A homomorphism between Bott-Samelson bimodules

In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.


Introduction
In recent development of representation theory of algebraic reductive groups, the Hecke category plays central role.Here, the Hecke category means a categorification of the Hecke algebra of Coxeter groups.One can find the importance of the Hecke category in representation theory in Williamson's survey [Wil18].
There are several incarnations of the Hecke category.They can be roughly divided into two types: geometric ones and combinatorial ones.The geometric Hecke category which appeared in representation theory first is the category of semisimple perverse shaves on the flag variety.This category is the Hecke category with a field of characteristic zero.Juteau-Mauter-Williamson [JMW14] introduced the notion of parity sheave.The category of parity sheaves on the flag variety is a geometric incarnation of the Hecke category with any field.When the characteristic of the ground field is zero, parity sheaves are the same as semisimple perverse shaves.
Soergel [Soe07] introduced a category which is now called the category of Soergel bimodules.Similar to the situation of the geometric ones, if the characteristic of the ground field is zero, this category is the Hecke category.Soergel's category is equivalent to the category of semisimple perverse sheaves on the flag variety over a filed of characteristic zero.This fact is used to prove the Kosuzl duality of the category O [BGS96].
Soergel's category does not behave well over a field of positive characteristic in general.As a generalization of Soergel's category, the author introduced a new combinatorial category and proved that this category is the Hecke category in more general situation than Soergel's theory [Abe19].There is also another combinatorial category defined by Elias-Williamson [EW16] which is defined earlier than [Abe19].The category is called the diagrammatic Hecke category and it is proved that the category is the Hecke category in general situation.We remark that these categories are equivalent to each others when they behave well [RW18,Abe19,Abe20].
It is proved that these theories works well very general, including most cases over a field of positive characteristic.However, we still need some assumptions.The situation is subtle.In [Abe19], we need one non-trivial assumption which we recall later.One problem is that this assumption is not easy to check.In [Abe19], a sufficient condition for this assumption which we can check easier is given.However the author thought that the assumption holds in more general.The aim of this paper is to prove this assumption under a mild condition.(The situation is also subtle for the diagrammatic Hecke category.See [EW20, 5.1].We do not discuss about it in this paper.)1.1.Soergel bimodules.We recall the category introduced in [Abe19] and the assumption.Let (W, S) be a Coxeter system such that #S < ∞ and K a commutative integral domain.We fix a realization [EW16, Definition 3.1] (V, {α s } s∈S , {α ∨ s } s∈S ) of (W, S) over K. Namely V is a free K-module of finite rank with an action of W , α s ∈ V , α ∨ s ∈ Hom K (V, K) such that (1) s(v) = v − α ∨ s , v α s for any s ∈ S and v ∈ V .(2) α ∨ s , α s = 2. (3) Let s, t ∈ S (s = t) and m s,t the order of st.If m s,t < ∞ then the two-colored quantum numbers [m s,t ] X , [m s,t ] Y attached to {s, t} are both zero.(See 3.1 for the definition of these numbers.)We also assume the Demazure surjectivity, namely we assume that α s : Hom K (V, K) → K and α ∨ s : V → K are both surjective for any s ∈ S. We define the category C as follows.Let R = S(V ) be the symmetric algebra and Q the field of fractions of R.
We also assume that M is flat as a right R-module.A morphism ϕ : (M, ( We often write M for (M, (M x Q )).For M, N ∈ C, we define M ⊗N ∈ C as follows.As an R-bimodule, we have for s 1 , . . ., s l ∈ S and n ∈ Z is called a Bott-Samelson bimodule.Let BS denote the category of Bott-Samelson bimodules.
In [Abe19], we proved that BS gives a categorification of the Hecke algebra assuming the following.We refer it as [Abe19, Assumption 3.2].
Let s, t ∈ S, s = t such that m s,t is finite.Then there exists a morphism We introduce the following assumption.
Assumption 1.1.For any s, t ∈ S such that m s,t < ∞, the two-colored quantum binomial coefficients ms,t k X and ms,t k Y are both zero for any k = 1, . . ., m s,t − 1.
For the definition of two-colored quantum binomial coefficients, see 2.1 and 3.3.This assumption is related to the existence of Jones-Wenzl projectors.(See Proposition 3.4 and [EW20, Conjecture 6.23].)The main theorem of this paper is the following.Theorem 1.2 (Theorem 3.9).Under Assumption 1.1, [Abe19, Assumption 3.2] holds.
Note that Assumption 1.1 is a very mild condition.For example, if a realization comes from a root system then it is always satisfied (Proposition 3.7).
