Group actions, and D equivalences of categories of coherent sheaves of symplectic resolutions
Abstract
Let G be a reductive group over an algebraically closed field k of characteristic . Let be its Lie algebra. Let be the variety of Borel subalgebras in . There are two questions that motivates this work. One is the question of constructing an action of a group on the category  the derived category of coherent sheaves of symplectic resolution X. The second question is understanding equivalence functors between these categories for different symplectic resolution of the same symplectic singularity. In this paper, we answer the first question by constructing such an action for the case , P parabolic subgroup. This extends the affine braid group action on category of coherent sheaves, in the case of the well known springer resolution . We construct the group action by constructing a local system of categories on a topological space called , with value  the category hence obtaining an action of . We hope to prove a generalization of this construction is well defined for arbitrary symplectic resolution. In [2] we further explain how a refinement of this local system construction, gives an answer to the second question, showing that these equivalence functors, are parametrized by homotopy classes of maps between certain points in the base space . We also lift the result to the case that the field k is of characteristic zero.
Contents
I’m very thankful to Prof. Bezrukavnikov, for his great help and support.
1 Notations and Basic Facts
The symbols below have the following meaning in this paper:
Let G be a reductive group over an algebraically closed field k, of characteristic .
Fix a maximal torus T. Let be the integral weight lattice. Assume all parabolic subgroups that we will mention, contain T. Let be a fixed Levi subgroup. Let . (The canonical embedding is given by restriction from L to T). Explicitly for all roots with weight space
Remark 1.
We summarize here the notations we will be using in the text, however, a more detailed explanation of them appears in the background section.
Definition 1.
Let W denote the Weyl group. is called pregular (We will also call it regular, when there is no risk of confusion between char 0 and p), if the stabilizer in W, of is trivial, (Under the action ). Equivalently, mod p for all coroots .
Most of of the weights in are pregular as elements of .
Let be a parabolic subgroup, with Levi L.
Definition 2.
Let . Let be the sheaf of twisted crystalline differential operators on G/P. (We will denote it as when there is no ambiguity). Let be the twisted D modules.
Definition 3.
Let be the global sections of the sheaf . Let be the category of finitely generated modules.
Observation 1.
Definition 4.
Working over k, we can think of modules as coherent sheaves on with extra data. Let F be the Forgetful functor . Let be the category of coherent sheaves, supported on the formal neighborhood of the zero section of .
Let be twisted Dmodules , s.t the support of is on the formal neighborhood of the zero section of .
Let be the category of finitely generated modules, with generalized p center, zero.
(Another notation that we will use for this category is )
Observation 2.
The derived localization theorem induces an equivalence on the full subcategories . ([10]).
Example.
Let be the enveloping algebra. Let P be a borel B subgroup. The category is modules, whose Harich Chandra center is and with whose generalized pcenter is zero.
Observation 3.
There is a surjective morphism , and .
Let be the nilpotent cone. Let be the moment map. (Also called parabolic springer map). Let be its image.
Remark: In the above notations and in the rest of the paper  we made a choice to work with coherent sheaves and D modules, supported on a formal neighborhood of , the zero section of . This can be generalized to support on formal neighborhoods of other parabolic springer fibers as well.
Notations: In this paper, we use the letters Q,P to denote associate Parabolic subgroups with the same Levi subgroup L.
2 Motivation and statement
Two questions form the motivation for this work: For G/k , as above, , B a Borel subgroup, the nilpotent cone in . The springer resolution is a symplectic resolution. Its derived category (0 stands for the support condition) carries a known action of the affine braid group ([27][11][8][9][6]. See also [1] for different realization of the action). One would like to find analogous actions for other symplectic resolutions. In this work we build a construction for the case (a symplectic resolution of a finite cover of ). It’s a construction of a local system of categories on a simple topological space , with value . This gives rise to an action of on the category . In the case of P=B, this recovers the action of on the category. More precisely,  the pure affine braid group. However in this case there is additional symmetry in the construction, that allows the action to extend to action of . In [2], we also explain how to lift this action to case of G over a char 0 field and generalize to the category of coherent sheaves without the support condition.
