A Hecke action on $G_1T$-modules

We give an action of the Hecke category on the principal block $\mathrm{Rep}_0(G_1T)$ of $G_1T$-modules where $G$ is a connected reductive group over an algebraically closed field of characteristic $p>0$, $T$ a maximal torus of $G$ and $G_1$ the Frobenius kernel of $G$. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.


Introduction
Let G be a connected reductive group over an algebraically closed field K of characteristic p > 0. One of the most important problems in representation theory is to describe the characters of irreducible representations.In the case of algebraic representations of G, Lusztig gave a conjectural formula on the characters of irreducible representations of G in terms of Kazhdan-Lusztig polynomials of the affine Weyl group for p > h where h is the Coxeter number.Thanks to the works of Kazhdan-Lusztig [KL93, KL94a, KL94b], Kashiwara-Tanisaki [KT95,KT96] and Andersen-Jantzen-Soergel [AJS94], this is proved for p large enough.An explicit bound on p is known by Fiebig [Fie12].
However, as Williamson [Wil17] showed, Lustzig's conjecture is not true for many p.Therefore we need a new approach for such p.Riche-Williamson [RW18] gave it and we explain their approach.Assume that p > h.Let Rep 0 (G) be the principal block of the category of algebraic representations of G.For each affine simple reflection s, we have the wall-crossing functor θ s : Rep 0 (G) → Rep 0 (G).The Grothendieck group of Rep 0 (G) is isomorphic to the anti-spherical quotient of the group algebra of the affine Weyl group.
Here the structure of a representation of the affine Weyl group is given by [M ](s + 1) = [θ s (M )] for M ∈ Rep 0 (G).Riche-Williamson [RW18] conjectured the existence of the categorification of this anti-spherical quotient.More precisely, they conjectured that there is an action of D on Rep 0 (G) where D is the diagrammatic Hecke category defined by Elias-Williamson [EW16].Assuming this conjecture, they proved that the anti-spherical quotient of D is a graded version of the category of tilting modules in Rep 0 (G).In particular, their result gives a character formula for indecomposable tilting modules in terms of p-Kazhdan-Lusztig polynomials.Recently this character formula was proved by Achar-Makisumi-Riche-Williamson [AMRW19] when p > h and for any p by Riche-Williamson [RW20].We note that if p ≥ 2h − 2 then a character formula for indecomposable tilting modules implies a character formula for irreducible modules.We also remark that Sobaje [Sob20] gave an algorithm to calculate the character of irreducible modules by the character of indecomposable tilting modules.
Achar-Makisumi-Riche-Williamson also proved a big part of the conjecture, but not a full statement.In the case of G = GL n , the original conjecture is proved by Riche-Williamson [RW18].Recently, the conjecture is proved by Bezrukavnikov-Riche [BR20].In this paper, we consider the G 1 T -version of this conjecture where T ⊂ G is a maximal torus and G 1 is the Frobenius kernel of G. Namely, we define an action of the category D on the principal block of G 1 T -modules.1.1.The category SBimod.We use the category SBimod [Abe19] instead of the category D. The category SBimod is equivalent to the category D. We recall the definition of SBimod.Let W aff be the affine Weyl group attached to G.An object we consider is a graded S-bimodule with a decomposition M ⊗ S Frac(S) = x∈W aff M Frac (S)   x such that mf = x(f )m for f ∈ S and m ∈ M Frac(S) x .Here x is the image of x in the finite Weyl group.For such objects M and N , we have the tensor product M ⊗ N = M ⊗ S N with the decomposition (M ⊗ N ) ⊗ S Frac(S) = x∈W aff (M ⊗ N ) Frac (S)   x where (M ⊗ N ) Frac (S)   x = yz=x M Frac(S) y ⊗ Frac(S) N Frac(S) z .A homomorphism M → N is a degree zero S-bimodule homomorphism which sends M Frac(S) x to N Frac(S) x for any x ∈ W aff .
Let X be the character group of T .An alcove is a connected component of X ⊗ Z R \ t H t where t runs through the affine reflections in W aff and H t is the fixed hyperplane of t.We fix an alcove A 0 and let S aff be the reflections with respect to the walls of A 0 .Then (W aff , S aff ) is a Coxeter system.For each s ∈ S aff , put S s = {f ∈ S | s(f ) = f }.Then the S-bimodule S ⊗ S s S(1) has the unique decomposition as described above such that (S ⊗ S s S(1)) Frac(S) w = 0 only when w = e, s.We denote this object by B s .Now SBimod consists of the objects M which is a direct summand of a direct sum of objects of a form B s 1 ⊗ • • • ⊗ B s l (n) where s 1 , . . ., s l ∈ S aff and n ∈ Z.It is proved in [Abe19] that the category SBimod is equivalent to the diagrammatic Hecke category defined by Elias-Williamson.As showed in [EW16,Abe19], this gives a categorification of the Hecke algebra of affine Weyl group, namely the split Grothendieck group of SBimod is isomorphic to the Hecke algebra.1.2.Another combinatorial category.We also give another realization of the category of Andersen-Jantzen-Soergel K AJS [AJS94].As in [Lus80], we use the combinatorics of alcoves to define the category.Let A be the set of alcoves.We fix a positive system ∆ + of the root system ∆ of G. Then this defines an order on A [Lus80].Recall that we have fixed A 0 ∈ A. The action of W aff on X ⊗ Z R induces the action of W aff on A. The map w → w(A 0 ) gives a bijection W aff → A.
Set S ∅ = S[(α ∨ ) −1 | α ∈ ∆].We define the category K as follows: An object of K is a graded S-bimodule M with a decomposition S ∅ ⊗ S M = A∈A M ∅ A such that mf = x(f )m for m ∈ M ∅ A , f ∈ S ∅ , x ∈ W aff such that A = x(A 0 ) and x the image of x in the finite Weyl group.A morphism f : M → N is a degree zero S-bimodule homomorphism such that f (M ∅ A ) ⊂ A ≥A N ∅ A .We will also define some subcategories of K .Especially the category denoted by K P plays an important role in our construction.Since it is technical, we do not say anything about its definitions in the introduction.We only note that for each A ∈ A the module We define an action B ∈ SBimod on K as follows.Note that we have a submodule Theorem 1.2 (Theorem 2.35, 2.40).We have the following.