1.2.Diagrammatic category.Let D be the diagrammatic Hecke category defined in [EW16].We assume that the category D is "well-defined" [EW20, 5.1].In [EW16], under some assumptions [EW20, 5.3], a functor F from D to BS is constructed.The construction of F is deeply related to [Abe19, Assumption 3.2] as we explain here.
The morphisms in the category D are defined by generators and relations.So to define F, we have to define the images of generators.Except the generators called 2m s,tvalent vertices (s, t ∈ S), the images of generators are given easily.For 2m s,t -valent vertices, the images should be morphisms in [Abe19, Assumption 3.2].Hence, to prove [Abe19, Assumption 3.2] is almost equivalent to the construction of F. Therefore as a consequence of our main theorem, we can prove the following.
Theorem 1.3.Under Assumption 1.1, the category D is equivalent to BS.

Localized calculus.
In the proof, we use localized calculus.Ideas of localized calculus are found in [EW16,Abe19] and more systematic treatment recently appeared in [EW20].
Let C Q be the category of (P x ) x∈W where We check that ϕ Q gives a desired homomorphism by calculations.One of the things which we need to prove is the following.Let s, t ∈ S such that m s,t < ∞.For simplicity, assume that be a reduced expression of the longest element in the group s, t generated by {s, t}.Then for any g ∈ s, t , we have .
(If V comes from a root system, then this formula can be proved by applying the localization formula to the Bott-Samelson resolution of the flag variety.The author learned this from Syu Kato.) In Section 2, we calculate the left hand side of 1.1.Moreover, we give an explicit formula of the left hand side for any sequence (s 1 , s 2 , . ..) of {s, t}.The hardest part of this calculation is to find a correct result.Once we find the correct formulation, the proof is done by induction.
For a general element m ∈ M , we first give a formula to express ϕ Q (m) using the left hand side of (1.1) (with any s 1 , s 2 , . ..).We also have an algorithm to check ϕ Q (m) ∈ N (Lemma 3.8).In Section 3, using this algorithm and an explicit formula obtained in Section 2, we prove the main theorem.
1.4.On Assumption 1.1.In [Abe19], a sufficient condition for [Abe19, Assumption 3.2] was given.In [EW16], a sufficient condition for the existence of F was given.Both conditions are stronger than Assumption 1.1.It was expected that these theorems are proved under the weaker condition [EW20, Remark 5.6] but concrete conditions were not known.
In this paper, we prove these theorems under Assumption 1.1.Moreover, we prove that the theorems are almost equivalent to Assumption 1.1.More precisely, we prove the following.Let ϕ Q : M Q → N Q be the morphism in C Q introduced above and ψ Q : N Q → M Q the morphism obtaining by the same way as ϕ Q .Then ϕ Q and ψ Q give desired morphisms if and only if Assumption 1.1 holds (Proposition 3.10).Therefore the author thinks that Assumption 1.1 is the final form in this direction Acknowledgments.The author thanks Syu Kato for giving many helpful comments.The author was supported by JSPS KAKENHI Grant Number 18H01107.

A calculation in the universal Coxeter system of rank two
Since our main theorem is concerned with a rank two Coxeter system, in almost all part of this paper, we only consider a Coxeter system of rank two.In this section, we give an explicit formula of the left hand side of (1.1).Such formula can be proved in a universal form.Hence we work with the universal Coxeter system of rank two in this section.
2.1.Two-colored quantum numbers.In this subsection we introduce two-colored quantum numbers [Eli16,EW16].Let Z[X, Y ] be the polynomial ring with two variables over Z.
We prove some properties of these polynomials which we will use later.Some of them are known well or immediately follow from known results.We give proofs for the sake of completeness.
Proof.The first two statements follow from the definition using induction.For the third, if n is odd then it follows from (1).If n is even then it is obvious.We also have An obvious consequence of (1) (2) which will be used several times in this paper is the following.For k 1 , . . ., k r , l 1 , . . ., Proof.We prove by induction on n.The cases of n = 0 and n = 1 follow from the definitions.Assume that the lemma holds for n − 1, n − 2. Then In the last we used Lemma 2.2 (4) again.Similarly we have Proof.By the previous lemma, we have The first formula of the lemma follows from Lemma 2.3.The second follows from the first and Lemma 2.2 (4).The third formula follows from a similar calculation.
). Hence again the lemma follows from Lemma 2.2.

A formula.