A central role in the construction is played by the topic of quantizations in characteristic p. The base of the construction is subset of the universal parameter space of quantizations of symplectic resolutions. The quantizations of a symplectic resolution X, which are parametrized by points in , give rise to various t structures on the category . The local system that we construct enables a description of the variation of the t structures along the parameter space . ([3])
A second question that motivated the work is Kawamata’s ’K equivalence implies D equivalence conjecture’: two smooth projective varieties X,X’ over algebraically closed field, which are birationally equivalent, are called K equivalent, if there is a birational correspondence , for Z smooth projective variety, such that the pullbacks to Z of the canonical divisors are linear equivalent. The conjecture is that K equivalence implies the equivalence of the bounded derived categories of coherent sheaves. One caveat of this conjecture, is that one do not expect to get a canonical equivalence. Indeed, a special case where K equivalence is satisfied  is two symplectic resolutions X,X’ of the same symplectic singularity Y. In this case, the D equivalence was proved by Kaledin. The key to his construction of equivalence functor , is a choice of a tilting generator of , which is a noncanonicaly defined object. Hence, the second question is understanding the family of equivalences between the derived categories. In [2], we explain that a refinement of the above local system, gives a parametrization for natural equivalence functors between by homotopy classes of maps between certain points in the base space .
3 Background
D modules and Coherent sheaves in characteristic p
Let k be a perfect field of characteristic . Let X be a smooth variety over k.
Let be the sheaf of crystalline differential operators on X. We recall the relations of with the structure sheaf of the cotangent space and recall the relations between modules and coherent sheaves on . Two main ideas are 1. is a quantization of the sheaf . 2. is a quasi coherent sheaf of algebras on because the center of is canonically isomorphic to the sheaf . The superscript (1) stands for Frobenius twist. For a variety over a perfect field k of char p, which is defined over , the variety and its Frobenius twist are isomorphic as k schemes, hence we usually omit the superscript from the notation and identify .
Definition 5.
Let be the sheaf of crystalline differential operators on X. It assigns for an affine open subset , an algebra .
is defined to be generated by and (Vector fields). The relations are:
Let , .
Let . .
is a subalgebra.
Remark: if k was an algebraically closed field of characterstic zero, this is the definition by generators and relations of the ordinary sheaf of differential operators in characteristic zero.
Twisted differential operators. Let be a line bundle. Then the sheaf of twisted differential operators is well defined.
Notation: Let let , where corresponds to under the equivalence . Similarly for where
Notation: Let be the category of twisted D modules on X. (For X:=G/P we also denote it or when there is no ambiguity)
Claim 1.
It is still true, as in characteristic 0, that is a quantization of . That is, there is a natural filtration on this sheaf and .
Claim 2.
In contrary to the characteristic zero situation, the an element in is not uniqly identified by its action on . The morphism isn’t injective. The basic example to keep in mind, is where X is the affine line, and the differential operator is . It’s not zero as an element of , yet it acts by zero on .
Theorem 1.
The center of is big. It’s canonically isomorphic to the sheaf of rings
Theorem 2.
, and are both Azumaya algebras on .
Given an Azumaya algebra over a variety Y, we can consider the category of coherent modules.
When an Azumaya algebra is split over Y. That is, , then the category of coherent modules is equivalent to Coh(Y).
Consider Y:=.
and are Morita equivalent. There is a canonical equivalence between the category of modules and the category of modules.
Theorem 3.
doesn’t split over , but it does splits on a formal neighborhood of the zero section of .
The splitting implies the equivalence:
(1) 
Where in both side we restrict to sheaves with support on the formal neighborhood of the zero section .
In the special case that (Or ) this can be generalized.
Theorem 4.
Consider
Consider the springer map . splits on the formal neighborhood of every fiber of the springer resolution . Hence obtain an equivalence
(2) 
Where the subscript stands for a support condition on a formal neighborhood of for .
We won’t use this generality though in this work.
