The regular case

Fix integersd, e >0 and fixμ∈ P. Assume that the weights o_μ,−, o_φ,− have the level

−e−N and−d−N respectively.

Letw∈I_μ,+ and putu=w⁻¹, v =w⁻¹₋ andz =w₋. Note thatu∈I_φ,+^μ , v∈I_φ,−^μ andz∈I_μ,−.

The first step is to compare the algebras^wA_μ,+ = End^w_Oμ,+(^wP_μ,+)^op and ^vA¯^μ_φ,− = ExtvO^μ_φ,−(^vL^μ_φ,−)^op, and, then, the algebras ^wA¯_μ,+ = Ext^w_Oμ,+(^wL_μ,+)^op and ^vA^μ_φ,− = EndvO^μ_φ,−(^vP^μ_φ,−)^op. More precisely, we prove the following.

Proposition 5.1. If x ∈ ^wI_μ,+ and y = x⁻₋¹, then y ∈ ^vI_φ,−^μ . We have a k-algebra iso-morphism ^wA_μ,+ =^vA¯^μ_φ,− such that 1_x → 1_y for each x∈^wI_μ,+. We have a k-algebra isomorphism^wA¯_μ,+=^vA^μ_φ,−such that1_x→1_y for eachx∈^wI_μ,+. The gradedk-algebras

wA¯μ,+ and^vA¯^μ_φ,−are Koszul and are Koszul dual to each other. Further, we have1x= 1^!_y for each x∈^wIμ,+.

Proof. ByCorollaries 4.34, 4.49, composingH,Vk yields k-algebra isomorphisms

wA_μ,+=^w_Aμ,+=^vA¯^μ_φ,−, (5.1) which identify the idempotents 1_y ∈ ^vA¯^μ_φ,− and 1_x ∈^wA_μ,+,^w_Aμ,+ for each x∈ ^wI_μ,+

andy=x⁻¹₋ .

Now, we claim that^vA^μ_φ,− has a Koszul grading. ByLemma 2.2and Section3.5, it is enough to check that^vA_φ,− has a Koszul grading. This follows from the matrix equation inProposition 4.29and from [5, thm. 2.11.1], because^vA_φ,− =^v_Aφ,− as k-algebras by Corollary 4.34.

Equip^vA^μ_φ,− with the Koszul grading above. Then, Lemma 2.1implies that

vA^μ,!_φ,−= Ext^v_O^μ_φ,−v L^μ_φ,−op

=^vA¯^μ_φ,−. (5.2) Thus, the graded k-algebra ^vA¯^μ_φ,− is also Koszul. Since ^wAμ,+ =^vA¯^μ_φ,− as a k-algebra, this implies that thek-algebra^wAμ,+ has a Koszul grading.

ApplyingLemma 2.1once again, we get

wA^!_μ,+= Ext^w_Oμ,+w L_μ,+op

=^wA¯_μ,+. (5.3)

In particular, the gradedk-algebra^wA¯_μ,+ is also Koszul.

Finally, using(5.2),(5.1)and(5.3)we getk-algebra isomorphisms

vA^μ_φ,−=^vA¯^μ,!_φ,−=^wA^!_μ,+=^wA¯μ,+.

They identify the idempotent 1y∈^vA^μ_φ,− with the idempotent 1x∈^wA¯μ,+. 2

Remark 5.2. The Koszul grading on ^vA^μ_φ,− can also be obtained using mixed perverse sheaves on the ind-schemeX as in[5, thm. 4.5.4],[1]. Note that there is no analogue of [5, lem. 3.9.2] in our situation, because^vR_φ,− is not Koszul self-dual.

The second step consists of comparing the k-algebras ^zA¯_μ,− = Ext^z_Oμ,−(^zL_μ,−)^op,

uA^μ_φ,+ = EnduO^μ_φ,+(^uP^μ_φ,+)^op and the k-algebras ^zA_μ,− = End^z_Oμ,+(^wP_μ,−)^op, ^uA¯^μ_φ,+ = ExtuO^μ_φ,+(^uL^μ_φ,+)^op.

We cannot argue as in the previous proposition, because we have no analogue of the localization functorΦinProposition 4.47for positive levels. Hence, we have no analogue of Proposition 4.48andCorollary 4.49.

To remedy this, we’ll use the technic of standard Koszul duality. To do so, we need the following crucial result.

Lemma 5.3.The quasi-hereditary k-algebra^wA_μ,+ has a balanced grading.

Now we can prove the second main result of this section.

Proposition 5.4. If x∈^zIμ,− andy =x⁻¹₊ then y ∈^uI_φ,+^μ . We have a k-algebra isomor-phism^zA¯μ,−=^uA^μ_φ,+ such that1x→1y for eachx∈^zIμ,−, and ak-algebra isomorphism

zA_μ,− =^uA¯^μ_φ,+ such that 1_x →1_y for each x∈^zI_μ,−. The graded k-algebras^uA¯^μ_φ,+ and

zA¯_μ,− are Koszul and are Koszul dual to each other. Further, we have1_x= 1^!_y for each x∈^zI_μ,−.

Proof. ByProposition 3.9, the Ringel dual of^wAμ,+ is^wA_μ,+=^zAμ,−. Thus, since the k-algebra^wAμ,+ has a balanced grading by Lemma 5.3, we deduce that the k-algebra

zAμ,− has a Koszul grading.

We equip ^zA_μ,− with this grading. Then,Lemma 2.1implies that

zA^!_μ,−= Ext^z_Oμ,−z L_μ,−op

=^zA¯_μ,−. (5.4)

Hence, the gradedk-algebra^zA¯μ,− is Koszul and balanced.

Therefore, [29, thm. 1], Proposition 3.9and (5.4)yield ak-algebra isomorphism

zA¯_μ,− =^zA^!_μ,−=_z

A_μ,−!

=_w A^!_μ,+

.

