The level-rank duality

In this section, we’ll relate the map K above to the level-rank duality. Consider the Lie algebrasl(e). We identify the set of weights ofsl(e) withC^e/C1^e. Thus, elements of C^e can be viewed as weights ofsl(e), or equivalently, as level 0 classical affine weights of the affine Kac–Moody algebra sl(e). Let {εi; i ∈ [1, e]} be the canonical basis of C^e. For each λ ∈ P and each k ∈ [1, N], we decompose the k-th entry of λ+ρ as the sum λk +ρk = ik +e rk with ik ∈ [1, e] and rk ∈ Z. Then, we can view the sum wt_e(λ) = N

k=1ε_ik + (N

k=1r_k)δ as a level 0 affine weight of sl(e), and the sum wt_e(λ) =N

k=1ε_ik as a weight of sl(e).

The vector spaces Ve =C^e⊗C[z⁻¹, z] carries an obvious level zero action of sl(e).

It induces an action of sl(e) on the space ν

(Ve) = p=1

νp

(Ve) of tensor

prod-uct of wedge powers. Write vi+er = εi ⊗z^r ∈ Ve for each i ∈ [1, e], r ∈ Z. Note

μfor thesl(e)-action is isomorphic to thesl()-module μ

(V), and the weight subspace of weight ¯ν for thesl()-action is isomorphic to the sl(e)-module ν

(V_e). (1,1) to (, ν) in lexicographic order. Next, there is an obvious isomorphism of sl(e) ×sl()-modules τ : N

(Ve,) → N

(V,e) which exchanges the vector spaces C^e and C. Then, we define LR = (fν,μ)⁻¹ ◦τ ◦fμ,ν. We’ll call LR the level-rank duality.

Our next result compares the isomorphisms LR and K. To do so, we consider the C-linear isomorphism θ : [O^ν_−e]→ ν

Proof. Let trivμ, sgn_μ be the idempotents of the C-algebra of the parabolic subgroup Wμ⊂Wassociated with the trivial representation and the signature.

For each tuple a ∈ Z^N we write v(a) = N

k=1v_ak. Let (V_e)^⊗N_μ be the subspace of (Ve)^⊗N of weight ¯μrelatively to thesl(e)-action. The elementv(a) of (Ve)^⊗N belongs to

(Ve)^⊗_μ^N if and only ifa∈W·e1μ. Therefore, the assignmentx·trivμ →v(x·e1μ) with x∈W yields aC-linear isomorphism CW·trivμ →(Ve)^⊗N_μ .

Since x • oμ,− = (x·e1μ−ρ)e, we have x∈I_μ,−^ν if and only if the N-tuple x·e1μ

lies inP^ν+ρ. We deduce that the isomorphism above factors to aC-linear isomorphism Bν,μ : sgn_ν·CW ·trivμ →ν

(Ve)μ such that sgn_ν·x·trivμ → ∧^ν(x·e1μ). Under this isomorphism, the basis {∧^ν(a); a∈(P^ν+ρ)∩(W ·e1_μ)}ofν

(V_e)_μ is identified with {sgn_ν·x·triv_μ; x∈I_μ,^ν₋}.

ByCorollary 3.3, we have a bijectionI_μ,^ν₋→I_ν,^μ₋ such thatx→x⁻¹. We claim that, under the isomorphismsB_ν,μ andB_μ,ν, the mapLRis identified with theC-linear map sgn_ν·CW·triv_μ →sgn_μ·CW·triv_ν such that sgn_ν·x·triv_μ →sgn_μ·x⁻¹·triv_ν. Since, by Remark 3.13, the isomorphismK takes the element [V^ν(x•o_μ,−)] to [V^μ(x⁻¹•o_ν,−)], this finishes the proof of the proposition.

The claim is a direct consequence of the definition of the map LR. 2 6.3. The CRDAHA

Let Γ ⊂ C^× be the group of the-th roots of 1. Fixν ∈Z(N) and fix an integer d >0.

Let Γd =SdΓ^d. It is a complex reflection group. LetPd be the set of partitions of d. Write |λ| = d and let l(λ) be the length of λ. Let P_d be the set of -partitions of d, i.e., the set of-tuplesλ= (λp) of partitions with

p|λp|=d. There is a bijection between the set of irreducible representations ofΓd andP_d, see e.g.[35, sec. 6].

