On the stable moment graph of an affine Kac--Moody algebra

In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph.


Introduction
Finding the characters of the irreducible modular representations of semisimple algebraic groups is a fundamental problem in the representation theory of finite groups. In [36] Lusztig stated a conjecture which related these characters to the affine Kazhdan-Lusztig polynomials and which was similar to a conjecture by Kazhdan and Lusztig (cf. [30]) on the composition series of Verma modules for semisimple complex Lie algebras. Kazhdan-Lusztig's conjecture has been solved using geometric methods and the theory of D-modules (cf. [4], [8]), while Lusztig's conjecture has not been completely settled yet. Indeed, by [1], [32], [28] it is possible to recover it as a limit from the characteristic zero case, but this means that Lusztig's conjecture is not known to be true yet for all the characteristics it is supposed to hold.
The attempt of finding a new approach to these character conjectures brought Fiebig to apply moment graph techniques in representation theory. In particular, he associated a particular moment graph to any complex symmetrizable Kac-Moody algebra g: its Bruhat graph, that is an oriented graph with labeled edges and whose set of vertices is given by the elements of the Weyl group W of g. A central role in this theory is played by indecomposable Braden-MacPherson sheaves on the Bruhat graph of g, which correspond to indecomposable projective objects admitting a Verma flag in an equivariant version of the representation category O of g (cf. [15]). Moreover, moment graphs and Braden-MacPherson sheaves can be constructed also in the positive characteristic setting, in which case the moment graph is attached to a semisimple reductive algebraic group G. In this way, Fiebig could relate Kazhdan-Lusztig and Lusztig conjectures to a question on the graded rank of the stalks of indecomposable Braden-MacPherson sheaves. In particular, he stated a conjectural formula connecting this rank explicitly to Kazhdan-Lusztig polynomials. Such a formula is known to hold in characteristic zero and, in this case, to be equivalent to the character formula conjectured by Kazhdan and Lusztig (cf. [15]), while in positive characteristic it is expected to be true for characteristics bigger than the Coxeter number of G.
Fiebig's conjectural formula motivated our paper [34], where we lifted at the categorical level of sheaves on moment graphs certain properties of Kazhdan-Lusztig polynomials and parabolic Kazhdan-Lusztig polynomials. The methods we developed in [34] work in any characteristic under certain technical assumptions, even in cases in which the formula has been proven to fail. In [21] Fiebig and Williamson related indecomposable Braden-MacPherson sheaves on a Bruhat graph to parity sheaves on the corresponding flag variety, a modular counterpart of intersection cohomology complexes introduced by Juteau, Mautner and Williamson (cf. [26]). Once this connection is established, by [21,Theorem 9.2] the failure of the formula corresponds to the failure of a parity sheaf to be perverse. In this setting, our results proved that even in the cases in which parity sheaves are not perverse, they still satisfy certain elementary properties that intersection cohomology complexes have in characteristic zero. Anyway, giving a categorical version of properties of Kazhdan-Lusztig polynomials is not at all trivial and rather interesting, also in the case in which the connection between Braden-MacPherson sheaves and these polynomials is known. Indeed, the lifting at a categorical level provides us with extra structure which may help in the understanding of objects related to these ubiquitous polynomials.
While in [34] we considered rather basic equalities involving Kazhdan-Lusztig polynomials, in this paper the result we want to deal with is much more complex. It concerns a stabilisation property of the affine Kazhdan-Lusztig polynomials, which has been proven by Lusztig in [37]. More precisely, for any affine Weyl group W, Lusztig defined in [37] its periodic module M, that is a free Z[q ± 1 2 ]-module with a standard basis indexed by the set A of alcoves of W and also equipped with a structure of a module over the Hecke algebra H of W. By extending methods of [30], he found a nice basis of M and showed that the change of basis matrix from the standard basis to this one is given by certain polynomials in Z[q]: the generic polynomials, which we will denote by {Q A,B } A,B∈A . A fundamental property of these polynomials is that they are invariant under finite coweight translation, that is Q A,B = Q A+λ,B+λ for any finite coweight λ. On the other hand, we have another family of polynomials which are attached to the affine Hecke algebra: the Kazhdan-Lusztig polynomials {P x,y } x,y∈ W . Since it is possible to identify the Weyl group W with its set of alcoves, we may index any affine Kazhdan-Lusztig polynomial by a pair of alcoves and ask whether P A,B and Q A,B are related. This is precisely the point of the theorem by Lusztig we want to lift at the level of sheaves on moment graphs. Indeed, what he proved is that if the alcoves A, B are far enough in the fundamental chamber C + , the generic polynomial Q A,B coincides with the corresponding affine Kazhdan-Lusztig P A,B . The goal of this paper is hence to describe how the stalks of Braden-MacPherson sheaves on certain finite intervals of the Bruhat graph behave and in particular to show this stabilisation property.
Before outlining the structure of the paper and illustrating more in details in what way we obtain this categorical analogue of Lusztig's theorem, we want to motivate our interest in such a result. By linking the affine Hecke algebra and its periodic module, the moment graph version of the stabilisation property we mentioned above should allow us to investigate both H and M from a categorical point of view. Let g be an affine Kac-Moody algebra and W the corresponding (affine) Weyl group. The affine Hecke algebra of W controls the representation theory of g at a non-critical level and in this case moment graph techniques have already been applied (cf. [15]), while the periodic module, according to a conjecture by Feigin and Frenkel (cf. [13]), governs the the representation theory of g at a critical level, where a moment graph picture has been missing so far. We believe that the stable moment graph G stab we associate with g in this paper will be relevant in this direction. From the geometric point of view, M is related to the semi-infinite flags (cf. [12]) -also referred to as periodic affine Schubert varieties by Lusztig (cf. [39])-and we expect G stab to be connected to these objects too. It would be very interesting to understand the link between the stable moment graph and this geometric counterpart. Finally, we want to mention that the definition of G stab arises from rather intriguing combinatorics, which have been our starting point and which we are going to address in this work.
The paper is organized as follows.
The aim of the second section is to recollect the basics of the theory of affine Kac-Moody algebras, in order to fix the notation.
In Section 3 we recall the definition of the category of k-moment graphs on a lattice, stand k for a local ring with 2 ∈ k × , we introduced in [35]. Let us fix once and for all an affine Kac-Moody algebra g, a Borel subalgebra b and a Cartan subalgebra h. Let us moreover denote by W the Weyl group of g and S the set of simple reflections corresponding to the data we fixed. For any quadruple ( g ⊇ b ⊇ h, J), where J ⊆ S, we define the Bruhat moment graph G J (see §3.2). Moreover, we associate with g ⊇ b ⊇ h another k-moment graph: the periodic one, that we denote by G per and which coincides with G ∅ up to the orientation of the edges (see §3. 3).
In Section 4 we focus on the parabolic Bruhat graph G par , that is the one corresponding to the choice of J being S, the set of finite simple reflections. The vertices of this graph are given by the alcoves in the fundamental chamber and the goal of this section is to study the behaviour of intervals G par | [A,B] , with A and B far enough in C + . Our first hope was that the stabilisation phenomenon described by Lusztig would have been visible also at the moment graph level. This is unluckily not completely true. Indeed, for any fixed pair of alcoves A, B in the fundamental chamber it is possible to translate them in such a way that G par restricted to the corresponding interval and considered as unlabeled graph is invariant under translation by positive multiples of ρ, the coweight defined as half the sum of the finite positive coroots (see Lemma 4.2). Since the label function is a fundamental data a moment graph comes with, next step is to discuss the labeling of the edges of such intervals. It turns out that the set of edges is in this case bipartite in stable and non-stable 1.1. Acknowledgements. I wish to thank Peter Fiebig for encouraging me to look at the stable moment graph and for sharing his ideas with me. I would like to acknowledge Rocco Chirivì, whose suggestions were always very useful. I owe many thanks to Vladimir Shchigolev too, for his careful reading of a very preliminary version of this paper.