Hence, using (5.3)and Propositions 5.1, 3.9, we get ak-algebra isomorphism

zA¯_μ,−=^wA¯_μ,+=^vA^μ,_φ,−=^uA^μ_φ,+

such that 1_x→1_y. So, thek-algebra^uA^μ_φ,+ has a Koszul grading which is balanced.

We equip ^uA^μ_φ,+ with this grading.Lemma 2.1implies that

uA^μ,!_φ,+ = Ext^u_O^μ_φ,+u L^μ_φ,+op

=^uA¯^μ_φ,+. (5.5)

Hence, the gradedk-algebra^uA¯^μ_φ,+ is Koszul and balanced.

Therefore,[29, thm. 1],Proposition 3.9yield

uA¯^μ_φ,+=^uA^μ,!_φ,+=_u

A^μ,_φ,+!

=_v A^μ,!_φ,−

.

So, using(5.2)andPropositions 5.1, 3.9, we get ak-algebra isomorphism

uA¯^μ_φ,+=_vA¯^μ_φ,−

=^wA_μ,+=^zAμ,−. 2 Now, let us proveLemma 5.3.

Proof of Lemma 5.3. We equip ^wAμ,+ with the grading given by ^vA¯^μ_φ,−, see Proposi-tion 5.1. Since ^vA¯^μ_φ,− is Koszul, we must prove that ^vA¯^μ,!_φ,− is quasi-hereditary and that the grading on^vA¯^μ,_φ,− is positive, see Section2.6.

We have ^vA¯^μ,!_φ,− =^wA¯_μ,+ byProposition 5.1. Hence, it is quasi-hereditary. Next, the grading of^z_A¯_μ,− is positive byRemark 4.30. Thus, to prove the lemma, it is enough to check that we have a gradedk-algebra isomorphism ^vA¯^μ,_φ,− =^z_A¯_μ,−.

First, we check that^wA_μ,+=^zAμ,− as (ungraded)k-algebras.

To do this, note thatProposition 3.9(c) yields ak-algebra isomorphism^wA_μ,+=^zAμ,−. Further, we have ^zA_μ,− = End^z_Oμ,−(^zP_μ,−)^op and, by Proposition 3.9(b), the Ringel equivalence takes^zO^Δ_μ,− to ^zO^Δ_μ,+. Thus, Propositions 2.4(b),4.33(c) yield

wA_μ,+= End^zOμ,+

_z Tμ,+

op

=kEnd^z_OS0,μ,+z

T_S0,μ,+op

=kEnd^z_ZS0,μ,+z

C_S0,μ,+op

=^z_Aμ,−, where the last equality isDefinition 4.28.

Now, we must identify the gradings of ^vA¯^μ,_φ,− and ^z_A¯_μ,− under the isomorphism

wA_μ,+=^z_Aμ,− above.

First, we consider the grading on^vA¯^μ,_φ,−. Let^wT¯μ,+be the graded^vA¯^μ_φ,−-module equal to ^wTμ,+ as an ^wAμ,+-module, with the natural grading (this is well-defined, because

vA¯^μ_φ,− is Koszul). Then, by Section2.6, we have^vA¯^μ,_φ,−= End^wAμ,+(^wT¯μ,+)^op.

Next, we consider the grading on ^z_A¯_μ,−. By Corollary A.6, we have ^z_A¯_μ,− = kEnd^w_ZS,μ,+(^wC¯_S,μ,+) = End^w_Zμ,+(^wC¯_μ,+).

Let us identify^wAμ,+=^wAμ,+ via(5.1). We must prove the following.

Claim 5.5.There is a graded k-algebra isomorphism End^w_Aμ,+wT¯μ,+

= End^wZμ,+

_wC¯μ,+

. To do that, we’ll need some new material.

Consider the graded categories given by ^wO¯μ,+ = ^w_A¯_μ,+-gmod and ^wO¯S,μ,+ =

w_A¯_S,μ,+-gmod. Since we have ^wT¯μ,+ ∈ ^vA¯^μ_φ,−-gmod and ^w_A¯_μ,+ = ^vA¯^μ_φ,− by Corol-lary 4.49, we may view^wT¯μ,+ as an object of ^wO¯μ,+.

We have an isomorphism of graded S-algebras ^w_A¯

S,μ,+ = End^w_ZS,μ,+(^wB¯_S,μ,+)^op. Consider the pair of adjoint functors ( ¯VS,ψ¯_S) between^wO¯_S,μ,+and^wZ¯_S,μ,+-gmodgiven by

V¯S =^wB¯_S,μ,+⊗^w_A^¯S,μ,+•:^wO¯_S,μ,+→^wZ¯_S,μ,+-gmod, ψ¯_S = Hom^w_ZS,μ,+(^wB¯_S,μ,+,•) :^wZ¯_S,μ,+-gmod→^wO¯_S,μ,+. We consider the object ^wT¯S,μ,+ of^wO¯S,μ,+ given by^wT¯S,μ,+= ¯ψS(^wC¯S,μ,+).

Recall the functor VS0 :^wO^Δ_S0,μ,+ →^wZ^Δ_S0,μ,+ studied in Proposition 4.33. It yields anS0-algebra isomorphism End^wOS0,μ,+(^wPS0,μ,+)^op→^wAS0,μ,+. Thus, since^wPS0,μ,+

is a pro-generator of ^wO_S0,μ,+, the functor φ_S0 = Hom^w_OS0,μ,+(^wP_S0,μ,+,•) identifies

wOS0,μ,+ with the category^wAS0,μ,+-mod.

Consider the functor ψS0 : ^wZS0,μ,+-mod → ^wAS0,μ,+-mod given by ψS0 = Hom^wZS0,μ,+(^wBS0,μ,+,•). We have the commutative triangle

wO^Δ_S0,μ,+ ^V_∼^S0

φS0

wZ^Δ_S0,μ,+

ψS0

wAS0,μ,+-mod,

(5.6)

such that VS0(^wTS0,μ,+) = ^wCS0,μ,+. Now, we define the module ^wT_S0,μ,+ = ψS0(^wCS0,μ,+) in ^wAS0,μ,+-mod. It is identified with ^wTS0,μ,+ by φS0, because (5.6) is commutative. ByProposition 4.33(c), the functor ψ_S0 yields ak-algebra isomorphism End^wZS0,μ,+(^wCS0,μ,+)→End^w_AS0,μ,+(^wTS0,μ,+). Note thatφS0andψS0 are both exact.

