THE DIMENSION OF SOME AFFINE DELIGNE–LUSZTIG VARIETIES
B
YE
VAVIEHMANN
ABSTRACT. – We prove Rapoport’s dimension conjecture for affine Deligne–Lusztig varieties forGLh
and superbasicb. From this case the general dimension formula for affine Deligne–Lusztig varieties for special maximal compact subgroups of split groups follows, as was shown in a recent paper by Görtz, Haines, Kottwitz, and Reuman.
©2006 Elsevier Masson SAS
RÉSUMÉ. – On démontre la conjecture de Rapoport sur la dimension des variétés de Deligne–Lusztig affines pourGLhetbsuperbasique. Ce cas implique la formule générale pour la dimension des variétés de Deligne–Lusztig affines pour des sous-groupes compacts maximaux de groupes déployés, résultat démontré dans un article récent de Görtz, Haines, Kottwitz et Reuman.
©2006 Elsevier Masson SAS
1. Introduction
Letkbe a finite field withq=prelements and letkbe an algebraic closure. LetF=k((t)) and letL=k((t)). LetOFandOLbe the valuation rings. We denote byσ:x→xqthe Frobenius ofkoverkand also ofLoverF.
Let G be a split connected reductive group over k. LetA be a split maximal torus of G andW the Weyl group ofAinG. Forμ∈X∗(A)lettμ be the image oft∈Gm(F)under the homomorphismμ:Gm→A. LetBbe a Borel subgroup ofGcontainingA. We writeμdomfor the dominant element in the orbit ofμ∈X∗(A)under the Weyl group ofAinG.
We recall the definitions of affine Deligne–Lusztig varieties from [6,1]. LetK=G(OL)and letX=G(L)/K be the affine Grassmannian. The Cartan decomposition shows thatG(L)is the disjoint union of the setsKtμKwhereμ∈X∗(A)is a dominant coweight. For an element b∈G(L) and dominant μ∈X∗(A), the affine Deligne–Lusztig variety Xμ(b) is the locally closed reducedk-subscheme ofXdefined by
Xμ(b)(k) =
g∈G(L)/K|g−1bσ(g)∈KtμK .
Left multiplication byg∈G(L)induces an isomorphism betweenXμ(b)andXμ(gbσ(g)−1).
Thus the isomorphism class of the affine Deligne–Lusztig variety only depends on the σ-conjugacy class ofb.
There is an algebraic group over F associated to Gandb whoseR-valued points (for any F-algebraR) are given by
J(R) =
g∈G(R⊗FL)|g−1bσ(g) =b .
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
There is a canonicalJ(F)-action onXμ(b).
Let ρ be the half-sum of the positive roots of G. By rkF we denote the dimension of a maximalF-split subtorus. LetdefG(b) = rkFG−rkFJ. Letν∈X∗(A)Qbe the Newton point ofb, compare [3]. For nonempty affine Deligne–Lusztig varieties the dimension is given by the following formula. Note that there is a simple criterion by Kottwitz and Rapoport (see [5]) to decide whether an affine Deligne–Lusztig variety is nonempty.
THEOREM 1.1. – Assume thatXμ(b)is nonempty. Then dim
Xμ(b)
=ρ, μ−ν −1
2defG(b).
Rapoport conjectured this in [7], Conjecture 5.10 in a different form. For the reformulation compare [4]. In [9], Reuman verifies the formula for some small groups andb= 1. ForG=GLn, minusculeμand overQp rather than over a function field, the Deligne–Lusztig varieties have an interpretation as reduced subschemes of moduli spaces ofp-divisible groups. In this case, the corresponding dimension formula is shown by de Jong and Oort (see [2]) ifbσis superbasic and in [10] for generalbσ. In [1] 2.15, Görtz, Haines, Kottwitz, and Reuman prove Theorem 1.1 for allb∈A(L). They also show in 5.8 that if there is a Levi subgroupM ofGsuch thatb∈M(L) is basic inM and if the formula is true forM, bandμM in a certain subset of the set of all M-dominant coweights, then it is also true for(G, b, μ). Thus it is enough to consider superbasic elementsb, that is elements for which noσ-conjugate is contained in a proper Levi subgroup ofG. They show in 5.9 that it is enough to consider the case thatG=GLhfor somehand thatb is basic withm=vt(det(b))prime toh. In this paper we prove Theorem 1.1 for this remaining case.
The strategy of the proof is as follows: We associate to the elements of Xμ(b) discrete invariants which we call extended semi-modules. This induces a decomposition of each connected component ofXμ(b)into finitely many locally closed subschemes. Their dimensions can be written as a combinatorial expression which only depends on the extended semi-module.
By estimating these expressions we obtain the desired dimension formula.
For minusculeμ, and over Qp, the group J(Qp) acts transitively on the set of irreducible components of Xμ(b). As an application of the proof we show that for nonminuscule μ, the action ofJ(F)on this set may have more than one orbit.
2. Notation and conventions
From now on we use the following notation: LetG=GLhand letAbe the diagonal torus.