We seth= 1/eandhp=νp+1/e−p/for eachp∈Z/Z. LetH^ν(d) be the CRDAHA of Γd with parameters hand (hp). It is the quotient of the smash product of CΓd and the tensor algebra of (C²)^⊕d by the relations

[yi, xi] = 1−k

j=i

γ∈Γ

s^γ_ij−

γ∈Γ\{1}

cγγi,

[yi, xj] =k

γ∈Γ

γs^γ_ij ifi=j, [xi, xj] = [yi, yj] = 0.

The parameters are such thatk=−hand−cγ =₋₁

p=0γ^−p(hp−hp−1) forγ= 1.

Let O^ν_−e{d}be the category O of H^ν(d). It is a highest weight category with set of standard modules Δ(O^ν_−e{d}) ={Δ(λ); λ∈P_d}, see[38, secs. 3.3, 3.6],[40, sec. 3.6].

To avoid confusions we may write Δ^ν_e(λ) = Δ(λ). LetS_e^ν(λ) be the top of Δ^ν_e(λ).

We have the block decomposition O^ν₋e{d} =

μO^νμ{d}, where μ = (μ₁, . . . , μ_e) is identified with ¯μ=e

i=1μ_iε_i and it runs over the set of all integral weights ofsl(e). By [28], these blocks are determined by the following combinatorial rule.

For each λ∈P_d, an (i, ν)-node inλis a triple (x, y, p) with x, y >0 andy (λp)x

such that y−x+νp =i moduloe. Let n^ν_i(λ) be the number of (i, ν)-nodes inλ. Then, we have Δ^ν_e(λ)∈ O^νμ{d}if and only ifλ∈Λ^ν_μ{d}, where

Λ^ν_μ{d}=

λ∈P_d; p=1

ωνp−

e−1

i=1

n^ν_i(λ)−n^ν₀(λ) αi= ¯μ

. (6.1)

The expression above should be regarded as an equality of integral weights of sl(e).

More precisely, the symbols ω1, ω2, . . . , ωe−1 and α1, α2, . . . , αe−1 are respectively the fundamental weights and the simple roots ofsl(e), and the subscript ν_p in(6.1)should be viewed as the residue class ofν_p in [1, e). See[38, lem. 5.16]for details.

By[36, rem. 7.2], the condition(6.1)is equivalent to Λ^ν_μ{d}=

λ∈Pd; wt_e(λ+ρ_ν−ρ) = ¯μ .

Note that an integral weight ofsl(e) can be represented by an element ofZ^e(k) if and only if it lies inωk+eZΠ. Thus, sinceν∈Z(N), ifΛ^ν_μ{d} =∅thenμcan be represented by ane-tuple inZ^e(N). Indeed, it is not difficult to check that ifΛ^ν_μ{d} =∅thenμ∈C_N^e. For each μ ∈ CN^e and a ∈ N, we write O^νμ[a] = Oμ^ν{d}and Λ^ν_μ[a] = Λ^ν_μ{d}, where d=a e+

p=1ω_νp−μ:ρ. For eachλ∈P, we haveλ∈Λ^ν_μ[a] if and only ifn^ν₀(λ) =a and wt_e(λ+ρ_ν−ρ) = ¯μ.

We are interested by the following conjecture[9, conj. 6].

Conjecture 6.2.The blocksO^νμ[a]andO^μν[a]have a (standard) Koszul grading. The Koszul dual ofO^νμ[a] is equivalent to the Ringel dual ofOν^μ[a].

6.4. The Schur category

Fix a compositionν ∈CN. Let _Pd^ν ={λ∈P_d; l(λ_p)ν_p}. There is an inclusion P_d^ν ⊂P^ν ⊂Z^N such that

λ→

λ10^ν1−l(λ1), λ20^ν2−l(λ2), . . . , λ0^ν−l(λ) , where the partitionλp is viewed as an element in Z^l(λp).

Consider theν-dominant weightρ_ν = (ν₁, ν₁−1, . . . ,1, ν₂, ν₂−1, . . . , ν, . . .1)∈P^ν. For eachλ∈P^ν, we abbreviateV_e^ν(λ) =V^ν((λ+ρ_ν−ρ)_e) andL^ν_e(λ) =L^ν((λ+ρ_ν−ρ)_e).