(1) For each A ∈ A we have an indecomposable module Q(A) ∈ K P such that Q(A) {A} S and Q(A) {A } = 0 implies A ≥ A.
(2) Any object in K P is isomorphic to a direct sum of Q(A)(n) for A ∈ A and n ∈ Z.
(3) The split Grothendieck group [ K P ] is isomorphic to a certain submodule P 0 of the periodic Hecke module.(The submodule was introduced in [Lus80].) 1.3.A relation with a work of Fiebig-Lanini.Fiebig-Lanini [FL15] had a similar work (earlier than this work) and defined a certain category.Logically, results in this paper does not depend on their work.However, in the proof in this paper, we borrow many ideas from their work.Moreover, in subsection 2.10, we prove that our category K P is equivalent to the category of Fiebig-Lanini.The author thinks it is possible to establish the theory on top of the theory of Fiebig-Lanini, but the existence of a Hecke action does not soon follow from their theory.
1.4.Relations with representation theory.The category K P is not the category we really need.We modify this category as follows.Objects in K P are the same as those in K P and the space of homomorphisms is defined by We can prove that the action of B ∈ SBimod on K P is well-defined.
(1) The object Q(A) is also indecomposable as an object of K P .
(2) We have We also define a functor F : K P → K AJS .Recall that we have the wall-crossing functor ϑ s : K AJS → K AJS for each s ∈ S aff .
Let K AJS,P be the essential image of F. One of the main results in [AJS94] says that K ⊗ S K f AJS,P Proj(Rep 0 (G 1 T )).Since the action of SBimod on K P K AJS,P gives an action on K ⊗ S K f AJS,P , we now get the action of SBimod on Proj(Rep 0 (G 1 T )).We can extend this action to Rep 0 (G 1 T ), see 3.7.
Let A 0 be the alcove containing ρ/p where ρ is the half sum of positive roots.We have an equivalence where λ w(A 0 ) = pw(ρ/p) − ρ for w ∈ W aff and P (λ A ) is the projective cover of the irreducible representation with the highest weight λ A .Let Z(µ) ∈ Rep(G 1 T ) be the baby Verma module with the highest weight µ and (P (λ) : Z(µ)) the multiplicity of Z(µ) in a Verma flag of P (λ).By the constructions, we have the following.

Theorem 1.5 (Corollary 3.36). The multiplicity (P (λ
In the final part, we discuss Lusztig's conjecture on irreducible characters of algebraic representations.We give a proof of the conjecture based on the theory developed in this paper.
Acknowledgment.The question treated in this paper was asked by Masaharu Kaneda.The author had many helpful discussions with him.He also thank the referees giveng helpful comments and pointing out errors.The author was supported by JSPS KAK-ENHI Grant Number 18H01107.

Our combinatorial category
We use a slightly different notation from the introduction.In particular, we do not fix the alcove A 0 .So we distinguished two actions (from the right and left) of W aff on A as in [Lus80].
2.1.Notation.Let (X, ∆, X ∨ , ∆ ∨ ) be a root datum.Let A the set of alcoves, namely the set of connected components of X R \ α∈∆,n∈Z {λ ∈ X R | λ, α ∨ = n} where X R = X ⊗ Z R. Let W f be the finite Weyl group and W aff = W f Z∆ the affine Weyl group with the natural surjective homomorphism W aff → W f .For each α ∈ ∆ and n ∈ Z, let s α,n : X → X be the reflection with respect to {λ ∈ X R | λ, α ∨ = n}.As in [Lus80], let S aff be the set of W aff -orbits on the set of faces.Then for each s ∈ S aff and A ∈ A, we set As as the alcove = A which has a common face of type s with A. The subgroup of Aut(A) (permutations of elements in A) generated by S aff is denoted by W aff .Then (W aff , S aff ) is a Coxeter system isomorphic to the affine Weyl group.The Bruhat order on W aff is denoted by ≥.The group W aff acts on A from the right.
We give related notation and also some facts.If we fix an alcove A 0 , then W aff A via w → wA 0 and W aff acts on A by (w(A 0 ))x = wx(A 0 ).This gives an isomorphism W aff W aff .The facts stated below are obvious from this description.
Let Λ be the set of maps A → X such that λ(xA) = xλ(A) for any x ∈ W aff and A ∈ A where x ∈ W f is the image of x.We write λ A = λ(A) for λ ∈ Λ and A ∈ A. For each A ∈ A, λ → λ A gives an isomorphism Λ ∼ − → X and the inverse of this isomorphism is denoted by ν → ν A .The group W aff acts on Λ by (x(λ))(A) = λ(Ax).
Let λ ∈ Λ and A, A ∈ A and assume that A, A are in the same Λ aff -orbit.Namely there exists µ ∈ Λ aff such that A = A µ = A + µ A .Since elements in Λ are constant on Z∆-orbits, we get λ A = λ A .Namely the isomorphism λ → λ A only depends on Λ afforbit in A. Hence we also denote the isomorphism by λ → λ Ω where Ω ∈ A/Λ aff .The inverse is denoted by λ → λ Ω .The Λ aff -orbit through A is equal to {A + λ | λ ∈ Z∆}.We denote this by A + Z∆.
The following lemma is obvious from the definitions.
Fix a positive system ∆ + ⊂ ∆.Let α ∈ ∆ + and n ∈ Z.We say A ≤ s α,n (A) if for any λ ∈ A we have λ, α ∨ < n.The generic Bruhat order ≤ on A is the partial order generated by the relations A ≤ s α,n (A).The following lemma is obvious from the definition.

Lemma 2.2. Let
Proof.We assume λ ∈ Z ≥0 ∆ + and prove that A ≤ A .We may assume λ = α ∈ ∆ + .Take n ∈ Z such that n − 1 < µ, α ∨ < n for any µ ∈ A. For µ ∈ A, we have On the other hand, assume that A ≤ A .Take ν ∈ A. Then by Lemma 2.2, we have This defines a topology on A. The following lemma is an immediate consequence of the previous lemma and it plays an important role throughout this paper.
where n 1 , . . ., n r ∈ Z.(In this paper, graded free means graded free of finite rank.)We set grk where v is the indeterminate.
