On the Induction of p-Cells

We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.


Introduction
Let (W, S) be a Coxeter system and let H (W,S) denote its Hecke algebra.In [KL79], Kazhdan and Lusztig introduced a preorder on the elements of W whose equivalence classes are known as right Kazhdan-Lusztig cells.These Kazhdan-Lusztig cells can be used to construct representations of H (W,S) , called cell modules.Since their introduction, cells have been extensively studied and there is a very rich theory describing them.
If the Coxeter group is crystallographic, we can replace the Kazhdan-Lusztig basis of H (W,S) with the p-canonical basis (see [JW17]) which can be obtained as characters of the indecomposable parity complexes on a suitable Kac-Moody flag variety with coefficients in a field of characteristic p.For p = 0, the p-canonical basis specializes to the Kazhdan-Lusztig basis by results of Härterich [Här99].
Working with the p-canonical basis, we obtain a positive characteristic analogue of the Kazhdan-Lusztig cells, called p-cells.The motivation for studying p-cells is that, other than providing a first approximation of the multiplicative structure of the p-canonical basis, they allow one to construct representations of the Hecke algebra (called cell modules) that naturally come with a canonical basis.

INTRODUCTION
Moreover, according to a recent conjecture by Achar, Hardesty and Riche [AHR18] there is a deep relation between p-cells in affine Weyl groups and the geometry of nilpotent orbits.
The theory of p-cells has been initially developed by the first author in [Jen19], where several properties of Kazhdan-Lusztig cells are generalized to p-cells.The following theorem is inspired by [BV83,Proposition 3.11] and [Roi98, Proposition 1] and follows immediately from a key result in [Jen19, Theorem 3.9].In this paper we deal with the generalization of another compatibility result with respect to parabolic subgroups.This property is known as parabolic induction of cells.The following is the main result of the present paper.

Theorem
Theorem (Parabolic induction for right p-cells).Let I ⊆ S be a finitary subset and W I ⊆ W the corresponding standard parabolic subgroup.Denote by I W the set of minimal length representatives of cosets in W I \W .Let C be a right p-cell of W I and H C the corresponding right cell module of Parabolic induction is well-known for Kazhdan-Lusztig cells.It was first proven by Barbasch and Vogan in [BV83, Proposition 3.15] (see also [Lus84,(5.26.1)]) for a finite Weyl group W in the setting of primitive ideals for complex semi-simple Lie algebras.For arbitrary Coxeter groups, it was first conjectured by Roichman in [Roi98,§5].Finally, Geck proved it in the general setting of cells for Hecke algebras with unequal parameters in [Gec03].
In the following we will use the notation of the theorem.One may introduce the I-hybrid basis, which combines the p-canonical basis of H (WI ,I) and the standard basis elements corresponding to I W to get a Z[v, v −1 ]-basis of H (W,S) .Geck's proof is completely elementary and algebraic, combinatorial in nature.It is based on the study of the I-hybrid basis of the Hecke algebra (for p = 0) and its relation with the Kazhdan-Lusztig basis.
In contrast, our proof is based on a geometric argument.We use the categorification of the p-canonical basis via indecomposable parity complexes on the flag variety of a Kac-Moody group.First, we prove that the base change coefficients between the p-canonical and the I-hybrid basis are Laurent polynomials with non-negative coefficients.To achieve this, we relate the base change between these two bases to Braden's hyperbolic localization functors [Bra03].This works because parity sheaves are well behaved with respect to the hyperbolic localization functors (see [JMW12]).We remark that our argument is an adaptation of the work of Grojnowski and Haiman [GH07], which deals with mixed Hodge modules in characteristic 0, to the setting of parity sheaves.
The rest of the proof follows along the lines of Geck's proof.It uses a hybrid order which combines the right p-cell preoder in W I with the Bruhat order on I W .The crucial ingredient here is Proposition 3.11 which shows that the base change coefficients between the p-canonical basis and the I-hybrid basis are unitriangular with respect to the hybrid order.