Since taking Hom’s commutes with localization, we have ε(^w_A¯_S,μ,+) = ^wAS0,μ,+. Thus, forgetting the grading and taking base change yield a functor ε : ^wO¯S,μ,+ →

wO_S0,μ,+ which is faithful and faithfully exact, see Section4.2. Let^wO¯^Δ_S,μ,+⊂^wO¯_S,μ,+

be the full subcategory of the modules taken to^wO^Δ_S0,μ,+ byε.

We identify ^wOS0,μ,+ with^wAS0,μ,+-modbyφS0. We have the following.

Claim 5.6. We have the following commutative diagram

wO¯^Δ_S,μ,+ ^V¯S

ε

wZ¯^Δ_S,μ,+

ε ψ¯S

wO¯^Δ_S,μ,+

ε wO^Δ_S0,μ,+ ^{VS0 w}Z^Δ_S0,μ,+

ψS0 wO^Δ_S0,μ,+.

Since^wAS0,μ,+ is Noetherian, any finitely generated module has a finite presentation.

ByProposition 4.33(c), the functorVS0 is exact and we haveVS0(^wPS0,μ,+) =^wBS0,μ,+. We identify ^wOS0,μ,+ with ^wAS0,μ,+-mod as above. By the five lemma, the canonical morphism ^wBS0,μ,+⊗^wAS0,μ,+ • → VS0 is an isomorphism. We deduce that the left square inClaim 5.6is commutative.

Next, by definition we haveε◦ψ¯S =ψS0◦ε. From the diagram5.6, we deduce that ψ_S0 maps into^wO^Δ_S0,μ,+ and is a quasi-inverse toVS0.Claim 5.6follows.

FromClaim 5.6 we deduce thatε(^wT¯S,μ,+) =^wTS0,μ,+ and^wT¯S,μ,+∈^wO¯^Δ_S,μ,+. Next, since ε : S-gmod → S₀-mod is faithfully exact by Section 4.2 and ψ_S0 gives an isomorphism End^wZS0,μ,+(^wCS0,μ,+)→End^w_AS0,μ,+(^wTS0,μ,+), we deduce from Claim 5.6 that ¯ψ_S gives a graded k-algebra isomorphism End^w_ZS,μ,+(^wC¯_S,μ,+) → End^w_AS,μ,+(^wT¯S,μ,+).

Finally, since the functor ψ_S0 is exact and ε is faithfully exact by Section 4.2, we deduce fromClaim 5.6 that ¯ψS is exact on^wZ¯^Δ_S,μ,+.

Now, by Definition 4.28, we have ^w_A¯

μ,+ = k^w_A¯

S,μ,+. Thus, the specialization atk gives the modulek^wT¯S,μ,+ in^wO¯μ,+=^w_A¯_μ,+-gmod.

Claim 5.7.We have k^wT¯S,μ,+=^wT¯μ,+.

The specialization gives a graded ring homomorphism kEnd^w_AS,μ,+(^wT¯_S,μ,+) → End^w_Aμ,+(k^wT¯S,μ,+). Since ε(^wT¯S,μ,+) = ^wTS0,μ,+, forgetting the grading it is taken to the obvious map kEnd^w_OS

0,μ,+(^wT_S0,μ,+) → End^w_Oμ,+(k^wT_S0,μ,+). The latter is an isomorphism by Proposition 2.4(b). Since k^wT¯S,μ,+ = ^wT¯μ,+, we deduce that kEnd^w_AS,μ,+(^wT¯_S,μ,+) = End^w_Aμ,+(^wT¯_μ,+).

Since we have kEnd^wZS,μ,+(^wC¯S,μ,+) = End^wZμ(^wC¯μ,+) and End^wZS,μ,+(^wC¯S,μ,+) = End^w_AS,μ,+(^wT¯_S,μ,+), this impliesClaim 5.5.

Now, we proveClaim 5.7. The grading of^wT¯μ,+is characterized in the following way.

First, we have a decomposition ^wT¯_μ,+ =

xwT¯(x•o_μ,+) where ^wT¯(x•o_μ,+) is the natural graded lift of ^wT(x•oμ,+). Since^wT(x•oμ,+) is indecomposable in^wOμ,+, it admits at most one graded lift in ^wO¯_μ,+, up to grading shift. Let ¯V(x•o_μ,+) be the graded^vA¯^μ_φ,−-module equal toV(x•oμ,+) as an^wAμ,+-module, with the natural grading (this is well-defined, because^vA¯^μ_φ,− is Koszul). The natural grading is characterized by the fact that there is an inclusion ¯V(x•oμ,+)⊂^wT¯(x•oμ,+) which is homogeneous of degree 0.

The graded object^wT¯S,μ,+ satisfies a similar property. Indeed, by Lemma 4.24, the graded S-sheaf ^wC¯_S,μ,+(x) is filtered by shifted Verma-sheaves, and the lower term of this filtration yields an inclusion ¯VS,μ(x)l(x) ⊂^wC¯S,μ,+(x). Further, consider the object

wT¯_S(x•o_μ,+) in^wO¯_S,μ,+given by^wT¯_S(x•o_μ,+) = ¯ψ_S(^wC¯_S,μ,+(x)). Then, we have the decomposition^wT¯S,μ,+=

xwT¯S(x•oμ,+).