Let B be the Borel subgroup of lower triangular matrices. For μ, μ ∈X∗(A)Q we say that μμ ifμ−μis a non-negative linear combination of positive coroots. As we may identify X∗(A)QwithQh, this induces a partial ordering on the latter set. An elementμ= (μ1, . . . , μh)∈ X∗(A)∼=Zhis dominant ifμ1· · ·μh.
LetN=Lhand letM0⊂N be the lattice generated by the standard basise0, . . . , eh−1. Then K=GLh(OL) = Stab(M0)andg→gM0defines a bijection
Xμ(b)(k)∼=
M⊂Nlattice |inv
M, bσ(M)
=tμ . (2.1)
We define the volume ofM=gM0∈Xμ(b)to bevt(det(g)).
We assumebto be superbasic. The Newton pointν∈X∗(A)Q∼=Qhofbis then of the form ν= (mh, . . . ,mh)∈Qh with(m, h) = 1. Fori∈Zdefineei byei+h=tei. We choose bto be
the representative of itsσ-conjugacy class that mapseitoei+mfor alli. For superbasicb, the condition that the affine Deligne–Lusztig variety is nonempty, namelyνμ, is equivalent to μi=m. From now on we assume this.
For each centralα∈X∗(A)there is the trivial isomorphism Xμ(b)→Xμ+α
tαb .
We may therefore assume that allμi are nonnegative. For the lattices in (2.1), this implies that bσ(M)⊆M.
In the following we will abbreviate the right-hand side of the dimension formula forXμ(b)by d(b, μ).
The set of connected components of X is isomorphic to Z, an isomorphism is given by mappingg∈GLh(L)tovt(det(g)). LetXμ(b)ibe the intersection of the affine Deligne–Lusztig variety with thei-th connected component ofX. Letπ∈GLh(L)withπ(ei) =ei+1for alli∈Z.
Thenπcommutes withbσ, and defines isomorphismsXμ(b)i→Xμ(b)i+1 for alli. Thus it is enough to determine the dimension ofXμ(b)0.
For superbasicb, an element ofJ(F)is determined by its value ate0. More precisely,J(F) is the multiplicative subgroup of a central simple algebra overF. HencedefG(b) =h−1. If vt(det(g)) =ifor someg∈J(F), thenginduces isomorphisms betweenXμ(b)jandXμ(b)j+i for allj. OnXμ(b)0, we have an action of{g∈J(F)|vt(det(g)) = 0}=J(F)∩Stab(M0).
Remark 2.1. – To a vectorψ= (ψi)∈Qhwe associate the polygon inR2that is the graph of the piecewise linear continuous functionf: [0, h]→Rwithf(0) = 0and slopeψion[i−1, i].
One can easily see that d(b, μ) is equal to the number of lattice points below the polygon corresponding toνand (strictly) above the polygon corresponding toμ.
3. Extended semi-modules
In this section we describe the combinatorial invariants which are used to decomposeXμ(b)0. DEFINITION 3.1. – (1) Let m and hbe coprime positive integers. A semi-module for m, his a subset A⊂Z that is bounded below and satisfies m+A⊂A and h+A⊂A. Let B=A\(h+A). The semi-module is callednormalizedif
a∈Ba=h(h2−1).
(2) Let ν = (mh, . . . ,mh)∈Qh. Let μ = (μ1, . . . , μh)∈Nh not necessarily dominant with νμ. A semi-module A for m,h is of type μ if the following condition holds: Letb0= min{b∈B}and let inductively bi=bi−1+m−μih∈Zfori= 1, . . . , h. Thenb0=bh and {bi|i= 0, . . . , h−1}=B.
Remark 3.2. – Semi-modules are also used by de Jong and Oort in [2] to define a stratification of a moduli space of p-divisible groups whose rational Dieudonné modules are simple of slopemh. In this caseμis minuscule, and they use semi-modules form, h−mto decompose the moduli space.
LEMMA 3.3. –If Ais a semi-module, then its translate −Σa∈hBa +h−12 +Ais the unique normalized translate ofA. It is called the normalization ofA. There is a bijection between the set of normalized semi-modules form,hand the set of possible typesμ∈Nhwithνμ.
Proof. –For the first assertion one only has to notice that the fact that thehelements ofBare incongruent modulohimplies that
a∈Ba−h(h−1)2 is divisible byh. For the second assertion letAbe a normalized semi-module, letb0= min{a∈B}and let inductivelybi=bi−1+m−μih where μi is maximal with bi ∈A. Then bh=b0 and {bi |i= 0, . . . , h−1}=B. From
b0< bi0 fori0= 1, . . . , h−1we obtaini0
i=1(m−μih)>0for alli0< h. Similarly,b0=bh
implies h
i=1μi =m. This shows ν μ. As m+A⊂A, the μi are nonnegative. Given μ as above, the corresponding normalized semi-moduleA can be constructed as follows: Let b0= 0, and inductivelybi=bi−1+m−μih. ThenAis the normalization of{bi+αh|α∈N, 0i < h}. 2
DEFINITION 3.4. – Let m and h be as before and let μ= (μi)∈Nh be dominant with μi=m. An extended semi-module (A, ϕ) for μ is a normalized semi-module A for m, htogether with a functionϕ:Z→N∪ {−∞}with the following properties:
(1) ϕ(a) =−∞if and only ifa /∈A.