Localization theorem in characteristic p
Let be the universal Cartan subalgebra of . Let be any borel subgroup of G, and be its lie algbera, then there is a canonical isomorphism . Let be a maximal torus contained in B. Let be its lie algebra.
Let W be the Weyl group. W acts on the weight lattice by the dot action.
Define to be the center of . The center consists of two parts. Let be the HarichChandra center. . is the symmetric algebra of , and we take the invariants for the action. The second part is the Frobenius center , also called pcenter. is an algebra generated by the elements of the form , where is the restricted power map, which is characterized by . Maximal ideals of are in bijection with points of . . That is functions on , (the frobenius twist of ).
is build from these two parts. When , . In other words, to give a central character of , we need to give a pair of elements , which are compatible.
In particular lets restrict to the case that e is nilpotent and integral weight.
Let . There is a natural map mod W. Hence defines a maximal ideal of . We define .
Let be the category of finitely generated modules.
is called pregular, if the stabilizer in W of is trivial.
Given a compatible pair . We consider mod. The category of finitely generated modules with generalized p character . That is, modules for which there exists a power of the maximal ideal e in which kills them. We also denote this category by .
Derived localization in char p
Let .
Theorem 5.
There is a natural isomorphism .
Recall that the notation stands for the category of twisted D modules.
Theorem 6.
Let be pregular, then derived global sections functor is an equivalence of categories. [9]
Theorem 7.
Let be a compatible pair. The derived localization induces an equivalences on the subcategories
Note that combining the last theorem with the equivlanece in equation (1), the following equivalence is obtained: .
Theorem 8.
Similarly, Let , P a parabolic. Let . Let be p regular. then the derived global sections functor is an equivalence of categories, [10] , and .
The moment map
[17]
Let G be a reductive over algebraically closed field k. Let . Let Z/k be a variety with an action of G. The varieties and are poisson varieties. The G action on Z, gives rise to a G equivariant morphism of Poisson varieties . We compose with the natural isomorphism from the killing form, and refer to as the moment map.
In this paper, we consider the moment map for the case Z=G/P, where P is a parabolic subgroup. The map .
In the case P=B a Borel, the image of this map is the nilpotent cone , and the map is a symplectic resolutionthe springer resolution. More generally Let L be the Levi subgroup of P. The image of this map is the closure of a nilpotent orbit, which we denote . (The notation is the emphasize that it only depends on the Levi.) The map is not quite a symplectic resolution but rather generically a finite cover.
Symplectic resolutions
A symplectic singularity Y, is a normal variety, with an algebraic symplectic 2 form, , on the smooth locus of Y, s.t for some smooth projective resolution of Y, the pullback of from the smooth locus extends to a 2 form on all of X, possibly degenerate.
Beaville showed that if extends to one smooth resolution, then it extends to any other resolution.[4]
A symplectic resolution X of a symplectic singularity Y, is a resolution as above, where the 2 form on X is a symplectic form.
When the symplectic singularity Y is affine, Y must be the affinization of X. . Hence in such case it’s enough to specify X, without specifying Y.
Observation: A symplectic manifold, is in particular a Poisson variety.
Observation: The canonical bundle of a symplectic resolution is trivial .
Standard examples of symplectic resolutions include:

The springer resolution , or more generally the restriction of this resolution to a Slodowy slice.
Slodowy slice:[28] Let be a nilpotent element.
Slodowy slice, is a subvariety of , s.t the restriction of the springer resolution to is another symplectic resolution. In other words, the preimage of under the springer resolution is a smooth manifold, and the restriction of the symplectic 2 form on to is non degenerate.
To define , use Jacobson Morozov theorem to get triple that includes e. And define to be the intersection of e plus the centralizer of f in , with the nilpotent cone. One place where the Slodowy slice plays a role is, using the symplectic resolution given by restricting the springer resolution to the slodowy slice, one generalizes the known Braid group action on the category (coherent sheaves with support on formal neighhbhorhod of the zero section ), to an action on the bounded derived category of coherent sheaves with support on for any . ([8],[9])

Quiver varieties,[25]. These varieties have important applications in representation theory. For example, in Nakajima’s geometric construction of the representations and crystals of Kac Moody algebras.