Affine Kac-Moody algebras
We want to fix some notation relative to affine Kac-Moody algebras. The main reference is [27, §6].
2.1. Basic notation. Let us consider a finite-dimensional simple complex Lie algebra g and b ⊇ h a Borel and a Cartan subalgebras. This data uniquely determines a set of simple roots Π = Π(b, h) and a root system ∆ = ∆ + ⊔ ∆ − , where ∆ + , resp. ∆ − , denotes the set of positive, resp. negative, roots.
We want now to consider the (untwisted) affine Kac-Moody algebra g. As a vector space, it is defined as ] ⊕ Cc ⊕ CD. If κ : g × g → C denotes the Killing form of g, then the commutation relations are as follows Moreover, there is a Cartan subalgebra of g, which corresponds to h, namely Thus the set of real roots ∆ re of g has a nice description in terms of the root system of g. Indeed, we have (cf. [27,Proposition 6.3]) while the set of positive real roots is given by is the set of (affine) simple roots.
2.2. The affine Weyl group and its set of alcoves. For any α ∈ ∆ re , let us denote by s α ∈ GL( h * ) the reflection whose action on v ∈ h * is defined by The affine Weyl group W is then generated by the s α 's with α ∈ ∆ re + , while we may identify the finite Weyl group W with the subgroup of W generated by reflections which are indexed by finite positive real roots.
Let us denote by the set of simple reflections of W. Then ( W, S) is a Coxeter system and the set T of reflections of W can be also obtained by conjugating S, i.e.
Moreover, we denote by ℓ : W → Z ≥0 the length function and by ≤ the Bruhat order on W.
Let h * R and h * R be the R-span of Π and of Π, respectively. We shall now recall another realization of W, as group of affine transformations of h * R . This is obtained by identifying h * R with the affine there is a bilinear form (·, ·) : g × g → C that induces an isomorphism ν : h → h * such that we may identify α ∨ and 2α (α,α) . Then, the action of W on λ ∈ h * , is given by Denote by Q ∨ the coroot lattice of g and by T µ the translation by µ ∈ Q ∨ , that is the transformation defined as T µ (λ) = λ + µ for any λ ∈ h * R . This is an element of the affine Weyl group, since T −nα ∨ = s α+nδ s α . It is easy to check that for any w ∈ W and for any µ ∈ Q ∨ we have wT µ w −1 = T w(µ) , so the group of translations by an element of the coroot lattice turns out to be a normal subgroup. A well known fact is that W = W ⋉ Q ∨ (cf. [23,Proposition 4.2]).
Denote by and observe that the affine reflection s α+nδ fixes pointwise such a hyperplane. We call alcoves the connected components of h * R \ α+nδ∈ ∆ re + H α,n and write A for the set of all alcoves. The fundamental (Weyl) chamber is C + := {λ ∈ h * R | λ, α ∨ > 0 ∀α ∈ Π} and an element λ ∈ C + is called dominant weight. We denote by A + the set of all alcoves contained in C + .
We state now a 1-1 correspondence between W and A (cf. [23,Theorem 4.8]). In order to do that, we fix an alcove A + , that is the unique alcove in A + which contains the null vector in its closure. A + is usually called the fundamental alcove and it has the property that every element λ ∈ A + is such that 0 < (λ, α) < 1 for all α ∈ ∆ + (cf. [23, §4.3]).
The affine Weyl group W acts on the left (by (1)) simply transitively on A (cf. [23, §4.5]) and so we obtain Let us observe that each wall of A + is fixed by exactly one reflection s ∈ S. We say that such a wall is the s-wall of A + . In general every A ∈ A has one and only one wall in the W-orbit of the s-wall of A + . This is called s-wall of A.
The affine Weyl group acts on itself by right multiplication, so it makes sense to define a right action of W on A. It is of course enough to define such an action for the generators of the group. Thus for each alcove A let As be the unique alcove having in common with A the s-wall.

Moment graphs
Let us fix once and for all a local ring k such that 2 is an invertible element. In the first part of this section we recall the definition of the category of k-moment graphs on a lattice, while in the second part we focus on certain moment graphs, which are relevant for the representation theory of affine Kac-Moody algebras.
3.1. Category of k-moment graphs on a lattice. Definition 3.1 (cf. [17]). Let Y be a lattice of finite rank. A moment graph on the lattice Y is given by (V, E, , l), where: (MG1) (V, E) is a directed graph without directed cycles nor multiple edges, is a partial order on V such that if x, y ∈ V and E : x → y ∈ E, then x y, (MG3) l : E → Y \{0} is a map called the label function. Following Fiebig's notation (cf. [17]), we will write x −−− y if we are forgetting about the orientation of the edge.
For any lattice Y of finite rank, let us denote the extended lattice by Y k = Y ⊗ Z k.
Definition 3.2 (cf. [34]). Let G be a moment graph on the lattice Y . We say that G is a k-moment graph on Y if all labels are non-zero in Y k , that is the image l(E) := l(E) ⊗ 1 is a non-zero element of Y k .
By abuse of notation, we will often denote by l(E) also its image in Y k .
This property is very important and, in the next sections, it will give a restriction on the ring k.
Definition 3.4 (cf. [34]). A morphism between two k-moment graphs , the following two conditions are verified: Given two morphisms of k-moment graphs . It is an easy check that this composition is well-defined and associative.
Definition 3.5. We denote by MG(Y k ) the category whose objects are the k-moment graphs on Y and corresponding morphisms.

3.1.2.
Isomorphisms of k-moment graphs. In [34], we gave an abstract definition of isomorphisms but we did not notice that the notion we were considering actually coincided with the one of invertible morphisms. The following lemma proves this fact.
f is an isomorphism in the categorical sense if and only if the following two conditions hold: (ISO1) f V is bijective (ISO2) for all u → w ∈ E ′ , there exists exactly one x → y ∈ E such that f V (x) = u and f V (y) = w.
Proof. To begin with we show that a morphism satisfying (ISO1) and (ISO2) is invertible. Denote by f −1 : . We have to verify that f −1 is well-defined, that is we have to check conditions (MORPH2a) and (MORPH2b). Suppose there exists an edge F : u → w ∈ E ′ , then, by (ISO2), there is an edge E : x → y ∈ E such that f V (x) = u and f V (y) = w. Since f satisfies (MORPH2a), f l,x (l(E)) = h · l ′ (F ) for h ∈ k × and we get is not satisfied, then f V , and hence f , is not invertible. Moreover, (ISO1) implies that for all u → v ∈ E ′ , there exists at most one x → y ∈ E such that f V (x) = u and f V (y) = v (otherwise f V would not be injective). Now, let f be the following homomorphism (we do not care about the f l,x 's): Isomorphisms between k-moment graphs were an important tool in [34] in order to obtain a categorical analogue of certain equalities between Kazhdan-Lusztig polynomials.
3.2. Bruhat moment graphs. From now on we fix g ⊇ b ⊇ h and keep the notation of Section 2. For any subset J ⊂ S we define the (affine) Bruhat moment graph Definition 3.6. G J is the moment graph on the affine coroot lattice Q ∨ which is given by If J = ∅, we will simply write G instead of G ∅ and call it the regular Bruhat graph of g, while, if J = S := {s α | α ∈ Π}, we will denote G par := G S and call it the parabolic Bruhat graph.
Example 3.1. Let g = sl 2 . In this case, ∆ re where α is the (unique) positive root of sl 2 and (α, α) = 2. The corresponding regular Bruhat graph G is an infinite graph, whose vertices are given by the words in two letters (s 1 := s α and s 0 ) without repetitions. Two elements are connected if and only if the difference between their lengths is odd and in this case the edge is oriented from the shorter to the longer one. Thanks to the correspondence (2), we may identify the set of vertices with the set of alcoves of g. If we restrict G to the interval [A + , s 1 s 0 s 1 A + ], we get the following graph The periodic moment graph. We want now to associate another moment graph with g. In order to do this, we need to recall the notion of generic order on the set of alcoves.
3.3.1. Two partial orders on the set of alcoves. Following [37], we provide the set of alcoves with two partial orders. First of all, the Bruhat order on W induces a partial order on A. Indeed, for all alcoves A, B ∈ A with A = xA + , B = yA + , x, y ∈ W we may set We still call it Bruhat order.
We observe that in general if we look at two alcoves it is not obvious at all if they are comparable with respect to the Bruhat order without knowing the corresponding elements in W.
Next, we recall Lusztig's definition of a nicer partial order on A, in the sense that for all pair of alcoves we will be able to say whether they are comparable and, in case, to establish which one is the bigger one. Each where H + is the half space that intersects every translate of C + . Let A ∈ A, if H is the reflecting hyperplane between A and As, s ∈ S, we consider the partial order generated by Notice that it is not clear in general how ≤ and are related. Let us denote by X ∨ the lattice of (finite) integral coweights, that is (again under the identification of h R and h * R ) (3) . Far enough inside A + , ≤ and coincide, that is for all λ ∈ X ∨ ∩ C + , A, B ∈ A the following are equivalent: (1) A B; (2) nλ + A ≤ nλ + B for n >> 0.
Because of this result Lusztig called generic Bruhat order. Remark that is invariant under translation by finite coweights.
Definition 3.7. The periodic moment graph G per = G per ( g ⊇ b ⊇ h) = (V, E, , l) is a moment graph on Q ∨ and it is given by Let us observe that we identified W and A by (2) and therefore G and G per coincide as labeled unoriented graphs, as we can see by comparing Example 3.1 with the following one.
Example 3.2. Let g = sl 2 . If we restrict the corresponding periodic moment graph to the interval [s 1 s 0 s 1 A + , s 0 s 1 A + ], we get the following moment graph.

Finite intervals of G par far enough in the fundamental chamber
This section is devoted to the description of certain intervals of the parabolic Bruhat graph G par , that is the Bruhat graph corresponding to the data ( g ⊇ b ⊇ h, S).