Let (W, S) be the universal Coxeter system of rank two, namely the group W is generated by the set of two elements S = {s, t} and defined by relations s 2 = t 2 = 1.The length function is denoted by and the Bruhat order is denoted by -module of rank two with a basis {α s , α t }.We define an action of W on V by Let Φ = {w(α s ), w(α t ) | w ∈ W } be the set of roots and the set of positive roots Φ + is defined by Φ + = {w(α s ) | ws > w} ∪ {w(α t ) | wt > w}.For each α ∈ Φ, we have the reflection s α ∈ W .This is defined as s αs = s, s αt = t, s w(α) = ws α w −1 for α ∈ {α s , α t } and w ∈ W .
The following formula can be proved by induction.
We define some elements which will be needed for our main formula.We use the following notation for sequences in S. A sequence in S will be written with the underline like w = (s 1 , . . ., s l ).We write Let w = (s 1 , . . ., s l ) ∈ S l be a sequence of elements in S and g ∈ W .For a real number r, let r be the integral part of r.We define where Let R be the symmetric algebra of V and R ∅ = Φ −1 R the ring of fractions.We define an element a w (g) of R ∅ by Lemma 2.10.If s i = s i−1 for some i, namely if w is not a reduced expression, then a w (g) = 0.
Proof.Set A = {e ∈ {0, 1} l | w e = g}.Define f : A → A by f (e) = (e 1 , . . ., e l ) where The aim of this section is to prove the following theorem.
Theorem 2.11.For w ∈ S l , we have From the above lemma, we may assume s i−1 = s i for any i.By definitions, we also may assume g ≤ w, otherwise both sides are zero.

Proof of Theorem 2.11.
In this subsection we prove Theorem 2.11.
We split the sum in the definition of a w (g) to e l = 0 part and e l = 1 part.If e l = 0, then We change the notation slightly and we get the following lemma.
Lemma 2.12.Let w ∈ S l and u ∈ S. Then we have To prove the theorem we need the following lemmas.
(3) We have X w g \ {β} = X w gu \ {γ} and X wu gu = X w g ∪ {γ}.Proof.Since our Coxeter system has rank two, for x ∈ W , there exists α ∈ Φ + such that s α = x if and only if (x) is odd.One of elements in wg −1 , swg −1 has the odd length.Hence there exists β ∈ Φ + such that g and we get (1).The proof of (2) is similar.
We prove (3).Let δ ∈ X w g .Then s δ g ≤ w.Since our Coxeter system is of rank two, if (s δ gu) ≤ (w) − 1, we have s δ gu ≤ w.Hence δ ∈ X w gu .Therefore if (s δ g) ≤ (w) − 2, then since (s δ gu) ≤ (s δ g) + 1, we have δ ∈ X w gu .Let u be the element in S which is not u.Then we have sw < w, wu < w.
Hence s δ gu = wu u ≤ w.Therefore δ ∈ X w gu .If w = u then u = s since sw < w.We have (s δ g) = (w) − 1 = 0, hence s δ g = 1.Since g ≤ w, we have g = u or g = 1.Since (s δ ) is odd, by s δ g = 1, we have g = u and In any case, if δ ∈ X w g , then δ = β or δ ∈ X w gu .Hence X w g \ {β} ⊂ X w gu .If δ = γ, the element s δ g is wu or swu.Since wu > w, we have s δ g ≤ w only when s δ g = swu = w.Therefore δ = β.Hence X w g \ {β} ⊂ X w gu \ {γ}.By replacing g with gu, we get the reverse inclusion.
Lemma 2.15.Let w = (s 1 , . . ., s l ) ∈ S l such that s i−1 = s i for any i and g ∈ W . Set u = s l .
(3) X w g = X w gu .Proof.We may assume g < gu by replacing g with gu if necessary.We also may assume that s 1 = s by swapping s with t if necessary.(1) follows from Lemma 2.12 and a (w,u) (g) = 0.
For (2), first we assume sg > g and g = 1.Then the reduced expression of g has a form g = t • • • u where u ∈ S is the element which is not u, namely the reduced expression starts with t and ends with u .Since w = (s, . . ., u) and s i−1 = s i for any i, we have (g) ≡ (w) (mod 2).Hence the lemma follows from the definition of k w g .The proof in the case of sg < g, g = 1 is similar.
Assume g = 1.If u = s, then s 1 = s l = s, hence (w) is odd.If u = t, then s 1 = s and s l = t.Hence (w) is even.In both cases, we can confirm k w g = k w gu by the definition.Since wu < w, by Property Z in [Deo77] we have s γ g ≤ w if and only if s γ gu ≤ w. (3) follows.