(2) ϕ(a+h)ϕ(a) + 1for alla.
(3) ϕ(a)max{n|a+m−nh∈A}for alla∈A. Ifb∈Afor allba, then the two sides are equal.
(4) There is a decomposition ofAinto a disjoint union of sequencesa1j, . . . , ahj withj∈N and the following properties:
(a) ϕ(alj+1) =ϕ(alj) + 1.
(b) Ifϕ(alj+h) =ϕ(alj) + 1, thenalj+1=alj+h. Otherwisealj+1> alj+h.
(c) Theh-tuple(ϕ(al0))is a permutation ofμ.
An extended semi-module such that equality holds in (3) for alla∈Ais calledcyclic.
LetAbe a normalized semi-module form,hand letμ be its type. Letμ=μdom. Letϕbe such that (1) holds and that we have equality in (3) for alla∈A. Then in (2) the two sides are also equal for alla∈A. A decomposition ofAas in (4) is given by putting all elements into one sequence that are congruent moduloh. Hence(A, ϕ)is a cyclic extended semi-module forμ, called thecyclic extended semi-module associated toA.
Example 3.5. – We give an explicit example of a noncyclic extended semi-module form= 4, h= 5, and μ= (0,0,0,2,2). Let A be the normalized semi-module of type (0,0,1,2,1).
ThenB =A\(5 +A)consists of −2,−1,2,5, and6. Let ϕ(−1) = 0andϕ(a) = max{n| a+m−nh∈A}ifa∈A\ {−1}. See also Fig. 1 that shows elements ofAmarked by crosses and the corresponding values ofϕ. A decomposition ofAis given as follows: Three sequences are given by the elements of A congruent to−2, 2, and 5 modulo5, respectively. The forth sequence is given by all elements congruent to4modulo5and greater than−1. The last sequence consists of the remaining elements−1and6,11,16, . . ..
LEMMA 3.6. – If(A, ϕ)is an extended semi-module forμ, and ifμ0 is the type ofA, then μ0domμ. Ifμ0dom=μ, then(A, ϕ)is a cyclic extended semi-module.
Proof. –Let(A, ϕ0)be the cyclic extended semi-module associated toA. Let {x1, . . . , xn}=
a∈A|ϕ(a+h)> ϕ(a) + 1
ϕ(a) · · · −∞ 0 0 −∞−∞ 0 1 2 2 1 1 2 · · ·
a −3 −2 −1 0 1 2 3 4 5 6
· · · · · × × · · × × × × × × × · · ·
Fig. 1. A noncyclic extended semi-module.
withxi> xi+1for alli. Fori∈ {1, . . . , n}let
ϕi(a) =
⎧⎨
⎩
−∞ ifa /∈A, ϕ(a) ifaxi, ϕi(a+h)−1 else.
We show that (A, ϕi) is an extended semi-module for some μi with μi−1dom μidom and μi−1dom =μidom for alli1. Asϕn=ϕ, it then follows that μ0domμndom=μwith equality if and only ifn= 0, that is ifϕis cyclic.
The decomposition of (A, ϕi) is defined as follows: For a < xi, the successor of a is a+h. Otherwise it is the successor from the decomposition of (A, ϕ). From the properties of the decompositions forϕ0 andϕone deduces that the decomposition satisfies the required properties. Letni0be maximal withxi−nih∈Aand letαi=ϕ(xi+h)−1−ϕ(xi)>0.
Thusϕi is obtained fromϕi−1by subtracting αi from the values atxi, xi−h, . . . , xi−nih.
From μi−1 we obtain μi by replacing the two entries ϕi−1(xi−nih) =ϕi−1(xi)−ni and ϕi−1(xi)−αi+ 1(which is the value ofϕof the successor ofxiin the sequence corresponding toϕi) byϕi−1(xi)−αi−niandϕi−1(xi) + 1. As
ϕi−1(xi)−ni, ϕi−1(xi)−αi+ 1∈
ϕi−1(xi)−αi−ni, ϕi−1(xi) + 1 , we haveμidom−1 μidomandμidom−1 =μidom. 2
COROLLARY 3.7. –Ifμis minuscule, then all extended semi-modules forμare cyclic.
Proof. –Let(A, ϕ)be such an extended semi-module. Letμbe the type ofA. Thenμdomμ, thusμdom=μ. Hence the assertion follows from the preceding lemma. 2
LEMMA 3.8. – There are only finitely many extended semi-modules(A, ϕ)for eachμ.
Proof. –Letμbe the type of the semi-moduleA. Asμdomμ, there are only finitely many possible types and corresponding normalized semi-modules. For fixedA, the third condition for extended semi-modules determines all but finitely many values ofϕ. For the remaining values we have0ϕ(a)max{n|a+m−nh∈A}. Thus for eachAthere are only finitely many possible functionsϕsuch that(A, ϕ)is an extended semi-module forμ. 2
4. The decomposition of the affine Deligne–Lusztig variety
Let M ∈Xμ(b)0 be a lattice inN. In this section we associate to M an extended semi- module forμ. This leads to a paving ofXμ(b)0by finitely many locally closed subschemes. For minusculeμ, this decomposition of the set of lattices is the same as the one constructed by de Jong and Oort in [2], compare also [10, Section 5.1].