Symplectic resolutions of quotient singularities. [15] A basic example is symplectic resolution of Kleinian singularities. Let be a finite subgroup. Consider the quotient . This is a poisson variety, where the poisson structure is induced from the canonical symplectic structure on . The canonical minimal resultion of is a symplectic resolution. ([14], [20],[21])
More generally, we can consider any finite dimensional symplectic vector space, and a symplectic subgroup . Consider the Poisson variety . where the Poisson structure is induced from the symplectic form on W. Symplectic resolutions of such varieties (when exists), have been studied.
Remark: Slodowy slice and Kleinian singularities McKay correspondence gives a correspondence between finite subgroups as above , and types A,D,E Dynkin graphs. Recall  Let be a finite subgroup. Let I:=Irreps() parametrize the irreducible representations , of . the corresponding Dynkin diagram is defined by letting , where is the standard representation, restricted to . (Observe that )
Given , take to be of the corresponding type. Let be a subregular nilpotent. Then there is an equivalence of Poisson varieties between the quotient singularity and the Slodowy slice:.
Another quotient singularity that has been studied is the following. Let be a finite dimensional semi simple lie algebra. Assume is of type A, B, or C. Let be its cartan. When taking the vector space W, to be the cotangent bundle of the cartan, the weyl group , is embedded in the symplectic group . In this setup, the quotient has a symplectic resolution. (In fact the condition that was of type A,B,or C is also necessary). More generally one could take any coxeter group rather than .
Moreover, specializing the above setup to type A, a symplectic resolution of the quotient space is well known. This is the Hilbert Chow morphism. .
Quantizations of symplectic resolutions
Examples for filtered quantizations of a structure sheaf of a (poisson) variety, include: as the filtered quantization for , symplectic reflection algebras as quantizations of quotient singularities for symplectic finite dimensional vector space and finite subgroup, and the sheaf of differential operators for a cotangent space .
Given a symplectic resolution , The universal parameter space for quantizations of X is [12][13][22]. The starting point of the proof is the fact that formal locally there exists only one quantization.
For every let be the corresponding quantization of the structure sheaf. Observe  For any symplectic resolution . When these quantizations for come from the sheaf of twisted differential operators on G/P. (Recall that )
For different symplectic resolutions of the same symplectic singularity, the birational isomorphism between induces an isomorphism on the Picard groups . Hence it makes sense to compare the quantizations and parametrized by the same
Derived Localization in characteristic p
Fix a symplectic resolution . Let . One can ask when does derived localization hold, when is the global sections functor from sheaves over , to modules over the global sections an equivalence. e.g  Let , then derived localization holds when is regular. This fact is a derived characteristic p version of Beilinson Berenstein localization theorem[5].
More generally, for any symplectic resolution, it’s a conjecture that derived localization holds away from a discrete set of hyperplanes . Let me call these hyperplanes the ’walls’ in .
Let be the complement of these walls.
D equivalence of symplectic resolutions
Definition 6.
two smooth projective varieties X,X’ over algebraically closed field, which are birationally equivalent, are called K equivalent, if there is a birational correspondence , for Z smooth projective variety, such that the pullbacks to Z of the canonical divisors are linear equivalent.
Kawamata conjectures that K equivalence implies an equivalence of the bounded derived categories.
One case in which this holds is for two symplectic resolutions. K equivalence holds, since the canonical bundles of symplectic resolutions are trivial. It’s a proof of Kaledin that there is also an equivalence of the derived categories.
Theorem 9.
Given X,X’ symplectic resolutions of a fixed symplectic singularity Y, there is an equivalence .
Definition 7.
A tilting generator for a smooth variety X/k, is a locally free sheaf on X, s.t 1. , for all , 2. The functor is an equivalence of categories.
Theorem 10.
A symplectic resolution has a tilting generator.
Proof.