Following [40], let A^ν_−e{d} ⊂ O^ν_−e be the Serre subcategory consisting of the finite lengthg-modules of level−e−N whose constituents belong to the set{L^ν_e(λ); λ∈P_d^ν}. By[40], it is a highest weight category with the set of standard modules Δ(A^ν_−e{d}) = {V_e^ν(λ); λ∈P_d^ν}.

We have the following conjecture[40, conj. 8.8].

Conjecture 6.3.There is a quotient functor O_−e^ν {d} → A^ν_−e{d}taking S_e^ν(λ) to L^ν_e(λ) if λ∈P_d^ν and to 0 else. If νp dfor each p, then we haveP_d^ν =Pd and the functor above is an equivalence of highest weight categories.

A proof ofConjecture 6.3 is given in[36].

6.5. Koszul duality of the Schur category

In view of Conjecture 6.3, the following claim can be regarded as an analogue of Conjecture 6.2.

Theorem 6.4. The category A^ν_μ[a] has a (standard) Koszul grading. The Koszul dual of A^ν_μ[a]is equivalent to the Ringel dual of A^μ_ν[a].

Proof. Recall theC-linear mapK: [O^ν_μ,−e]→[O^μ_ν,−]. First, we claim thatK([A^ν_μ,−e]) = [A^μ_ν,−]. ByProposition 6.1 and the definition of the Schur category, we must compute the element LR(∧^ν(a)), for eacha=λ+ρ_ν such that λ∈P^ν and∧^ν(a)∈ν

(V_e)_μ. First, we consider theC-vector spaceLR(ν

(V_e)_μ). The mapf_μ,νtakes the monomial

Now, to finish the proof of the theorem, it is enough to observe thatθ([A^ν_−e[a]]) is the sum, over all affine weights ˆμ∈P+aδ, of the intersection ofθ([A^ν_−e]) with the subspace of ν

(Ve) of weight ˆμfor thesl(e)-action. 2 The following is obvious.

Corollary 6.5.Conjecture 6.3 impliesConjecture 6.2. 2 6.6. Koszulity of the q-Schur algebra

Now, we consider the case = 1.

By the Kazhdan–Lusztig equivalence[27, thm. 38.1], the categoryA^(N)_−e{d}is equiv-alent to the module category of theq-Schur algebra. This equivalence is an equivalence of highest-weight categories. This follows from the fact that an equivalence of abelian

categoriesC1→C2 such thatC1,C2are of highest weight and that the induced bijec-tion Irr(C1)→Irr(C2) is increasing is an equivalence of highest weight categories, i.e., it induces a bijection Δ(C₁)→Δ(C₂). See also[40, thm. A.5.1]for a more detailed proof.

Therefore,Theorem 6.4implies the following.

Corollary 6.6. The q-Schur algebra is Morita equivalent to a Koszul algebra which is balanced. 2

See also[9]for a different approach to this result. Note that our proof gives an explicit description of the Koszul dual of theq-Schur algebra in term of affine Lie algebras.

Appendix A. Finite codimensional affine Schubert varieties

By a scheme we always mean a scheme over the fieldk, and by a variety we’ll mean a reduced scheme of finite type which is quasi-projective. LetT be a torus. AT-scheme is a scheme with an algebraicT-action.

Fix a contractible topologicalT-spaceET with a topologically freeT-action. For any T-scheme X, we set XT =X ×T ET. There are obvious projections p:X ×ET →X andq:X×ET →X_T.

We’ll identifyS with theT-equivariant cohomologyk-spaceHT(•) of a point.

A.1. Equivariant perverse sheaves on finite dimensional varieties

Fix a T-variety X. Let D^b_T(X) be the T-equivariant bounded derived category. It is the full subcategory of the bounded derived categoryD^b(X_T) of sheaves ofk-vector spaces on XT (for the analytic topology on X), consisting of the complexes of sheaves F with an isomorphismq^∗F p^∗FX for someFX ∈D^b(X).

The cohomology of an object F ∈ D^b_T(X) is the graded S-module H(F) =

i∈ZHom_D^b

T(X)(kX,F[i]). For each E,F ∈ D^b_T(X) we have an object RHom(E,F) inD^b(X_T) such that Ext_D^b

T(X)(E,F) =H(RHom(E,F)).

IfY ⊂X is aT-equivariant embedding, let ¯Y be its closure inX. LetIC_T(Y) be the minimal extension ofkY[dimY] (the shifted equivariant constant sheaf). It is a perverse sheaf onX supported on ¯Y.