Let mandh be as in Section 2. Letv∈N and recall that tei=ei+h. Then we can write v=
i∈Zαieiwithαi∈kandαi= 0for smalli. Let I:N\ {0} →Z,
v→min{i|αi= 0}.
For a latticeM∈Xμ(b)0we consider the set A=A(M) =
I(v)|v∈M\ {0}
.
ThenA(M)is bounded below andh+A(M)⊂A(M). Asbσ(M)⊂M, we havem+A(M)⊂ A(M), thusA(M)is a semi-module form,h. We have
vol(M) =N\(A∩N)−A\(N∩A)= 0.
This implies that
a∈Ba=h−1
i=0 i, thusAis normalized.
Let further
ϕ=ϕ(M) :Z→N∪ {−∞}, a→
max{n| ∃v∈MwithI(v) =a, t−nbσ(v)∈M} ifa∈A(M),
−∞ else.
Note that by the definition ofA(M), the set on the right-hand side is nonempty. Asbσ(M)⊂M, the values ofϕare indeed inN∪ {−∞}.
LEMMA 4.1. – LetM ∈Xμ(b)0. Then(A(M), ϕ(M))is an extended semi-module forμ.
Proof. –We already saw that A(M) is a normalized semi-module. We have to check the conditions onϕ. The first condition holds by definition. Letv∈M withI(v) =abe realizing the maximum forϕ(a). Then tv∈M withI(tv) =a+himplies thatϕ(a+h)ϕ(a) + 1, which shows (2). Let v∈M with I(v) =a and t−ϕ(a)bσ(v)∈M. Then I(t−ϕ(a)bσ(v)) = a+m−ϕ(a)h∈A(M), whence the first part of (3). Letb∈Afor allba. Letn0= max{n| a+m−nh∈A}. Letv∈M withI(v) =a+m−n0hand letv= (bσ)−1(tn0v)∈N. Then I(v) =a, thusv=
baαbebfor someαb∈k. Asb∈Afor allba, we also haveeb∈Mfor all suchb. Thusv∈Mwitht−n0bσ(v) =v∈M. Henceϕ(a) =n0. It remains to show (4). For a∈Zandϕ0∈Nlet
Va,ϕ0=
v∈M|v= 0orI(v)a, t−ϕ0bσ(v)∈M
andVa,ϕ0 =Va,ϕ0/Va,ϕ0+1. ThenVa0,ϕ0 is ak-vector space of dimension |{aa0|ϕ(a) = ϕ0}|. We construct the sequences by inductively sorting all elementsa∈Awithϕ(a)ϕ0for someϕ0: Forϕ0= min{ϕ(a)|a∈A}we take each elementawith this value ofϕas the first element of a sequence. (At the end we will see that we did not construct more thanhsequences.) We now describe the induction step from ϕ0 to ϕ0+ 1: If v1, . . . , vi is a basis of Va,ϕ0 for somea, then thetvj are linearly independent inVa+h,ϕ0+1. ThusdimVa,ϕ0dimVa+h,ϕ0+1 for everya. Hence there are enough elementsa∈Awithϕ(a) =ϕ0+ 1to prolong all existing sequences such that conditions (a) and (b) are satisfied. We take thea∈Awithϕ(a) =ϕ0+ 1 that are not already in some sequence as first elements of new sequences. Inductively, this constructs sequences with properties (a) and (b). To show (c), leta < b0. Then
{i|μi=n}= dimkVa,n−dimkVa−h,n−1
=al0|ϕ al0
=n. This also shows that we constructed exactlyhsequences. 2
For each extended semi-module(A, ϕ)forμlet SA,ϕ=
M⊂Nlattice|A(M) =A, ϕ(M) =ϕ
⊂X.
LEMMA 4.2. – The sets SA,ϕ are contained in Xμ(b)0. They define a decomposition of Xμ(b)0 into finitely many disjoint locally closed subschemes. Especially, dimXμ(b)0= max{dimSA,ϕ}.
Proof. –The last property in the definition of an extended semi-module shows that (A, ϕ) determinesμ. ThusSA,ϕ⊆Xμ(b)0. Using Lemmas 3.8 and 4.1 it only remains to show that the subschemes are locally closed. The condition that a∈A(M) is equivalent to dim(M ∩ ea, ea+1, . . .)/(M ∩ ea+1, ea+2, . . .) = 1. This is clearly locally closed. If ais sufficiently large, it is contained in all extended semi-modules for μ and if a is sufficiently small, it is not contained in any extended semi-module forμ. Thus fixingAis an intersection of finitely many locally closed conditions onXμ(b)0, hence locally closed. Similarly, it is enough to show that ϕ(a)< n for some a∈A andn∈Nis an open condition on {M ∈X |bσ(M)⊂M, A(M) =A} ⊂X. But this condition is equivalent to
ei|ia ∩M∩tn(bσ)−1(M)
/ei|ia+ 1= (0), which is an open condition. 2
Let(A, ϕ)be an extended semi-module forμ. Let V(A, ϕ) =
(a, b)∈A×A|b > a, ϕ(a)> ϕ(b)> ϕ(a−h) . (4.1)
THEOREM 4.3. –
(1) LetAandϕbe as above. There exists a nonempty open subschemeU(A, ϕ)⊆AV(A,ϕ) and a morphism U(A, ϕ)→ SA,ϕ that induces a bijection between the set ofk-valued points ofU(A, ϕ)andSA,ϕ. Especially,dim(SA,ϕ) =|V(A, ϕ)|.