2.2.The categories.Fix a noetherian integral domain K.We define Λ ∨ using X ∨ exactly in the same way as we defined Λ using X.We put The algebra R is equipped with a grading such that deg(Λ ∨ K ) = 2. Assumption 2.5.In the rest of this section, we assume the following.
(1) We have 2 ∈ K × and any α ∨ = β ∨ ∈ (∆ ∨ ) + are linearly independent in X ∨ K/m for any maximal ideal m ⊂ K.This condition is called the BKM condition.
Lemma 2.6.The representation X ∨ K of W f is faithful.Proof.If w ∈ W f fixes any element in X ∨ K , it fixes any image of α ∈ ∆.By the assumption, ∆ ∨ → X ∨ K is injective.Therefore w fixes any coroot.Hence w is identity.The image of α ∨ ∈ ∆ ∨ in X ∨ K is denoted by the same letter.We also put S = S(X ∨ K ).We give a grading to S via deg(X ∨ K ) = 2. Let S 0 be a commutative flat graded S-algebra.
For an S-module M , we denote M ∅ = S ∅ ⊗ M .Let S 0 be a graded S-algebra.We consider the category • M is a graded (S 0 , R)-bimodule which is finitely generated torsion-free as a left S 0 -module.
. This is a graded (S 0 , R)-bimodule.For M ∈ K (S 0 ), we put supp The action of W aff on A/Λ aff factors through W aff → W f and W f acts on A/Λ aff simply transitively.We have A is determined by the (S 0 , R)-bimodule structure.Hence any (S 0 , R)-bimodule homomorphism M → N sends A∈Ω M ∅ A to A∈Ω N ∅ A .We will often use this fact.Remark 2.8.Here we do not assume that a morphism M → N sends M ∅ A to N ∅ A .Therefore a submodule M ∅ A ⊂ M ∅ is not functorial For each closed subset I ⊂ A, we define By the following lemma, M I ∈ K (S 0 ).Hence M → M I is an endofunctor of K (S 0 ).
Lemma 2.9.The module M I is a submodule of M and we have We also have Proof.The first part is obvious and for the second part, the left hand side is contained in the right hand side.Take m from the right hand side and let f ∈ S such that f m ∈ M .Then we have f m ∈ M I and m is in the left hand side.The last assertion is obvious.
(S) M I 1 ∪I 2 = M I 1 + M I 2 for any closed subsets I 1 , I 2 .(LE) For any α ∈ ∆, there exist M Ω for all Ω ∈ W α,aff \A with an injective morphism for any α ∈ ∆, M satisfies (LE).The converse is not true.The correct statement is that M satisfies (LE) if and only if for any α ∈ ∆ there exists N ∈ K (S α 0 ) which is isomorphic to M α and satisfies Then M satisfies (S).In particular, if M satisfies (LE), then M α satisfies (S).
Proof.Set Ω = W α,aff A and let I 1 , I 2 ⊂ A be closed subsets.We have We may assume I 1 ∩ Ω ⊂ I 2 ∩ Ω.We can take closed subsets I 1 and I 2 such that Then we have Hence we may assume I 1 = I 1 and I 2 = I 2 .In this case (S) obviously holds.
Let K ⊂ A be a locally closed subset, namely K is the intersection of a closed subset I with an open subset J.It is easy to see that M I /M I\J M I /M I \J naturally for a closed subset I and an open subset J such that K = I ∩ J.We define M K = M I /M I\J for M ∈ K(S 0 ).By Lemma 2.9, we have By putting (M K ) ∅ A as the image of M ∅ A by this isomorphism, we have an object M K of K (S 0 ).The following lemma is obvious.
Lemma 2.11.We have supp A (M K ) = supp A (M ) ∩ K for any locally closed subset Proof.The proof is divided into several steps.
(1) Assume that both K 1 , K 2 are closed.Then the lemma follows from the definitions.
(2) Assume that Therefore the canonical embedding is surjective.We get the lemma.
(3) Assume that K 2 is closed.Take a closed subset I 1 and an open subset J 1 such that (4) Now we prove the lemma in general.Let I i be a closed subset and J i be an open subset such that K i = I i ∩ J i and put , we get the lemma.
Proof.Take a closed subset I and an open subset J such that K = I ∩ J.
We prove M K satisfies (S).Let I 1 , I 2 be closed subsets.Since (M K ) We prove M K satisfies (LE).We may assume is locally closed.We say that an object M of K(S 0 ) admits a standard filtration if M {A} is a graded free S 0 -module for any A ∈ A. Let K ∆ (S 0 ) be a full subcategory of K(S 0 ) consisting of an object M which admits a standard filtration and supp A (M ) is finite.By Lemma 2.12, if M ∈ K ∆ (S 0 ) then M K ∈ K ∆ (S 0 ) for any locally closed subset K ⊂ A.
Lemma 2.14.Let M 1 , . . ., M l ∈ K(S 0 ) and assume that supp A (M 1 ), . . ., supp A (M l ) are all finite.Let I ⊂ A be a closed subset and A ∈ I such that I \ {A} is closed.Then there exist closed subsets Then it is easy to see that I i = I 0 ∪{A 1 , . . ., A i } is closed and satisfies the conditions of the lemma.
Lemma 2.15.Let M ∈ K ∆ (S 0 ) and K a locally closed subset.Then M K is graded free as a left S 0 -module.
Proof.Since M K ∈ K ∆ (S 0 ), we may assume K = A. Take closed subsets Finally we define the category K P (S 0 ) which plays an important role later.The definitions are taken from [FL15].
Definition 2.16.We say a sequence We define the category K P (S 0 ) ⊂ K ∆ (S 0 ) as follows: M ∈ K P (S 0 ) if and only if for any sequence Proof.Replacing M i with (M i ) K for i = 1, 2, 3, we may assume K = A. We can take closed subsets 2, 3 and j = 0, . . ., r, as in Lemma 2.14.Then the exactness of 0 → (M 1 ) I j → (M 2 ) I j → (M 3 ) I j → 0 follows from the induction on j and a standard diagram argument.
The lemma follows from Lemma 2.12.

Base change.
