Parabolic inclusion and truncation

Leti=i_ν,φ be the canonical inclusion^wO^ν_μ,± ⊂^wO_μ,±. It is a fully faithful functor and it admits a left adjoint τ = τφ,ν which takes an object of ^wO_μ,± to its largest quotient which lies in ^wO^ν_μ,±. We’ll call i the parabolic inclusion functor and τ the parabolic truncation functor.

Sinceiis exact, the functorτ takes projectives to projectives. We have (a) i(L(x•oμ,±)) =L(x•oμ,±) forx∈^wI^ν_μ,±,

(b) τ(^wP(x•o_μ,±)) =^wP^ν(x•o_μ,±) forx∈^wI^ν_μ,±, (c) τ(^wP(x•oμ,±)) = 0 for x∈^wIμ,±\^wI^ν_μ,±.

The same argument as in[5, thm. 3.5.3], using[17, thm. 5.5], implies that the functor iis injective on extensions in ^wO^ν_μ,−. See the proof ofProposition 4.47for more details.

By Lemma 2.6, for each w∈ I_μ,−^ν , the Ringel equivalence yields equivalences of tri-angulated categories D^b(^vO_μ,+) → D^b(^wO_μ,−) and D^b(^vO^ν_μ,+) → D^b(^wO^ν_μ,−), where v=w₊:=w_νww_μ, such that the diagram below commutes

D^b(^vOμ,+) D^b(^wO_μ,−)

D^b(^vO^ν_μ,+)

i

D^b(^wO^ν_μ,−).

i

Thus, the parabolic inclusion functor iis also injective on extensions in^vO^ν_μ,+. 3.6. The main result

Fix integers e, f > 0 and parabolic typesμ, ν ∈ P. We choose o_μ,± of level ±e−N and o_ν,± of level ±f −N. Fix an element w ∈ W. Assume that w ∈ I_μ,+^ν , and set v=w⁻¹₋ ∈I_ν,−^μ .

The main result of this paper is the following.

Theorem 3.12.We haveC-algebra isomorphisms^wA^ν_μ,+=^vA¯^μ_ν,− and^wA¯^ν_μ,+ =^vA^μ_ν,− such that 1x→1y withy =x⁻¹₋ for eachx∈^wI^ν_μ,∓. The gradedC-algebras ^wA¯^ν_μ,+ and ^vA¯^μ_ν,−

are Koszul and balanced. They are Koszul dual to each other, i.e., we have a graded C-algebra isomorphism (^wA¯^ν_μ,+)^! = ^vA¯^μ_ν,− such that 1^!_x = 1_y for each x ∈ ^wI^ν_μ,+. The categories^wO^ν_μ,+,^vO^μ_ν,− are Koszul and are Koszul dual to each other. 2

We’ll abbreviate ^wA¯^ν,!_μ,±= (^wA¯^ν_μ,±)^!.

Remark 3.13. Since ^wA¯^ν_μ,+ and ^vA¯^μ_ν,− are balanced, the Koszul duality commutes with the Ringel duality. In particular, for w∈I_μ,−^ν andv =w⁻¹∈I_ν,−^μ , composing (•)^! and (•)we get a gradedC-algebra isomorphism (^wA¯^ν,_μ,−)^!=^vA¯^μ_ν,−such that (1_x)^!= 1y with y =x⁻¹ for eachx∈^wI^ν_μ,−.

4. Moment graphs, deformed categoryOand localization

First, we introduce some notation. Fix integerse, f >0 and parabolic typesμ, ν∈ P. LetV be a finite dimensionalC-vector space, letS be the symmetric C-algebra over V and let m⊂S be the maximal ideal generated byV. We may regard S as a graded C-algebra such thatV has the degree 2.

Let R be a commutative, Noetherian, integral domain which is a (possibly graded) S-algebra with 1.

LetS0be the localization ofS at the idealmand letkbe the residue field ofS0. Note thatk=Cand thatS0 has no natural grading.

4.1. Moment graphs

Assume thatR=S, viewed as a graded C-algebra.

Definition 4.1.Amoment graph overV is a tuple G= (I, H, α) where (I, H) is a graph with a set of vertices I, a set of edges H, each edge joins two vertices, and αis a map H →P(V),h→kαh.

We say that the moment graphG is a GKM-graph if we have kαh1 =kαh2 for any edgesh1=h2 adjacent to the same vertex.

Definition 4.2.An order onG is a partial orderonIsuch that the two vertices joined by an edge are comparable.

Given an orderonGleth, hdenote theoriginand thegoal of the edgeh. In other wordsh andhare two vertices joined byhand we haveh ≺h.

Remark 4.3.We use the terminology in[21]. In[14], a moment graph is always assumed to be ordered. We’ll also assume thatG isfinite, i.e., that the setsI andH are finite.