For each x ∈ ^wI_μ,+, we consider also the object ¯V_S(x•o_μ,+) in ^wO¯_S,μ,+ given by V¯S(x•oμ,+) = ¯ψS( ¯VS,μ(x))l(x). Since ¯ψS is exact, we deduce that ^wT¯S(x•oμ,+)

is filtered by ¯VS(y•oμ,+)’s, and the lower term of this filtration yields an inclusion V¯S(x•oμ,+)⊂^wT¯S(x•oμ,+) which is homogeneous of degree 0.

Now, by Proposition 4.33(c), we haveVS0(VS0(x•oμ,+)) =VS0,μ(x). From the proof of Claim 5.6, we deduce thatψS0(VS0,μ(x)) =VS0(x•oμ,+). Thus

εV¯S(x•oμ,+)

=εψ¯S

V¯S,μ(x)

=ψS0εV¯S,μ(x)

=VS0(x•oμ,+).

SinceVS0(x•oμ,+) is free overS0, we deduce that ¯VS(x•oμ,+) is free overSby Section4.2.

Therefore, the inclusion ¯VS(x•oμ,+)⊂^wT¯S(x•oμ,+) gives an inclusionkV¯S(x•oμ,+)⊂ k^wT¯S(x•oμ,+) which is homogeneous of degree 0.

Hence, to prove Claim 5.7 we are reduced to check that we have ¯V(x•oμ,+) = kV¯_S(x•o_μ,+) in^wO¯_μ,+. Note that ¯V(x•o_μ,+) and kV¯_S(x•o_μ,+) are both graded lifts of the Verma module V(x•o_μ,+), which is indecomposable. Thus they coincide up to a grading shift.

To identify this shift, recall that byLemma 4.24we have a surjection ^wB¯_S,μ,+(x)→ V¯S(x)l(x). We define ^wP¯S(x•oμ,+) = ¯ψS(^wB¯S,μ,+(x)). Since ¯ψS is exact, we have a surjection^wP¯S(x•oμ,+)→V¯S(x•oμ,+).

Now, since ^vA¯^μ_φ,− is Koszul, we can consider the natural graded lift ^wP¯(x•oμ,+) of ^wP(x•oμ,+) in ^wO¯μ,+. By definition of the natural grading, we have a surjection

wP(x¯ •oμ,+)→V¯(x•oμ,+).

So, we must prove that ^wP¯(x•o_μ,+) = k^wP¯_S(x•o_μ,+) in ^wO¯_μ,+. This is obvious, because ^wP¯(x•o_μ,+) = ^w_A¯

μ,+1_x by definition of the natural grading of^wP(x•o_μ,+), and becauseDefinition 4.28 yields the chain of isomorphisms

wP¯_S(x•o_μ,+) = ¯ψ_SwB¯_S,μ,+(x)

= End^w_ZS,μ,+wB¯_S,μ,+op

1_x=^w_A¯

μ,+1_x. 2 We can now prove the following graded analogue of Corollary 4.34. See also Corol-lary 4.49.

Corollary 5.8.Assume that u∈I_φ,+^μ . Let z =u⁻¹₋ ∈Iμ,−. We have an isomorphism of gradedk-algebras^uA¯^μ_φ,+ →^z_A¯_μ,− such that 1y→1x withx=y₋⁻¹ for each y∈^vI_φ,+^μ . Proof. By[29, thm. 1]and Lemma 5.3, the gradedk-algebra^vA¯^μ,_φ,− =^z_A¯_μ,− is Koszul.

By Proposition 5.4, the graded k-algebra^uA¯^μ_φ,+ is Koszul and is isomorphic to ^wA_μ,−

as a k-algebra. By Proposition 5.4, we have ^vA¯^μ_φ,− =^wAμ,+ as ak-algebra. Finally, by Proposition 3.9, the Ringel dual of^wAμ,+ is ^wA_μ,+ =^zAμ,−. Thus we have a k-algebra isomorphism ^zAμ,− = ^zAμ,−, which lifts to a graded k-algebra isomorphism ^uA¯^μ_φ,+ =

z_A¯_μ,− by unicity of the Koszul grading[5, cor. 2.5.2]. 2 5.2. The general case

We can now complete the proof ofTheorem 3.12. We first prove a series of preliminary lemmas.

Fix parabolic typesμ, ν ∈ P and integersd, e, f > 0. Choose integral weights o_μ,−, o_ν,−, o_φ,− of level−e−N,−f −N and −d−N respectively.

Fix an element w ∈ W. Let τφ,ν : ^wOμ,+ → ^wO^ν_μ,+ be the parabolic truncation functor, see Section3.5. Applying the functorτφ,νto the module^wPμ,+, we get ak-algebra homomorphismτφ,ν :^wAμ,+ →^wA^ν_μ,+.

Lemma 5.9.Assume that w∈I_μ,+^ν . Then, thek-algebra homomorphism τ_φ,ν :^wA_μ,+ →

wA^ν_μ,+ is surjective. Its kernel is the two-sided ideal generated by the idempotents1xwith x∈^wIμ,+\^wI^ν_μ,+. Further, we have τφ,ν(1x) = 1x for each x∈^wI^ν_μ,+.

Proof. By Section3.5, the functorτ_φ,νtakes the minimal projective generator of^wO_μ,+

to the minimal projective generator of^wO^ν_μ,+. Leti_ν,φbe the right adjoint of τ_φ,ν. For anyM the unitM →i_ν,φτ_φ,ν(M) is surjective. Hence, for any projective module P we have a surjective map

Hom^w_Oμ,+(P, M)→Hom^w_Oμ,+

P, i_ν,φτ_φ,ν(M)

= Hom^w_O^ν_μ,+

τ_φ,ν(P), τ_φ,ν(M) . Thus, thek-algebra homomorphismτ_φ,ν is surjective.

LetI ⊂^wA_μ,+ be the 2-sided ideal generated by the idempotents 1_x such that x∈

wI_μ,+ and τ_φ,ν(1_x) = 0. By Section 3.5(c), the latter are precisely the idempotents 1_x withx∈^wIμ,+\^wI^ν_μ,+.