Structure of the Paper
Section 2 We recall the definition of the Hecke algebra of a crystallographic Coxeter system.Then we discuss its geometric categorification via parity sheaves.We also recall the definition of and the main result about Braden's hyperbolic localization functors.Finally, we introduce the p-canonical basis and p-cells and mention some of their elementary properties.
Section 3 We introduce a new basis, called I-hybrid basis, for the Hecke algebra, combining the p-canonical basis for a parabolic subgroup and the standard basis.Then we study positivity properties of the p-canonical basis with respect to the I-hybrid basis.Finally, we prove that induction for p-cells still works.

Acknowledgements
We .2]for the definition).Then X is a finitely generated free abelian group, and for i ∈ J we have elements α i ∈ X and α ∨ i ∈ X ∨ = Hom Z (X, Z) respectively satisfying a i,j = α ∨ i (α j ) for all i, j ∈ J.We require our Kac-Moody root datum to satisfy the following assumptions (see [Mar18,§7.3.1]):(i) free, i.e. the set {α i : i ∈ J} is linearly independent over Z in X, (ii) cofree, i.e. the set {α ∨ i : i ∈ J} is linearly independent over Z in X ∨ , (iii) cotorsion free, i.e.X ∨ / i∈J Zα ∨ i is torsion-free, (iv) X is of rank 2|J| − rk(A).
Remark 2.1.These assumptions were originally imposed by Mathieu and Kumar (see [Mar18, Example 7.10] for the relation between a realization of A and a Kac-Moody root datum associated to A satisfying the assumptions (i) -(iv)).Even though they are not necessary for the construction of Kac-Moody groups (see [Rou16, Remarque 3.5 and §3.19] and [Mar18, §8.7]), we will need the first assumption in our proof.
To A we associate a crystallographic Coxeter system (W, S) as follows: choose a set of simple reflections S of cardinality |J| and fix a bijection S ∼ → J, s → i s .For s = t ∈ S we define m s,t to be 2, 3, 4, 6, or ∞ if a is,it a it,is is 0, 1, 2, 3, or 4 respectively.The group W is called the Weyl group of A.
The Hecke algebra H = H (W,S) associated to (W, S) is the free Z[v, v −1 ]-algebra with {H w | w ∈ W } as basis, called the standard basis, and multiplication determined by: There is a unique Z-linear ring involution (−) on

Parity Complexes on Kac-Moody Flag Varieties
In this section we want to recall some results about Kac-Moody flag varieties and parity complexes on them.The standard references for Kac-Moody flag varieties are [Mar18] and [Kum02].
To our Kac-Moody root datum with Cartan matrix A we can associate a (maximal) Kac-Moody group G with a canonical torus T ⊂ G. Mathieu actually constructs group ind-schemes over Z in [Mat88] and [Mat89], but for our purposes it is enough to consider their complex points as in [Kum02, §6.1] (see [Mar18,Exercise 8.123] and [Rou16, §3.20] for comparisons of the two constructions).
Let W be the Weyl group of G, i.e. the Weyl group of the Cartan matrix A. Let Φ denote the set of roots of G and Φ + the set of positive roots.We denote by Φ re the set of real roots, i.e. roots that can be written as w • α i , where w ∈ W and α i is a simple root.
Let U + be the (pro-)unipotent group obtained as the image of the positive part of the associated Kac-Moody Lie algebra under the exponential map defined in [Kum02, §6.1.1](also denoted U ma+ in [Rou16, p. 3.1]).For every positive real root α ∈ Φ re + there exists a one parameter subgroup U α ⊂ U + isomorphic to the additive group G a (see [Kum02, Example 6.1.5]).For every α we fix an isomorphism u α : G a ∼ −→ U α .Such an isomorphism is unique up to multiplication by a unit in C (acting on G a ).The subgroup U + is normalized by the torus T .Moreover, every U α for α a positive real root is normalized by T and we have The Borel subgroup B is a subgroup of G isomorphic to the semidirect product B = T ⋉ U + .The set G/B may be equipped with the structure of an ind-projective ind-variety and it is called Kac-Moody flag variety (see [Kum02, §7.1]).
For any finitary subset I ⊆ S we have the corresponding standard parabolic subgroup P I containing B (see [Kum02, Definitions 6.1.13(4) and 6.1.18]).It admits a Levi decomposition P I = L I ⋉ U I , where L I is a connected (finite dimensional) reductive group associated to A I , the sub-Cartan matrix of A obtained by taking the rows and the columns indexed by I. We denote by Φ I the roots of L I .The group T is a connected algebraic torus, whereas G, B, U + , P I and U I are all pro-algebraic groups.The following two examples cover the most important cases: Example 2.2.If A is a Cartan matrix, then the Kac-Moody root datum is equivalent to a root datum in the ordinary sense, and G is the associated complex simply connected algebraic group, B a Borel subgroup, T ⊂ B a maximal torus and U + the unipotent radical of B. In this case, P I is a standard parabolic subgroup and L I the corresponding Levi.Example 2.3.Assume that A = (a i,j ) is a Cartan matrix of size l − 1 and that the Kac-Moody root datum is simply connected (see [Mar18,Example 7.11]).Let G be the corresponding semisimple simply connected algebraic group.Moreover, one can add an l-th row and column to obtain a generalized Cartan matrix A as follows where the α i are the simple roots of G and θ the highest root.In this case, the Kac-Moody group G associated to A is of so-called untwisted affine type (see [Kum02 be the subgroup generated by U α , for α ∈ Φ(w).Applying [Kum02, Lemma 6.1.3]gives the following isomorphism (of varieties): The multiplication induces an isomorphism (see [Kum02, Exercise 6.2.E (1)] and [Kum02, Proposition 7.1.15]): u −→ uwB.
In this paper, we will view F l G := G/B as ind-variety equipped with the algebraic statification coming from the B-orbits.Fix a field k of characteristic p.We will consider D b B (F l G ), the Bequivariant bounded constructible derived category of sheaves of k-vector spaces on F l G .
Recall the notion of a parity complex introduced by Juteau