Definition 4.4.A gradedR-sheaf overG is a tupleM= (Mx,Mh, ρx,h) with

• a gradedR-moduleMx for eachx∈I,

• a gradedR-moduleMh for eachh∈H such thatα_hMh= 0,

• a gradedR-module homomorphismρ_x,h:Mx→ Mhforhadjacent tox.

A morphism M → N is an IH-tuple f of graded R-module homomorphisms fx : Mx→ Nx andfh:Mh→ Nh which are compatible with (ρx,h).

TheR-modulesMx are called thestalksofM. ForJ ⊂I we set M(J) =

(mx)_x∈J; mx∈ Mx, ρh,h(mh) =ρh,h(mh), ∀hwithh, h∈J . Note thatM({x}) =Mx. The space of global sections of the gradedR-sheafMis the gradedR-moduleΓ(M) =M(I).

We say thatM is of finite type if all Mx and allMh are finitely generated graded R-modules. IfMis of finite type, then the gradedR-moduleΓ(M) is finitely generated, becauseR is Noetherian.

The graded structural algebra of G is the graded R-algebra ¯Z_R = {(a_x) ∈ R^⊕I; ah−ah∈αhR, ∀h}.

Definition 4.5.Let ¯FR be the category consisting of the gradedR-sheaves of finite type overGwhose stalks are torsion freeR-modules. Let ¯ZRbe the category consisting of the graded ¯ZR-modules which are finitely generated and torsion free overR.

The categories ¯FR and ¯ZR are Krull–Schmidt gradedR-categories (because they are hom-finiteR-categories and each idempotent splits). Taking the global sections yields a functor Γ : ¯F_R→Z¯_R.

We’ll call again graded R-sheaves the objects of ¯Z_R, hoping it will not create any confusion.

The global sections functorΓ has a left adjointL called thelocalization functor [14, thm. 3.6],[21, prop. 2.16, sec. 2.19].

We say that M is generated by global sections if the counit LΓ(M) → M is an isomorphism, or, equivalently, if M belongs to the essential image of L. This implies that the obvious map Γ(M) → Mx is surjective for each x. The functor Γ is fully faithful on the full subcategory of ¯FR of the graded R-sheaves which are generated by global sections, see[14, sec. 3.7].

For each subsetsE⊂H andJ ⊂I, we setME=

h∈EMhand ρJ,E =

x∈J

h∈E

ρx,h :M(J)→ ME.

Let dx be the set of edges with goal x, ux be the set of edges with origin x and e_x=d_x

u_x. Given an orderonG, we setM∂x= Im(ρ_≺x,dx),M^x= Ker(ρ_x,ex) and M[x] = Ker(ρ_x,dx), where we abbreviate ρ_x,E =ρ_{x},E. Following[16, sec. 4.3], we call M^x thecostalk ofMat x. Note thatM^x,M[x],Mx are graded ¯ZR-modules such that M^x⊂ M[x] ⊂ Mx.

Assume from now on that the moment graph G is a GKM graph. Let us quote the following definitions, see [21, lem. 3.2]and[14, lem. 4.8].

Definition 4.6.Assume thatM ∈F¯_R is generated by global sections. Then, (a)Misflabby if Im(ρ_x,dx) =M∂xfor eachx∈I,

(b)Mis Δ-filtered if it is flabby and if the graded ¯ZR-moduleM[x] is a free graded R-module for eachx∈I.

Now, we can define the following categories.

Definition 4.7.Let ¯F^Δ_R,be the full subcategory of ¯FRconsisting of the Δ-filtered objects.

Let ¯Z^Δ_R, be the essential image of ¯F^Δ_R, byΓ.

We may abbreviate ¯F^Δ_R = ¯F^Δ_R, and ¯Z^Δ_R = ¯Z^Δ_R,when the order is clear from the con-text. The categories ¯F^Δ_R and ¯Z^Δ_R are Krull–Schmidt gradedR-categories. Since Δ-filtered objects are generated by global sections, from the discussion above we deduce thatL,Γ are mutually inverse equivalences of gradedR-categories between ¯F^Δ_R and ¯Z^Δ_R.

For eachM ∈ Z¯^Δ_R we write M^x = L(M)^x, M[x] =L(M)[x], Mx =L(M)x, M_≺x = L(M)_≺x,M∂x=L(M)∂x andMh=L(M)h.

The categories ¯F^Δ_R and ¯Z^Δ_R are exact categories with the exact structures defined as follows. Following[14, lem. 4.5], we say that a sequence 0→M→M→M→0 in ¯Z^Δ_R is exact if and only if the sequence of (free)R-modules 0→M_[x] →M_[x]→M_[x] →0 is exact for eachx∈I. This yields an exact structure on ¯Z^Δ_R. We transport it to an exact structure on ¯F^Δ_R via the equivalence (L, Γ).