A sketch of the D equivalence proof  Let be a tilting generator for X. (That’s a choice). X and X’ agree on an open subset whose complement is of codim . can be extended to a vector bundle on X’, s.t for . (yet is not necessarily the tilting generator of X). agree on an open set whose complement is of codim , and the algebras are isomorphic. Hence can consider the functor . That’s the functor that is proved by Kaledin to be an equivalence. The proof uses a trick related to being a CalabiYau category. [16]
∎
The caveat is that the equivalence constructed is highly non canonical, since a tilting generator isn’t. The local system constructed below has a refinement that leads to a better understanding of family of natural D equivalences between this categories.
The classical action on Grothendick group of
For simplicity we work over the complex numbers in this section. Let be the springer resolution. Let , Let be the springer fiber over e. set of Borels s.t .
There is a known action of (affine) braid group on , (The subscript e stands for restricting the support to be on the formal neighborhood of ). One of the goals of the local system we build, with value the category (P parabolic), is to generalize this action.
The action on is a categorification of an extension of a known action of the weyl group on the cohomology of the springer fiber . That is, at the level of Grothendieck group, , there is an equivalence . The action induced on is dual to the classical action on the cohomology of springer fiber by the Weyl group. We briefly recall that action.
Properties of the springer fiber
Let , = set of borels s.t . e.g Let , V/k n dimensional vector space. Identifing Borel subgroups with complete flags in V, is flags that e preserves. ()
Properties of : where is the adjoint orbit of e in . (note that the codimension of nilpotent orbit is always even, so the formula makes sense). In addition is always connected, in interesting cases irreducible, and often singular.
Extreme cases for springer fibers are , for which . and regular nilpotent e, for which (indeed the regular locus of , is exactly where the birational morphism is an isomorphism). Explicitly for , regular nilpotent e is (up to conjugation) a single Jordan block in some basis , and the single Borel in the fiber of the springer map over e, is the upper triangular matrices in this basis. The corresponding flag is that which is determined by this basis
Explicit examples:
Example 1. Let , e be of Jordan type (2,1). Then . (two projective lines, glued at a point). Lets write e in a basis as a matrix which has all zeros except for upper right corner. Then explicitly in terms of flags, the first is flags of the form , where is any 2 dimensional sub vector space between the fixed and V. The second is flags of the form and again, the free component is which is one dimensional vector space between 0 and . These sets of flags indeed have one point in common, and each is isomorphic to . dim
Example 2. Let , let e be of Jordan type (2,2). Then there are two irreducible components of : and ( stands for projectivization). They are glued along a that is embedded in diagonally and as the zero section of the other component. dim
The use of Grothendieck resolution in the construction of the action
The action at the level of Grothendieck groups was constructed using the Grothendieck resolution. In fact, the action at the level of categories, on , using correspondences as in [11], also used the Grothendieck resolution. There is an underlying reason for that.
Grothendieck resolution is obviously closely related to . At the level of k points s.t , hence at that level, restricting to at the basis, gives you preimage . (Remark:That’s not true at the level of schemes. The preimage of is a non reduced scheme. Its associated reduced scheme is ). (See [24], [7] for more)
is very useful because it is small whereas the springer map is only semismall. That’s the reason that some constructions work better using , and then with more work one may make them work for .
The Weyl group action on
Let W be the Weyl group. For every nilpotent there is an action on W on the cohomology of the springer fiber . In general, there is no action of W on the space .
Considering the extreme cases. The case of regular e is not of interest since, is a point. Case e=0: For e=0 one gets an interesting action on . In this case, the action coincide with an older known action that is described in algebraic terms: The cohomology ring of G/B is well understood. There is an isomorphism . The natural action of W on , coming from the action of W on the cartan, is the W action.
The construction of the action for any : (See [23]). Let be the Grothendieck resolution. One works in the setup of constructible and perverse sheaves. Let be the constant sheaf. One first constructs an action of W on : Since is small, is a perverse sheaf, moreover it is of the form , . (Pushforward of constant sheaf restricted to . stands for regular semi simple elements, those over which is an etale cover of rank ). Since is an etale cover of rank , W acts on . By functoriality of IC extension, W acts on . Then the restriction to a fiber gives the action on .