The p-Canonical Basis and p-Cells
In this section, we recall the definition of the p-canonical basis in the geometric setting.For this we will need to define the character map: We will need the following positivity property of the p-canonical basis which can be found in [JW17, Proposition 4.2 and its proof]: Proposition 2.8.For all x ∈ W we have: The next result is obtained from [Jen19, Corollary 3.10] by applying the anti-involution ι of H and using Proposition 2.8 (ii).
Proposition 2.9.Let I ⊆ S be a finitary subset.Then for x, y ∈ W I and z ∈ I W we have: Next, we recall the definition of p-cells from [Jen19, §3.1].This notion is an obvious generalization of the notion of cells introduced by Kazhdan  For more properties and results about p-cells, we refer the reader to [Jen19].

Hyperbolic Localization
Let T be a complex torus and X a complex variety with an action of T .In this section, we let k be a field and D(X) denote the constructible derived category of sheaves of k-vector spaces on X.Moreover, we make the following assumption:

X has a covering by T -stable affine open subvarieties. (C)
Note that this assumption is automatically satisfied if X is normal by Sumihiro's theorem (see [Sum74;KKLV89]).Let χ : G m −→ T be a cocharacter of T .We want to understand the hyperbolic localization with respect to the G m action induced by χ.
Denote by Z := X χ ⊆ X the variety of χ-fixed points in X and let Z 1 , . . ., Z m be its connected components.For 1 i m we will denote the attracting and repelling varieties of the component Z i by respectively.Let Z + (resp.Z − ) be the disjoint, disconnected union of the Z + i (resp.Z − i ) and define f ± : Z −→ Z ± and g ± : Z ± → X via the component-wise inclusions.The attracting and repelling maps p ± : Z ± −→ Z are defined via p + (x) = lim s→0 χ(s)x and p + (x) = lim s→∞ χ(s)x.It follows from [Hes81, Proposition 4.2] that these are algebraic maps.
Braden defines the hyperbolic localization functors (−) !* , (−) * !: D(X) −→ D(Z) associated to the cocharacter χ for F ∈ D(X) as follows: We will use the following result (see [Bra03, Theorem 1]): Theorem 2.11.For any F ∈ D(X) there is a natural morphism ι F : F * !−→ F ! * .If F is weakly equivariant (e.g.comes from an object in the equivariant derived category), then Using this result, Braden proves for k = Q that the hyperbolic localization of the intersection cohomology complex IC(X; Q) is a direct sum of shifted intersection cohomology complexes.