Remark 4.8. The forgetful functor ¯Z^Δ_R → Z¯R is exact by [15, lem. 2.12]. Here ¯ZR is equipped with the exact structure naturally associated with its structure of abelian category. To avoid confusion we’ll call it thestupid exact structure.

Remark 4.9. Let M ∈ Z¯^Δ_R. Then, we have M ΓL(M) and Mx = L(M)x. For each J ⊂I, letMJ be the image of the obvious mapM=ΓL(M)→

x∈JMx, see also[15, def. 2.7]. It is a graded R-module, and we haveM_J =L(M)(J) by[14, lem. 3.4]. Since Z¯_R,x =R for all x, the set ( ¯Z_R)_J is a graded R-subalgebra ofR^⊕J. So M_J is a graded ( ¯ZR)J-module.

We’ll use two different notions of projective objects in the exact category ¯Z^Δ_R, compare [21, sec. 3.8].

Definition 4.10.(a) A moduleM ∈Z¯^Δ_R is projectiveif the functor HomZ¯R(M,•) on ¯Z^Δ_R maps short exact sequences to short exact sequences.

(b) A module M ∈ Z¯_R is F-projective if L(M) is flabby, M_x is a projective graded R-module for eachx,Mh=Mh/αhMh andρh,his the canonical map for eachh.

Remark 4.11.IfM ∈Z¯^Δ_R is F-projective, then it is projective[14, prop. 5.1]. Note that [14, prop. 5.1]usesreflexivegradedR-sheaves. However, Δ-filtered gradedR-sheaves are automatically reflexive by[14, sec. 4].

We say that an object M ∈ Z¯^Δ_R istilting if the contravariant functor HomZ¯R(•, M) on ¯Z^Δ_R maps short exact sequences to short exact sequences.

For each graded R-module M, let M^∗ be its dual graded R-module, i.e., we set M^∗ =

i(M^∗)ⁱ with (M^∗)ⁱ = hom_R(M, Ri). Here hom_R is the Hom space of gradedR-modules. Since ¯Z_Ris commutative, the graded dualR-moduleM^∗of a graded Z¯R-moduleM is a graded ¯ZR-module.

As above, let the symbols Tilt and Proj denote the sets of indecomposable tilting and projective objects. By[14, sec. 4.6], the following holds.

Proposition 4.12. The duality D : ¯ZR → Z¯R such that M → M^∗ restricts to an exact equivalence Z¯^Δ_R, → ( ¯Z^Δ_R,)^op. In particular, it yields bijections Proj( ¯Z^Δ_R,) → Tilt( ¯Z^Δ_R,)andTilt( ¯Z^Δ_R,)→Proj( ¯Z^Δ_R,). 2

We define the support of a gradedR-sheafM onG to be the set supp(M) ={x∈I;

Mx= 0}. We can now turn to the following important definition.

Proposition-Definition 4.13. (See[14].) Let (G,) be an ordered GKM-graph. There is a unique object B¯R,(x)in Z¯R which is indecomposable, F-projective, supported on the coideal {x}and with B¯R,(x)x=R. We callB¯R,(x)a graded BM-sheaf.

We may abbreviate ¯BR(x) = ¯B_R,(x) and ¯CR(x) = ¯C_R,(x) =D( ¯B_R,(x)).

Remark 4.14.The existence and unicity of graded BM-sheaves is proved in[14, thm. 5.2]

using the Braden–MacPherson algorithm[7, sec. 1.4]. See also[16, thm. 6.3]. The con-struction of ¯BR(x) is as follows.

• Set ¯B_R(x)_y = 0 fory x.

• Set ¯B_R(x)_x=R.

• Letyxand suppose we have already constructed ¯BR(x)zand ¯BR(x)hfor anyz, h such thatyz, h, hx. Forh∈dy set

– ¯B_R(x)_h= ¯B_R(x)_h/α_hB¯_R(x)_h andρ_h,his the canonical map, – ¯B_R(x)_∂y = Im(ρ_≺y,dy)⊂B¯_R(x)_dy,

– ¯BR(x)y is the projective cover of the gradedR-module ¯BR(x)∂y,

– ρy,h is the composition of the projective cover map ¯BR(x)y →B¯R(x)∂y with the obvious projection ¯B_R(x)_dy →B¯_R(x)_h.

Proposition 4.15. An F-projective object in Z¯R is a direct sum of objects of the form B¯R(x)jwith x∈I andj∈Z.

Proof. See[21, prop. 3.9]. 2

Next, following [14, sec. 4.5], we introduce the following.

Proposition-Definition 4.16. There is a unique objectV¯R(x)∈Z¯R that is isomorphic to R as a gradedR-module and on which the tuple(z_y)∈Z¯_Racts by multiplication withz_x. We call it a graded Verma-sheaf.

Proposition 4.17. (a) The graded Verma-sheaves are Δ-filtered and self-dual.