4 The parameter space
To construct the local system for a symplectic resolution we need a base: Let . Let be the set of hyperplanes where derived localization doesn’t hold. Let to be the complement of that set of hyperplanes. Let be the complexification. That’s the topological space which is the base for our local system construction.
Observe that for the springer resolution , is the space of regular weights. Hence,  affine pure braid group.
Another interpretation of the base space
Recall that for all symplectic resolutions of Y, the Picard groups are canonically identified. Lets denote this group Pic. One can take the ample cones of different symplectic resolutions of Y. There is a Coxeter group, , which is attached to Y by Namikawa [26] . In the case of this is the ordinary Weyl group . This group has an action on . Consider all the ample cones of different symplectic resolutions of Y, and their transitions via the action of . Let be the complement of the boundaries of these cones in . There is a conjecture that there is an isomorphism between .
There is a local system of categories on the parameter space with the value the category the category , which for the case P=B, generalizes the well known weak action of the affine Braid group on this category. Let denote the groupoid of the space . (Remark: By a local system we mean a weak local system. Attaching categories to points in , attaching functors up to isomorphism to homotopy classes of paths between points, s.t concatenation of paths corresponds to composition of the functors. We do not discuss higher compatibilities).
We first build a local system with value . The restriction of the functors attached to paths, to the subcategory will give the required local system.
In order to build it, it’s convenient to use the presentation by generators and relations for the groupoid given by Selvatti.
5 The groupoid of the base space generators and relations
Salvetti’s generators and relations for groupoid of complexification of completion of real hyperplane arrangement
Salvetti idea
Given a real hyperplane arrangement in . one looks at the complexification of the complement , and wants to find a presentation for the groupoid of that space. By constructing a CW complex embedded in , whose embedding is a homotopy equivalence, Salvetti is able to describe the generators and relations for the groupoid in a combinatorial way. Generators are the 1 cells, relations are given by the boundaries(attaching maps) of the 2 cells. The zero cells in the CW complex, correspond to the real alcoves in .
Generators
The generators of the groupoid are the positive half loops, going between alcoves A,A’ that share a codim 1 face. Let be the generator for the path from A to A’
Relations:
To express the relations we need to define a notion: notion: length of path Given a path in consisting of generators, its length is the number of generators involved.
First set of relations one way to describe the relations is: for each two alcoves A,A’: all positive minimal length orbit between A and A’ are homotopic.
Smaller set of relations:
However, It’s sufficient to take a smaller set of relations:
for each codim 2 face F, and alcove A, that has F in its boundary, there are exactly two positive minimal paths between A and its opposite with respect to F, . The relation we impose is that these paths are equivalent.
6 Construction of the local system
We specialize the setting of to the following case: Fix a maximal torus . Fix a levi . Let P be any parabolic with the levi L. with Levi L. We construct a local system with value
This category depends on P only up to the levi.
In this section we discuss the parameter space . Then we build a functor from the groupoid to Cat, attaching a category to each alcove, and a functor between the categories of the alcoves for each path generator of the groupoid. To prove it’s indeed a functor from the groupoid, we check the relations that guarantee that given two homotopy equivalent paths, the composition of the corresponding functors gives isomorphic functors.
The parameter space, , in the case
In this case, the parameter space , has an interpretation in terms of the root space of G.
Claim 3.
, regular parabolic weights
Proof.
To see that , recall that there are canonicals equivalence , and .
Moreover, the walls in are defined as the hyperplanes where derived localization do not hold. It’s known that derived localization do not hold exactly for singular parabolic weights. (that is s.t mod p for some coroots with not subset of Lie(L)).
∎
For convenience denote , , (= in this case.)
Denote the walls of singular weights, by
These affine root hyperplanes is the hyperplane arrangement in in this case.
is the topological space which is the base for our local system construction. Observe that for P=B, a Borel,  affine pure braid group.
The categories attached to the alcoves
Key Lemma
In order to construct a local system with value  the category , we have to use different associated parabolic subgroups P,Q. The following lemma about the global sections of the sheaves of differential operators on these spaces will be key in that construction.