Let S 1 be a flat commutative graded S 0 -algebra.For M ∈ K (S 0 ), We put K = K (S), K = K(S), K ∆ = K ∆ (S) and K P = K P (S).We also put ( K 2.5.Hecke action.Let s ∈ S aff and we define α s ∈ Λ K and α ∨ s ∈ L K as follows: let A ∈ A and α ∈ ∆ + such that s α,n = As for some n ∈ Z. Then we put α s = α A and α ∨ s = (α ∨ ) A .These depend on a choice of A and α.For each s ∈ S aff we fix such A and α and define α s , α ∨ s .
Lemma 2.19.The pair (α s , α ∨ s ) does not depend on A, α up to sign.Proof.Let A ∈ A and take Recall that we have an object The decomposition does not depend on a choice of δ s .
Let SBimod be the category defined in [Abe19] for (W aff , S aff ) and the representation Lemma 2.20.Let B ∈ SBimod.Then there exists a decomposition x is the Frac(R)-bimodule as in the definition of SBimod.
Proof.Assume that B 1 ∈ SBimod is a direct summand of B ∈ SBimod and let e ∈ End SBimod (B) be the idempotent such that B 1 = e(B).If B satisfies the lemma, then by putting (B 1 ) ∅ x = e(B ∅ x ), we see that B 1 also satisfies the lemma.Therefore we may assume Note that for B = B s , the lemma holds as we have seen in the above.Hence it is sufficient to prove that if B 1 , B 2 satisfies the lemma then B 1 ⊗ B 2 also satisfies the lemma. For x .
For M ∈ K (S 0 ) and B ∈ SBimod, we define M * B ∈ K (S 0 ) by x is free as a left R ∅ -module.The following lemma follows.Lemma 2.21.We have supp (2.1) The isomorphism is given by m ⊗ f → (mf, ms(f )).Note that the last isomorphism is an isomorphism as left We know that if ϕ is an (S 0 , R)-bimodule homomorphism, ψ is also an (S 0 , R)-bimodule homomorphism and it induces a bijection between the spaces of (S 0 , R)-bimodule homomorphisms.(See, for example, [Lib08, and the same for N by (2.1) for A ∈ A.
Therefore ψ is a morphism in K (S 0 ).On the other hand, assume that ψ is a morphism in K (S 0 ).Consider the map Φ : where α s , δ s = 1.
Proof.We have If Ωs = Ω, then in the second direct sum, we can replace As with A. Therefore Assume that Ωs = Ω and take By calculations using this, we have Hence the right action of (2) Fix α ∈ ∆.By replacing M α with an object which is isomorphic to M α , we may assume The argument of the proof of (1)(b), we have Hence M * B s satisfies (LE).
2.6.An example.We give an example of our category.Let (X = Z, ∆ = {α = 2}, X ∨ = Z, ∆ ∨ = {α ∨ = 1}) be the root system of type A 1 .The Weyl group W f is {e, s α }.Let s 1 ∈ S aff (resp.s 0 ∈ S aff ) be the element corresponding to W aff {0} (resp.W aff {1}).Then S aff = {s 0 , s 1 }.The set of alcoves is given by Define Q An ∈ K = K (S) as follows.As an (S, R)-bimodule, we define (This object will be denoted by Q An,α later.) We have supp In the below, by an isoimorphism We define p 1 : In the definition of p 2 , we note that bs(f ) ≡ ag, bs(g) ≡ af (mod α ∨ ) since a ≡ b, s(f ) ≡ f, s(g) ≡ g, f ≡ g (mod α ∨ ).These are K[α ∨ ]-bimodule homomorphisms and from the above description, p 1 is a morphism in K .We have Therefore p 2 is also a morphism in K .We define i 1 : -bimodule homomorphism.We can also check that i 1 is a morphism in K .We also define i 2 : Note that the decomposition is not compatible with respect to the decomposition over K[α ∨ ] ∅ since i 1 is not compatible with the decomposition.2.7.Hecke actions preserve K ∆ .We assume that K is local.Then since any direct summands of any graded free S-module is also graded free, a direct summand of an object in K ∆ is also in K ∆ .The aim of this subsection is to prove the following proposition.
We fix M ∈ K ∆ and s ∈ S aff in this subsection and prove M * B s ∈ K ∆ .The most difficult part is to prove that M * B s satisfies (S).First we remark that, since M * B s satisfies (LE) by Lemma 2.23, (M * B s ) α satisfies (S) by Lemma 2.10.as left S-modules.
(2) First we prove that there exists an embedding (M * B s ) I /(M * B s ) I\J → M {A,As} (−1).We may assume J = {A ∈ A | A ≤ A} since I \ J is not changed.Then J is sinvariant.Put I 1 = I ∪ Is.Then I 1 is an s-invariant closed subset and I 1 ∩ J = (I ∩ J) ∪ (Is ∩ J) = (I ∩ J) ∪ (I ∩ J)s = {A, As}.We have I 1 \ {As} ⊃ I. Hence we have an embedding M {A,As}(−1) .We prove that this embedding is surjective.
First we assume that K is a field.Take a sequence of closed subsets By (1) and the existence of an embedding we proved, dim K (N By replacing A i with A i s in the second and fourth sum, we have Since Hence the embedding has to be a bijection Now let K be a general Noetherian integral domain.Assume that we can prove that (N {A i ,A i s} is finitely generated as a K-module, by Nakayama's lemma, (N is surjective where (•) m means the localization at m.
Since this is true for any maximal ideal m, the map (N I i /N I i−1 ) l → M l {A i ,A i s} is surjective for any l ∈ Z, hence it is an isomorphism.Therefore it is sufficient to prove (N In the rest of the proof, we omit the grading. To prove this, we need some properties on the base change to K/m.Let L ∈ K .Then we have L ⊗ K (K/m) is a (S/mS, R/mR)-bimodule and we have Therefore it defines an object in K K/m , here the suffix K/m means that in the definition of K we replace K with K/m.Let K ⊂ A be a closed subset.Then we have a map (1) The map is surjective.