(b) We have V¯_R(x)^y = ¯V_R(x)_y = ¯V_R(x)_[y] =R if x=y and 0 else.

(c) For M∈Z¯R we have

HomZR

V¯R(x)i, M

=M^x−i, HomZR

M,V¯R(y)j

= (My)^∗j, homZ¯R

V¯_R(x)i,V¯_R(y)j

=δ_x,yR^i−j.

Proof. Obvious, see e.g., [21, sec. 3.11]. 2

Remark 4.18. A Δ-filtered graded R-sheaf M has a filtration 0 = M0 ⊆ M1 ⊆. . . ⊆ M_n =M by graded ¯Z_R-submodules with[14, rk. 4.4, sec. 4.5]

n r=1

Mr/Mr−1=

y

M_[y], Mr/Mr−1= ¯VR(yr)ir, rs ⇒ ysyr.

In particular, a Δ-filtered graded ¯ZR-module is free and finitely generated as a graded R-module.

Remark 4.19.From Proposition 4.17(c) we deduce that (M_y)^∗= (M^∗)^y. 4.2. Ungraded version and base change

Assume thatR is the localization ofS with respect to some multiplicative set. Then, we define anR-sheaf overG, its space of global sections, the structural algebraZR, the categoriesF_R,Z_R, etc., in the obvious way, by forgetting the grading in the definitions above. In particular, we define as above an exact categoryZ^Δ_R.

Now, assume thatR=S. Forgetting the grading yields a faithful and faithfully exact functor ¯Z^Δ_R → Z^Δ_R. See, e.g., [20], for details on faithfully exact functors. Let VR(x), BR(x),CR(x) be the images of ¯VR(x), ¯BR(x), ¯CR(x). We callVR(x) a Verma-sheaf and BR(x) a BM-sheaf.

Next, for each morphism of S-algebras R→R we consider the base change functor

• ⊗RR:R-mod→R-modgiven byM→RM =M⊗RR.

Assume that R is the localization of R with respect to some multiplicative subset.

Then, the base change yields functors • ⊗R R : FR → FR and • ⊗RR : ZR → ZR. These functors commute with Γ, L and the canonical map RHomZR(M, N) → Hom_ZR(RM, RN) is an isomorphism, see[21, secs. 2.7, 2.18, 3.15]. Further, the base change commutes with the functorM→M_[x], it preserves the flabby sheaves, the Verma sheaves, the F-projective ones and it yields an exact and faithful functorZ^Δ_R →Z^Δ_R, see [21, sec. 3.15].

Assume now thatR is the localization of S with respect to some multiplicative set.

Then, we defineV_R(x) = RV_S(x), B_R(x) =RB_S(x) and C_R(x) = RC_S(x). Note that B_R(x), C_R(x) are F-projective by[21, sec. 3.8], and that if R=S₀ then B_R(x), C_R(x) are indecomposable by[21, sec. 3.16]. Further, anyF-projectiveR-sheaf inFRis a direct sum of objects of the formBR(x) withx∈I, see [21, prop. 3.13].

Forgetting the grading and taking the base change, we get the functorε:S-gmod→ S₀-mod. It is faithful and faithfully exact. Indeed, ifε(M) = 0, then there is an element f ∈ S \m such that f M = 0. Since M is graded, this implies that M = 0. Hence ε is faithful, and by[20, thm. 1.1] it is also faithfully exact. Finally, note that a graded S-moduleM is free overS (as a graded module) if and only ifε(M) is free overS0.

Similarly, consider the functor ε : ¯ZS → ZS0 given by forgetting the grading and taking the base change. We have the following lemma.

Lemma 4.20. (a) The functorε commutes with the functorM →M_[x]. (b) The functor ε: ¯Z_S →Z_S0 is faithful and faithfully exact.

(c) For each M∈Z¯_S, we have M ∈Z¯^Δ_S if and only if ε(M)∈Z^Δ_S0. 2 4.3. The moment graph of O

Fix a parabolic typeμ∈ P and fixw∈W.

Unless specified otherwise, we’ll setV =t. In this section, we define a moment graph

wGμ,± over twhich is naturally associated with the truncated categories^wOμ,±. Then, we consider its category of (graded) R-sheaves over ^wGμ,±. In the graded case we’ll assume that R = S, with the grading above. In the ungraded one, we’ll assume that R=S orS₀, see the previous section for details.

Definition 4.21. Let ^wGμ,± be the moment graph over t whose set of vertices is ^wIμ,±, with an edge between x, y labeled by kαˇ if there is an affine reflection sα ∈ W such thatsαy∈xWμ. We equip^wGμ,+ with the partial ordergiven by the opposite Bruhat order, and we equip ^wGμ,− with the partial ordergiven by the Bruhat order.