Positivity Properties
Recall that for our fixed finitary subset I ⊆ S we denote the parabolic subgroup generated by I by W I and the minimal coset representatives by I W .The multiplication induces a bijection  The main idea is to replace the purity arguments in [GH07, Theorem 3.2] by parity arguments.Before explaining the proof, we need to introduce some more notation.
Recall the decomposition P I = L I ⋉ U I from §2.2.To simplify notation, we will write L for L I and P for P I from now on.Denote by B L the Borel subgroup of L induced by our choice of positive roots, so that we have B L = B ∩ L. Lemma 3.3.For any y ∈ I W the stabilizer y B of yB satisfies B L = y B ∩ L.
Proof.The group y B ∩ L is the Borel subgroup of L corresponding to the system of positive roots {α ∈ Φ I | y −1 (α) ∈ Φ + I } ⊂ Φ I .Since y ∈ I W , we have y −1 s > y −1 for all s ∈ I, so y −1 (α s ) ∈ Φ + .Hence, for all α ∈ Φ + I we have y −1 (α) ∈ Φ + and similarly for α ∈ Φ − I we have y (α) We fix a dominant co-character γ : C * −→ T whose stabilizer in W is W I (this is possible because our realization is free).For each w ∈ W the connected components of the fixed locus X γ w are precisely the intersections X w ∩ i y (F l L ) for y ∈ I W (note that γ acts trivially by conjugation on a root subgroup U α if and only if α lies in Φ I ).
Lemma 3.5.The attracting map π y : P yB/B −→ LyB/B is induced by the group homomorphism P → L. The attracting variety to i y (F l L ) is P yB/B = x∈WI Y xy .Moreover, π y is an affine bundle, with fiber isomorphic to C ℓ(y) .Proof.By the previous lemma we have ByB ⊂ π −1 y (yB).Since W I stabilizes γ, the Levi subgroup L commutes with the image of γ.We can write any element p ∈ P as p = lu, with l ∈ L and u ∈ U ⊂ B. Now we have The following diagram of L-equivariant maps is commutative and the horizontal arrows are bijections since π y is an L-equivariant fiber bundle and Denote by j y : P yB/B −→ Fl G the inclusion.The following result gives a geometric interpretation of the coefficients occuring when expressing the class of an object in D b B (F l G ) in terms of the I-hybrid basis.Lemma 3.6.For x ∈ W I and y ∈ I W we have: In this equation ch(F ) ∈ H (WI ,I) is viewed as an element in H by identifying H (WI ,I) as the subalgebra of H generated by H s for s ∈ I.
(ii) The map π y is a topological fibration with fibers isomorphic to C ℓ(y) .Since F is constant along the fibers of π y we have The flag variety F l G is the disjoint union of P yB/B, for y where we used the first part for the last equality.We conclude by expressing ch i * y (π y ) ! j * y F in the p-canonical basis of H (WI ,I) .
The following result crucially relies on Braden's hyperbolic localization: Proposition 3.7.For any y ∈ I W the functor i * y (π y ) ! j * y preserves parity complexes and thus restricts to a functor Parity B (F l G ) −→ Parity BL (F l L ).
Proof.Theorem 2.11 implies that y∈ I W (π y ) ! j * y is isomorphic to Braden's hyperbolic localization functor (−) !* : ). Juteau, Mautner and Williamson develop in [JMW16, §2.2] a general framework to show that the hyperbolic localization of the pushforward of the constant sheaf on a resolution of singularities satisfying several properties (see [JMW16, §2.2 (1) -(3)]) is a parity complex.In [JMW16, Proof of Theorem 1.6] they verify that these conditions are satisfied for Bott-Samelson resolutions of Schubert varieties of a Kac-Moody group.Since any indecomposable parity complex E w occurs as a direct summand of the push-forward of the constant sheaf on a suitably chosen Bott-Samelson resolution, it follows that (E w ) ! * is a parity complex.Finally, a complex F on F l γ G is parity if and only if its restriction to any connected component is a parity complex.
Proof of Proposition 3.2.By Lemma 3.6(ii), the polynomial p r I xy,w is equal to v ℓ(y) the coefficient of p H x in ch i * y (π y ) ! j * y E w .By Proposition 3.7, the complex i * y (π y ) ! j * y E w is a parity complex on F l L , hence its character has positive coefficients when written in the p-canonical basis of H (WI ,I) .