Two descriptions.
There are actually two descriptions of the graph G par : one identifies the set of vertices with the finite coroot lattice Q ∨ , while the other identifies the set of vertices with A + , the set of alcoves in the fundamental chamber.
As the affine Weyl group acts on h * R , we can consider the W-orbit of 0, that is the finite coroot lattice Q ∨ . Moreover Stab W (0) = W = S , the finite Weyl group, and hence W/W is in bijection with the coroot lattice via the mapping w → w(0). Clearly, for any pair of minimal length representatives x, y ∈ W/W, there exist w ∈ W and α + nδ ∈ ∆ re + such that y = s α+nδ xw if and only if On the other hand, W \ W is clearly in bijection with W/W via the mapping x → x −1 . The set of minimal representatives for the equivalence classes, under the correspondence (2), is given by the set A + of the alcoves in the fundamental chamber. Moreover, we will connect xA + , yA + ∈ A + if and only if there exist an element of the finite Weyl group w ∈ W and an affine positive root α ∈ ∆ re + such that x = wys α , that is x −1 = s α y −1 w −1 . (i) Description via the finite coroot lattice As we can see in the previous example, in the description of G par via the alcoves in the fundamental chamber, the set of edges seems to have a very complex structure, while in the other one the order on the set of vertices is hard to understand. Since we are interested in the study of intervals, the description via the finite coroot lattice turns out to be not that useful for our purposes, unless g = sl 2 . 4.1.1. The sl 2 case. If g = sl 2 , it is actually possible to give a very explicit description of G par . In this case we may identify the finite root lattice with the finite coroot lattice and then the set of vertices is V = Zα. For any pair n, m ∈ Z, it is immediate to check that then G par is a fully connected graph. Notice that, even if (4) holds for any pair of integers n and m, we do not allow loops, so n = m always. It follows Finally, observe that α = s 0 (0) and −α = s α s 0 (0); so, for any pair of n = m ∈ Z, nα < mα if and only if either |n| < |m| or n = −m > 0.
The second part of this section is devoted to showing that finite intervals of G par "far enough" in C + have surprisingly a very regular structure.

4.2.
Nice behaviour of finite intervals of G par . In this paragraph, we will consider only the description of G par in which the set of vertices coincides with A + .
Remark that the fundamental chamber C + is a fundamental domain with respect to the left action of the finite Weyl group (cf. [23, §1.12]), so the element in point (ii) is unique.
Proposition 4.1. There exists a K > 0, depending only on the root system ∆, such that if λ ∈ C + and d λ is the minimum of distances from λ to the borders of C + , then all µ ∈ C + linked to λ and such that |λ − µ| < K · d λ are strongly linked to λ.
Proof. For any λ ∈ C + and any finite positive root α ∈ ∆ + we denote by r λ,α the line {λ + αx | x ∈ R} ⊆ h + R . It is clear that the set of finite dominant weights strongly linked to λ corresponds to ( α∈∆+ r λ,α ) C + . On the other hand, we may describe the set of µ ∈ C + linked to λ as follows. Fix α ∈ ∆ + and consider the line r λ,α . Each time that such a line hits a wall of C + , it reflects off the wall and goes on this way. Let us denote by r λ,α the piecewise linear path inside of C + so obtained. Now α∈∆+ r λ,α is the set of finite dominant weights linked to λ.
Thus it is enough to show that there exists a K > 0 such that if µ ∈ r λ,α and |λ−µ| < K ·d λ , then µ ∈ r λ,α . Notice that the finite Weyl group acts on h * R as a group of orthogonal transformations, hence we may reduce to show that for all w ∈ W \ {e, s α }, the distance of the weight w(λ) from the line r λ,α is not less than K · d λ . Moreover, one may think of this reduction as an "unfolding" back r λ,α to r λ,α and considering the conjugates of λ instead of λ.
Since the distance of w(λ) from the line r λ,α is the minimum of the distances of w(λ) from λ+ xα Computing the square norm, and denoting λ w := λ − w(λ), we have: To start with, let us notice that D sαw = D w , since λ sαw = λ − w(λ) + w(λ), α ∨ α = λ w + w(λ), α ∨ α, hence: is a finite negative root, then clearly (s α w) −1 (α) ∈ ∆ + , hence, using the invariance property just proved, in what follows we may assume that w ∈ W \{e, s α } is such that Denote now by ∆ w + the set of positive roots sent to negative roots by w −1 , let C w be the (closed convex rational) cone ∆ w + R + generated by the elements of ∆ w + and notice that α is not in ±C w . Indeed, α is not in C w since all elements of this cone are sent to non-negative linear combination of finite negative roots by w −1 and, on the other hand, α is a finite positive root while all elements in −C w are non-negative linear combinations of finite negative roots.
Let L w be the set of weights λ w , where λ runs in C + and fix a reduced expression s i1 . . . s ir , with the angle between them; clearly this symbol depends only on the lines generated by u and v up to sign to change and up to supplementary angles. In particular, . Hence the map π(C w \ {0}) → R sending π(µ) → cos 2 [µ, α] achieves a maximal value M α,w and this maximal value is less than 1 since π(α) ∈ π(C w \ {0}). In particular we have cos 2 Finally, since there are only a finite number of pairs (α, w), we have M := max M α,w < 1. Now notice that w(λ) ∈ C + , because w = e, so |λ w | ≥ d λ , as the segment from λ to w(λ) must cross a wall of C + .
We have to show D w ≤ 0. Since This finishes the proof.
Let ρ be half the sum of the finite positive coroots, that is ρ = 1 2 α∈∆+ α ∨ . Moreover, for any alcove A ∈ A, let us denote by c A its centroid.
By using Proposition 4.1, together with the identification α ∨ = 2α/(α, α) for all α ∈ ∆, we get the following characterisation of finite intervals of G par which are far enough from the walls of the dominant chamber. By the previous proposition, we know that in G par |[A+mρ,B+mρ] edges adjacent to a vertex corresponding to a given alcove D are of the desired type if for any E ∈ [A + mρ, B + mρ] it holds Observe that for any n ≥ 0 and for any Moreover, if n > 0, d cF +nρ > d cF and therefore it makes sense to consider Finally, we may set We say that the edges of type (i), that is given by reflections, are stable, while the ones of type (ii), that is given by translations, are non-stable. We denote the corresponding sets E S , resp. E N S .
It is enough to translate the interval of (α + β) ∨ = α + β to get the structure described in Lemma 4.1.
For any pair A, B ∈ A + , B ≤ A and for any pair are isomorphic as oriented graphs.
Proof. Set µ := λ 2 − λ 1 . The isomorphism we are looking for is given by C → C + µ. Observe that, by Proposition 3.1, the Bruhat order coincides in the fundamental chamber with the generic one and so it is invariant by finite coweight translation; then the map we have just defined is an isomorphism of posets.
Remark 4.1. We want to stress the fact that in Lemma 4.2 we are not proving the existence of an isomorphism of moment graphs, but only between the underlying oriented graphs, that is we are not considering labels. Our first hope was that we could find a collection of {f l,C } C∈[A+λ1,B+λ1] satisfying condition (MORPh2a) and (MORPH2b). In the next two paragraphs, we will see that it is not the case. In particular, it turns out that the labels of stable edges are invariant by finite coroot translation (cf. Lemma 4.1), while the ones of non-stable edges are not (cf. Lemma 4.4).
From now on we will denote by w ∈ W the corresponding alcove wA + ∈ A, thanks to the identification (2) of the affine Weyl group with its set of alcoves. In particular, if wA + is contained in the fundamental chamber, we will write w ∈ A + . 4.2.1. Stable edges. Let | S| = n and fix a numbering of the simple reflections. We define the permutation σ A,µ ∈ S n , for A ∈ A and µ ∈ X ∨ , in the following way: σ A,µ (i) = j if the image under the translation by µ of the s i -th wall of A is the s j -th wall of A + µ (cf. §2.2). Let W the extended affine Weyl group, that is W = W ⋉ Ω, where Ω := X ∨ /Q ∨ (cf. [38]). Lemma 4.3. For any µ ∈ X ∨ the permutation defined above is independent on A ∈ A, i.e. there exists σ µ ∈ S n such that σ A,µ = σ µ for any alcove A.
Proof. We know that T kα ∨ = s −α+kδ s α for k ∈ Z ≥0 and α ∈ ∆. Since we are reflecting twice in the same direction (orthogonal to α), the walls of A + kα ∨ have the same numbering as the ones of A.
Let us denote (by abuse of notation) also by σ µ : Q ∨ → Q ∨ the automorphism of the affine coroot lattice induced by the map α i → α σµ(i) for α i corresponding to the simple reflection s i ∈ S. Let us observe that σ µ preserves the positive cone Π R ≥0 by definition, and the following result is straightforward.

