] ∈ M λ (1) and Q is a vertex of type − ( e + e + e ) ∈ Z / (cid:27) (Section 4.1). For any such λ , we have explicit description of M λ (1) (Lemma 4.1)and H acts on P λ (1) by left multiplication on the vertex Q . Set K = GL ( O F ),Λ = Λ k and K = t GL ( O F ) t − , Λ = t O ⊕ O ⊕ O ,K = t t 00 0 1 GL ( O F ) t − t − 00 0 1 , Λ = t O ⊕ t O ⊕ O . Let [ µ ] = [ e , e , e ] ∈ M λ (1), and let i ∈ { , , } be the representative of − ( e + e + e ) ∈ Z / 3. Note that these groups are subgroups of H ⊂ J =GL ( F ), and K i is the stabilizer of ([Λ i ] , [ µ ]) with respect to the action of H on P λ (1). Finally, we define K [ µ ] as K i and set m µ = min { e − e , e − e } .30 heorem 6.1. Let notation be as above. Then the irreducible components of X λ (1) are parameterized by the elements in F [ µ ] ∈ M λ (1) J /K [ µ ] , and J acts onthe set of irreducible components by left multiplication on this set. Moreover,their geometric structures of ¯ k -varieties are given as follows:(i) Let λ = ( m , , m ) ∈ X ∗ ( T ) + with X λ (1) = ∅ . For any [ µ ] = [ e , e , e ] ∈ M λ (1) with e = e > e or e > e = e (resp. e > e > e ), theirreducible component corresponding to gK [ µ ] ∈ J /K [ µ ] is an affine bundleof rank m − m − m − m − 3) over P \ P ( k ) (resp. X I ). If[ µ ] = [0 , , ∈ M λ (1) (and hence λ = (0 , , gK [ µ ] ∈ J /K [ µ ] is a point.(ii) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m < X λ (1) = ∅ . For any [ µ ] =[ e , e , e ] ∈ M λ (1) with m µ = − m (resp. m µ > − m ), the irreduciblecomponent corresponding to gK [ µ ] ∈ J /K [ µ ] is isomorphic toΩ × A m − m − (resp. Ω × G m × A m − m − ) . In this case, the irreducible components are pairwise disjoint.(iii) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m > X λ (1) = ∅ . For any[ µ ] = [ e , e , e ] ∈ M λ (1) with m µ = m (resp. m µ > m ), the irreduciblecomponent corresponding to gK [ µ ] ∈ J /K [ µ ] is isomorphic toΩ × A m − m − (resp. Ω × G m × A m − m − ) . In this case, the irreducible components are pairwise disjoint. Proof. Fix λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ (1) = ∅ . The case where λ = (0 , , 0) is well-known. So we may assume λ = (0 , , X Sλ (1)(¯ k ) = [ ( Q, [ µ ]) ∈P λ (1) ( X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ))by Proposition 4.3. We show that for any ( Q, [ µ ]) ∈ P λ (1), the subset X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) is an irreducible component of X λ (1)(¯ k ). By Proposition 3.4, we have X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = G r I[ µ ] ( Q ) ( m = 0) G r II , − m [ µ ] ( Q ) ( m < G r III ,m [ µ ] ( Q ) ( m > . So, by Lemma 4.2 and Proposition 5.5, it is an irreducible closed subset ofdimension m − m , which is also the dimension of X λ (1) (see for example[9, Theorem 4.17]). Thus, using Proposition 2.7, X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) is anirreducible component of X λ (1)(¯ k ). 31ince the action of H on P λ (1) induces H \P λ (1) ∼ = M λ (1), we have X λ (1) ∼ = G J /H [ F [ µ ] ∈ Mλ (1) H /K [ µ ] G r u,j [ µ ] ( Q ) = [ F [ µ ] ∈ Mλ (1) J /K [ µ ] G r u,j [ µ ] ( Q ) , where J acts on the set of the irreducible components by left multiplicationon the index set. Again by Proposition 5.5, the geometric structure of eachirreducible component is given as above.Let V be an irreducible component of X Sλ (1). Then V is quasi-compact (cf.