We have ak-algebra isomorphism^wAμ,+/I ^wA^ν_μ,+, because^wO^ν_μ,+ is the Serre sub-category of^wOμ,+generated by the simple modules killed byI. Under this isomorphism, the mapτφ,ν is the canonical projection^wAμ,+→^wAμ,+/I.

The last claim in the lemma follows from Section3.5(b). 2 Now, letv∈I_ν,− andw=v⁻₊¹. We havew∈I_φ,+^ν .

We equip thek-algebras ^vAν,−, ^vAφ,− with the gradings^wA¯^ν_φ,+, ^wA¯φ,+, see Proposi-tion 5.4. Consider the categories^vO¯_ν,−=^wA¯^ν_φ,+-gmodand^vO¯_φ,−=^wA¯φ,+-gmod, which are graded analogues of the categories^vO_ν,− and^vO_φ,−.

To unburden the notation, we’ll abbreviateLν =^vLν,−andLφ=^vLφ,−. Let ¯Lν, ¯Lφbe the natural graded lifts in^vO¯ν,−,^vO¯φ,−of the semi-simple modulesLν,Lφrespectively.

Forv∈Iν,− andd+N > f, we’ll use the following graded analogue of the translation functorTφ,ν:^vOφ,− →^vOν,− inProposition 4.36.

Lemma 5.10.Assume that v ∈I_ν,− and d+N > f. Then, there is an exact functor of graded categories T¯φ,ν : ^vO¯φ,− → ^vO¯ν,− such that T¯φ,ν( ¯Lφ) = ¯Lν and T¯φ,ν coincides withTφ,ν when forgetting the grading.

Proof. First, note that the translation functor T_φ,ν : ^vO_φ,− → ^vO_ν,− is well-defined, becaused+N > f andv∈I_ν,−. See Section4.6for more details.

Now, byCorollary 4.34, we have^vO_ν,− =^vAν,−-mod and^vO_φ,− =^vAφ,−-mod. By Corollary 4.50, we can view^vBν,−,^vBφ,−as right modules over^vAν,−,^vAφ,−respectively.

Consider the functors Vk : ^vO_ν,− → ^vZ_ν,− and Vk : ^vO_φ,− → ^vZ_φ,− given by Vk =

vB_ν,−⊗^vAν,−•andVk =^vB_φ,−⊗^vAφ,−•.

By Proposition 4.33, the functors Vk on ^vOν,−, ^vOφ,− are exact and we have Vk(^vAν,−) = ^vBν,−, Vk(^vAφ,−) = ^vBφ,−. Further, since ^vAν,+, ^vAφ,+ are Noetherian, any finitely generated module has a finite presentation. Thus, by the five lemma, the obvious morphism of functorsV_k→Vk is invertible.

Next, by Corollary 5.8, we have ^wA¯^ν_φ,+ = ^v_A¯_ν,− and ^wA¯_φ,+ = ^v_A¯_φ,− as graded k-algebras. Thus, we have ^vO¯_ν,− = ^v_A¯_ν,−-mod and ^vO¯_φ,− = ^v_A¯_φ,−-mod. Consider the functors on ^vO¯_ν,−, ^vO¯_φ,− given by ¯Vk = ^vB¯_ν,−⊗^v_A^¯_ν,− •, ¯Vk = ^vB¯_φ,−⊗^v_A^¯φ,− • respectively. We have the commutative squares

vO¯_ν,−

¯Vk

vZ¯_ν,−

vO_ν,− ^{Vk v}Z_ν,−

vO¯_φ,−

V¯k

vZ¯_φ,−

vO_φ,− ^{Vk v}Z_φ,−.

Now, let^vO¯^proj_φ,− ⊂^vO¯_φ,− be the full subcategory of the projective objects. We define

vO¯^proj_ν,−,^vO^proj_ν,− and^vO^proj_φ,− in a similar way.

The functor Vk is fully faithful on ^vO^proj_ν,− and ^vO^proj_φ,− byCorollary 4.50. Hence, the functor ¯Vk is fully faithful on ^vO¯^proj_ν,− and ^vO¯^proj_φ,−. Therefore, we may identify ^vO¯^proj_ν,−,

vO¯^proj_φ,− with some full subcategories^vZ¯^proj_ν,−,^vZ¯^proj_φ,− of^vZ¯ν,−,^vZ¯φ,− via ¯Vk

The functor ¯θφ,ν inRemark 4.42 gives an exact functor^vZ¯^proj_φ,− → ^vZ¯^proj_ν,−. We define the functor ¯Tφ,ν : ^vO¯^proj_φ,− → ^vO¯^proj_ν,− so that it coincides with ¯θφ,ν under ¯Vk. It gives a functor of triangulated categories K^b(^vO¯^proj_φ,−) → K^b(^vO¯^proj_ν,−), where K^b denotes the bounded homotopy category.

Since^vO¯^proj_ν,− and^vO¯^proj_φ,− have finite global dimensions, there are canonical equivalences K^b(^vO¯^proj_φ,−) D^b(^vO¯_φ,−) and K^b(^vO¯^proj_ν,−) D^b(^vO¯_ν,−). Thus, we can view ¯T_φ,ν as a functor of triangulated categoriesD^b(^vO¯_φ,−)→D^b(^vO¯_ν,−).

By Proposition 4.36(b), the functor T_φ,ν gives an exact functor ^vO^proj_φ,− → ^vO^proj_ν,−. ByRemark 4.42, the functor ¯T_φ,ν coincides withT_φ,ν when forgetting the grading. The functor Tφ,ν yields a functor of triangulated categoriesD^b(^vO_φ,−)→K^b(^vO_ν,−) which is t-exact for the standard t-structures and which coincides with Tφ,ν when forgetting the grading. Hence, the functor ¯Tφ,ν is also t-exact. Thus, it gives an exact functor T¯φ,ν :^vO¯φ,−→^vO¯ν,− which coincides with Tφ,ν when forgetting the grading.

Now, we concentrate on the equality ¯Tφ,ν( ¯Lφ) = ¯Lν.