(2) If(A, ϕ)is a cyclic extended semi-module, thenU(A, ϕ) =AV(A,ϕ).
Proof. –We denote the coordinates of a pointxof AV(A,ϕ) byxa,b with(a, b)∈ V(A, ϕ).
To define a morphismAV(A,ϕ)→X, we describe the imageM(x)of a pointx∈AV(A,ϕ)(R) whereRis ak-algebra. For eacha∈Awe define an elementv(a)∈NR=N⊗kRof the form v(a) =
baαbeb withαa= 1. TheRJtK-moduleM(x)⊂NR will then be generated by the v(a). We want thev(a)to satisfy the following relations: Fora∈h+Awe want
v(a) =tv(a−h) +
(a,b)∈V(A,ϕ)
xa,bv(b).
(4.2)
Lety= max{b∈B}. Ifa=ywe want
v(a) =ea+
(a,b)∈V(A,ϕ)
xa,bv(b).
(4.3)
For all other elementsa∈B, we want the following equation to hold: Leta∈A be minimal witha+m−ϕ(a)h=a. Thenv=t−ϕ(a)bσ(v(a))∈NRwithI(v) =a. Let
v(a) =v+
(a,b)∈V(A,ϕ)
xa,bv(b).
(4.4)
CLAIM 1. –For every x∈AV(A,ϕ)(R) there are uniquely determined v(a)∈NR for all a∈Asatisfying(4.2)to(4.4).
We set
v(a) =
j∈N
αa,jea+j
withαa,j∈Randαa,0= 1for alla. We solve the equations by induction onj. Assume that the αa,jare determined forjj0and such that the equations forv(a)hold up to summands of the formβjejwithj > a+j0. To determine theαa,j0+1, we writea≡y+im(modh)and proceed by induction oni∈ {0, . . . , h−1}. Fori= 0anda=y, the coefficientαa,j0+1is the uniquely determined element such that (4.3) holds up to summands of the form βjej withj > j0+ 1.
Note that by induction onj and asb > a, the coefficient ofey+j0+1on the right-hand side of the equation is determined. Fora=y+nhwithn >0, the coefficients are similarly defined by (4.2). Fori >0anda∈Aminimal in this congruence class, the coefficient is determined by (4.4). Here, the coefficient ofea+j0+1on the right-hand side of each equation is determined by induction oniandj. For largerain this congruence class we use again (4.2). By passing to the limit onj, we obtain the uniquely definedv(a)∈NRsolving the equations.
CLAIM 2. –LetM(x) =v(a)|a∈ARJtK. Then at each specialization ofxto ak-valued pointywe haveA=A(M(y))andϕ(M(y))(a)ϕ(a)for alla.
From the definition of M we immediately obtain A⊆A(M(y)). To show equality con- sider an element v =
aαav(a) ∈M(y) = M. Write v =
i∈Zbiei with bi ∈k. Let i0 = min{I(αav(a))}. If bi0 = 0, then I(v) = i0 ∈ A. Otherwise we consider
{a|I(αav(a))=i0}αav(a). Note thatI(v(a))≡i0(modh)for allaoccurring in the sum. Then (4.2) shows that this sum can be written as a sum ofv(b)withb > i0. Thus we may replacei0by a larger number. Asi∈Afor all sufficiently largei, this shows thatI(v)∈A, soA(M) =A.
Letx∈AV(A,ϕ)(k)and letM =M(x). We show thatt−ϕ(a)bσ(v(a))∈M for all a. This means that ϕ(M)(a)ϕ(a) for all a. Consider the elements a∈A that are minimal with a+m−ϕ(a)h=afor somea∈B\ {y}. For these elements, the assertion follows from (4.4).
If ais minimal with a+m−ϕ(a)h=y, thenI(t−ϕ(a)bσ(v(a))) =y. As all ei withiy are inM, this element is also contained inM. Ifϕ(a) =ϕ(a−h) + 1thenv(a) =tv(a−h) and the assertion holds fora−hif and only if it holds for h. From this, we obtain the claim for alla∈Awithϕ(a) = max{n|a+m−nh∈A}. Especially, it follows for all sufficiently large elements of A. It remains to prove the claim for the finitely many elementsa∈Awith max{n|a+m−nh∈A}> ϕ(a). We use decreasing induction ona: Letabe in this set, and assume that we know the assertion for alla> a. From (4.2) we obtain that
t−ϕ(a)bσ v(a)
=t−ϕ(a)−1bσ tv(a)
=t−ϕ(a)−1bσ
v(a+h)−
b>a+h,ϕ(a+h)>ϕ(b)ϕ(a)+1
xa+h,bv(b)
.