Proof of Theorem 2.11.We prove the theorem by induction on (w).If (w) = 0, then this is trivial.Let u ∈ S and we prove that the theorem is true for (w, u) assuming that the theorem is true for w.If (w, u) is not a reduced expression, then both sides of the theorem are zero.Hence we may assume (w, u) is a reduced expression.By the previous lemma, we also may assume gu > g.If g ≤ w, then by Property Z [Deo77], g ≤ wu.
Hence both sides are zero.
Take s 1 , . . ., s l ∈ S such that w = (s 1 , . . ., s l ).If g ≤ w and gu ≤ w, then a w (gu) = 0.By Lemma 2.12, inductive hypothesis and Lemma 2.14, As in the proof of Lemma 2.14, we have g = w or g = s 1 w (the latter does not happen when l = 0).Hence k w g = k (w,u) g = 1 from the definitions.Therefore the theorem holds in this case.
We assume g, gu ≤ w.Then (w) > 0. We may assume s 1 = s by swapping (s, X) with (t, Y ) if necessary.By Lemma 2.12 and inductive hypothesis, we have Take β, γ ∈ Φ + as in Lemma 2.13.Then by Lemma 2.13, the right hand side is 1 Hence it is sufficient to prove that k w g γ − k w gu δ = k (w,u) g g(α u ).Since gu > g, the reduced expression of g ends with the simple reflection which is not u.Hence the reduced expression of gug −1 can be obtained by concatenating the reduced expressions of g, u and g −1 .Therefore we have (gug −1 ) = (g) + (u) + (g −1 ) = 2 (g) + 1.Moreover, if sg > g, then we have sgug −1 > gug −1 .
First we assume sg > g and g = 1.Then ss g(u) = sgug −1 > gug −1 .Hence by Lemma 2.9.Since gu > g and wu > w, the reduced expressions of g and w ends with the same simple reflection.Namely if u ∈ S is the element which is not u, then the reduced expression of w is w = s • • • u and the reduced expression of g is g = t • • • u since we assumed sg > g.Since g ≤ w, the last (g)-letters of the reduced expression of w is the reduced expression of g.Hence (wg −1 ) + (g) = (w) and the reduced expression of wg −1 starts with s and ends with s.Therefore twg −1 > wg −1 and s β = wg −1 .Hence by Lemma 2.9, we have A calculation of γ is similar.We have (wug −1 ) = (g) + (u) + (w −1 ) and the reduced expression of wug −1 starts with s and ends with t.Therefore (swug −1 ) = (wug −1 ) − 1, s γ = swug −1 and s(swug −1 ) > swug −1 .Hence by Lemma 2.9, we have Put m = ( (w) − (g) − 1)/2 and n = (g).Then we have Therefore we have By the definition, we have Hence, By Lemma 2.5, this is equal to Hence it is sufficient to prove This follows immediately from Lemma 2.7.The case of tg > g is similar.By Lemma 2.9, we have The reduced expressions of w and g end the same reflection, hence (wg −1 ) = (w)− (g).The reduced expression of g starts with s.Hence the reduced expression of wg −1 starts with s, ends with t.Hence s β = swg −1 , s(swg −1 ) > swg −1 and (s β ) = (w) − (g) − 1.
Hence by Lemma 2.9, we have We have (wug −1 ) = (w) + (g) + 1 and the reduced expression starts with s and ends with s.Hence s γ = wug −1 , ts γ > s γ , and (s γ ) = (g) + (w) + 1. Therefore by Lemma 2.9, we have Put m = ( (w) − (g))/2 − 1 and n = (g) + 1.Then We have Therefore, by Lemma 2.5, we have Therefore it is sufficient to prove gu which is again an immediate consequence of Lemma 2.7.We assume g = 1 and u = t.Then one can check that formulas for g(α u ), β, γ in the case of sg > g, g = 1 hold.Hence the theorem follow from the calculations in this case.If g = 1 and u = s, then one can use the calculations in the case of tg > g, g = 1.

A homomorphism between Bott-Samelson bimodules
3.1.Finite Coxeter group of rank two and a realization.We add the tilde to the notation in the previous section, namely ( W , S) is the universal Coxeter system of rank 2, V is the free The notation without tilde will be used for non-universal version.Let (W, S) be a Coxeter system such that S = {s, t}, s = t.We assume that the order m s,t of st is finite.Let K be a commutative integral domain and (V, Let R (resp.R) be the symmetric algebra of V (resp.V ).We regard R as a graded K-algebra via deg(V ) = 2.We put As some of them are appeared already, objects related to the universal Coxeter system is denoted with the tilde and the corresponding letter without the tilde means the image in the finite Coxeter system.For example, if w = (s 1 , s2 , . ..) is a sequence of elements in S, then w = (s 1 , s 2 , . ..) is the corresponding sequence in S. As we have already explained, a sequence is denoted with the underline and removing the underline means the product of elements in the sequence.Hence w = s1 s2 For each root α ∈ Φ, we have sα ∈ W and s α ∈ W .