Since ¯T_φ,ν coincides withT_φ,ν when forgetting the grading, by Proposition 4.36(d), for each x∈^wI_ν,− there is an integerj such that ¯T_φ,ν( ¯L(xw_ν•o_φ,−)) = ¯L(x•o_ν,−)j. We must check thatj = 0.

Let ¯T_ν,φbe the left adjoint to ¯T_φ,ν, seeRemark 5.11below. ByCorollary 5.8we have

vO¯_ν,− = ^v_A¯_ν,−-mod and ^vO¯_φ,− = ^v_A¯_φ,−-mod. The graded k-algebras ^wA¯^ν_φ,+, ^wA¯φ,+

(= ^v_A¯_ν,−, ^v_A¯_φ,−) are Koszul. The natural graded indecomposable projective modules

are of the form^v_A¯_ν,−1x,^v_A¯_φ,−1xwithx∈^vI_ν,−,^vI_φ,− respectively. We must check that T¯ν,φ(^v_A¯_ν,−1x) =^v_A¯_φ,−1xwν for eachx∈^wI_ν,−.

By definition of ¯Tφ,νwe have an isomorphism ¯θφ,ν◦V¯kV¯k◦T¯φ,νof functors on^vO¯^proj_φ,−. Further, by definition of ¯Vk, we have ¯Vk(^v_A¯_ν,−1x) = ^vB¯ν,−(x) and ¯Vk(^v_A¯_φ,−1xwν) =

vB¯φ,−(xwν). Therefore, it is enough to check that ¯θν,φ(^vB¯ν,−(x)) =^vB¯φ,−(xwν).

ByRemark 4.42, this follows from Proposition 4.41(f) by base change. 2

Remark 5.11.General facts imply that the functor ¯T_φ,νhas a left adjoint ¯T_ν,φ:^vO¯_ν,− →

vO¯_φ,−, see the proof ofProposition 4.36. By definition ofT_ν,φand the unicity of the left adjoint, the functor ¯Tν,φcoincides withTν,φwhen forgetting the grading.

Now, we prove the following lemma which is dual toLemma 5.9.

Lemma 5.12.Assume that v ∈I_ν,− and d+N > f. Then, the functor T_φ,ν : ^vO_φ,− →

vO_ν,− induces a surjective graded k-algebra homomorphism T_φ,ν : ^vA¯^μ_φ,− → ^vA¯^μ_ν,−. Its kernel contains the two-sided ideal generated by{1_x; x∈^vI^μ_φ,−, x /∈I_ν,+}. Further, for x∈^vI^μ_φ,−∩Iν,+ we have xwν ∈^vI^μ_ν,− andTφ,ν(1x) = 1xwν.

Proof. First, note that, sincev∈Iν,−andd+N > f, the functorTφ,ν:^vOφ,− →^vOν,−

is well-defined and it takes^vO^μ_φ,− into^vO^μ_ν,− byProposition 4.36(c).

We’ll abbreviate L_φ = ^vL^μ_φ,− and L_ν = ^vL^μ_ν,−. By Proposition 4.36(f), we have T_φ,ν(L_φ) =L_ν. Thus, sinceT_φ,ν is exact, it induces a graded k-algebra homomorphism T_φ,ν:^vA¯^μ_φ,−= ExtvO^μ_φ,−(L_φ)^op→^vA¯^μ_ν,−= ExtvO^μ_ν,−(L_ν)^op.

ComposingT_φ,νwith its left adjointT_ν,φ, we get the functorΘ=T_ν,φ◦T_φ,ν. To prove that Tφ,ν is surjective, we must prove that the counit Θ → 1 yields a surjective map Ext^v_O^μ_φ,−(Lφ)→Ext^v_O^μ_φ,−(Θ(Lφ), Lφ).

The parabolic inclusion^vO^μ_φ,−⊂^vOφ,−is injective on extensions by Section3.5. So we must prove that the counit yields a surjective map Ext^vO_φ,−(Lφ)→Ext^vO_φ,−(Θ(Lφ), Lφ).

Let us consider the graded analogue of this statement. Set ¯Θ = ¯Tν,φ◦T¯φ,ν, where T¯ν,φ, ¯Tφ,ν are as inLemma 5.10 andRemark 5.11. We have

Ext^vO_φ,−(Lφ) =

j

ExtvO¯φ,−

L¯φ,L¯φj , Ext^vO_φ,−

Θ(Lφ), Lφ

=

j

ExtvO¯φ,−

Θ( ¯¯ Lφ),L¯φj .

Thus we must prove that for eachi, j the counitη: ¯Θ→1yields a surjective map ExtⁱvO¯φ,−

L¯φ,L¯φj

→ExtⁱvO¯φ,−

Θ( ¯¯ Lφ),L¯φj

. (5.7)

ByLemma 5.10, we have ¯T_φ,ν( ¯L_φ) = ¯L_ν. Further, since ^vO¯_ν,− = ^wA¯^ν_φ,+-gmod and

vO¯_φ,−=^wA¯φ,+-gmod, the gradings on^vO¯_φ,− and^vO¯_ν,− are Koszul byProposition 5.4.

Hence, since the right hand side of(5.7)is ExtⁱvO¯ν,−( ¯Lν,L¯νj), it is zero unlessi=j.

Now, we define the integer = min{d; ¯Θ( ¯Lφ)^d = 0}. Recall that the grading on

wA¯φ,+ is positive. Further, since^vO¯φ,−=^wA¯φ,+-gmod, we can view ¯Θ( ¯Lφ) as a graded

wA¯φ,+-module. Thus ¯Θ( ¯Lφ) is a quotient of ¯Θ( ¯Lφ) which is killed by the radical of

wA¯φ,+. We deduce that ¯Θ( ¯Lφ)⊂top( ¯Θ( ¯Lφ)).