4.2.2.
Non-stable edges. Now we describe how labels of non-stable edges change. In order to do so, we need the following result In the proof of Lemma 4.4 we will also need the following equality (ii) Let y = T aα ∨ x, for some a ∈ Z and α ∈ ∆ + . Let moreover µ ∈ X ∨ , ω ∈ Ω and γ ∈ Q ∨ be such that T µ = ωT γ . Then, if y, T µ x, T µ y ∈ A + , Clearly, to get a reflection, we have to choose u = s α .
The result then follows from (7) and the fact that w(α) (this expression in unique) and we may apply Lemma 4.3 and point (i) of this lemma with T γ x instead of x to get The stable moment graph. We are now ready to define the main character of this paper: the stable moment graph G stab . This is the moment graph having as set of vertices the alcoves in the fundamental chamber (that we identify with the corresponding elements of the Weyl group), equipped with the Bruhat order (that here coincides with the generic one); we connect two vertices if and only if there exists a real positive root α ∈ ∆ re + such that y = xs α , and in this case we set Then we have: For any interval [y, w] and for any µ ∈ X ∨ there exists an isomorphism of k-moment Proof. Since the order on the set of vertices of G stab is invariant by finite coweight translation, we have an isomorphism of posets given by the map f V : x → x + µ. This map induces also a bijection between set of edges, as we have already seen in the proof of Lemma 4.2.
For any x ∈ [y, w] we set f l,x = σ µ (see Lemma 4.3) and the data (f V , {f l,x }) gives us an isomorphism of k-moment graphs for any k.

Sheaves on moment graphs
The notion of sheaf on a moment graph is due to Braden and MacPherson (cf. [7]) and it has been used by Fiebig in several papers (cf. [14], [15], [19], [17], [18]). In the first part of this section, we recall the definition of category of sheaves on a k-moment graph and we present two important examples, namely, the structure sheaf and the canonical -or BMP-sheaf. In the second part, for any homomorphism of k-moment graphs f , we define the pullback functor f * and the push-forward functor f * . These two functors turn out to have the same adjointness property as in classical sheaf theory (see Proposition 5.1).

5.1.
The category of sheaves on a k-moment graph. As in the previous sections, let Y be a lattice of finite rank and k a local ring (with 2 ∈ k * ). Let us denote the symmetric algebra of Y by S = Sym(Y ) and set S k := S ⊗ Z k its extension. As a polynomial ring, S k has a natural Z-grading, but we keep the convention (coming from geometry) of doubling it, that is we set (S k ) {2} = Y k . From now on, all the S k -modules will be finitely generated and equipped with this Z-grading. Moreover, we will consider only degree zero morphisms between them. Finally, for j ∈ Z and M a graded S k -module we denote by M {j} the graded S k -module obtained from M by shifting the grading by j, that is M {j} {i} = M {j+i} .
Definition 5.1 (cf. [7]). Let G = (V, E, , l) ∈ MG(Y k ), then a sheaf F on G is given by the x,E }) be two sheaves on it. A morphism ϕ : F −→ F ′ is given by the following data such that, for any x ∈ V on the border of E ∈ E, the following diagram commutes We denote by Sh k (G) the category, whose objects are the sheaves on G and whose morphisms are as in Definition 5.2.

5.2.
Combinatorial sheaf theory. In this paragraph we recall or introduce notions as space of sections, flabbiness, pullback and pushfoward functors, which mimic the corresponding ones in classical sheaf theory.

5.2.1.
Sections of sheaves. Let us consider a sheaf F ∈ Sh k (G) and let I be a subset of the set of vertices V of G. The the set of local sections of F over I is defined as follows.
We write Γ(F ) for Γ(V, F ) and we call it the set of global sections of F .

5.2.2.
The structure sheaf and the structure algebra. . With any k-moment graph, it is possible to associate its structure sheaf Z , that is the sheaf on G given by Z x = S k for all x ∈ V, the canonical quotient map, for any vertex x ∈ V on the edge E ∈ E. Then the structure algebra Z of G is the set of global sections of Z , namely The symmetric algebra S k acts on the structure algebra via diagonal action and it is easy to check that Z is actually an algebra under componentwise addiction and multiplication.

Flabby sheaves.
Once obtained the analogue of the spaces of sections, we would like to define the concept of flabby sheaves. Clearly, in order to do so, the notion of open set is needed. We declare I ⊆ V to be open if it is upwardly closed, that is if and only if whenever x ∈ I and y ≥ x, then also y ∈ I.
From now on, for any ϕ ∈ Aut k (Y k ), we will denote by ϕ also the automorphism of S k that it induces.
We need a lemma, in order to make consistent the definitions we are going to give.
(ii) the twisted actions of S k on H F defined via s n F := f l,x (s) · n F and s n F := f l,y (s) · n F coincide on H F /l(E) H F (· denotes the action of S k on F E before the twist). Moreover, l(E) H F = {0} in both cases.
Proof. It is enough to prove the claim for s ∈ (S k ) {2} = Y k , since S k is a k-algebra generated by Y k . If ϕ is an automorphism of S k , for any S k -module M , we will denote Tw ϕ : M → M the map sending M to M and twisting the action of S k on M by ϕ.

Definitions.
and the action of S k is twisted in the following way: , where x is on the border of E (PUSH3) for all u ∈ V ′ and F ∈ E ′ , such that u is in the border of the edge F ,(f * ρ) u,F is defined as the composition of the following maps: . We call f * direct image or push-forward functor.
for all x ∈ V and E ∈ E, such that x is in the border of the edge E, We call f * inverse image or pullback functor.
Example 5.1. Let G ∈ MG(Y k ) and let p : G → {pt} be the homomorphism of k-moment graphs having p l,x = id Y k for all x, vertex of G. Then, for any F ∈ Sh k (G) p * (F ) = Γ(F ). Moreover p * (S k ) = Z , the structure sheaf of G.

5.3.2.
Adjuction formula. Although the following result will not be used in the rest of the paper, we include it for completeness.
, then f * is left adjoint to f * , that is for all pair of sheaves F ∈ Sh k (G) and H ∈ Sh k (G ′ ) the following equality holds such that for all x ∈ V and E ∈ E, with x is on the border of E, the following diagram commutes We start by verifying that this map is well-defined. We have to show that for any h ∈ H u , From Diagram (11), we get the following commutative diagram (12) ( x,E by definition (they are both the canonical projection) and we obtain To conclude our proof, we have to show the surjectivity of γ. Suppose where, for all u ∈ V ′ and F ∈ E ′ such that u is on the border of F , the following diagram commutes For any x ∈ V, let us consider u := f V (x) and define ϕ x as the composition of the following maps , that is there exists an edge F ∈ E ′ such that f E (E) = F , we define ϕ E as the composition of the following maps , then ψ u = 0 and the claim is trivial. Otherwise, u ∈ f V (V) and we get the following diagram, with Cartesian squares As application of the previous proposition, we get the following corollary.
In particular, we get the following isomorphism of S k -modules Proof. Consider the homomorphism p : G → {pt}, where we set p l,x = id Y k for all x, vertex of G.
The structure sheaf of {pt} is just a copy of S k and, for all F ∈ Sh k (G), by Prop. 5.1, we get Bu we have already noticed in Example 5.1 that p * S k ∼ = Z and p * F = Γ(F ). Moreover, that Hom S k (S k , Z) ∼ = Z and we get the claim.

BMP-sheaves.
The following definition, due to Fiebig and Williamson, generalises the one of canonical sheaves given by Braden and MacPherson in [7]. These sheaves will play a fundamental role in the rest of this paper.
Definition 5.7 (cf. [21]). Let G ∈ MG(Y k ) and let B be a sheaf on it. We say that B is a Braden-MacPherson sheaf if it satisfies the following properties: (BMP1) for any x ∈ V, B x ∈ S-mod is free (BMP2) for any E : x → y ∈ E, ρ y,E : B y → B E is surjective with kernel l(E) · B y (BMP3) B is flabby (BMP4) for any x ∈ V, the map Γ(B) → B x is surjective.
Hereafter, Braden-MacPherson sheaves will be referred to also as BMP -sheaves or canonical sheaves.
An important theorem, characterizing Braden-MacPherson sheaves, is the following one.
Theorem 5.1 (cf. [21], Theorem 6.3). Let G ∈ MG(Y k ) (i) For any w ∈ V, there is up to isomorphism unique Braden-MacPherson sheaf B(w) ∈ Sh k (G) with the following properties: (ii) Let B be a Braden-MacPherson sheaf. Then, there are w 1 , . . . , w r ∈ V and l 1 . . . l r ∈ Z such that We want to quote also a result by Fiebig, that will be used later, which tells us that the structure sheaf is not in general flabby, but that if it is the case, then it is isomorphic to an indecomposable BMP-sheaf.

5.4.1.
Pullback of BMP-sheaves. We conclude this section by recalling a result from our paper [34].
Lemma 5.2 (cf. [34]). Let G and G ′ be two k-moment graphs on Y , both with a unique maximal vertex, w resp. w', and let f : G −→ G ′ be an isomorphism. If B w and B ′ w ′ are the corresponding canonical sheaves, then B w ∼ = f * B ′ w ′ as k-sheaves on G.