Let^wZ¯_S,μ,±be the graded structural algebra of^wGμ,±. Let^wZ¯_S,μ,±be the category of graded^wZ¯S,μ,±-modules which are finitely generated and torsion free overS. Let^wZ¯^Δ_S,μ,±

be the category of Δ-filtered gradedS-sheaves on ^wGμ,±,. Let ¯VS,μ,±(x)∈^wZ¯S,μ,± be the (graded) Verma-sheaf whose stalk atxisS. We’ll use a similar notation for all other objects attached to^wGμ,±.

We’ll use the same notation as in Section4.2. For example^wB_S,μ,±(x) is the nongraded analogue of^wB¯_S,μ,±(x), and^wB_R,μ,±(x) =R^wB_S,μ,±(x).

We also set ^wZ¯_k,μ,± =k^wZ¯_S,μ,±, ^wZ_k,μ,± =k^wZ_S0,μ,±, ^wZ¯_k,μ,± =^wZ¯_k,μ,±-gmod and

wZ_k,μ,± =^wZ_k,μ,±-mod. In the graded case, we put ^wB¯_k,μ,±(x) =k^wB¯_S,μ,±(x). In the non-graded case, we put ^wB_k,μ,±(x) =k^wBS0,μ,±(x). To unburden the notation we may omit the subscript kand write^wZμ,±=^wZk,μ,±,^wZμ,±=^wZk,μ,±, etc.

Proposition 4.22. The (graded) BM-sheaves on ^wGμ,± areΔ-filtered.

Proof. The (graded) BM-sheaves on ^wGμ,− are Δ-filtered by [14, thm. 5.2]. We claim that the (graded) BM-sheaves on^wGμ,+ are also Δ-filtered.

By Section4.2, a gradedS-sheaf M is Δ-filtered if and only if theS0-sheaf ε(M) is Δ-filtered. Now, by Proposition 4.33(c) below, the BM-sheaves on^wGμ,+ are Δ-filtered forR=S₀. This proves our claim. 2

Letl(x) be the length of an elementx∈W. FromPropositions 4.13, 4.22, we deduce the following.

Proposition 4.23. For each x ∈ ^wI_μ,±, there is a unique Δ-filtered graded S-sheaf

wB¯S,μ,±(x) ∈ ^wZ¯^Δ_S,μ,± which is indecomposable, F-projective, supported on the coideal {x}and whose stalk at xis equal toS±l(x). 2

Assume thatw∈Iμ,+ and let v =w₋ ∈I_μ,−. From the anti-isomorphism of posets

wIμ,+ ^vIμ,−inCorollary 3.3(c), we deduce that^wGμ,+and^vGμ,− are the same moment graph with opposite orders. In particular, we have^wZ¯S,μ,+ =^vZ¯S,μ,−as graded algebras.

Further, byProposition 4.12, there is an exact equivalence D:^wZ¯^Δ_S,μ,+→_vZ¯^Δ_S,μ,−op

.

The same is true if + and − are switched everywhere. We define ^wC¯S,μ,±(x) = D(^vB¯_S,μ,∓(y)) with y = x_∓, v = w_∓ for each x ∈ ^wI_μ,±. It is a Δ-filtered graded S-sheaf on ^wGμ,± which is indecomposable, tilting, supported on the ideal {x} and whose costalk at xis equal to S±l(x). We abbreviate ^wB¯_S,μ,± =

xwB¯_S,μ,±(x) and

wC¯_S,μ,±=

xwC¯_S,μ,±(x).

ByRemark 4.18we have the following lemma.

Lemma 4.24. (a) The graded S-sheaf ^wB¯_S,μ,±(x) is filtered by graded Verma-sheaves.

The top term in this filtration is V¯S,μ,±(x)±l(x), which is a quotient of ^wB¯S,μ,±(x).

The other subquotients are of the formV¯S,μ,±(y)jwith yxandj∈Z.

(b) The gradedS-sheaf^wC¯S,μ,±(x)is filtered by graded Verma-sheaves. The first term in this filtration is the sub-object V¯_S,μ,±(x)±l(x). The other subquotients are of the formV¯_S,μ,±(y)jwith y≺xandj∈Z.

Since ^wZ¯_S,μ,+ = ^vZ¯_S,μ,− we may also regard ^vC¯_S,μ,−(x₋) = D(^wB¯_S,μ,+(x)) as a graded^wZ¯_S,μ,+-module. The following proposition says that^wB¯_S,μ,+(x) is self-dual.

Proposition 4.25. For each x ∈ ^wIμ,+, we have ^wB¯S,μ,+(x) = ^vC¯S,μ,−(x₋) as graded

wZ¯_S,μ,+-modules.

Proof. Ifμ=φis regular then the claim is[15, thm. 6.1].

Assume now that μ is not regular. We abbreviate ¯B_μ(x) = ^wB¯_S,μ,+(x). We must prove that D( ¯B_μ(x)) = ¯B_μ(x). The regular case implies that D( ¯B_φ(x)) = ¯B_φ(x). We can assume thatw∈I_μ,− and thatx∈^wI_μ.