Lemma 1.
Let G be an algebraic groups over algebraically closed field, k, characteristic . Let P,Q be parabolic subgroups with same levi L. Let (parabolic integral weight). Let
Then, for a parabolic integral weight, There is an isormophism .
Proof.
There is a morphism . It is the morphism [[1]] induced from the natural map
It is surjective for ([9]) Denote its kernel by . It’s enough to show as subgroups of . The proof of this equality for reduces to the same claim in the setup where the base field is .
Claim 4.
Let be affine open subset of G/P. Let be a module on U. Let . Let . consider it as module under [[1]]. Then .
Proof.
(of claim) Let Consider the composition . Both maps are injectives. the first by D affiness of G/P, the second since Weyl algebra is simple. Hence .
∎
Proof of Lemma 1: By the claim, it’s enough to find modules, on G/P and on G/Q respectively that comes from D module on affine open, and have the same kernel under the action of .
The required modules are  on G/P, let [P]:=1P be the trivial coset. Let . On G/Q Let [Q]:=1Q be the trivial coset. Let U:=P.[Q] be the orbit under the action of P. is affine open. Let .
∎
The categories attached to alcoves
For any parabolic Q with the Levi L, for each the sheaf is well defined. (using the canonical equivalence )
By the ’key lemma’, for two parabolic subgroups with the levi L, there is a canonical isomorphism of the global sections algbera . Denote this algebra
To each real alcove . attach the category using some
Independence of The construction we give will be independent of the choice of in an alcove A, in the sense that there is the canonical equivalences between the categories for , (given by localization, tensoring with and taking global sections). And the functors that we construct for different paths, will sit in a square commutative diagram.
for and in adjucent alcoves.
Functors attached to the generators of the groupoid
We now construct the functors to be attached to paths. For the definition we need the following notion:
Partial order on alcoves, defined by a cone
Definition 8.
Given a real hyperplane arrangement, fix a cone . Define: two alcoves A,A’ have relation ”A’ is above A wrt ” if . Notation is .
Remark: an equivalent notion is: the positive half loop from A to A’ belongs to .
Definition of the functor , assigned to the generator
Given adjacent alcoves , there is a parabolic P, with the property that . Equivalently, where is the positive weights cone for P in .
Let , be weights in these alcoves. Define the associated functor to be
(3) 
Where , are global sections and localization functors respectively.
Lemma 2.
The functor (1) is independent of a choice of parabolic P, for which .
The proof of this lemma will be given in a later section. This claim is the reason, we denote this functor by a symbol , omitting P from the notation.
Proof that the relations of the groupoid hold
To prove our construction is a functor from the groupoid to Cat, we need to check the relations hold. For this we use the following concrete claim.
Claim 5.
Consider our hyperplane arrangement . Let F be a codimension 2 face. Let be an alcove which contains F in its boundary. Let denote the opposite alcove with respect to F. Let C be the cone in , defined by the hyperplanes in the boundary of the alcove , that intersect the face F. Then the two minimal paths in from to are going up with respect to the cone C.
That’s easy to verify.
Claim 6.
Given an alcove , and a codimension 2 face F in its boundary. There is a parabolic P(with the levi L), such that the 2 positive minimal orbits from to are orbits that always go up with respect to the partial order on alcoves given by the positive cone of P.
Proof.
By claim 3, it works for P whose positive cone corresponds to a cone defined by the alcove and face F. ∎
Finally, It will follow that the construction is indeed a functor from the groupoid if we prove the following claim
Claim 7.
satisfies the following relation: Any two paths between two alcoves, that go through increasing alcoves according to a fixed parabolic P, have isomorphic corresponding functors.
Proof.
Follows from definition since , and
∎
Proof of Lemma 2: is well defined
Recall  Let be adjacent alcoves in with shared codim 1 face. Let P and Q be two parabolic subgroups that have the Levi L, and satisfy and . Lemma 2 is the claim that there is an isomorphism of functors , where