[14, Corollary 6.5]), and hence V (¯ k ) ∩ G B [ µ ] ( Q ) = ∅ for all but finitely many ( Q, [ µ ]) ∈ P λ (1). Indeed, using [9, Lemma 2.4], wecan show that the function V (¯ k ) → N which maps P ∈ V (¯ k ) to the distancefrom P to [Λ ¯ k ] is bounded. Since V is an irreducible component, we have V = X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) for some ( Q, [ µ ]) ∈ P λ (1). On the other hand, by thesimilar argument as below, we can easily show that X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) = X Sλ (1)(¯ k ) ∩ G B [ µ ′ ] ( Q ′ )unless ( Q, [ µ ]) = ( Q ′ , [ µ ′ ]). So all of the irreducible components of X λ (1) areparameterized by the elements in F [ µ ] ∈ M λ (1) J /K [ µ ] .Finally, let us consider the cases (ii) and (iii). Let ( Q, [ µ ]) , ( Q ′ , [ µ ′ ]) ∈ P λ (1),then we have X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ) ∩ G B [ µ ′ ] ( Q ′ ) = ∅ unless ( Q, [ µ ]) = ( Q ′ , [ µ ′ ]). To show this, recall that we can identify Flag(¯ k )with the set of chambers containing [Λ ¯ k ] (Section 5.2). If C is a chamber in X II ∪ X III and if D is a chamber in B containing [Λ ¯ k ], then C and D alwayshave relative position I. This can be checked as in the first part of the proof ofProposition 5.5. Thus, by Proposition 3.4, the vertex Q is the unique nearestone in B to any vertex in X Sλ (1)(¯ k ) ∩ G B [ µ ] ( Q ). This completes the proof. Remark 6.2. In the case (i) of Theorem 6.1, the irreducible components arenot disjoint in general. For example, there exists a vertex P of type 0 in B ∞ suchthat { P, [Λ ] , [Λ ] } is a chamber and P is not a vertex in B . Then P belongsto both G r I[0 , , − ([Λ ]) and G r I[1 , , ([Λ ]), which are irreducible components of X (1 , , − (1). On the other hand, the disjoint decomposition in (ii) and (iii) isan example of J -stratification introduced by Chen and Viehmann in [4].Next, let us consider the superbasic case. In this case, we need some notationin addition to those in Section 4.2. Let λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b i ) = ∅ ( i = 1 , M λ ( b ) ′ = ( M λ ( b ) ( m ≤ { [ µ ] ∈ M λ ( b ) | m µ, II ≥ m } ( m > , M λ ( b ) ′ = ( M λ ( b ) ( m ≥ { [ µ ] ∈ M λ ( b ) | m µ, III ≥ − m + 1 } ( m < , where m µ, I = min { e − e , e − e } , m µ, II = min { e − e + 1 , e − e } , m µ, III =min { e − e , e − e + 1 } for any [ µ ] = [ e , e , e ] ∈ X ∗ ( T ) ′ + . Finally, notethat H b i stabilizes [Λ ] , [Λ ] and [Λ ] under the action on B ∞ . In fact, for any g ∈ H b i , the equation v L (det( g )) = 0 implies that g ∈ O × ( t ) ( t ) O O × ( t ) O O O × . Theorem 6.3. Let notation be as above. Then the irreducible components of X λ ( b i ) ( i = 1 , 2) are parameterized by the elements in ( J b i /H b i ) × M λ ( b i ) ′ , and J b i acts on the set of irreducible components by left multiplication on this set.Moreover, their geometric structures of ¯ k -varieties are given as follows:(i) Let λ = ( m , , m ) ∈ X ∗ ( T ) + with X λ ( b ) = ∅ . For any [ µ ] = [ e , e , e ] ∈ M λ ( b ) with e = e > e or e > e = e (resp. e > e > e ), theirreducible component corresponding to ( gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b )is isomorphic to A m − m − (resp. G m × A m − m − ) . If [ µ ] = [0 , , 0] (and hence λ = (1 , , gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b ) is a point.