Statement of the main result and proof of the subgeneric case
In order to provide a new approach to Kazhdan-Lusztig amd Lusztig conjectures, Fiebig applied the theory of sheaves on moment graphs to the representation theory of complex symmetrisable Kac-Moody algebras and of semisimple reductive algebraic groups over a field of positive characteristic and formulated a conjectural formula relating Kazhdan-Lusztig polynomials to the stalks of indecomposable Braden-MacPherson sheaves (cf. [15], [17], [19]). More precisely, if G J is the Bruhat graph we defined in §3.2, thus for any w ∈ W J we can consider the subgraph G J w := G J |{≤w} . It is a finite k-moment graph (for any k) with highest vertex w, hence we may build the corresponding indecomposable Braden-MacPherson sheaf B J (w) ∈ Sh k (G J ). For any free and finitely generated S k -module M = n i=1 S{l i }, its graded rank is Finally, let us denote by P J,−1 y,w Deodhar's analogue of Kazhdan-Lusztig polynomials at the parameter u = −1 (cf. [9]). Question 6.1 (Cf. [17], Conjecture 4.4.). Under which assumptions on the characteristic of the base field, if y ≤ w then rk (B J (w)) y = P J,−1 y,w (q)? This question motivated our paper [34], where we were able to give the moment graph analogue of several properties of Kazhdan-Lusztig polynomials. The interpretation of certain equalities of these polynomials at a categorical level has the advantage of furnishing a deeper understanding of several phenomena, since they are lifted to a setting where there is extra structure. The problem we address in this paper concerns a stabilisation property of the affine Kazhdan-Lusztig polynomials, proved by Lusztig in [37], while he was trying to find a support for his conjecture on modular representations.
Let W denote a finite Weyl group, W its affinisation and S the set of finite simple reflections. Then, as we have already noticed, the set of minimal representatives of W \ W is in bijection with the set of alcoves in the fundamental chamber and the corresponding parabolic Kazhdan-Lusztig polynomials may be indexed by pairs of alcoves A, B ∈ A + . The result that Lusztig proved can be reformulated in terms of these polynomials. In particular, quoting Soergel's reformulation (cf. [40, Theorem 6.1]), the parabolic Kazhdan-Lusztig polynomials P J,−1 A,B indexed by pairs of alcoves far enough in the fundamental chamber stabilize, in the sense that, for any pair of alcoves A, B, there exists a polynomial Q A,B with integer coefficients such that The Q A,B 's are called generic polynomials and turn out to have a realization very similar to the one of the regular Kazhdan-Lusztig polynomials. Indeed, Lusztig in [37] associated to every affine Weyl group W its periodic module M, that is the free Z[q ± 1 2 ]-module with set of generators -or standard basis-indexed by the set of all alcoves A. It is possible to define an involution and to prove that there exists a self-dual basis of M: the canonical basis. In this setting, the generic polynomials are the coefficients of the change basis matrix. Our interest in the periodic module is motivated by the fact that M governs the representation theory of the affine Kac-Moody algebra, whose Weyl group is W, at the critical level.
The aim of this section is to study the behaviour of indecomposable Braden-MacPherson sheaves on finite intervals of the parabolic Bruhat graph G par far enough in C + (see §4). More precisely, let I = [A, B] be an interval far enough in the fundamental chamber. Inspired by [37,Proposition 11.15], we claim that for any C ∈ [A, B] and for all µ ∈ X ∨ ∩ C + , We showed in §4 that G par | [A,B] is in general not isomorphic to G par | [A+µ,B+µ] as a k-moment graph, so unluckily we cannot use Lemma 5.2 to get the isomorphism of S k -modules above. On the other hand, we proved in Lemma 4.5 that, for all µ ∈ X ∨ , there is an isomorphism of k-moment graphs . For any finite interval I far enough in the fundamental chamber, consider the monomorphism g I : G stab |I ֒→ G par |I , given by g I,V = id V and g I,l,x = id for all x ∈ I. We obtain the functor defined by the setting F → F stab := g * I (F ). The goal of the rest of this paper is to prove the following result. Theorem 6.1. Let I be a finite interval far enough in the fundamental chamber and let k be such that ( G par |I , k) is a GKM-pair. Then the functor · stab : Sh k (G par |I ) → Sh k (G stab |I ) preserves indecomposable Braden-MacPherson sheaves.
We will prove this theorem via explicit calculations in the sl 2 -case, while for the general case we will need results and methods developed by Fiebig in [19].
Once Theorem 6.1 is proven, we will obtain Equality (14) by applying Lemma 4.5.
6.1. The subgeneric case. Using the same terminology as Fiebig in [20], we denominate subgeneric the case in which g = sl 2 .
In the subgeneric case, for any finite interval I and any k such that ( G par |I , k) is a GKM-pair, the corresponding indecomposable BMP-sheaf is isomorphic to the structure sheaf A of G par |I . There are several ways to prove this fact.
The first one comes from the topology of the underlying variety. We try to give a roughly idea of this argument. To start with, let us suppose k = Q. In the case of a finite interval of a Bruhat moment graph, its structure algebra and the space of global sections of a BMP-sheaf describe the T -equivariant cohomology and intersection cohomology, respectively, of the corresponding Richardson variety. Since all the Richardson varieties are in the subgeneric case rationally smooth, the intersection cohomology coincides with the usual cohomology and then the structure sheaf with the canonical sheaf. It is possible to perform the same argument, using parity sheaves and results from [21] in the positive characteristic setting.
An alternative proof comes from Fiebig's multiplicity one result. Indeed, using the inductive formula (2.2.c) of [30], it is easy to show that, P J−,1 A,B = 1 for all A, B ∈ A + with A ≤ B. We then have to apply the following theorem A third proof is presented in the appendix of this paper and it makes use only of some combinatorial arguments together with intrinsic properties of indecomposable BMP-sheaves.
In this paragraph, G stab , resp. G par , denotes the parabolic moment graph, resp. the stable moment graph, for the A 1 root system.
We have already seen that for any two vertices x, w with x ≤ w the stalk of the Braden-MacPherson sheaf on G par ≤w is B(w) x ∼ = S k . This fact, by Proposition 5.2, is equivalent to the flabbiness of the structure sheaf on G par ≤w . Therefore in order to show that the functor stab preserves indecomposable canonical sheaves, again by Proposition 5.2, it is enough to verify that, for any vertex w, the structure sheaf A on G stab ≤w is still flabby. Recall that the set of vertices of G par (and so of G stab ) can be identified with the finite (co)root lattice, that is Zα, where α = α ∨ is the positive (co)root of A 1 . Moreover, G par is a complete graph and the label function is given, up to a sign, by l(hα −−− kα) = −α + (h + k)c. By definition, we get G stab from G par by deleting the non-stable edges, then hα −−− kα ∈ E stab if and only if sgn(h) = −sgn(k) (where, by convention, we set sgn(0) = −). Lemma 6.1. Let r ∈ Z >0 . If n ∈ Z, set, for any h ∈ Z, with hα ≤ nα, Proof. We verify that, for any h, k ∈ Z such that hα, kα ≤ nα, if hα −−− kα is an edge, then (15) e z r nα,hα − e z r nα,kα ≡ 0 (mod − α + (h + k)c) We may clearly suppose h > 0 and k ≤ 0. Let at first consider h ∈ [|n| − r + 1, n]. If −k ∈ [|n| − r + 1, n], then e z r nα,hα = e z r nα,kα = 0 and there is nothing to prove. Otherwise, k ∈ [r − |n|, 0] and (16) e z r nα,hα − e z r nα, Let consider the case h ∈ (0, |n| − r]. If −k ∈ [|n| − r + 1, n], then The proof is very similar to the one of the previous lemma and therefore we omit it.
Define e z 0 nα := (1) hα≤nα . Lemma 6.3. Let r ∈ Z ≥0 , n ∈ Z and m ∈ Z be such that mα ≤ nα. If ( G stab ≤nα , k) is a GKM-pair, for all z ∈ Γ([mα, nα], A) {r} , there exist o s i k , e s j k ∈ (S k ) {i} , with i ∈ [0, r], j ∈ (0, r] and p such that pα ∈ [mα, nα], such that Proof. Let hα be the maximal vertex in [mα, nα] such that z hα = 0. We prove the statement by induction on l = ♯[mα, hα]. If such a vertex does not exists, that is l = 0, then z = (0) and there is nothing to prove. We should consider four cases: n > 0 and l > 0; n > 0 and l ≤ 0; n ≤ 0 and l > 0; n ≤ 0 and l ≤ 0. Actually, we will verify only the first case, since the others can be proven in a very similar way.
Let n > 0 and h > 0. If h = n, then we set z ′ = z − z e nα z 0 nα and the result follows from the inductive hypothesis. Otherwise, h < n and then {r} has the property that z ′ pα = 0 for all p ∈ [hα, nα] and we obtain the statement from the inductive hypothesis. By Lemma 6.1 and Lemma 6.2, z is a sum of extensible sections, and so it is extensible as well.
Finally, we get the following theorem. Theorem 6.3. Let g = sl 2 , I be a finite interval far enough in the fundamental chamber and let k be such that ( G par |I , k) is a GKM-pair. Then for every finite interval I, the functor · stab preserves indecomposable canonical sheaves.