FromProposition 4.41(e), (f) below, we deduce that

y∈Wμ

DB¯μ(x)

2l(y)−l(wμ) ⊕D(M) =

y∈Wμ

B¯μ(x)

l(wμ)−2l(y) ⊕M,

where M is a direct sum of objects of the form ¯B_μ(z)j with z < x. We have also D( ¯B_μ(e)) = ¯B_μ(e). Thus, by induction, we may assume thatD( ¯B_μ(z)) = ¯B_μ(z) for all z < x, and the equality above implies thatD( ¯B_μ(x)) = ¯B_μ(x). 2

Remark 4.26. The graded sheaf ^wB¯_S,μ,−(x) is not self-dual, i.e.,^wB¯_S,μ,−(x) is not iso-morphic to^vC¯S,μ,+(x+) in general.

Let P_y,x^μ,q(t) =

iP_y,x,i^μ,q tⁱ be Deodhar’s parabolic Kazhdan–Lusztig polynomial “of typeq” associated with the parabolic subgroup Wμ ofW.

LetQ^μ,−1_x,y (t) =

iQ^μ,−1_x,y,itⁱ be Deodhar’s inverse parabolic Kazhdan–Lusztig polyno-mial “of type −1”.

We use the notation in[26, rk. 2.1], which is not the usual one. We abbreviateP_y,x= P_y,x^φ,q andQ_y,x=Q^φ,_y,x⁻¹.

Proposition 4.27. For each x, y∈^wIμ,+, we have graded S-module isomorphisms (a) ^wB¯S,μ,+(x)y =

i0(Sl(x)−2i)^{⊕Py,x,iμ,q} , (b) ^wB¯_S,μ,+(x)_[y] =

i0(S2l(y)−l(x) + 2i)^{⊕Py,x,iμ,q} .

Proof. Any finitely generated projective S-module is free, and that the S-module

wB¯_S,μ,+(x)_y is projective because ^wB¯_S,μ,+(x) is F-projective. Hence, part (a) follows from Proposition 4.43 below and [26, thm. 1.4]. Part (b) follows from (a) and Proposi-tion 4.25as in [15, prop. 7.1] (where it is proved for μ regular). More precisely, write V¯(x) = ¯VS,φ(x), ¯B(x) =^wB¯S,μ,+(x) andZ=^wZS,μ,+. Then, we have

B(x)¯ ^y= Ker(ρ_y,ey)⊂B(x)¯ _[y]= Ker(ρ_y,dy)⊂B(x)¯ _y. In particular, for α_uy =

h∈uyα_h, we haveα_uyB(x)¯ _[y]⊂B(x)¯ ^y.

Next, the graded S-module ¯B(x)_y is free and ¯B(x)_h = ¯B(x)_y/α_hB(x)¯ _y for h∈ u_y because ¯B(x) is F-projective. Thus we have ¯B(x)^y⊂αuyB(x)¯ y, becauseαh1 is prime to αh2 inS ifh1=h2.

We claim that ¯B(x)^y=αuyB(x)¯ _[y]. Letb∈B(x)¯ ^y. Writeb=αuyb withb∈B(x)¯ y. If ρy,h(b) = 0 with h ∈ dy then ρy,h(b) = αuyρy,h(b) = 0, because the αuy-torsion submodule of ¯B(x)_h is zero (we have ¯B(x)_h = ¯B(x)_h/α_hB(x)¯ _h, the S-module ¯B(x)_h

is free, and α_h is prime toα_uy). This implies the claim.

In particular, we have ¯B(x)_[y] = ¯B(x)^y2l(y) because u_y = l(y). Hence, by Re-mark 4.19 andProposition 4.25 we have a gradedS-module isomorphism

B(x)¯ _y∗

2l(y) = ¯B(x)^y

2l(y) = ¯B(x)_[y]. 2

Definition 4.28. Consider the graded S-algebra ^w_A¯_S,μ,± = End^w_ZS,μ,±(^wB¯_S,μ,±)^op and the graded k-algebra ^w_A¯_μ,± = k^w_A¯_S,μ,±. Set also ^w_AS0,μ,± = End^w_ZS

0,μ(^wB_S0,μ,±)^op and let^w_Aμ,± be the k-algebra underlying^w_A¯_μ,±.

For each x ∈ ^wIμ,±, let 1x ∈ ^w_A¯_μ,± be the idempotent associated with the direct summand ^wB¯_S,μ,±(x) of^wB¯_S,μ,±.

Proposition 4.29. The graded k-algebra^w_A¯_μ,± is basic. Its Hilbert polynomial is

P(^w_A¯_μ,+, t)x,x = and the sets of indices of the matrices in the left and right factors are identified through the mapx→x⁻¹.