By induction, the right-hand side is inM and Claim 2 is shown.
As allμi are nonnegative, we constructed a morphism from AV(A,ϕ)to the subscheme XA
ofX defined byXA(k) ={M|A(M) =A, bσ(M)⊆M}.
CLAIM 3. –There is a nonempty open subschemeU(A, ϕ)ofAV(A,ϕ)that is mapped toSA,ϕ. If(A, ϕ)is cyclic, thenU(A, ϕ) =AV(A,ϕ).
In general we do not have ϕ(M)(a) =ϕ(a) for all a. The proof of Lemma 4.2 shows that ϕ(M)(a)ϕ(a)is an open condition onXA, and thus on AV(A,ϕ). LetU(A, ϕ)be the corresponding open subscheme, which is then mapped to SA,ϕ. We have to show that it is nonempty, thus to construct a point inAV(A,ϕ)where the corresponding functionϕ(M)is equal toϕ. Ifϕ(a) = max{n|a+m−nh∈A}, thenϕ(M)(a) =ϕ(a). Especially, the two functions are equal for allaif(A, ϕ)is cyclic. In this caseU(A, ϕ) =AV(A,ϕ). Ifϕ(a) + 1 =ϕ(a+h) and if ϕ(M)(a+h) =ϕ(a+h), then ϕ(M)(a+h)−1ϕ(M)(a)ϕ(a) implies that ϕ(M)(a) =ϕ(a). Thus it is enough to find a point whereϕ(M)(a) =ϕ(a)for alla∈Awith
ϕ(a+h)> ϕ(a) + 1. For each suchaletbabe the successor in a decomposition of(A, ϕ)into sequences. Then (a+h, ba)∈ V(A, ϕ). Let xa+h,ba= 1 for these pairs and choose all other coefficients to be0. Then for this point andaas before we have thatϕ(M)(a) =ϕ(ba)−1 = ϕ(a). ThusU(A, ϕ)is nonempty.
CLAIM 4. –The mapU(A, ϕ)→ SA,ϕdefines a bijection onk-valued points.
More precisely, we have to show that for eachM ∈ SA,ϕthere is exactly onex∈U(A, ϕ)(k) such thatM contains a set of elementsv(a)fora∈AwithI(v(a)) =aand satisfying (4.2) to (4.4) for thisx. The argument is similar as the construction ofv(a)for givenx: By induction on jwe will show the following assertion: There exist xj= (xja,b)∈U(A, ϕ)(k)andvj(a)∈M for allawitht−ϕ(a)bσ(vj(a))∈M and which satisfy Eqs. (4.2) to (4.4) forxjup to summands of the formβnenwithn > a+j. Furthermore thexja,bwithb−ajand the coefficients ofen invj(a)forna+jwill be chosen independently ofjand only depending onM.
Forj= 0choose anyx0∈U(A, ϕ)(k)andv0(a)∈M withI(v0(a)) =a, first coefficient1 andt−ϕ(a)bσ(v0(a))∈M. The existence of thesev0(a)follows fromM∈Xμ(b). Assume that the assertion is true for somej0. Fornj0letxja,a+n0+1 =xja,a+n0 . We proceed again by induction onito define the coefficients fora≡y+im(modh). Leta=y. Choose the coefficientsxjy,y+n0+1 withn > j0such that
vj0+1(y) =ey+
(y,y+n)∈V(A,ϕ)
xjy,y+n0+1 vj0(y+n)
satisfies t−ϕ(y)bσ(vj0+1(y))∈M. The definition of ϕ=ϕ(M) shows that such coefficients exist and fromϕ(y+n)< ϕ(y)it follows that they are unique. For the other elementsv(a)we proceed similarly: For those witha−h /∈A we use Eq. (4.4), on the right-hand side with the values from the induction hypothesis, to define the newvj0+1(a). Fora∈h+Awe use (4.2).
As we know thatt−ϕ(a−h)−1bσ(tvj0(a−h))∈M, it is sufficient to consider the b > awith ϕ(a−h)< ϕ(b)< ϕ(a). At each step the coefficient ofea+j0+1of the right-hand side is already defined by the induction hypothesis. It only depends on thexja,a+n0 and the coefficients ofeb+n ofvj0(b)withnj0, hence only onM. The coefficients ofxj0+1 are given by requiring that t−ϕ(a)bσ(vj0+1(a))∈M. 2
5. Combinatorics
In this section we estimate|V(A, ϕ)|to determine the dimension of the affine Deligne–Lusztig varietyXμ(b).
Remark 5.1. – For cyclic extended semi-modules we haveϕ(a+h) =ϕ(a) + 1for alla∈A.
Thus
V(A, ϕ) =
(bi, b)|bi∈B, b∈A, b > bi, ϕ(b)< ϕ(bi) whereB=A\(h+A).
PROPOSITION 5.2. – Let (A, ϕ) be the cyclic extended semi-module associated to the normalized semi-moduleAof typeμ. Then|V(A, ϕ)|=d(b, μ).