Main Result
The proof of the main result draws inspiration from [Gec03].Throughout, we are working with the right p-cell preorder instead of the left Kazhdan-Lusztig cell preorder.The analogous version for left p-cells can be obtained by applying the anti-involution ι.The following result is the analogue of [Gec03, Lemma 2.2].Its proof works in our setting after replacing all Kazhdan-Lusztig related notions by the corresponding p-canonical analogues.We will rewrite the proof here for the sake of completeness.
Lemma 3.9.Let J ⊆ W I be a subset such that Proof.Since H is generated by H s for s ∈ S as an algebra, it is enough to check that p H x H y H s ∈ M for all x ∈ W I and y ∈ I W . Deodhar's lemma (see [GP00, Lemma 2.1.2])shows that there are three cases to consider: (i) ys ∈ I W and ℓ(ys) > ℓ(y).In this case, we have p H x H y H s = p H x H ys ∈ M as desired.
Corollary 3.12.Let J ⊆ W I and M be as in Lemma 3.9.Then we have Proof.Let x ∈ J and y ∈ I W . Proposition 3.11 shows that p H xy lies in M.
Moreover, Proposition 3.11 shows that in any total order refining the preorder the base change matrix between the p-canonical basis and the I-hybrid basis is uni-triangular Since its inverse is of the same form, the claim follows.
The proof of our main result is analogous to [Gec03,§4].We will give a complete proof for the reader's convenience: Theorem 3.13.Let x, u ∈ W I and y, w ∈ I W . Then we have: In particular, the following holds: Corollary 3.12 shows that p H xy lies in M. Since M is a right ideal, the element p H xy H s also lies in M. From Corollary 3.12 it follows that we can write The Hasse-diagrams of the cell preorders look as follows.We display Kazhdan-Lusztig right cells on the left and right p-cells on the right.In these diagrams the cells that are depicted at one height form a two-sided cell.
We will choose I = {1, 2} so that W I is a finite Coxeter group of type B 2 and we get (Parabolic restriction of right p-cells).Let I ⊆ S be a finitary subset and W I ⊆ W the corresponding standard parabolic subgroup.Denote by W I the set of minimal length representatives of cosets in W/W I .Let D ⊆ W be a right p-cell of W and H D the corresponding right cell module of H (W,S) .Then D is a disjoint union of sets of the form xC, where x ∈ W I and C is a right p-cell for W I .The direct sum of the associated right p-cells modules for H (WI ,I) is isomorphic to the restricted module H D | H (W I ,I) .
, §13.2]) and some Kac-Moody flag varieties associated to G admit an alternative description as partial affine flag varieties.Denote by K = C((t)) the field of complex Laurent series and by O = C[[t]] the ring of complex power series.The Iwahori subgroup I is defined as the inverse image of a Borel subgroup B ⊂ G(C) under the evaluation map G(O) → G(C), t → 0. Then the affine flag variety G(K)/I and the affine Grassmannian G(K)/G(O) are isomorphic to the Kac-Moody flag variety G/ P I for I = ∅ and I = {1, . . ., l − 1} respectively.We will mostly be interested in the orbits of P I on F l G = G/B.Fix a finitary subset I ⊆ S and the standard parabolic subgroup W I = I ⊆ W generated by I.The group P I can also be realized as P = BW I B. Each left W I -coset contains a unique element of minimal length.We denote by I W the set of these minimal coset representatives.The action of P I on G/B induces the Bruhat decomposition G/B = w∈ I W I Y w where I Y w := P I wB/B.In the following, we will usually denote ∅ Y x by Y x .If I = ∅ each ∅ Y w is isomorphic to an affine space of dimension ℓ(w) (see [Kum02, 7.4.16Proposition]).For w ∈ I W the decomposition of I Y w into B-orbits gives a cell decomposition , Mautner and Williamson in [JMW14, §2.2].We will denote by Parity B (F l G ) the full subcategory of D b B (F l G ) whose objects are parity complexes.Theorem 2.4.For each w ∈ W there exists up to isomorphism a unique indecomposable parity complex E(w) ∈ D b B (G/B) with support Y w and restriction E(w)| Yw = k Yw [dim Y w ].Every indecomposable parity complex in D b B (G/B) is up to shift isomorphic to E(w) for some w ∈ W .The uniqueness up to isomorphism follows from [JMW14, Proposition 4.3 and Theorem 2.12].The existence of the indecomposable parity complexes E(w) is shown in [JMW14, Theorem 4.6]. 1 In the proof, the complex E(w) is constructed via the push-forward of the constant sheaf on a suitably chosen Bott-Samelson resolution.
H x where F x denotes the stalk of F in xB/B.The map ch is called the character map and restricts to an isomorphism ch : [Parity B(F l G )] ∼ −→ H of Z[v, v −1 ]-algebras where [Parity B (F l G )] denotes the split Grothendieck group of Parity B (F l G ). Definition 2.6.Define p H w = ch([E(w)]) for all w ∈ W .The set { p H w | w ∈ W } is a Z[v, v −1 ]-basis of the Hecke algebra H, called the p-canonical or p-Kazhdan-Lusztig basis.Remark 2.7.Using the transposed generalized Cartan matrix as input for the diagrammatic category of Soergel bimodules and the definition of the p-canonical basis given in [Jen19, Definition 2.4], one obtains the same basis of H (W,S) .This follows from the main result in [RW18, Part 3].In other words, the various definitions of the p-canonical basis are consistent (up to taking Langlands' dual input data).
y) −→ xy and we have ℓ(xy) = ℓ(x) + ℓ(y) if (x, y) ∈ W I × I W (see [BB05, Proposition 2.4.4]).The following I-hybrid basis interpolates between the standard basis for I = and the p-canonical basis for I = S. 2 Lemma 3.1.The set Since the base change matrix with the standard basis is unitriangular, the set { p H x H y | x ∈ W I , y ∈ I W } is also a basis.We introduce the following notation for the base change coefficients between the p-canonical and the I-hybrid bases p H w = x∈WI , y∈ I W p r I xy,w p H x H y with p r I xy,w ∈ Z[v, v −1 ] for w ∈ W , x ∈ W I and y ∈ I W .The following result shows that these polynomials have in fact non-negative coefficients: Proposition 3.2.For all w ∈ W , x ∈ W I and y ∈ I W the polynomial p r I xy,w lies in Z 0 [v, v −1 ].
Remark 3.8.For z ∈ W , x ∈ W I and y ∈ I W writep H x H y • p H z = u∈WI , w∈ I W p d I,uw xy,z p H u H w with p d I,uw xy,z ∈ Z[v, v −1 ].Grojnowski and Haiman show in [GH07, Corollary 3.9] that the Laurent polynomials p d I,uw xy,z have non-negative coefficients for p = 0.Recently Williamson [Wil19] obtained, after specializing v to 1, a more general results which holds for a larger class of reflection subgroups of W .