Next, we claim that for any simple graded^wA¯φ,+-module ¯Lsuch that ¯Tφ,ν( ¯L)= 0, the map η( ¯L) : ¯Θ( ¯L)→L¯ yields an isomorphism top( ¯Θ( ¯L))→L. Indeed,¯ η( ¯L) is surjective because it is non-zero, and for any simple quotient ¯Θ( ¯L)→L¯ we have

0= HomvO¯φ,−

Θ( ¯¯ L),L¯

= HomvO¯ν

T¯_φ,ν( ¯L),T¯_φ,νL¯ .

ByProposition 4.36(d), (e) this implies that ¯T_φ,ν( ¯L) = ¯T_φ,ν( ¯L), and that it is non-zero.

Therefore, we have ¯L= ¯L, proving the claim.

Recall that ¯Lφ is a semi-simple module, see Section 3.4. Applying the claim to the simple summands ¯L⊂L¯φsuch that ¯Tφ,ν( ¯L)= 0, we get that top( ¯Θ( ¯Lφ)) = Imη( ¯Lφ). In particular top( ¯Θ( ¯Lφ)) is pure of degree zero. This implies that we have= 0. Therefore, the kernel of η( ¯Lφ) lives in degrees>0. Hence, by Koszulity of the grading of ^vO¯φ,−, we get

ExtⁱvO¯_φ,−

Kerη( ¯L_φ),L¯_φi

= 0.

Hence the surjectivity of (5.7) for i =j follows from the long exact sequence of Ext’s groups associated with the exact sequence

0→Ker η( ¯Lφ)

→Θ( ¯¯ Lφ)→L¯φ.

This proves that the map Tφ,ν is surjective, proving the first part of the lemma.

Next, we have I_ν,^μ₋ ={xw_ν; x∈I_φ,−^μ ∩I_ν,+}by Corollary 3.3(a). Further, we have T_φ,ν(1_xwν) = 0 forx /∈I_ν,−byProposition 4.36(e), andxw_ν∈I_ν,+if and only ifx∈I_ν,−. This proves the second claim of the lemma. Finally, the last claim of the lemma follows from Proposition 4.36(d). 2

Next, we prove the following.

Lemma 5.13. Assume thatw∈I_μ,+^ν . Letv =w⁻¹₋ . We have v∈I_ν,−^μ . Assume also that d+N > f and e+N > d. There is a k-algebra isomorphism p_μ,ν : ^wA^ν_μ,+ → ^vA¯^μ_ν,−

such that p_μ,ν(1_x) = 1_y for eachx∈^wI^ν_μ,+, where y=x⁻¹₋ , and such that the following square is commutative

wνwAμ,+

(5.1) τφ,ν

vA¯^μ_φ,−

Tφ,ν

wA^ν_μ,+ ^{pμ,ν v}A¯^μ_ν,−.

(5.8)

Proof. We havev=wμw⁻¹wν. Note thatwνw∈I_φ,−^ν ∩Iμ,+, because byLemma 3.2we havew⁻¹∈I_ν,+^μ , hencew⁻¹wν ∈I_φ,+^μ ∩Iν,− andwνw= (w⁻¹wν)⁻¹∈Iμ,+∩I_φ,^ν₋.

Let π_μ,ν : ^wνwA_μ,+ → ^vA¯^μ_ν,− be the composition of the k-algebra homomorphism Tφ,ν:^vA¯^μ_φ,−→^vA¯^μ_ν,−inLemma 5.12and of thek-algebra isomorphism^wνwAμ,+=^vA¯^μ_φ,− in(5.1).

Note that ^wνwA^ν_μ,+ = ^wA^ν_μ,+. We must construct a k-algebra isomorphism pμ,ν :

wA^ν_μ,+→^vA¯^μ_ν,− such thatπμ,ν=pμ,ν◦τφ,ν.

Letx∈^wνwI_μ,+. Thusw_μx⁻¹∈^vI^μ_φ,−. ByLemma 5.12, we have

π_μ,ν(1_x)= 0 ⇔ T_φ,ν(1_wμx−1)= 0 ⇔ w_μx⁻¹∈I_φ,^μ₋∩I_ν,+. ByLemma 5.9, we have

τ_φ,ν(1_x)= 0 ⇔ x∈I_μ,+^ν . Again byLemma 3.2, we have

wμx⁻¹∈I_φ,−^μ ∩Iν,+ ⇔ xwμ∈I_φ,+^ν ∩Iμ,− ⇔ x∈I_μ,+^ν .

Hence, we have τ_φ,ν(1_x) = 0 if and only if π_μ,ν(1_x) = 0. Thus, we have Ker(τ_φ,ν) ⊂ Ker(πμ,ν), because the left hand side is generated by the 1x’s killed byτφ,ν and the right hand side contains the 1x’s killed byTφ,ν.

This proves the existence of ak-algebra homomorphismp_μ,νsuch thatπ_μ,ν =p_μ,ν◦τ_φ,ν andpμ,ν(1x) = 1_wμx−1 for eachx∈^wI^ν_μ,+. The mappμ,ν is surjective, because the map T_φ,ν is surjective byLemma 5.12.

Now, we prove thatpμ,ν is injective. By Section3.5, the parabolic inclusion functor i_μ,φ :^vO^μ_ν,− → ^vO_ν,− yields a gradedk-algebra homomorphismi_μ,φ : ^vA¯^μ_ν,− → ^vA¯_ν,−. Setz=v⁻¹=wνwwμ. The following is proved below.

Claim 5.14.We have the commutative diagram

wνwA_μ,+

(5.1) Tμ,φ

vA¯^μ_φ,−

Tφ,ν

iμ,φ

vA¯^μ_ν,−

iμ,φ

zAφ,+

(5.1)

vA¯φ,− Tφ,ν

vA¯ν,−.

(5.9)

Note thatz∈Iμ,−, hence the mapTμ,φ is well-defined.