General case
In order to prove Theorem 6.1, we have to show that, for any interval I far enough in the fundamental chamber, if B is an indecomposable Braden-MacPherson sheaf on G par |I , then B stab is indecomposable and satisfies properties (BMP1), (BMP2), (BMP3), (BMP4). Proof. To begin with, let us observe that (BMP1) and (BMP2) follow immediately from the definition of the pullback functor, as · stab = i * I . Moreover G stab |I is obtained from G par |I by deleting the non-stable edges and hence there is an inclusion By assumption there is a surjective map Γ(I, F ) → F x = (F stab ) x and, from the inclusion above, it follows that the map Γ(I, F stab ) → (F stab ) x is surjective too.
Therefore it is only left to show that B stab is a flabby indecomposable sheaf on G stab |I .
7.1. Flabbiness. Let G denote the regular Bruhat graph of g, while let G per be the periodic graph of §3.3. It is possible to define a functor · per : Sh G → Sh G per in a very easy way. Let F = ({F x }, {F E }{ρ x,E }), then we set (F per ) x = F x for any x ∈ V, (F per ) E = F E for any E ∈ E and ρ per x,E = ρ x,E . A fundamental step in the proof of the flabbiness of B stab consists in showing that · per maps canonical sheaves to flabby sheaves.

Translation functors.
Here we recall the definition of (left and right) translation functors and of the corresponding categories of special modules.
Let us denote by Z the structure algebra of the (affine regular) Bruhat graph G. To any simple reflection s ∈ S, we associate the two following automorphisms of the structure algebra.
First, let σ s be defined as follows.
Secondly, if τ s is the automorphism of the symmetric algebra S k induced by the mapping λ → s(λ) for all λ ∈ h * , we set: Let Z − mod f be the category of Z-modules which are finitely generated and free as S k -modules. Moreover, let us denote by Z s and s Z the S k -sub-modules of Z consisting of σ s -and s σ-invariants, respectively. To start with, let us observe that Z is a Z s -and s Z-module under pointwise multiplication.. Let c s = (c s x ) x∈ W be the element of Z given by c s x = x(α s ), while α s ∈ Z denotes the constant section whose components are all equal to α s . Thus the following lemma tells us that Z decomposes in a nice way as Z s -module, resp. s Z-module Lemma 7.2.
(i) There is a decomposition Z = Z s ⊕ c s Z s of Z s -modules (cf. [14], Lemma 5.1) (ii) There is a decomposition Z = s Z ⊕ α s Z s of Z s -modules (cf. [35], Lemma 4.1) Definition 7.1.
(i) -The (right) translation on the wall is the functor θ s,on : Observe that this functor is well-defined due to Lemma 7.2.
-By composition, we get a functor θ s := θ s,out • θ s,on : Z − mod f → Z − mod f that we call (right) translation functor. (cf. [14]). Remark 7.1. The advantage of defining left translation functors is that they can be applied also in the parabolic setting. Indeed, if Z J is the structure algebra of the Bruhat graph is still well-defined for any finitary J ⊆ S (cf. [35], where the definition of left translation functors lead to a categorification of a parabolic Hecke module). In the case of J = S, the set of finite simple reflections, we will denote the corresponding translation functor by s θ par . First of all we need to set some notation. Let Q be the quotient field of S k and, for any M ∈ Z − mod f , let us write M Q := M ⊗ S k Q. If K is a Bruhat graph, then (see §3.1 of [15]) there is a decomposition M Q := M ∩ x∈V M x Q and so a canonical inclusion M ⊆ x∈V M x Q . For all subsets of the set of vertices Ω ⊆ V, we may define: For any vertex x ∈ V, we set For m = (m x , m y ) ∈ M (E), let us set π x ((m)) = m x , π y ((m)) = m y and define L (M ) E as the push-out in the following diagram of S k -modules: In this way, we also get the maps ρ x,E and ρ y,E . Finally, from [15,Theorem 3.5], the functor L is left adjoint to the global section functor From now on, we will deal again with Bruhat moment graphs and we will denote once again by Z the structure algebra of the regularaffine Bruhat graph G. First of all, let us notice that, being the structure algebra of a moment graph independent from the partial order on the set of vertices, Z coincides with the structure algebra of the periodic moment graph G per . Therefore we may summarise this setting as follows.
Since for a BMP-sheaf F on G it holds L • Γ(F ) ∼ = F (cf. [15]), our claim is equivalent to the fact that L (Γ(F per )) = L (Γ(F )) per is a flabby sheaf on G per if F is a Braden MacPherson sheaf. We will prove it using translation functors. 7.1.3. Three properties of the Bruhat order. Let us consider the following properties of the Bruhat order.
(PO1) The elements w and tw are comparable for all w ∈ W and t ∈ T . The relations between all such pairs w, tw generate the partial order. (PO2) We have [w, ws] = {w, ws} for all w ∈ W and s ∈ S such that w < ws. (PO3) (i) For x, y ∈ W such that x < xs and y ≤ xs we have ys ≤ xs.
(ii) For x, y ∈ W such that xs < x and xs ≤ y we have xs ≤ ys.
From now on, will denote a partial order on W having the same properties as above. And let G be the moment graph on Q ∨ obtained from the regular Bruhat graph G by changing the orientation of the edges according to this new partial order. This graph still satisfies the property of not having oriented cycles, thanks to (PO1) above. In [14] Fiebig proved that translation functors preserve ≤-flabby object, while he was proving them to preserve a stronger property of objects in Z − mod f . In what follows, we are going to recall the main steps of his argument and to point out that, whenever a proof depends on the Bruhat order on the set of vertices, it actually depends only on the properties we listed before 7.1.4. The moment graph G s . A first fundamental observation made by Fiebig is that it is possible to identify the set of invariants Z s with the structure algebra of the Bruhat moment graph G s = G ( g ⊇ b ⊇ h, {s}) (cf. [14,  §5.1] ). We were able to define Bruhat moment graphs, since the Bruhat order is preserved by the quotient map (see [5,Proposition 2.5.1]). We want now to show that it makes sense to define a quotient graph of G too, if J consists of only one simple reflection. Let be a partial order having the above properties, then for any s ∈ S, we will denote W s the set of minimal representatives wrt of the equivalence classes of W/{e, s}. Moreover, for any x ∈ W let us set x := min{x, xs}, where the minimum is taken with respect to . This leads us to the definition of the moment graph G s on Q ∨ , having as set of vertices W s , equipped with the partial order , edges x → y if either x −−− y or x −−− ys was an edge of G, in which case the label is the same. It is clear that Z s coincides with the structure algebra of this graph Z ′ . Again, we say that N ∈ Z ′ −mod f is -flabby if L (N ) is a flabby sheaf on G s . For all I ⊆ W, let us denote by I the image of I in W s under the map u → u . The following two results do not depend on the partial order. By applying once again Lemma 7.4, θ s,on (Γ • L (M I )) = θ s,on (Γ(I, L (M ))) and hence the statement follows from the assumption of M being -flabby, since θ s,on is right-exact. 7.1.5. Local definition of translation out of the wall. In order to prove that also θ s,out preservesflabby objects, we have to give a realization of it in terms of sheaves. Let us consider N ∈ Z ′ −mod f such that it θ s,out (N ) is isomorphic to Γ(L (θ s,out (N ))). Let us set Let us denote by p s : G → G s the morphism of k-moment graphs defined as p s,V : x → x and p s,l,x = id for all x ∈ W.
We may now reformulate Lemma 5.4 of [14] in this notation. The proof only needs (PO1), (PO2), (PO3) of . Lemma 7.7. The affine Weyl group W equipped with the partial order is a directed poset.
Proof. By induction on ℓ(u) + ℓ(v). The base step is u = v = e and in this case we can take z = e. Now let u = s 1 s 2 · · · s r with s 1 , s 2 , . . . , s r simple reflections and r ≥ 1. Let us set u ′ := us r ; since ℓ(u ′ ) < ℓ(u) there exists z ′ be such that u ′ , v z ′ . By (PO2), we know that u and u ′ are comparable, so we have two cases.
If u ⊳ u ′ then we take z = z ′ and our claim follows from the inductive hypothesis. Otherwise, u ′ ⊳ u and we have again to consider two cases for z ′ , z ′ s r . Suppose first that z ′ ⊳ z ′ s r . Let x = z ′ and y = xs r and notice that x = z ′ < z ′ s r = xs r and y = us r z ′ = zs r , so we may apply (PO3) (i) and find u = ys r xs r = z ′ s r . So in this case we take z = z ′ s r since we have also v z ′ < z ′ s r = z.
Finally suppose z ′ s r ⊳z ′ . Let x = z ′ s r and y = us r and notice that x⊳xs r and y = us r z ′ = xs r , so we may again apply (PO3) (i) and find u = ys r xs r = z ′ . So we take z = z ′ and our claim is proved.
For u, v ∈ W we will write u ⊳ v if u v and [u, v] = {u, v}. The following lemma generalises [5, Corollary 2.2.8].
Lemma 7.8. Let s ∈ S and t ∈ T be such that s = t. Let moreover w ∈ W be such that w ⊳ ws, wt. Then ws, wt ⊳ wts.
Proof. The claim follows by applying repeatedly (PO3).
To start with, by (PO3) (ii) with x = ws and y = wt, we get w wts. Since s = t, it holds wts = w and hence wts ⊳ wt. Indeed, if it were not the case, wts ∈ [w, wt] \ {w, wt} but by assumption [w, wt, ] = {w, wt}. From (PO2) we deduce that wts ⊳ wt. Now we apply (PO3) (i) with x = wt and y = w and it follows ws wts. It is only left to show that [ws, wts] = {ws, wts}.
Suppose there were a z ∈ [ws, wts] \ {ws, wts}. By (PO1), either z ⊳ zs or zs ⊳ z. In the first case, by applying twice (PO3) (i) (once with y = ws and x = z and once with y = z and x = wts) we obtain zs ∈ [w, wt] = {w, wt}, which is a contradiction, since we supposed zs = ws, wts. On the other hand, if zs ⊳ z, we repeat the same argument as before, once z has been substituted by zs, and we get z ∈ [w, wt]. It follows that either z = w or z = wt, which is not possible because w ⊳ ws and wt ⊳ wts, while we assumed zs ⊳ z.
Lemma 7.9. Let u, v ∈ W be such that u ⊳ v and let s ∈ S be such that vs ⊳ v and us ⊳ u. Then for any z ∈ (us, v] \ { u} which is minimal, any maximal chain from z to v of length k induces a maximal chain from us to v of the same length. Proof. By the minimality assumption, us ⊳ z = z 0 and, (PO1), there exists a t 0 ∈ T such that z 0 = ust 0 . From Lemma 7.8 with w = us and t = t 0 , we get z 0 , u ⊳ z 0 s. If z 1 = z 0 s, we obtain a maximal chain us ⊳ z 0 s ⊳ . . . ⊳ z k−1 ⊳ z k = v Otherwise, we apply again Lemma 7.8 with z 0 = w and t = t 1 . Once again, if z 2 = z 1 s, we have done and if not we proceed as before. In this way we get a chain from us to v of length k.
Observe that all the z i s are still in the interval, since z i s v, by (PO3) (i) with x = vs and y = z i .
For any u, v ∈ W with w v let us denote by d(u, v) the supremum of the lengths of all maximal chains from u to v. Corollary 7.1. Let u, v ∈ W be such that u ⊳ v and let s ∈ S be such that vs ⊳ v and us ⊳ u. Let z ∈ (us, v] \ { u} be not minimal, then d(z, v) < d(u, v). Proposition 7.2. Let N ∈ Z ′ −mod f be such that the adjunction morphism N → Γ(L (N )) is an isomorphism. If N is -flabby, then also θ s,out (N ) ∈ Z − mod f is -flabby.
Proof. Let us keep the same notation as before. We claim that, for any set I ⊆ W upwardly closed wrt the canonical map Γ(M )) → Γ(I , M ) is surjective.
If Is = I, the proof is very similar to the one of Lemma 7.5, therefore we omit it. Let us set { u} = {v ∈ W | v u}, and analogously for {⊲u} then by [15,Proposition 4.2] it is enough to demonstrate the surjectivity of the following map for all u ∈ W Let us now assume us ⊳ u and consider the following set Observe that supp(M ) is finite, since otherwise Γ(M ) would not be a finitely generated S k -module. Therefore, by Lemma 7.7, we find a v ∈ W such that supp(M ) ⊆ { v}. Let us observe that we may assume vs ⊲ v without loss of generality.
Let us now proceed by induction on d(u, v). The base step is u = v and it is trivial. Otherwise, by Corollary 7.9, we may suppose by induction that m extends to any vertex z ∈ (us, v] \ {us, u} which is not minimal. We get in this way a section m ′ . Next, let us suppose z ⊳ us and z = u. In this case z ⊳ zs. Indeed, suppose it were not the case. By (PO1) z and zs are comparable and by (PO3) (ii) with x = u and y = z, it holds us zs. This contradicts the assumptions z ⊳ us and z = u. We have already proven that it is possible to extend m ′ |⊲z to the vertex z. Since the minimal vertices are not comparable, there are not edges between them and hence we extended m ′ to a section m ′′ ∈ Γ({ us} \ {u, us}, M ). But { us} \ {u, us} is now an s-stable set and therefore m ′′ extends to a global section and in particular to the vertex u.
Finally, as θ s = θ s,out • θ s,on and since Lusztig proved that the generic order has also properties (PO1), (PO2) and (PO3) (cf. [37]), we get the following theorem. (ii) The category of parabolic special modules is the full subcategory H par of Z par − mod f whose objects are isomorphic to a direct summand of a direct sum of modules of the form si 1 θ par • . . . • si r θ par (B J e ) n , where s i1 , . . . , s ir ∈ S and n ∈ Z. A fundamental characterisation of special modules is the following one. Thanks to the above result, we are now able to prove the following, which will make clear why we are dealing with such objects. Proof. We want to show that F = Γ(F ) is a flabby object in Z per − mod f . By Theorem 7.2, we know that F ∈ H, so we may prove our result by induction. If F = B e , there is nothing to prove. We have to show now that, if the claim is true for M ∈ H, then it holds also for θ s (M ) ∈ H, that, again by Theorem 7.2, is still isomorphic to the global sections of a Braden-MacPherson sheaf on G. But now by the inductive hypothesis we get that M is a flabby object in Z per − mod f and so, by applying Theorem 7.1, θ s (M ) = θ per s (M ) is also a flabby object in Z per − mod f . 7.1.7. Decomposition of the functor · stab . The functor · stab may be obtained by composing the five following functors.