(2) If L/L K is graded free, then this map is an isomorphism.We prove (1) first.By the exact sequence 0 In particular, if L satisfies (S), then L ⊗ K (K/m) also satisfies (S).Indeed, let K 1 , K 2 be closed subsets.Then we have a commutative diagram Here the horizontal maps are surjective by (1) in the above and the left vertical map is surjective since L satisfies (S).Hence the right vertical maps is surjective and it means that L ⊗ K (K/m) satisfies (S).We also have that if L satisfies (LE) then L ⊗ K (K/m) satisfies (LE).Let α ∈ ∆ and decompose L α as By the right exactness of the tensor product, we have (L We return to the proof of the lemma.We have Note that the bottom homomorphism is an isomorphism since the lemma is proved if K is a field. We prove that the left vertical map is an isomorphism by backward induction on i.By inductive hypothesis, N I i /N I i −1 M {A i ,A i s} for any i > i and in particular it is graded free.Hence N/N I i is also graded free.Therefore we have N I i ⊗ K (K/m) (N ⊗ K (K/m)) I i .Now we get the desired result by applying the five lemma to the following commutative diagram with exact columns 0 We can take a sequence of closed subsets Then by Lemma 2.27, where ε(A i ) ∈ {±1} is as in the proof of Lemma 2.27.In particular this is graded free and hence M I 1 /M I 2 = M I r /M I 0 is also graded free.
Proof of Proposition 2.24.Set N = M * B s .We prove that N satisfies (S).Let I 1 , I 2 are closed subsets and we prove the surjectivity of be the localization at the prime ideal (ν).Then N (ν) = S (ν) ⊗ S N satisfies (S).Hence this embedding is surjective after applying S (ν) ⊗ S .We denote L (ν) = S (ν) ⊗ S L for a left S-module L. Since N I 1 /N I 1 ∩I 2 is graded free by Lemma 2.28, we have We get the surjectivity.Now N {A} is well-defined and isomorphic to M {A,As} (ε(A)) where ε(A) ∈ {±1} is as in the proof of Lemma 2.27.Hence N {A} is graded free, namely N admits a standard filtration.
As a consequence of Lemma 2.27, we get the following corollary.
Corollary 2.29.If M ∈ K ∆ , then we have Therefore we have for each A ∈ A and s ∈ S aff .
The action of SBimod preserves K P too.
Proof.Let M ∈ K P and s ∈ S aff .We prove M * B s ∈ K P .We have already proved that 2.8.Indecomposable objects.Assume that K is complete local.For M, N ∈ K , Hom • S (M, N ) is finitely generated as an S-module since M, N are finitely generated and S is Noetherian.Hence Hom • K (M, N ) ⊂ Hom • S (M, N ) is also finitely generated.Therefore, Hom K (M, N ) is finitely generated K-module.Hence K has Krull-Schmidt property.This is also true for K P .
Set (R∆) int = {λ ∈ R∆ | λ, ∆ ∨ ⊂ Z} be the set of integral weights.For λ ∈ (R∆) int , let Π λ be the set of alcoves A such that λ, α ∨ − 1 < a, α ∨ < λ, α ∨ for any a ∈ A and simple root α.The set Π λ is called a box and each We define Q λ ∈ K as follows.Consider the orbit As an (S, R)bimodule, it is given by where the right action of R is given by (z A )f = (f A z A ).We have λ and 0 otherwise.The definition of Q λ comes from the structure sheaf of the moment graph associated to W f .The structure sheaf is defined by λ is a bijection which preserves orders and by this bijection we have Z Q λ .The following are well-known.(See [Abe20b] for example.) ∈ K} and the same for Z K\{w} .Then Z K /Z K\{w} S(−2 (w 0 w)) as a left S-module.Let d : A × A → Z be the function defined in [Lus80,1.4].From the second property we get the following.

Lemma 2.31. Let
In particular, we have Q λ ∈ K ∆ .Lemma 2.32.Let S 0 be a commutative flat graded S-algebra.We have Hom Proof.Since S 0 is flat, we have Any (S 0 , R)-bimodule is regarded as an S 0 ⊗ R-module.Let M ∈ K ∆ (S 0 ) and m ∈ M .According to the decomposition Then we have the following.
• For A ∈ A and f ∈ S W f , f A dose not depend on A.
) and by the property of Z we have remarked, this is an isomorphism.Therefore Q λ is a free S ⊗ S W f R-module of rank one with a basis q.We also remark that q ∈ S 0 ⊗ S Q λ = (S 0 ⊗ S Q λ ) I .Therefore ϕ → ϕ(q) gives an embedding Therefore the lemma follows from the following lemma.Lemma 2.33.
the right hand side is contained in the left hand side.Let A be in the left hand side.Take x ∈ W λ and µ ∈ Z∆ such that A = x(A − λ ) and Let A ∈ Π λ and take w ∈ W aff such that A = A − λ w.As in the proof of [Lus80, Proposition 4.2], for any x < w and A Lemma 2.34.We have the following. ( Proof.The second one is obvious from what we mentioned before the lemma.We prove (1) by induction on l.
(1) For any A ∈ A, there exists an indecomposable object Proof.Fix s 1 , . . ., s l as in the above.By Lemma 2.34, there is the unique indecomposable module Then by Corollary 2.29, ch is a [SBimod] H-module homomorphism.
Theorem 2.40.We have ch : , the image of ch is contained in P 0 and it surjects to P 0 .The H-module Hence ch is injective.

A relation with a work of Fiebig-Lanini.
In [FL15], Fiebig and Lanini constructed a category denoted by C and proved that this is an exact category.They also constructed a wall-crossing functor θ s for s ∈ S aff on C and proved that projective objects are preserved by wall-crossing functors.In this subsection, we prove the following.We identify W aff W aff and S R by using A + 0 .Theorem 2.41.The category K P is equivalent to the category of projective objects in C. The action of B s on K P corresponds to θ s for s ∈ S aff .
Let M ∈ K P and J ⊂ A an open subset.Then M J is an R-bimodule (as we identify S R) and the left action of f ∈ R W f is equal to the right action of f .Hence M J is an R ⊗ R W f R-module.The algebra R ⊗ R W f R is isomorphic to the structure algebra Z on the moment graph attached to W f .Hence we get a functor F from K P to the category of Z-coefficient presheaves on A.