Proof. –Recall that by b0 we denote the minimal element of A or B. Let bi be as in the definition of the type ofAand letbh=b0. First we show that
V(A, ϕ)→Z,
(bi, b)→b−bi+bh
induces a bijection betweenV(A, ϕ)and{a /∈A|a > bh}. Letb∈Afor some b > bi. Then b−bi+bi+1∈/Aif and only if(bi, b)∈ V(A, ϕ). Letbi0= max{bi∈B}. We haveb∈Afor all bbi0. Thus for everyb > bhwithb /∈A, there is an element(bi, b−bh+bi)∈ V(A, ϕ)for some h > ii0. Hence{a /∈A|a > bh}is in the image of the map. To show that it is injective and that its image is contained in{a /∈A|a > bh}, it is enough to show that(bi, b)∈ V(A, ϕ)implies that b−bi+bj∈/Afor allj∈ {i+ 1, . . . , h}. Indeed, this ensures that(bj, b−bi+bj)∈ V(A, ϕ)/ for all such j and that b−bi+bh∈/A. We writeb=bl+αhfor some l andα. Recall that ϕ(bi) =μi+1. As(bi, bl+αh)∈ V(A, ϕ), we haveμl+1+α < μi+1. Especially, l < i. This impliesμl+1+· · ·+μl+β+α < μi+1+· · ·+μi+βfor allβh−i. Using the recurrence for thebj, one sees that this impliesb−bi+bi+β∈/Afor allβh−i.
It remains to count the elements of{a /∈A|a > b0}. Ash+A⊆A, we have {a /∈A|a > b0}=
h−1
i=0
bi−b0−i
· 1 h.
From the construction ofAfrom its type we obtain
= h−1
i=0
i j=1
(m−μjh)−i
·1 h
= h−1
i=0
i j=1
m h −μj
−h−1 2
=d(b, μ). 2
THEOREM 5.3. – Let(A, ϕ)be an extended semi-module forμ. Then|V(A, ϕ)|d(b, μ).
Proof of Theorem 5.3 for cyclic extended semi-modules. –We writeB={b0, . . . , bh−1}as in the definition of the typeμofA. As the extended semi-module is assumed to be cyclic,μis a permutation ofμ. Using Remark 5.1 we see
V(A, ϕ)=(bi, a)∈B×A|a > bi, ϕ(a)< ϕ(bi)
=
{(bi,bj)∈B×B|bj>bi,μj+1<μi+1}
μi+1−μj+1
+(bi, bj+αh)|bj< bi< bj+αh, μi+1> μj+1+α. We refer to these two summands asS1andS2.
Let(˜b0,μ˜1), . . . ,(˜bh−1,μ˜h)be the set of pairs(b0, μ1), . . . ,(bh−1, μh), but ordered by the size ofbi. That is,˜bi<˜bi+1for alli. Let
f:B→B,
bi→bi+1=bi+m−μi+1h
where we identifybh with b0. This defines a permutation of B. From the ordering of the˜bi we obtaini0
i=0f(˜bi)i0
i=0˜bi for alli0. As f(˜bi) = ˜bi+m−μ˜i+1h, this is equivalent to i0+1
i=1 μ˜i(i0+ 1)mh for alli0. We thus haveνμ˜μ.
Recall the interpretation ofd(b, μ)from Remark 2.1. We show thatS1is equal to the number of lattice points aboveμand on or below μ. The second summand˜ S2will be less or equal to the number of lattice points aboveμ˜and belowν. Then the theorem follows for cyclic extended semi-modules.
We haveS1=
i<jmax{μ˜i+1−μ˜j+1,0}. Consider this sum for any permutationμ˜ofμ. If we interchange two entriesμ˜i andμ˜i+1 withμ˜i>μ˜i+1, the sum is lessened by the difference of these two values. There are also exactlyμ˜i−μ˜i+1lattice points on or belowμ˜and above the polygon corresponding to the permuted vector. Ifμ˜=μ, bothS1and the number of lattice points aboveμand on or belowμ˜are0. Thus by inductionS1is equal to the claimed number of lattice points.
The last step is to estimateS2. It is enough to construct a decreasing sequence (with respect to) ofψi∈Qhfori= 0, . . . , h−1withψ0= ˜μandψh−1=νsuch that the number of lattice points aboveψiand on or belowψi+1is greater or equal to the number of pairs(˜bi+1,˜bj+αh) contributing toS2. Note that theψiwill no longer be lattice polygons. Letfi:B→Bbe defined as follows: Forj > iletfi(˜bj) =f(˜bj). Let{fi(˜bj)|0ji}be the set off(˜bj), but sorted increasingly. Letψi= (ψij)be such thatfi(˜bj) = ˜bj+m−ψij+1h, i.e.
ψj+1i =˜bj+m−fi(˜bj)
h =m
h −fi(˜bj)−˜bj
h .