Where
• i : G par |I ֒→ G par and j : G per |I ֒→ G per are the morphisms of moment graphs induced by the corresponding inclusions of subgraphs • p par : G → G par is the quotient homomorphism defined by p par,V : x → x par , the minimal representative of xW, and p par,l,x = id Q ∨ for all x • · opp is the pullback of the isomorphism of moment graphs f : G stab |I → G per |I defined as f V = id Q ∨ and f l,x (λ) = x −1 (λ) for all x ∈ I and λ ∈ Q ∨ Some properties of the above functors are needed.
Proposition 7.4. The functors p * par and · opp preserve indecomposable BMP-sheaves. Proof. The fact that p * par maps BMP-sheaves to BMP-sheaves is just a particular case of Theorem 6.2 of [34] in the reformulation we gave in [35], Theorem 6.1.
In Lemma 5.2 of [34], we proved that f defined as above is an isomorphism and hence, by Lemma 5.2, the pullbak · opp preserves Braden-MacPherson sheaves. Proof. The statement follows by combining the results of this section, together with the fact that the functors i * and j * clearly map flabby sheaves to flabby sheaves. 7.2. Indecomposability. Here we prove the only step missing in the proof of Theorem 6.1. From now on, we will denote by W par the set of minimal representatives for the equivalence classes of W/W. 7.2.1. Localisation of special Z par -modules. Let β ∈ ∆ + , we consider the following localisation of the symmetric algebra S k : (21) S β k := S k [(α + nδ) −1 | α ∈ ∆ + \ {β}, n ∈ Z] Fiebig used this localisation in [19], in order to relate the category of regular special modules to a category introduced by Andersen, Jantzen and Soergel in [1].
Let us denote by W β the subgroup of W generated by the affine reflections s β,n , for n ∈ Z, and by W β the set of orbits for the left action of W β on W par . Remark that the group W β is isomorphic to A 1 . For any subset Ω ⊆ W par , let us write moreover Z par,β (Ω) := Z par (Ω) ⊗ S k S k,β . Lemma 7.10 (cf. [19], Lemma 3.1). Let Ω ⊂ W par be finite, then Proof. Omitted, since Fiebig's proof of [19,Lemma 3.1] works exactly the same in this parabolic setting too.
For M ∈ H par , we set M β := M ⊗ S k S k,β . Because any special module is a module on Z(Ω) for some Ω ⊂ W par finite (see [35,Lemma 4.3]), the decomposition of the previous lemma induces the following decomposition.
Next we are going to show that this localisation procedure preserves special modules. In particular, we prove that, under the localisation, a special module having support on a finite interval far enough in the fundamental chamber splits in a direct sum of special modules for the subgeneric parabolic structure algebra, that is the one corresponding to the case g = sl 2 .
Lemma 7.11. Let M ∈ H par such that Z par acts on it via Z par (I), for I a finite interval far enough in C + and M β = Θ∈W β M β,Θ , then, for any Θ ∈ W β , M β,Θ is isomorphic to a Z par ( sl 2 )-special module.
Proof. We prove by induction that any M β,Θ is a special module for the structure algebra of G par |Θ . If M = B e , there is nothing to prove. Suppose the lemma holds for M ∈ H par ; we have to show that it is true also for s θ par (M ) = Θ∈W β s θ par (M ) β,Θ . Thus it is enough to demonstrate it for one module M β,Θ .