Proof.We prove the lemma by backward induction on (g).If g = x, then by Theorem 2.11, we have a x(x) = ( δ∈ X x x δ) −1 .On the other hand, for e ∈ {0, 1} ms,t , we have xe = x if and only if e = (1, . . ., 1).Hence by the definition of a x(x), we have Next assume that g = sx.Define si = s if i is odd and si = t if i is even.Then x = (s 1 , . . ., sms,t ).For e ∈ {0, 1} ms,t , xe = g if and only if e = (0, 1, . . ., 1).Hence by the definition, . By Theorem 2.11, the left hand side of the lemma is (a x(g)π x ) −1 .Hence we get the lemma in this case.
Assume that g = x, sx.Then there exists ũ ∈ S such that x ≥ gũ > g.When g = 1, we take ũ = t.
• First assume that xũ < x.By Lemma 2.15, the left hand side is not changed if we replace g with gũ.We prove that the right hand side is also not changed.
Then this gives the lemma by inductive hypothesis.
-If sg > g, g = 1 or g = 1, then by the proof of Theorem 2.11, we have Therefore by the previous lemma, we . By inductive hypothesis, we get the lemma in this case.-Finally assume that sg < g.By the proof of Theorem 2.11, we have γ and we get the lemma.
Proof.By Theorem 2.11, the lemma follows from π x / γ∈ X w 1 γ ∈ R. If w = x, then it follows from Lemma 3.2.By swapping s with t, π y a ỹ(1) ∈ R. Since π y ∈ K × π x , we get the lemma for w = ỹ.In general, we have w The same discussion implies the lemma when w ≤ ỹ.

An assumption.
To prove the maim theorem, we need one more assumption.In this subsection, we discuss on the assumption.We start with the following proposition.
We need the following assumption to prove the main theorem.
We have a sufficient condition of Assumption 3.5.Proposition 3.6.If the action of W on Kα s + Kα t is faithful, then Assumption 3.5 holds.
then by [Eli16, before Claim 3.2, Claim 3.5], (st) k is the identity on Kα s + Kα t .This is a contradiction.Hence  which is zero as we have proved.On the other hand, if l is odd, then [l] X = 0.
Hence ms,t l X = 0.
Maybe more useful criterion is the following.
Proposition 3.7.If the realization comes from a root datum and W is the Weyl group, then Assumption 3.5 holds.
In general, for a sequence w = (s 1 , s 2 , . . ., s l ) ∈ S l of elements in S, we put The main theorem of this paper is the following.Theorem 3.9.Assume Assumption 3.5.There exists a morphism ϕ : We call the component corresponding to e the e-component of (B w ) Q .As an R-bimodule, The e-component of p We construct ϕ : B x → B y as follows.First we define ϕ By the same way, we also define Define r : W → W as follows.If w ∈ W is not the longest element, r(w) = s1 . . .sl where w = s 1 • • • s l is the reduced expression of w.If w is the longest element then r(w) = x.
Therefore we get the following.
Hence to prove ϕ Q (B x ) ⊂ B y , it is sufficient to prove that ((π x /ζ y (f ))a x(c) (r(y f ))y f (p)) f is in B y for any p ∈ R. To proceed the induction, we formulate as follows.
Proof.In [EW20], a functor Λ : D → C Q is defined and it is proved that Λ is well-defined.By the construction, we have Λ = (•) Q • F. Therefore (•) Q • F is well-defined and since (•) Q : BS → C Q is faithful, F is also well-defined.
Theorem 3.15.The functor F : D → BS gives an equivalence of categories.
Proof.The proof is the same as that of the corresponding theorem in [Abe19].It is obviously essentially surjective.In [EW16], for each object M, N ∈ D, elements in Hom D (M, N ) called double leaves are defined and proved that it is a basis of Hom D (M, N ) [EW16, Theorem 6.12].In [Abe19], the corresponding statement in BS is proved, namely double leaves in Hom C (F(M ), F(N )) are defined and proved that it is a basis.By the definition of F, F sends double leaves to double leaves.Hence F gives an isomorphism between morphism spaces.