We prove that F is fully-faithful.Since M = F (M )(A) as an R-module, F induces an injective map between space of morphisms, namely F is faithful.Let f : F (M ) → F (N ) be a morphism between sheaves.We define ϕ : be an open subset and J (resp.J ) be the largest (resp.smallest) s-invariant open subset which is contained in (resp.contains) J. Then we have morphisms such that j , j are surjective.We have (M * B s ) J M J * B s and (M * B s ) J M J * B s by Lemma 2.25.We have supp A (Ker j 1 ) ⊂ J \ J and supp A (Ker j 2 ) ⊂ J \ J .Hence, by [FL15, Lemma 2.8], (M * B s ) J satisfies the condition in [FL15, 8.3] and we get Finally we prove that the image of F is projective and the functor from K P to the category of projective objects in C is essentially surjective.Let K λ be a projective object in C defined in [FL15,Section 6].From the definitions, we have [FL15,Corollary 8.7], F (M ) is projective in C for any M ∈ K P .Moreover, by the proof of [FL15, Theorem 8.8], any projective object in C is a direct sum of direct summands of objects of a form θ s l • • • θ s 1 K λ .Since F is fully-faithful, the essential image of F is closed under taking a direct summand.Hence F is essentially surjective.

The category of Andersen-Jantzen-Soergel
Throughout this section, we assume that K is noetherian complete local ring.

Our combinatorial category.
In this subsection we introduced some categories using the categories introduced in the previous section.The categories will be related to the combinatorial categories of Andersen-Jantzen-Soergel.
Let S 0 be a flat commutative graded S-algebra.Let K (S 0 ) be the category whose objects are the same as those of K (S 0 ) and the spaces of morphisms are defined by We also define K(S 0 ) and K ∆ (S 0 ) by the same way.
Proof.Let ϕ and ψ as in the proof of Proposition 2.22.Then the proof of Proposition 2.22 shows that ϕ(M Hence M → M {A} defines a functor from K(S 0 ) to the category of graded S 0 -modules.Using this, we define as follows: A sequence For the definition of K P (S 0 ), we use the same condition to define K P (S 0 ).For M ∈ K ∆ (S 0 ), we say M ∈ K P (S 0 ) if for any sequence Note that this definition is not the same as that in the introduction.We will prove that two definitions coincide with each other later.Proposition 3.3.An indecomposable object in K (S 0 ) such that supp A (M ) is finite is also indecomposable as an object of K (S 0 ).
Proof.Let M ∈ K (S 0 ) and assume that supp and, since supp A (M ) is finite, this is nilpotent.Therefore the idempotent lifting property implies the proposition.
Lemma 3.4.Let K ⊂ A be a locally closed subset such that for any A ∈ K we have (A + Z∆) ∩ K = {A}.Then we have the following.
(1) For a morphism ϕ : M → N in K(S 0 ) which is zero in K(S 0 ), the homomorphism Proof.We may assume B = B s where s ∈ S aff .We take lifts of M 1 → M 2 and M 2 → M 3 in K(S 0 ) and we regard M 1 → M 2 → M 3 also as a sequence in K(S 0 ).As in Corollary 2.29, we have (M i * B s ) {A} (M i ) {A,As} (ε(A)) where ε(A) is as in the proof of Lemma 2.27.By the previous lemma, 0 Combining Proposition 3.2, we have K P (S 0 ) * SBimod ⊂ K P (S 0 ).Lemma 3.6.Let λ ∈ (R∆) int .The subset W λ A − λ is locally closed and we have a natural isomorphism Hom Proposition 3.7.The objects of K P are the same as those of K P .
Proof.First we prove that any M ∈ K P belongs to K P .By Theorem 2.35, we may assume for some λ ∈ (R∆) int , s 1 , . . ., s l ∈ S aff and n ∈ Z.By Proposition 3.2 and Lemma 3.5, we may assume M = Q λ .
We have Hom The object Q(A) is indecomposable by Proposition 3.3.Using the argument in the proof of Theorem 2.35, any object in K P is a direct sum of Q(A)(n).Hence the proposition is proved.
Hence our K P is the same as that in the introduction.
Corollary 3.8.Let M ∈ K P , N ∈ K ∆ and S 0 a flat commutative graded S-algebra.

Proof. We may assume
(1) By Proposition 3.2, we may assume M = Q λ .In this case, the corollary is equiv- . This is clear.
We can define ch : [K P ] → P 0 by the same formula as ch : [ K P ] → P 0 .By the previous proposition with Theorem 2.40, we get the following.Theorem 3.9.We have [K P ] P 0 .

A formula on homomorphisms.
Let m → m be a map from P 0 to P 0 defined in [Soe97,Theorem 4.3].For m ∈ P 0 and m ∈ P, take c x .Then we have (mh, m ) P = (m, m ω(h)) P where m ∈ P 0 , m ∈ P and h ∈ H.This easily follows from the definitions.Let w 0 ∈ W f be the longest element.Theorem 3.10.Let P ∈ K P and M ∈ K ∆ .Then Hom • K ∆ (P, M ) is graded free left S-module and the graded rank is given by R∆) int and s 1 , . . ., s l ∈ S aff , we may assume P has this form.Moreover, by Lemma 3.2 and the formula before the theorem, we may assume P = Q λ .In this case, we have Hom and this is graded free by the definition of K ∆ .Moreover, the graded rank of Let S λ be the set of reflections in W λ along the walls of A − λ .Then this is a generator of W λ and (W λ , S λ ) is a Coxeter system.The length function of this Coxeter system is denoted by λ .

The category
In this subsection, we analyze K α P = K P (S α ).First we define an object Q A,α where A ∈ A and α It is easy to see that Therefore it is sufficient to prove the following: 3 ) → 0 is exact.We can apply a similar argument of the proof of Proposition 3.7.
We can apply the argument in the proof of Theorem 2.35 and get the following proposition.
Proposition 3.12.Any object in K α P is a direct sum of Q α A,α (n) where A ∈ A and n ∈ Z. 3.4.The comibinatorial category of Andersen-Jantzen-Soergel.We recall the comibinatorial category of Andersen-Jantzen-Soergel [AJS94].We use a version of Fiebig [Fie11].We denote the category by K AJS .