7.2.2.
A property of indecomposable BMP-sheaves. Before proving our last result, we first have to recall an intrinsic property of indecomposable canonical sheaves, which has been used already by Braden and MacPherson in [7] to show the existence of these objects. We follow a reformulation due to Fiebig.
For any k-moment graph G = (V, E, , l) ∈ MG(Y k ) and for any x ∈ V, we denote (cf. [15, §4.2]) Consider F ∈ Sh k (G) and define F δx to be the image of Γ({⊲x}, F ) under the composition u x of the following maps Moreover, let us denote for some w ∈ V, then conditions (BMP3) and (BMP4) of Definition 5.7 may be replaced by the following condition (cf. [7, Theorem 1.4]) (BMP3') for all x ∈ V, with x ⊳ w, d x : B(w) x → B(w) δx is a projective cover in the category of graded S k -modules Since a (graded) finitely generated S k -module is projective if and only if it is free, B(w) x is isomorphic to the graded free S k -module with minimal number of generators which maps surjectively to B(w) δx . Proposition 7.5. Let I be a finite interval of G par far enough in C + and let B ∈ Sh G par | I be an indecomposable Braden-MacPherson sheaf. Then B stab is also indecomposable as sheaf on G stab |I . Proof. Since B is indecomposable, by Theorem 5.1, B = B(w) for some w ∈ I, that implies B(w) x = 0 = B stab,x for all x > w (x ∈ I) and B(w) w ∼ = S k ∼ = B stab,w . Suppose that B stab = C ⊕ D, then for what we have just observed, we may take C and D such that C x = D x = 0 for all x > w, C w ∼ = S and D w = 0. Let y ∈ I be a maximal vertex such that D y = 0. For any E : y → z ∈ E δy , by definition of Braden-MacPherson sheaf, ρ z,E : We now localise Γ(B) at a finite simple root β. Remark that, since we are representing the parabolic Bruhat graph using alcoves, we are taking the quotient of G opp instead of G. It means that we have to twist the action of S k on any vertex x by x −1 . However, once the action of the symmetric algebra is twisted, the two previous results still work in the same way. By combining Theorem 7.2 and Lemma 7.11 we know that L (Γ(B) β ) is a direct sum of Braden-MacPherson sheaves on certain moment graphs, each one of them is isomorphic to a finite interval of the parabolic Bruhat graph for A 1 . From the definition of L , it follows that L (Γ(B stab ) β ) = (L (Γ(B) β )) stab .
We have already proven that ρ y,E (D y ) = 0 for any E ∈ E δy ∩ E S and we want to show that ρ y,E (D y ) = for any E ∈ E δy . If it were not the case, there would be a non-stable edge F ∈ E δy ∩E N S such that ρ y,E (D y ) = 0. Let β ∈ ∆ + be such that l(F ) = β + nδ for some n ∈ Z. Localising at β, we would get ρ β y,F (D y,β ) = 0 and from the A 1 case, it follows that ρ β y,E (D y,β ) = 0 for all E ∈ E δy in β-direction, but we proved that this is not the case.
We are now ready to conclude. From what we showed, it follows that u y (C y ) = B δy and this implies D y = 0, since (B y , u y ) is a projective cover of B δy .

Appendix A. Finite parabolic intervals in the subgeneric case
In this appendix we want to study the behaviour of indecomposable BMP-sheaves on finite intervals of G par in the subgeneric case, that is g = sl 2 . In particular, we want to interpret in the moment graph setting the fact that the corresponding parabolic Kazhdan-Lusztig polynomials are all trivial. More precisely, we want to show that the structure sheaf A and the canonical sheaf B par are in this case isomorphic. Hence to prove our claim, it will be enough to define a surjective map S k → B par (w) δx for any pair of vertices x < w in G par .
Recall that the set of vertices is in this case totally ordered, so we may enumerate the vertices as follows, once identified the finite root α with the corresponding coroot α ∨ : v 0 = 0, v 1 = α, v 2 = −α, ... , v h = (−1) h+1 [ h+1 2 ]α, . . . . From now on, we denote the edges as E h,k : v h − − − v k and the labels as l h,k := l(E h,k ); we write moreover l h,k = α + n h,k c. Actually, the label of an edge E h,k is by definition ±l h,k ; however, there exists an isomorphic k-moment graph with same sets of vertices and edges, but this other label function and, by Lemma 5.2, the corresponding indecomposable canonical sheaves are isomorphic.
We will prove in several steps that, if v j ≤ v i and ( G par |[vj ,vi] , k) is a GKM-pair, then (B par (v i )) vj ∼ = S k by induction on i − j.
Lemma A.1. Let r ∈ N be such that r < i−j. If ( G par , k) is a GKM-pair, and z ∈ Γ(I, B par (v i )) {r} , then z is uniquely determined by its first r + 1 components, that is the restriction map Proof. Let z ∈ Γ(I, B par (v i )) {r} such that z vi = z vi−1 = . . . = z vi−r = 0. Observe that for any j + 1 ≤ h < i − r ≤ k ≤ i one has z v h ≡ z v k = 0 ( mod − α + n h,k c).
From the GKM-property it follows that GCD(−α+n h,k c, −α+n h,l c) = 1 for any i−r ≤ k = l ≤ i. Since S k is an UFD, z v h has to be divisible by (−α + n h,i−r c)(−α + n h,i−r+1 c) . . . (−α + n h,i c). This is a polynomial of degree r + 1 while z v h was a polynomial of degree r, so z v h = 0. We noticed in §4.1.1 that in the sl 2 case all the vertices are connected, so the number of edges is equal to the number of pairs of different elements in a set with r + 1 elements, that is r+1 2 . Then, dim k Γ({v i , v i−1 , . . . , v i−r }, B par (v i )) {r} = (r + 1) 2 − r + 1 2 = r + 2 2 .
It is left to show that the conditions are linearly independent. Let i − r ≤ h < k ≤ i and define the element (m (h,k) ) ∈ r 0 (S k ) {r} in the following way: Let us denote by m α , m c ∈ Γ(I, B par (v i )) {1} the constant sections m α,v = α, m c,v = c for all v ∈ I. Denote moreover by u vj := ⊕ρ v h ,E h,j , where ρ v h ,E h,j : S k → S k /(E h,j · S k ) are the canonical quotient maps.
Lemma A.3. Let r ∈ N and let ( G par , k) be a GKM-pair. The vector subspace of (B par vi ) vj generated by u vj (m r α ), u vj (m r−1 α m c ) . . . u vj (m α m r−1 c ), u vj (m r c ) has dimension equal to r + 1 if r < i − j or dimension equal to i − j otherwise.
Proof. As first notice that (B par (v i )) E j,k = S k /(l j,k · S k ) ∼ = k[c] by the mapping α → n j,k c. Then By the GKM-property it follows that n j,k = n j,h for all pair j + 1 ≤ k = h ≤ i and N is a Vandermonde matrix. In particular, such a matrix is not singular and so it has maximal rank, that is rk(N ) = t + 1 if t < i − j and rk(N ) = i − j otherwise. Finally, by definition m 0,v h = 0 for any v h ∈ I and u vj ((m 0 )) = 0.
Lemma A.5. Let r ∈ N be such that r < i − j. The collection of monomials {m l α m h c m k 0 | l, h, k ≥ 0, l + h + k = r} is a basis of Γ(I, B par (v i )) {r} as k-vector space.
Proof. Since the number of monomials in three variables of degree r is r+2 2 and by Lemma A.2 dim k Γ(I, B par (v i )) = r+2 2 as well, it is enough to prove that all monomial in m α , m c , m 0 are linearly independent. We prove the claim by induction on r.
Let r = 1. If xm α + ym c + zm 0 = 0, then clearly 0 = u vj (xm α + ym c + zm 0 ) = xu vj (m α ) + yu vj (m c ) + zu vj (m 0 ). By Lemma A.4 u vj (m 0 ) = 0, so xu vj (m α ) + yu vj (m c ) = 0. But by Lemma A.3 u vj (m α ) and u vj (m c ) generate a vector space of dimension 2, then x = y = 0. Finally, from zm 0 = 0 and Lemma A.4 it follows z = 0. Now let r > 1. Let z = l+m+n=r x l,m,n m l α m m c m n 0 = 0. We can write z = z 1 + z 0 m 0 , where z 1 is such that m 0 does not appear. Then by Lemma A.3 u vj (z) = u vj (z 1 ) + u vj (z 0 )u vj (m 0 ) = u vj (z 1 ) = 0. From Lemma A.3 we know that all u vj (m l α m r−l c ) are linearly independent and so 0 = u vj (z 1 ) = u vj ( Proof. We prove that (B par (v i )) δvi coincides with the u vi image of the ring generated by m α and m c . If r < i − j, by A.5, Γ(I, B par (v i )) {r} is generated by {m l α m h c m k 0 | l, h, k ≥ 0, l + h + k = r}. From A.4 it follows (B par (v i )) δvi = u vi (Γ(I, B par (v i )) {r} ) is contained in the ring generated by u vi ((m α )) and u vi ((m c )).
Otherwise, r ≥ i − j and E i,k ∈E δv i (B par (v i )) E i,k ∼ = k[c] i−j , having dimension i − j. Then by Lemma A.3 u vi (m α ) and u vi (m c ) generate (B par (v i )) δvi Thus we have a surjective map S k → E i,k ∈E δv i (B par (v i )) E i,k given by the mapping α → m α and c → m c . Then (B par (v i )) vj ∼ = S k .