Proof. First, note that if M, B ∈ Z¯^Δ_S and if B is projective, then the filtration (Mr) of M in Remark 4.18 yields a filtration of the graded S-module HomZS(B, M) whose associated graded is, seeProposition 4.17,

Therefore, for each x, x, there is a finite filtration of the graded S-module Hom^w_ZS,μ,+(^wB¯_S,μ,+(x),^wB¯_S,μ,+(x)) whose associated graded is, seeProposition 4.27,

Thus we have a gradedk-vector space isomorphism 1xw_A¯_μ,+1x =

Set Q_μ,−(t)x,y = Q^μ,_x,y⁻¹(t⁻²)t^l(y)−l(x) for each x, y ∈ ^wIμ,+. The matrix equation P(^w_A¯_μ,−, t) = Qμ,−(t)Qμ,−(t)^T is proved in Proposition A.5 below. Hence ^w_A¯_μ,− is also basic.

Next, for eachx, x, y∈^wIμ,+ anda=−1, q we have, see e.g.,[26, (2.39)]

xyx

(−1)^l(y)−l(x)Q^μ,a_x,y(t)P_y,x^μ,a(t) =δx,x.

Further, ifμ=φis regular then we havePy,x(t) =P_y,x^μ,q(t) andQx,y(t) =Q^μ,q_x,y(t). So, we have the matrix equationsQ_φ,−(t)P_φ,+(−t) = 1,P_φ,+(−t)Q_φ,−(t) = 1. We deduce that

P_wA¯_φ,−, t

P_wA¯_φ,+,−t

=Qφ,−(t)Qφ,−(t)^TPφ,+(−t)^TPφ,+(−t) = 1.

The matrix equation in the proposition follows easily, using the fact thatPy,x=P_y−1,x⁻¹

and Qx,y=Q_x−1,y⁻¹. 2

Remark 4.30.The grading on^w_A¯_μ,±is positive, becauseP_y,x,i^μ,q = 0 ifl(y)−l(x) + 2i >0 and P_y,x,i^μ,q = 0 ifl(y)−l(x) + 2i= 0 andy=x.

Remark 4.31.The results in this section have obvious analogues in finite type. In this case, we may omit the truncation andProposition 4.27,Corollary A.4yield

B¯_S,μ,+(x)_y=

i0

S

l(x)−2i ^⊕P

μ,q y,x,i

B¯S,μ,−(x)y

l(w0)

=

i0

S

l(w0x)−2i ^⊕Q

μ,−1 x,y,i

,

where w₀is the longest element in the Weyl groupW. Indeed, we have ω^∗B¯S,μ,+(w0xwμ) = ¯BS,μ,−(x)

l(w0wμ) ,

where ω : Gμ,− → Gμ,+ is the ordered moment graph isomorphism induced by the bijectionsIμ,+→Iμ,+,x→w0xwμandt→t,h→w0(h). Note that[26, props. 2.4, 2.6]

and Kazhdan–Lusztig’s inversion formula giveQ^μ,−1_x,y =P_w^μ,q_0ywμ,w0xwμ. 4.4. Deformed categoryO

Assume that R=S0 ork. IfR=k we may drop it from the notation.

We define OR,μ,± as the category consisting of the (g, R)-bimodulesM such that

• M=

λ∈t^∗M_λ withM_λ={m∈M; xm= (λ(x) +x)·m, x∈t},

• U(b) (R·m) is finitely generated overRfor eachm∈M,

• the highest weight of any simple subquotient is linked to oμ,±.

Here, for eachr∈ R and each elementm of anR-module M, the element r·m is the action ofr onm. Further, in the symbol λ(x) +xwe viewxas the element of R given by the image of x under the canonical map S = S(t) → R. The morphisms are the (g, R)-bimodule homomorphisms.

Next, consider the following categories

• ^wO_R,μ,− is the Serre subcategory ofO_R,μ,− consisting of the finitely generated mod-ules such that the highest weight of any simple subquotient is of the formλ=x•o_μ,−

withxwandx∈I_μ,−.

• ^wOR,μ,+ is the category of the finitely generated objects in the Serre quotient of O_R,μ,+ by the Serre subcategory consisting of the modules such that the highest weight of any simple subquotient is of the form λ = x•oμ,+ with x w and x∈Iμ,+. We’ll use the same notation for a module inOR,μ,+ and a module in the quotient category^wOR,μ,+, hoping it will not create any confusion.

For each λ∈t^∗ letR_λ be the (t, R)-bimodule which is free of rank one overR and such that the elementx∈tacts by multiplication by the image of the elementλ(x) +x by the canonical mapS →R. Thedeformed Verma modulewith highest weightλis the (g, R)-bimodule given byVR(λ) =U(g)⊗U(b)Rλ.

Proposition 4.32.(a) The category^wO_R,μ,±is a highest weightR-category. The standard objects are the deformed Verma modulesVR(x•oμ,±)with x∈^wIμ,±.