Now, by Proposition 4.36(g), the translation functor T_μ,φ yields a map ^wA^ν_μ,+ →

wwμA^ν_φ,+. Consider the diagram

wνwAμ,+ τφ,ν

Tμ,φ

wA^ν_{μ,+ p}

μ,ν

Tμ,φ

vA¯^μ_ν,−

iμ,φ

zAφ,+

τφ,ν

wwμA^ν_φ,+ ^{pφ,ν v}A¯_ν,−.

(5.10)

ByClaim 5.14, the outer rectangle in(5.9)is commutative. Thus, the outer rectangle in(5.10)is commutative. The left square in(5.10)is commutative, byProposition 4.36(c).

Thus, sinceτφ,ν is surjective, the right square in (5.10)is also commutative.

The middle vertical map in (5.10) is injective by Remark 4.38. Therefore, to prove thatp_μ,ν is injective it is enough to check thatp_φ,ν is injective.

Now, it is easy to see that the map pφ,ν is indeed invertible, because it is surjective by the discussion above (applied to the choice ν=φ) and dim(^wwμAφ,+ν

) = dim(^vA¯ν,−) byProposition 5.4.

Finally, to finish the proof ofLemma 5.13we must check thatp_μ,ν(1_x) = 1_y for each x∈^wI^ν_μ,+, wherey=x⁻¹₋ .

To do that, it suffices to observe that x₋=wνxwμ, and that, byProposition 5.1and Lemmas 5.9, 5.12, the square of maps in(5.8)gives the following diagram

1_x ^(5.1)

τφ,ν

1_wμx−1

Tφ,ν

1_x ^pμ,ν 1_wμx−1wν.

Now, we prove Claim 5.14. The right square in (5.9) is commutative, by Proposi-tion 4.36(c).

Let us concentrate on the left square. We must prove that the isomorphisms^wνwAμ,+=

vA¯^μ_φ,− and^wAφ,+=^vA¯φ,− inProposition 5.1yield a commutative square

wνwAμ,+

(5.1) Tμ,φ

vA¯^μ_φ,−

i zAφ,+

(5.1)

vA¯_φ,−.

(5.11)

By Proposition 4.36(g), the module T_μ,φ(^wνwP_μ,+) is a direct summand of ^zP_φ,+. By Proposition 4.41(f) and Remark 4.42, the sheaf θμ,φ(^wBμ,+) is a direct summand of

wBφ,+. Thus, we have the following diagram

wνwAμ,+

Vk

Tμ,φ

End^{wν w}Zμ,+(^wνwBμ,+)^op

θμ,φ

zA_φ,+ ^Vk End^z_Zφ,+(^zB_φ,+)^op.

Note that the horizontal maps are invertible byCorollary 4.50. This diagram is commu-tative by Remark 4.42, see also Proposition 4.41(a). Next, byProposition 4.48(c) and Corollary 4.49, we have a commutative diagram

Endwν wZμ,+(^wνwBμ,+)^op

θμ,φ

vA¯^μ_φ,−

H

i

End^zZφ,+(^zBφ,+)^{op H v}A¯φ,−.

Finally, the horizontal maps in(5.11)are equal to the composition ofHandVk. 2 Finally, we prove our main theorem.

Proof of Theorem 3.12. Since the highest weight categories^wO^ν_μ,+,^vO^μ_ν,−do not depend on e, f by Remark 3.10, we can assume that there is a positive integer d such that d+N > f ande+N > d. Thus the hypothesis ofLemma 5.13is satisfied.

By Propositions 5.1, 5.4 the k-algebra ^wA_μ,± has a Koszul grading. Thus, by Lemma 2.2and Section3.5, thek-algebra^wA^ν_μ,± has also a Koszul grading.

Let us equip ^wA^ν_μ,± with this grading. By Lemma 2.1, we have ^wA^ν,!_μ,± = ^wA¯^ν_μ,± as gradedk-algebras. Therefore, the gradedk-algebra^wA¯^ν_μ,± is Koszul and its Koszul dual is isomorphic to^wA^ν_μ,± as ak-algebra.

We have^wA¯^ν,!_μ,+ =^wA^ν_μ,+ as a k-algebra. By Lemma 5.13, we have also a k-algebra isomorphism^wA^ν_μ,+ =^vA¯^μ_ν,−. Thus, we have^wA¯^ν,!_μ,+ =^vA¯^μ_ν,− as k-algebras. By unicity of the Koszul grading, we deduce that^wA¯^ν,!_μ,+=^vA¯^μ_ν,− as gradedk-algebras.

The involutivity of the Koszul duality implies that we have also ^wA¯^ν,!_μ,− = ^vA¯^μ_ν,+ as gradedk-algebras, and^wA^ν_μ,− =^vA¯^μ_ν,+ ask-algebras.

Finally, we must check that under the isomorphism^wA¯^ν,!_μ,+=^vA¯^μ_ν,− we have 1^!_x= 1y

withy=x⁻¹₋ for eachx∈^wI^ν_μ,+.

Ifν=φthis isProposition 5.1. Ifμ=φthis is Proposition 5.4.

The isomorphism^wA¯^ν,!_μ,+ =^wA^ν_μ,+ above, which is given byLemma 2.1, identifies the idempotents 1^!_x and 1x for each x ∈ ^wI^ν_μ,+. Thus, by Lemma 5.13, the isomorphism

wA¯^ν,!_μ,+=^vA¯^μ_ν,− identifies the idempotents 1^!_x and 1y, wherey=x⁻₋¹. 2 6. Type A and applications to CRDAHA

Fix integerse, , N >0. Letg=gl(N).

6.1. Koszul duality in the type A case

Let b,t⊂ gbe the Borel subalgebra of upper triangular matrices and the maximal torus of diagonal matrices. Let (i) be the canonical basis ofC^N. We identifyt^∗ =C^N, t=C^N andW =S_N in the obvious way. Putρ= (0,−1, . . . ,1−N) andα_i=_i−_i+1 for eachi∈[1, N).

We define the affine Lie algebra gofgas in Section3.2. For any subsetX ⊂C and

We define the affine Lie algebra gofgas in Section3.2. For any subsetX ⊂C and