Let S 0 be a flat commutative graded S-algebra and we define the category which we denote K AJS (S 0 ).An object of ) is a collection of degree zero (S 0 ) ∅ -homomorphisms f A : M(A) → N (A) which sends M(A, α) to N (A, α) for any A ∈ A and α ∈ ∆ + .Put K AJS = K AJS (S) and K * AJS = K AJS (S * ) for * ∈ {∅} ∪ ∆.For each s ∈ S aff , the translation functor ϑ s : K AJS (S 0 ) → K AJS (S 0 ) is defined as We define F(S 0 ) : K P (S 0 ) → K AJS (S 0 ) as follows: first we put (F(S 0 )(M ))(A) = M ∅ A .To define (F(S 0 )(M ))(A, α), we take X ∈ K P (S α 0 ) and an isomorphism ϕ : X → M α in K P (S 0 ) such that X = Ω∈W α,aff \A (X ∩ A∈Ω X ∅ A ).Such X exists since M satisfies (LE).Then we have an isomorphism Then this isomorphism can be written as x → ϕ(x) A .Here we use the same letter ϕ for the induced map X ∅ → M ∅ .Now let (F(S 0 )(M ))(A, α) be the image of In other words, the image is the set of (ϕ(x A ) A , ϕ(x α↑A ) α↑A ) where x ∈ X ≥A .We may assume x ∈ A ∈W α,aff A X ∅ A .Of course we have to prove that this space does not depend on a choice of X.We use the following lemma.
A from the condition on x.
• f (x A ) A = 0 unless A ≥ A from the definition of morphisms in K P (S 0 ).Therefore, in the sum A ∈A f (x A ) A , we may assume However, by Remark 2.7, we have f (x A ) α↑A = 0. Hence f (x) α↑A = f (x α↑A ) α↑A .
We prove (2).We have f ( As ψ is a morphism, ψ(x) ∈ X ≥A .Hence the right hand side is in (F(S 0 )(M ))(A, α) determined by X .Therefore the space (F(S 0 )(M ))(A, α) determined by X is contained in the space (F(S 0 )(M ))(A, α) determined by X .By swapping X with X , we get the reverse inclusion and therefore the space (F(S 0 )(M ))(A, α) does not depend on a choice of X.
Let f : M → N be a morphism in K P (S 0 ) and take a lift f ∈ Hom K P (S 0 ) (M, N ) of f .Then we have a homomorphism (F(S 0 ) In other words, we put (F(S 0 )(f ))(A)(m) = f (m) A .It is easy to see that this does not depend on a lift f .We prove that the collection ((F(S 0 )(f ))(A)) A∈A preserves (F(S 0 )(M ))(A, α).Take X ∈ K P (S α 0 ) and ϕ : X ∼ − → M α as in the definition of (F(S 0 )(M ))(A, α).We also take ψ : ).This is given by ). Hence it is sufficient to prove that the image of On the other hand, let α↑A , m 2,(α↑A)s ).We get the proposition.
3.5.Some calculations of homomorphisms.In this subsection we fix a flat commutative graded S-algebra S 0 .We define some morphisms as follows.These will be used only in this subsection.Let A ∈ A and α ∈ ∆ + .
It is straightforward to see that these are morphisms in K.We also denote the images of these morphisms in K by the same letters.Lemma 3.15.We have End By the same argument, we also have Proof.Set M = S 0 ⊗ S Q A,α and let ϕ : M → M be a morphism.Since M(A ) = 0 for A = A, α ↑ A, we have ϕ A = 0 for such A .The morphism ϕ preserves M(β ↓ A, β) = 0 ⊕ S β 0 for any β ∈ ∆ + .Hence ϕ A (S β 0 ) ⊂ S β 0 .Therefore ϕ A (S 0 ) ⊂ S 0 and hence ϕ A = c id for some c ∈ S 0 .We also have ϕ α↑A = d id for some d ∈ S 0 .
Set Q λ = F(Q λ ).Let K AJS,P be the full-subcategory of K AJS consisting of direct summands of direct sums of objects of a form (ϑ s 1 • • • • • ϑ s l )(Q λ )(n) for s 1 , . . ., s l ∈ S aff , λ ∈ (R∆) int and n ∈ Z.By Proposition 3.14 and 3.26, we get the following theorem.Theorem 3.27.We have K P K AJS,P .In particular, the category SBimod acts on K AJS,P .3.7.Representation Theory.In the rest of this paper, we assume that K is an algebraically closed field of characteristic p > h where h is the Coxeter number.Let G be a connected reductive group over K and T a maximal torus of G with the root datum (X, ∆, X ∨ , ∆ ∨ ).The Lie algebra g of G has a structure of a p-Lie algebra.Let U [p] (g) be the restricted enveloping algebra.Let S be the completion of S at the augmentation ideal.For S 0 = S or K, let C S 0 be the category defined in [AJS94].The category C K is equivalent to the category of G 1 T -modules where G 1 is the kernel of the Frobenius morphism.Let Z S 0 (λ) ∈ C S 0 be the baby Verma module with the highest weight λ and P S 0 (λ) ∈ C S 0 the indecomposable projective module such that K ⊗ S 0 P S 0 (λ) is the projective cover of the irreducible module with the highest weight λ.Such objects exist by [AJS94,4.19Theorem] when S 0 = S.
We fix an alcove A 0 ∈ A and λ 0 ∈ X ∩ (pA 0 − ρ) where ρ is the half sum of positive roots and pA 0 = {pa | a ∈ A 0 }.For S 0 = S or K, let C S 0 ,0 be the full subcategory of C S 0 consisting of quotients of modules of a form w∈W aff P S 0 (w • p λ 0 ) nw where w • p λ 0 = pw((λ 0 + ρ)/p) − ρ and n w ∈ Z ≥0 .Then the cateogory C S 0 ,0 is a direct summand of C S 0 .Let Proj(C S 0 ,0 ) = {P ∈ C S 0 ,0 | P is projective}.
Let S 0 be a commutative S-algebra which is not necessary graded.We consider the following object: M = ((M(A)) A∈A , (M(A, α)) A∈A,α∈∆ + ) where M(A) is an (S 0 ) ∅module and M(A, α) ⊂ M(A) ⊕ M(α ↑ A) is a sub-(S 0 ) α -module.(We consider usual modules, not graded ones.)We denote the category of such objects by K f AJS (S 0 ).Starting from this, we can define the functor ϑ s and the category K f AJS,P (S 0 ) in a similar way.Andersen-Jantzen-Soergel [AJS94] proved the following.We modified the functor using [Fie11, Theorem 6.1].