Similarly as forνμ˜one can show that
νψi+1ψiμ˜
for alli. Asf0=f andfh−1= id, we haveψ0= ˜μandψh−1=ν. It remains to count the lattice points betweenψiandψi+1. To pass fromfitofi+1we have to interchange the valuef(˜bi+1) with all largerfi(˜bj)withji. Thus to pass from the polygon associated toψito the polygon ofψi+1 we have to change the value atj by(fi(˜bj)−f(˜bi+1))/h, and that for alljiwith fi(˜bj)> f(˜bi+1). Thus there are at least
ji,fi(˜bj)>f(˜bi+1)
fi(˜bj)−f(˜bi+1) h
=
ji,f(˜bj)>f(˜bi+1)
f(˜bj)−f(˜bi+1) h
lattice points above ψi and on or below ψi+1. For fixed i and j < i+ 1, the set of pairs (˜bi+1,˜bj+αh) contributing toS2 is in bijection with{α1|f(˜bj)−αh > f(˜bi+1)}. The cardinality of this set is at mostf(˜bj)−hf(˜bi+1) which proves thatS2 is not greater than the number of lattice points betweenμ˜andν. 2
Example 5.4. – We give an example of a cyclic semi-module (A, ϕ)where the type of A is not dominant but where|V(A, ϕ)|=d(b, μ). Letm= 4, h= 5, andμ= (0,0,1,1,2). Let (A, ϕ)be the cyclic extended semi-module associated to the normalized semi-module of type (0,0,1,2,1). Note thatAis the same semi-module as in Example 3.5. Then the dimension of the corresponding subscheme is
V(A, ϕ)=(−1,2),(5,6),(5,7)=d(b, μ).
Proof of Theorem 5.3. –Let (A, ϕ) be an extended semi-module for μ. Let ϕi and μi be the sequences constructed in the proof of Lemma 3.6. By induction on i we show that
|V(A, ϕi)|d(b, μi). Fori= 0, the extended semi-module(A, ϕ0)is cyclic, hence the assertion is already shown.
We use the notation of the proof of Lemma 3.6. The description of the difference betweenμi andμi−1given there shows that
d b, μi
−d b, μi−1
= h l=1
l j=1
μidom,j−1 −μidom,j
=μij−1∈
ϕi−1(xi)−αi−ni, ϕi−1(xi) + 1−1
×min{αi, ni+ 1}.
We denote this difference by Δ. To show that |V(A, ϕi)| − |V(A, ϕi−1)|Δ we use the decomposition into sequencesalj of the extended semi-module(A, ϕi−1). Using the definition of V(A, ϕ) and the description of the difference between ϕi and ϕi−1 from the proof of Lemma 3.6 one obtains
V(a, ϕi)−V(a, ϕi−1)=S1+S2+S3 where
S1=(xi+h, b)|b∈A, b > xi+h, ϕi−1(xi) + 1> ϕi−1(b)> ϕi−1(xi)−αi, S2=(b, xi−δh)|b∈B\ {xi−nih}, b < xi−δh, δ∈ {0, . . . , ni}
ϕi−1(xi)−δ−αi< ϕi−1(b)ϕi−1(xi)−δ,
S3=−(xi−nih, b)|b > xi−nih, ϕi−1(xi)−ni> ϕi−1(b)ϕi−1(xi)−ni−αi. Here we used thataxi implies thatϕi−1(a+h) =ϕi−1(a) + 1. For each sequence alj of the extended semi-module(A, ϕi−1)we use S1,l,S2,l, and S3,l for the contributions of pairs withb∈ {alj}to the three summands. Furthermore we write Sl=S1,l+S2,l+S3,l. We show the following assertions: Ifϕi−1(al0)∈/(ϕi−1(xi)−αi−ni, ϕi−1(xi) + 1)or ifal0=xi−nih, thenSl= 0. Otherwise,Slmin{αi, ni+ 1}. Then the theorem follows from property (4c) of extended semi-modules.
To determine theSl, we consider the following cases:
Case 1. – ϕi−1(al0)ϕi−1(xi) + 1. In this case it is easy to see thatS1,l=S2,l=S3,l= 0.
Case 2. – al0> xi. This implies that S2,l= 0. If ϕi−1(al0)ϕi−1(xi)−ni−αi, then S1,l+S3,l=αi−αi= 0. Let nowϕi−1(al0)∈(ϕi−1(xi)−αi−ni, ϕi−1(xi) + 1). Then
S1,l+S3,lalj|ϕi−1(xi) + 1> ϕi−1 alj
max
ϕi−1(xi)−αi+ 1, ϕi−1(xi)−ni. Asϕi−1(alj+1) =ϕi−1(alj) + 1for allj, the right-hand side is less or equal tomin{αi, ni+ 1}.
Case 3. – al0=xi−nih. This sequence starts withxi−nih, . . . , xi, xi+h. (Recall that the sequences{alj}forϕi−1are of this easy form with stepwidthhas long asaljxi< xi−1.) Note that within one sequencealj> alj impliesϕi−1(alj)> ϕi−1(alj). Hence this special sequence does not make any contribution, as inSlwe only consider pairs where both elements are in the sequence starting withxi−nih.
Case 4. – al0< xi, but not congruent toximoduloh. Againalj+1=alj+hifaljxi. We first assume thatϕi−1(al0)ϕi−1(xi)−ni−αi. ThenS2,l= 0. Assume thatb=aljcontributes
