The cohomology submodule of an indecomposable Soergel bimodule 35

For any Soergel bimodule B and any x ∈ W we define a Laurent polynomial hx(B) ∈ Z[v, v⁻¹]by

h_x(B)(v) = (grrk Γ≤xB/Γ_<xB)v^`(x), so that we havech(B) =P

x∈Whx(B)Hx.

IfBS(w)is a Bott-Samelson bimodule, from (3.2) we gethx(BS(w)) =P

e:w^e=xv^def(e). By Lemma 1.1.3 we havehx(BS(w))(v) =v^`(w)−`(x)+“terms of lower degree.”

If K=R it follows form Soergel’s conjecture that the polynomialsh_x(B_w) =h_x,w are the Kazhdan-Lusztig polynomials.

For a general K, the polynomials hx(Bw) can depend on the realization h^∗: these polynomials are called p-Kazhdan-Lusztig polynomials, where p = char(K). The p-KL polynomials are discussed in more detail in [JW17].

Lemma 3.2.6. We have hx(Bw) ∈ N[v] and hx(Bw)(v) = v^`(w)−`(x)+“terms of lower degree,” for anyx≤w.

Proof. We show this by induction on `(w). Ifwis reduced we have a decomposition:

BS(w) =Bw⊕M

y<w

By^⊕my

wheremy(v)∈Z[v, v⁻¹]is such that my(v) =my(v⁻¹). We have grrkBS(w) = grrkBw+X

y<w

my·grrkBy

wheregrrkis taken with respect to the leftR-module structures. Since 1^⊗_w ∈B_w, for any y < wthe bimodule By^⊕my lies in degree>−`(w)and we havedegmy < `(w)−`(y). The lemma now follows since for allx∈W we have

h_x(BS(w)) =h_x(B_w) +X

y<w

m_y·h_x(B_y).

Let us consider the following left gradedR-submodules of BS(w):

C_w = X

ecanonical

R·ll_w,e and D_w= X

enot canonical

R·ll_w,e. In general the left moduleCw is not stable under multiplication byR on the right.

Lemma 3.2.7. Let D_w as above. Then for any non-canonical 01-sequence e we have Im(LL

w,e)⊆D_w. Moreover, D_w is a gradedR-subbimodule ofBS(w).

Proof. We fix a non-canonical 01-sequence e and let x = w^e. The light leaf LL

w,e is a morphism fromBS(x) to BS(w) for some reduced expression xof x. Let x=s1s2. . . s_l.

It suffices to show that for any string basis element c_ε we have LL

w,e(c_ε) ∈ D_w. We define the following morphismφ_ε∈End^•(BS(x)):

ε₁ ε₂ ε₃ ε₄ ε_` 0 1

· · · where := and :=

φ_ε:=

(3.3) Clearly, we haveφ_ε(1^⊗_x) =c_ε. Therefore LL

w,e(c_ε) = (LL

w,e◦φ_ε)(1^⊗_x).

Let y be the expression obtained from x = s1s2. . . sl by removing the si for i such thatεi = 0. Then the morphism LL

w,e◦φε factorizes through BS(y), hence its image is contained inΓ_≤m(y)BS(w). By Proposition 3.2.4 we can write:

LL

w,e(cε) = X

f:w^f≤m(y)

hfllw,f. (3.4)

We have degLL

w,e(c_ε) = degll_w,e+ degc_ε=`(w)−2·Downs(e)−2`(y)< `(w)−2`(m(y)).

Recall that a 01-sequence f is canonical if and only if degllw,f = `(w)−2`(w^f). Since

`(w<sup>f</sup>)≤`(m(y)), no canonical01-sequence can appear in the sum in the RHS of (3.4). It follows immediately thatIm(LL

w,e)⊆D_w.

Now the last statement follows since for f ∈R we have ll_w,e·f =LL

w,e(1^⊗_x)f =LL

w,e(1^⊗_x ·f)∈D_w.

Actually, the same proof shows more generally that if φ ∈ Hom^•(BS(x), BS(w)) is such thatdegφ < `(w)−`(x)thenIm(φ)⊆Dw. In this way we deduce that the bimodule D_w does not depend on the choice of light leaves. Similarly, we also have that if φ ∈ End⁰(BS(w)), thenφ(Dw)⊆Dw.

Let nowwbe a reduced expression. Fix a decomposition ofBS(w)into indecomposable bimodules and let e_w ∈End⁰(BS(w)) be the primitive idempotent corresponding toB_w, i.e. BS(w) = Ker(ew)⊕Im(ew) andIm(ew)∼=Bw. For any x, the map

e^x_w : Γ≤xBS(w)/Γ_<xBS(w)→Γ≤xB_w/Γ_<xB_w induced bye_w is split surjective. In particular, we have that

grrk (Γ≤xBS(w)/Γ_<xBS(w)) = grrk (Γ≤xB_w/Γ_<xB_w) + grrk kere^x_w , that is

grrk kere^x_w

=v^{−`(x)</sup>h<sub>x</sub>(BS(w))−v<sup>−`(x)}h_x(B_w).