pC.C
H xy H s = a∈J , b∈ I W xy,s ∈ Z[v, v −1 ] for all a ∈ J , b ∈ I W . Observe that the right-hand side of (4) is the expansion of p H xy H s in the p-canonical basis.Therefore, our assumption that p H uw occurs with non-zero coefficient in p H xy H s means that u ∈ J .This in turn implies u p R x by the definition of J and finishes the proof of the theorem.Before we can state our main result, we should recall the definition of a cell module.Definition 3.14.For any right p-Cell C ⊆ W , write w p R C (resp.w p < R C) if there exists an element y ∈ C such that w p L y (and w ∈ C).By the definition of the right p-cell preorder, we can define right H-modules H , v −1 ] p H w and similarly H p < R Then the right p-cell module associated to C is defined as the right H-module given by the quotient H C := H p R 15 (Parabolic induction for right p-cells).Let C be a right p-cell of W I and H C the corresponding right cell module of H (WI ,I) .Then C • I W is a union of right p-cells of W .The right module of H (W,S) associated to C• I W is isomorphic to H C ⊗ H (W I ,I) H (W,S) .Proof.The first part is a corollary of Theorem 3.13 and the proof of the second part works exactly like the proof of [Gec05, Lemma 5.2].3.3An example: finite type C 3 for p = 2In this section, we illustrate Theorem 3.13 in finite type C 3 where Kazhdan-Lusztig cells do not decompose into p-cells for p = 2 (see [Jen19, §3.4.3]).We will use the following Cartan matrix as input and label the simple reflections accordingly.One may obtain a Kac-Moody root datum satisfying our assumptions from any based root datum of the corresponding connected semi-simple simply-connected algebraic group.It should be noted that our input is Langlands' dual to the one given in [Jen19, §3.4.3] as we work on the geometric side and not with diagrammatic Soergel bimodules.Explicit computer calculation gives the following Kazhdan-Lusztig right cells (as in [Jen19, §3.4.3]):C 0 = {Id} C 1 = {1, 12, 121, 123} C 2 = {2,21, 23, 212, 2123} C 3 = {3, 32, 321, 3212, 32123} C 4 = {13, 132, 1321} C 5 = {213, 2132, 21321} C 6 = {232, 2321, 23212} C 7 = {2121, 21213, 212132, 2121321, 21213213} C 8 = {1213, 12132, 121321} C 9 = {1232, 12321, 123212} C 10 = {13212, 132123, 1213212, 1232123, 12132123} C 11 = {21232, 212321, 2123212} C 12 = {232123, 232121, 2321213, 23212132} C 13 = {w 0 } For p = 2 these right Kazhdan-Lusztig cells exhibit the following decomposition behavior into right p-cells:C 2 = {2, 23} 2 C 2 ′ ∪ {21, 212, 2123} 2 C 2 ′′ C 3 = {3, 32} 2 C 3 ′ ∪ {321, 3212, 32123} 2 C 3 ′′ C 6 ∪ C 12 = {232} 2 C6 ∪ {2321,23212, 232123} 2 C 6/12 ∪ {232121, 2321213, 23212132} 2 C12 C i = 2 C i for i ∈ {0, . . ., 13} \ {2, 3, 6, 12}