(b) Assume that w ∈ I_μ,^ν_±. The Ringel equivalence gives an equivalence (•) :

wO^Δ_R,μ,±→^vO^∇_R,μ,∓ wherev=w_∓.

Proof. First, we consider the category^wO_R,μ,−. The deformed Verma modules are split, i.e., their endomorphism ring isR. Further, theR-category^wOR,μ,− is Hom finite. Thus we must check that^wO_R,μ,−has a projective generator and that the projective modules are Δ-filtered. Both statements follow from[12, thm. 2.7].

Next, we consider the category ^wO_R,μ,+. Once again it is enough to check that

wOR,μ,+ has a projective generator and that the projective modules are Δ-filtered. By [12, thm. 2.7], a simple moduleL(x•o_μ,+) inO_R,μ,+has a projective coverP_R(x•o_μ,+).

Note that the deformed category O in[12, thm. 2.7]is indeed a subcategory ofOR,μ,+

containing all finitely generated modules. SinceP_R(x•o_μ,+) is finitely generated, the functor HomOR,μ,+(PR(x•oμ,+),•) commutes with direct limits. Thus, since any module inO_R,μ,+is the direct limit of its finitely generated submodules, the moduleP_R(x•o_μ,+) is again projective inOR,μ,+. Now, a standard argument using[32, thm. 3.1]shows that the functor

HomOR,μ,+

x

PR(x•oμ,+),•

, x∈^wIμ,+,

factors to an equivalence of abelian R-categories ^wOR,μ,+ → A-mod, where A is the finite projectiveR-algebra

A= End_OR,μ,+

x

P_R(x•o_μ,+) op

.

Therefore by Proposition 3.9(a) and [35, thm. 4.15]the category ^wOR,μ,+ is a highest weight category over R. This finishes the proof of (a).

By [13, sec. 2.6] there is an equivalence of exact categories O^Δ_R,μ,+−→^∼ (O^Δ_R,μ,−)^op analogue to the one inLemma 3.4. It mapsVR(x•oμ,+) toVR(x₋•o_μ,−), andPR(x•oμ,+) toTR(x₋•oμ,−). We deduce that (^vOR,μ,−)=^wOR,μ,+forw∈Iμ,+andv=w₋∈Iμ,−. Now part (b) follows from generalities on Ringel equivalences, see Section2.5. 2

Let ^wP_R(x•o_μ,±) and ^wT_R(x•o_μ,±) be respectively the projective and the tilting objects in the highest weight categories^wO_R,μ,± as given byLemma 2.3. Set^wP_R,μ,± =

xwPR(x•o_μ,±) and^wT_R,μ,± =

xwTR(x•o_μ,±), where xruns over the set^wI_μ,±. Composing the Ringel duality and the BGG duality, we get an equivalence

D= d◦(•):^wO^Δ_R,μ,±→_v

O^Δ_R,μ,∓op

. (4.1)

Consider the base change functor ^wO_S0,μ,± → ^wO_k,μ,± such that M → kM.

By Proposition 2.4 we have k^wPS0(x•o_μ,±) = ^wP(x•o_μ,±) and k^wTS0(x•o_μ,±) =

wT(x•oμ,±) for eachx∈^wIμ,±. 4.5. The functor V

Assume that R=S0 ork. There is a well defined functor VR= Hom^wO_R,μ,−

_w

P(oμ,−),•

:^wO^Δ_R,μ,−→^wZR,μ,−, called the structure functor, see[13, sec. 3.4].

At the positive level, following[13, sec. before rem. 6], we define the structure functor VR as the composition

wO^Δ_R,μ,+ ^D (^vO^Δ_R,μ,−)^{op VR} (^vZ_R,μ,−)^{op D w}Z_R,μ,+. Proposition 4.33. LetR=S₀ ork. The following hold

(a) VR is exact on ^wOR,μ,− and on ^wO^Δ_R,μ,+, (b) VR commutes with base change,

(c) the functor VS0 on ^wO^Δ_S0,μ,± is such that (i) VS0◦D=D◦VS0,

(ii) VS0 is an equivalence of exact categories ^wO^Δ_S0,μ,±→^wZ^Δ_S0,μ,±,

(iii) for each x∈^wI_μ,± we have – VS0(VS0(x•oμ,±)) =VS0,μ,±(x), – VS0(^wPS0(x•oμ,±)) =^wBS0,μ,±(x), – VS0(^wT_S0(x•o_μ,±)) =^wC_S0,μ,±(x).

Proof. Parts (a),(b) and the first claim of (c) follow from the construction ofVR. The second claim of (c) follows from[14, thm. 7.1]. Note that we use here the exact

Proof. Parts (a),(b) and the first claim of (c) follow from the construction ofVR. The second claim of (c) follows from[14, thm. 7.1]. Note that we use here the exact

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