Theorem 3.28.There is an equivalence of the categories V : Proj(C S,0 ) ∼ − → K f AJS,P ( S).Note that the functor V is defined explicitly.Let K ⊗ S Proj(C S,0 ) be the category defined as follows.The objects of K ⊗ S Proj(C S,0 ) are the same as those of Proj(C S,0 ) and the space of homomorphism is defined by Proof.We consider the functor K ⊗ S Proj(C S,0 ) → Proj(C K,0 ) defined by P → K ⊗ S P .This is essentially surjective by [AJS94,4.19Theorem] and fully-faithful by [AJS94, 3.3 Proposition].
We also define K ⊗ S K f AJS,P ( S) and K ⊗ S K f AJS,P (S) by the same way.Lemma 3.30.We have the following.
(1) The category K f AJS,P (S) is equivalent to the category defined as follows: the objects are the same as K AJS,P and the space of homomorphisms is defined by Hom K f AJS,P = Hom It is easy to see that this gives an action of SBimod on Mod Z∆ (E), hence on C K,0 .
3.8.Characters.Any object P ∈ Proj(C S,0 ) has a Verma flag.We denote the multiplicity of Z S (w • p λ 0 ) in P by (P : Z S (w • p λ 0 )).The following lemma is obvious from the constructions.
The projective module P S (λ) is characterized by • P S (λ) is indecomposable.
The following corollary is obvious from the above proposition.H.For each w ∈ W aff , there exists an indecomposable object B(w) ∈ SBimod unique up to isomorphism such that ch(B(w)) ∈ H w + x<w Z[v, v −1 ]H x .We say that B(w) satisfies the Soergel conjecture if ch(B(w)) is a Kazhdan-Lusztig basis, namely ch(B(w)) ∈ H w + x<w vZ[v]H x .It is known that the Soergel conjecture is satisfied by any B(w) over a characteristic zero field, therefore, for a fixed w, if p is sufficiently large, B(w) satisfies the Soergel conjecture (cf.[EW14]).We fix λ ∈ (R∆) int and w ∈ W aff such that A + λ w ∈ Π λ here A + λ is the maximal element in W λ A − λ .Lemma 3.37.Let w λ ∈ W aff such that A + λ w λ = A − λ .Then we have S A + λ * B(w λ ) Q λ ( (w 0 )).
Proof.By the translation as in the proof of Lemma 2.31, we may assume λ = 0. Then W λ = W f and it is generated by S aff ∩ W f .Moreover, the element w λ is equal to the longest element w 0 .
It is sufficient to prove: B(w 0 ) {(z w ) ∈ R W f | z wt ≡ z w (mod α t )}( (w 0 )) where t runs through the set of reflections in W f and α t the corresponding element in Λ K [Abe19, 2.1].Let (G ∨ C , B ∨ C , T ∨ C ) be the reductive group over C, the Borel subgroup and the maximal torus with the root datum (X ∨ , ∆ ∨ , X, ∆) and the positive system ∆ + ⊂ ∆.Then the category of K-coefficient parity B ∨ C -equivariant sheaves on G ∨ C /B ∨ C is equivalent to the category of Soergel bimodules attached to (W f , X ∨ K ) [RW18].The object B(w 0 ) corresponds to the indecomposable parity sheaf such that the restriction to the big cell

x.
Let M ∈ K .Then we define M * B by M * B = M ⊗ S B as a graded S-bimodule and (M* B) ∅ w(A 0 ) = x∈W aff M ∅ wx −1 (A 0 ) ⊗ S ∅ B ∅ x for w ∈ W aff .We can prove the action preserves the subcategory K P (Proposition 2.24).Therefore the split Grothendieck group [ K P ] of K P has a structure of [SBimod]-module defined by [M ][B] = [M * B].Hence [ K P ] is a module of the Hecke algebra.This category satisfies the following.
As < A and I (resp.J) be an s-invariant closed (resp.open) subset such that I ∩ J = {A, As}.Set N = M * B s .Then we have N I\{As} /N I\{A,As} M {A,As} (−1), N I /N I\{As} M {A,As} (1).
1) is surjective and, by the above argument, the kernel is L ∩ (L ∅ A ⊕ 0) N I\{As} /N I\{A,As} .Therefore by the exact sequence (2.2), we have N I /N I\{As} M {A,As} (1).Lemma 2.27.Let A ∈ A such that As < A, I a closed subset and J an open subset.Then we have the following.(1)IfI ∩ J = {As}, then (M * B s ) I /(M * B s ) I\J M {A,As}(1) as left S-modules.(2) If I ∩ J = {A}, then (M * B s ) I /(M * B s ) I\J M {A,As} (−1) as left S-modules.Proof.Set N = M * B s ∈ K .(1) Put I 1 = {A ∈ A | A ≥ As}.This is s-invariant.Since I is closed and contains As, we have I 1 ⊂ I. Hence N I 1 /N I 1 \{As} → N I /N I\{As} .By Lemma 2.26, we have N I 1 /N I 1 \{As} M {A,As} (−1).Hence we have M {A,As} (−1) → N I /N I\{As} .
N A\J for any open subset J. Hence ϕ(M I ) ⊂ N I for any closed subset I ⊂ A. Therefore ϕ is a morphism in K P and therefore F is full.Next we prove that F (M * B s ) θ s (F (M )) for M ∈ K P .Let s ∈ S aff and s the functor defined in [FL15, 8.1].Then an argument of the proof in [Abe19, Proposition 5.3] gives s (M ) M ⊗ R B s as Z-modules.(Here, in the right hand side, we consider a Zmodule as an R-bimodule via or equivalently the image of (N * B s ) I where I = {A ∈ A | A ≥ As} \ {As}.Set I = {A ∈ A | A ≥ As}.Then I ⊃ I and I is s-invariant.Hence (N * B s ) I = N I ⊗ B s = N I ⊗ R s R by Lemma 2.25.Consider the projection (N * B s