(ii) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m < X λ ( b ) = ∅ . For any[ µ ] = [ e , e , e ] ∈ M λ ( b ) with m µ, I = − m (resp. m µ, I > − m ), theirreducible component corresponding to ( gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b )is isomorphic to A m − m − (resp. G m × A m − m − ) . (iii) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m > X λ ( b ) = ∅ . For any[ µ ] = [ e , e , e ] ∈ M λ ( b ) ′ with m µ, II = m (resp. m µ, II > m ), the irre-ducible component corresponding to ( gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b ) ′ isisomorphic to A m − m − (resp. G m × A m − m − ) . (iv) Let λ = ( m , , m ) ∈ X ∗ ( T ) + with X λ ( b ) = ∅ . For any [ µ ] = [ e , e , e ] ∈ M λ ( b ) with e = e > e or e > e = e (resp. e > e > e ), theirreducible component corresponding to ( gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b )is isomorphic to A m − m − (resp. G m × A m − m − ) . If [ µ ] = [0 , , 0] (and hence λ = (1 , , gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b ) is a point.33v) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m < X λ ( b ) = ∅ . For any [ µ ] =[ e , e , e ] ∈ M λ ( b ) ′ with m µ, III = − m + 1 (resp. m µ, III > − m + 1), theirreducible component corresponding to ( gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b ) ′ is isomorphic to A m − m − (resp. G m × A m − m − ) . (vi) Let λ = ( m , m , m ) ∈ X ∗ ( T ) + , m > X λ ( b ) = ∅ . For any[ µ ] = [ e , e , e ] ∈ M λ ( b ) with m µ, I = m − m µ, I > m − gH b , [ µ ]) ∈ ( J b /H b ) × M λ ( b )is isomorphic to A m − m − (resp. G m × A m − m − ) . In all cases, the irreducible components are pairwise disjoint. Proof. We prove the case for b , and the proof for b is similar. Let λ =( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b ) = ∅ . The case for λ = (1 , , 0) followsimmediately from the computation in the proof of Lemma 4.4. So we mayassume λ = (1 , , X λ ( b )(¯ k ) ∩ η − (0) = [ ( Q, [ µ ]) ∈P λ ( b ) (( X λ ( b )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ))by Proposition 4.6. It follows from Lemma 4.5 that each ( X λ ( b )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) is closed in X λ ( b )(¯ k ). The computation in the proof of Lemma 4.4shows that for each ( Q, [ µ ]) ∈ P λ ( b ), the subset ( X λ ( b )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q )is equal to G r II[ µ ] ,C M ( Q ), G r III[ µ ] ,C M ( Q ) or G r I[ µ ] ,D ( Q ) ( m = 0) G r II , − m [ µ ] ,D ( Q ) ( m < G r II ,m [ µ ] ,b ( Q ), G r III ,m [ µ ] ,b ( Q ) or G r II ,m [ µ ] ,b ( Q ) ⊔ G r III ,m [ µ ] ,b ( Q ) ( m > G r II[ µ ] ,C M ( Q ) or G r III[ µ ] ,C M ( Q ) appears only in the case where [ µ ] = [ e , e , e ]satisfies e > e = e or e = e > e ). In the case ( X λ ( b )(¯ k ) ∩ η − (0)) ∩ G C M [ µ ] ( Q ) = G r II ,m [ µ ] ,b ( Q ) ⊔ G r III ,m [ µ ] ,b ( Q ), it is easy to check that both G r II ,j [ µ ] ,b ( Q )and G r III ,j [ µ ] ,b ( Q ) are also closed in X λ ( b )(¯ k ). So, using Proposition 5.6, wehave a decomposition of X λ ( b ) into irreducible closed subvarieties of dimension m − m − 1. If m ≤ 0, then this decomposition is disjoint because C M and C have relative position I (compare the last part of the proof of Theorem 6.1).If m > 0, then we have G r II ,j [ µ ] ,b ( Q ) = G r III ,j [ µ ′ ] ,b ( b Q ) , where [ µ ] = [ e , e , e ] , [ µ ′ ] = [ e , e − , e ]. In this case, we always choosethe one for II. We can also check that this gives a disjoint decomposition of34 λ ( b ) into irreducible components. Thus all of the irreducible components areparameterized by the elements in ( J b /H b ) × M λ ( b ) ′ . Again by Proposition5.6, their geometric structures are given as above. Remark 6.4. The criterion for non-emptiness of X λ ( b ) is already known (cf.[9, Theorem 4.16]). Let λ = ( m , m , m ) ∈ X ∗ ( T ) + . Then X λ (1) = ∅ ⇔ m ≥ , m + m ≥ , m + m + m = 0 ,X λ ( b ) = ∅ ⇔ m ≥ , m + m ≥ , m + m + m = 1 ,X λ ( b ) = ∅ ⇔ m ≥ , m + m ≥ , m + m + m = 2 . A flat morphism of f : X → Y of varieties over ¯ k is called an A n -fibration,for some integer n , if for every y ∈ Y , the fiber f − ( y ) is isomorphic to A n .We will now consider an A n -fibration over a Deligne-Lusztig variety: Let G be a connected reductive group defined over F q , and let F : G → G be aFrobenius map over F q . Fix an F -stable Borel subgroup B containing an F -stable maximal torus T (such a pair always exists; see [2, 1.17]). Let W bethe Weyl group of T . For any w ∈ W , we denote by X ( w ) the Deligne-Lusztigvariety associated with w . Corollary 6.5. Let λ = ( m , m , m ) ∈ X ∗ ( T ) + with X λ ( b ) = ∅ .(i) Let b = 1. Then every irreducible component of X λ (1) is an A n -fibrationover a Deligne-Lusztig variety for some n if and only if λ is one of thefollowing forms: λ = (2 r, − r, − r ) , ( r, r, − r ) , (2 r + 3 , − r − , − r − , ( r + 2 , r + 1 , − r − r ≥ 0. In the first two cases, we have X (0 , , (1) ∼ = G J /K { pt } and X λ (1) ∼ = G J /K Ω × A r − ( r > k -varieties. In the last two cases, we have X λ (1) ∼ = ( G J /K Ω × A r +1) ) ⊔ ( G J /K Ω × A r +1) )as ¯ k -varieties.(ii) Let b = b . Then every irreducible component of X λ ( b ) is an A n -fibrationover a Deligne-Lusztig variety for some n if and only if λ is one of thefollowing forms: λ = (2 r +1 , − r, − r ) , ( r +1 , r +1 , − r − , ( r +1 , r, − r ) , (2 r +2 , − r, − r − r ≥ 0. In the first two cases, we have X λ ( b ) ∼ = G J b /H b A d as ¯ k -varieties, where d = 3 r, r + 1 respectively. In the last two cases, wehave X λ ( b ) ∼ = ( G J b /H b A d ) ⊔ ( G J b /H b A d )as ¯ k -varieties, where d = 3 r, r + 2 respectively.(iii) Let b = b . Then every irreducible component of X λ ( b ) is an A n -fibrationover a Deligne-Lusztig variety for some n if and only if λ is one of thefollowing forms: λ = ( r +1 , r +1 , − r ) , (2 r +2 , − r, − r ) , (2 r +1 , − r +1 , − r ) , ( r +2 , r +1 , − r − r ≥ 0. In the first two cases, we have X λ ( b ) ∼ = G J b /H b A d as ¯ k -varieties, where d = 3 r, r + 1 respectively. In the last two cases, wehave X λ ( b ) ∼ = ( G J b /H b A d ) ⊔ ( G J b /H b A d )as ¯ k -varieties, where d = 3 r, r + 2 respectively. Proof. In every case, one way