From Lemma 3.2.6 it follows that Kere^x_w is generated in degree < `(w)−2`(x) as a left R-module.

Lemma 3.2.8. The kernel of the morphism e_w is contained in D_w.

Proof. Fix an enumeration w1, w2, w3. . . of the elements of W which refines the Bruhat order. Let

L=X

i∈I

g_i·ll_w,ei

be an arbitrary element in Kere_w ⊆BS(w), with g_i ∈R. Let x =w_h be the element of maximal index in the set X := {w^ei | i∈ I}. We want to show by induction on h that there are no canonical light leaves appearing in the sumL.

For y ≤ w let F_y := X

e:w^e=y

R ·ll_w,e. Then the inclusion F_y ,→ BS(w) induces an isomorphism of leftR-modulesF_y ∼= Γ≤yBS(w)/Γ_<yBS(w).

For an integer k≥1 let us denote by Γ≤kB the submodule of elements supported on Gr({w₁, . . . , wk}). By Soergel’s hin-und-her (Lemma 3.1.9) we have an isomorphism:

Γ≤xBS(w)/Γ_<xBS(w)∼= Γ≤hBS(w)/Γ≤h−1BS(w).

LetLx=P

i∈Ixgi·llw,ei, whereIx={i∈I |w^ei =x}. Sincex is of maximal index in X, the projection of L and L_x to Γ≤hBS(w)/Γ≤h−1BS(w) coincide. Hence L_x ∈Kere^x_w andL−L_x ∈Γ_≤h−1BS(w).

TheR-module Kere^x_w is generated in degrees < `(w)−2`(x), so we can write L_x=X

j

h_jr_j,

withh_j ∈Randr_j ∈F_xsuch thatr_j ∈Kere^x_w anddegr_j < `(w)−2`(x). Notice the iff is the canonical01-sequence forx, thenll_w,f is a basis element ofF_x of degree`(w)−2`(x), hence it cannot appear inLx.

We have also dege_w(rj) < `(w)−2`(x) and e_w(rj) ∈ Γ<xBS(w). Since all canonical light leaves supported on an elementy smaller thanx have degree> `(w)−2`(x), we get e_w(rj)∈Dw for all j, and finally e_w(Lx)∈Dw.

Let now L⁰ = L−Lx+e_w(Lx). We have L⁰ ∈ Kere_w and L⁰ ∈ Γ≤h−1BS(w), so by induction it follows that there are no canonical light leaves appearing when we writeL⁰ in the light leaves basis. We have shown that there are no canonical light leaves inLx and in e_w(Lx), so the statement follows also for L.

It follows that Bw ∼= e_w(Cw) ⊕e_w(Dw) as left R-modules. Moreover, e_w(Dw) is a R-subbimodule of Bw and the restriction ofe_w to Cw is injective.

Definition 3.2.9. We define the singular cohomology submodule He_w ⊆BS(w) to be the orthogonal ofD_w with respect to the intersection formh−,−i_BS(w).

Consider the decomposition BS(w) = Ime_w ⊕Kere_w, with Ime_w ∼= B_w. Let e^∗_w ∈ End(BS(w))the adjoint of e^∗_w with respect to the intersection form. Sincedeg(e^∗_w) = 0 it preservesDw. Hencee_w preserves Hew, thus Hew splits as

Hew=

Hew∩Ime_w

⊕

Hew∩Kere_w

.

Recall from Lemma 3.1.10 that the restrictions of h−,−i_BS(w) to B_w and Kere_w is non-degenerate. SinceKere_w ⊆ D_w it follows that He_w ∩Kere_w = 0, hence He_w ⊆Ime_w and e_w restricts to the identity on Hew. We also obtainHew := e_w(Hew) = e_w(Dw)^⊥ ⊆Bw

where the orthogonal is taken with respect to the restriction of the intersection form to B_w. Finally, we can easily compute the graded rank ofHe_w:

grrkHew = grrkHew = grrkBS(w)−grrkDw= grrkCw =X

x≤w

v^2`(x)−`(w). (3.5)

3.3 Moment graphs of Coxeter groups

There exists a forth description of the equivariant cohomology of the flag variety, obtained by Goresky, Kottwitz and MacPherson [GKM98] using the localization theorem for torus actions.

As pointed out by Fiebig, one can generalize this construction to an arbitrary Coxeter group. Fiebig uses this to obtain a new realization of the category of Soergel bimodules.

We show in fact that also in the generality of an arbitrary Coxeter group this construction still returns the dual nil-Hecke ring (even if there is no flag variety of which they are the equivariant cohomology).