2 Background 2.1 Crystallographic Coxeter Systems and their Hecke Algebras Let
[Hum90]finite set and (m s,t ) s,t∈S be a matrix with entries in N ∪ {∞} such that m s,s = 1 and m s,t = m t,s 2 for all s = t ∈ S. Denote by W the group generated by S subject to the relations (st) ms,t = 1 for s, t ∈ S with m s,t < ∞.We say that (W, S) is a Coxeter system and W is a Coxeter group.The Coxeter group W comes equipped with a length function ℓ : W → N and the Bruhat order (see[Hum90]for more details).ACoxeter system (W, S) is called crystallographic if m s,t ∈ {2, 3, 4, 6, ∞} for all s = t ∈ S. We denote the identity of W by Id.From now on, fix a generalized Cartan matrix A = (a i,j ) i,j∈J (see [Kum02, §1.1.1]).Let (J, X, {α i : i ∈ J}, {α ∨ [Tit89, J}) be an associated Kac-Moody root datum (see[Tit89,  §1 [KL79]ig in[KL79].For h ∈ H we say that p H w appears with non-zero coefficient in h if the coefficient of p H w is non-zero when expressing h in the p-canonical basis.
p H y h (resp.h p H y ) for some h ∈ H. Right (resp.left) p-cells are the equivalence classes in W with respect to p R (resp. ty) and thusp H x H y H s = p H x H ys = p H x H ty = p H x H t H y .The definition of right p-cells implies that p H x H t is a linear combination of terms p H z with Our assumption then ensures that p H x H y H s is a linear combination of terms p H u H y is enough to consider the case where p H uw occurs with non-zero coefficient in p H xy H s for some s ∈ S. Consider the set J