follows from Theorem 6.1 or Theorem 6.3. Forthe converse, it suffices to show that if a variety V over ¯ k is an A n -fibrationover G m or P \ P ( k ), then V cannot be an A n -fibration over a Deligne-Lusztigvariety. To show this, we use the l -adic cohomology with compact support( l = char k ). If a ¯ k -variety V is an A n -fibration over a ¯ k -variety X , then wehave H qc ( V ) ∼ = H q − nc ( X ) as Q l -vector spaces (cf. [20, 5.5, 5.7]). In particular,the Euler characteristics of V and X are equal, i.e., χ ( V ) = χ ( X ). So, tocomplete the proof, we compare the Euler characteristics.If a variety V over ¯ k is an A n -fibration over G m , then we have χ ( V ) = χ ( G m ) = 0 . However, by [2, Theorem 7.5.1, Theorem 7.7.11] or [5, Theorem 7.1], the Eulercharacteristic of a Deligne-Lusztig variety is nonzero. So V cannot be an A n -fibration over a Deligne-Lusztig variety.If a variety V over ¯ k is an A n -fibration over P \ P ( k ), then we havedim Q l H qc ( V ) = dim Q l H q − nc ( P \ P ( k )) = q = 2 n + 2 , n + 4) q + q ( q = 2 n + 1)0 (otherwise)36nd | χ ( V ) | = | χ ( P \ P ( k )) | = q + q − . If q = 2, then | χ ( V ) | = 4. However, by [2, Theorem 7.5.1, Theorem 7.7.11], theEuler characteristic of a Deligne-Lusztig variety is odd if q = 2, and differentfrom χ ( V ). So we may assume q ≥ 3. We will compare the absolute value ofthe Euler characteristic of a Deligne-Lusztig variety and | χ ( V ) | = q + q − G, B , T , W be as above. Let N be the number of positive roots, andlet l be the number of simple roots. Further, let T be an F -stable maximaltorus of G obtained from a maximally split torus T by twisting by w ∈ W .Thus T = gT g − , where g − F ( g ) = ˙ w ( ˙ w is a representative of w ). Then, by[2, Theorem 7.5.1, Theorem 7.7.11], we have | χ ( X ( w )) | = | G F | q N | T F | . Moreover, by [6, Corollaire 3.3.22], we have H qc ( X ( w )) = 0 for 0 ≤ q < l ( w ) , where l ( w ) denotes the length of w ∈ W . Since dim X ( w ) = l ( w ) and H qc ( V ) = 0for q = 2 n + 1 , n + 4, it is enough to consider the case for l ( w ) ≥ 3. In thiscase, we obviously have N ≥ l ≥ N ≥ | χ ( X ( w )) | and | χ ( V ) | = q + q − q ≥ N ≥ l ≥ N ≥ 3. Using [2, Proposition 3.3.7], we may alsoassume that G is semisimple. Then, as in the proof of [2, Proposition 3.6.7], wehave | T F | = ( q − λ ) · · · ( q − λ l ) , where λ , . . . , λ l are roots of unity. For | G F | , as in [2, 2.9], we have | G F | = q N l Y i =1 ( q d i − ǫ i ) , where d + · · · + d l = N + l , d · · · d l = | W | and ǫ i is a root of unity. So we have | χ ( X ( w )) | ≥ Q li =1 ( q d i − q + 1) l . If N = l ≥ 3, then d = · · · = d l = 2. So if q > q = 3 , l > 3, we have | χ ( X ( w )) | ≥ ( q − l > ( q − q + 2) . If q = 3 and N = l = 3, then | χ ( V ) | = 10 and | W | = 8. The facts in [2, 1.18,2.9] imply easily that | G F | is divisible by 10 if and only if | W F | = 4, where W F is the subgroup of F -stable elements of W . If in this case, we have | G F | = 3 · · · (1 + 3 + 9 + 27) = 3 · · . 37n the other hand, we have | T F | ≤ ( q + 1) l = 4 with equality only if each λ i = − 1. However, if | W F | = 4, then the equality | T F | = 4 does not hold. 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