We recall the definition of moment graphs and their sheaves from [Fie08].⁴

The moment graphG :=G(W,h)is defined as follows: The set of vertices is given by the elementv ∈W. Two verticesv, ware connected by an edge if there exists a reflectiont∈ T such thatv =tw.⁵ We label this edge byα_t, where α_t is the positive root corresponding tot.

Definition 3.3.1. A sheaf Mon the moment graph ofW is given by

• a graded left R-module M^v for anyv∈W;

• for any edge v−tv a graded leftR-module M^v−tv such thatαt· M^v−tv = 0;

• for any v∈W,t∈ T a morphism of gradedR-modulesπv,tv :M^v → M^v−tv. We further assume that M^v is non-zero only for finitely many v ∈ W and that M^v is torsion free and finitely generated as aR-module.

4Notice that Fiebig uses a different convention: hisGr(x)corresponds to ourGr(x⁻¹).

5We do not ask here that`(v) =`(w)±1

Thespace of global sections of a sheafMis Γ(M) :=

(

(mv)∈ Y

v∈W

M^v

πv,tv(mv) =πtv,v(mtv)∀v∈W, t∈ T )

The structure algebra Zˆ is the space of global sections of the “constant sheaf” on the moment graph, thus it is defined by

Zˆ = (

(r_v)∈ Y

v∈W

R

r_v ≡r_tv (modα_t)∀v∈W, t∈ T )

.

Fori∈Nlet Zˆi the graded component of Z, that isˆ Zˆi :={(z_v)∈Zˆ|degzv =i}. We defineZ :=L

i∈ZZˆi. ThenZ is a subring of Z. We can also describeˆ Z as the subring of section in Zˆ with bounded degree, that is Z ={(z_v) ∈Zˆ | ∃i: degz_v ≤ifor all v ∈W}.

Notice that for an infinite Coxeter group we haveZ 6= ˆZ.

For any sheafM, the space of global sectionΓ(M)is in a natural way a graded module overZˆ, hence overZ, by point-wise multiplication.

For a subset Ω⊆W we defineZˆ^Ω to be the image of the composition Z ,ˆ → Y

v∈W

R Y

v∈Ω

R.

We defineZ^Ω similarly. Clearly, for any finite subset Ωwe have Z^Ω = ˆZ^Ω. A subsetΩis said to beupwardly closed if wheneverv∈Ωand w≥v, thenw∈Ω.

Definition 3.3.2. Let Z-mod^f be the full subcategory Z-mod whose objects are Z-modules M which are finitely generated and torsion free over S and such the Z-module structure factors throughZ^Ω for some finite Ω⊆W.

We define similarlyZ-modˆ ^f. The restriction functorZ-mod→ˆ Z-mod induces an equiv-alence of categoriesZ-modˆ ^f −→^∼ Z-mod^f.

Definition 3.3.3. We say that M ∈ Z-mod^f admits a Verma flag if for any upwardly closed subset Ω ⊆ W, the module M^Ω is free as a graded left R-module. We call V the full subcategory ofZ-mod^f ofmodules admitting a Verma flag.

Recall thatF_∇ is the category ofR-bimodules with a∇-flag.

Theorem 3.3.4. [Fie08, Theorem 4.3] There is an equivalence of categoriesV ∼=F_∇. We sketch now how this equivalence is obtained. We have two morphisms of rings τ, σ:R→Z defined by

(τ(f))x=f and(σ(f))x =x(f).

Hence we have a ring homomorphismR⊗_KR−−→^τ⊗σ Z. By restriction we obtain a functor F :Z-mod→R⊗R-mod which restricts to a functorF :V → F_∇.

In the other direction, we start with a R-bimodule B ∈ F∇. To B we associate the sheaf on the moment graphB such thatB^v = Γ^vB and such that

B^v−tv =B/(Ann(Gr(v)∩Gr(tv))B.

Since we haveΓ^vB =B/Ann(Gr(v))Bby [Soe07, Remark 6.2], there is a natural projection πv,tv : B^v → B^v−tv (the bimodules B^v and B^v−tv are the restrictions of B to Gr(v) and

to Gr(v)∩ Gr(tv) respectively). We can therefore define a functor G : F_∇ → V by G(B) = Γ(B).

The two functorsF and Gare inverse to each other. In particular, we have B ∼= Γ(B) as an R-bimodule, and since B is a sheaf on the moment graph we also get a natural structure ofZ-module on the bimodule B.

Finally, we describe aW-action on Z. Forz∈Q

v∈W R andx∈W we definex(z)_v = zvx. This action preserves Z: in factαt divides x(z)tv−x(z)v =ztvx−zvx for all t∈ T, x, v, w∈W.

Warning 3.3.5. We have two different structures ofR-modules on Z, given byτ andσ as defined above. To differentiate between them, we writeτ as the left action and σ as the right action. We always think ofZ as a R-algebra using the left action τ.

Im Dokument Hodge theoretic aspects of Soergel bimodules and representation theory (Seite 40-45)