LetIH_T( ¯Y) be thek-space of equivariant intersection cohomology of ¯Y and letH_T( ¯Y) be its equivariant cohomology. We haveIHT( ¯Y) =H(ICT( ¯Y)) and HT( ¯Y) =H(kY¯).

Forgetting theT-action we define IC( ¯Y),IH( ¯Y) andH( ¯Y) in the same way.

We’ll say thatX isgood if the following hold

• X has a Whitney stratificationX =

xXxbyT-stable subvarieties,

• X_x=A^l(x) with a linearT-action, for some integerl(x)∈N,

• there are integersnx,y,i∈Nsuch that

j_y^∗ICT( ¯Xx) =

wherej_x is the inclusionX_x⊂X. Equivalently, we have j_y^!IC_T( ¯X_x) =

We call the third property theparity vanishing.

Proposition A.1.If X is a goodT-variety, then the following hold (a) dim Extⁱ_Db(X)(IC( ¯Xx), IC( ¯Xy)) =

Proof. Part (a) is [5, thm. 3.4.1]. We sketch briefly the proof for the comfort of the reader. Set X_p =

wherezruns over the set of elements withl(z) =p. By(A.1),(A.2)the spectral sequence degenerates at E1, and we get

Therefore, since S vanishes in odd degrees, the spectral sequence degenerates at E₂. Thus, we have

The proof of (d) is as in[18, sec. 1],[6, sec. 3.3]. Since our setting is slightly different we sketch briefly the main arguments. Fix a partial order on the set of strata such that X¯x = Xx =

yxXy. Consider the obvious inclusions jx : Xx → X, i = j<x : X<x → Xx and j = jx : Xx → Xx. For each y x, we setF1 =j^∗_xIC( ¯Xx) and F2=j^∗_xIC( ¯Xy). Fors= 1,2 andp∈N, there are integersdsandds,y,p such that

j_y^∗Fs=

p

k_Xy[d_s−2p]^⊕ds,y,p, j_y^!Fs=

q

k_Xy

2l(y)−d_s+ 2q⊕ds,y,q

. (A.3)

Consider the diagram of gradedk-vector spaces given by

Ext_D^b_(X<x)(i^∗F1, i^!F2) Hom_H(X<x)(H(i^∗F1), H(i^!F2))

a

Ext_Db(X_x)(F1,F2) Hom_H(Xx)(H(F1), H(F2))

b

Ext_D^b_(Xx)(j^∗F1, j^∗F2) HomH(Xx)(H(j^∗F1), H(j^∗F2)).

In the right side sequence, the Hom’s are thek-spaces of graded module homomorphisms over the gradedk-algebras H(X_<x),H(X_x),H(X_x) respectively, which are computed in the category of non-graded modules. The horizontal maps are given by taking hyper-cohomology.

We’ll prove that the middle map is invertible by induction onx. This proves part (d).

The short exact sequence associated with the left side is exact by (A.3), see e.g., [16, lem. 5.3]and the references there. The lower map is obviously invertible, because j^∗Fs

are constant sheaves on Xx for each s = 1,2. The complexes i^∗Fs and i^!Fs on X<x

satisfy again(A.3)for any stratumXy⊂X<x. Thus, by induction, we may assume that the upper map is invertible. Then, to prove that the middle map is invertible it is enough to check thatais injective and that Im(a) = Ker(b).

By(A.3), the distinguished trianglesi_∗i^!→1→j_∗j^∗−−→⁺¹ and j!j^∗→1→i_∗i^∗ −−→⁺¹ yield exact sequences ofH(X_x)-modules

0→H i^!Fs

−−→αs H(Fs)−−→^βs H j^∗Fs

→0, 0→H_c

j^∗Fs

γs

−−→H(Fs)−−→^δs H i^∗Fs

→0,

where H_c(j^∗Fs) =H(j_!j^∗Fs) is the cohomology with compact supports ofj^∗Fs. Thus, the map a is injective, because a(φ) = α₂◦φ◦δ₁ for each φ, α₂ is injective andδ₁ is surjective.

Next, fix an element φ∈Ker(b). Sinceb(φ)◦β1 =β2◦φ by construction and β1 is surjective, we have β2◦φ = 0. Hence, there is an H(X_x)-linear map φ : H(F1) → H(i^!F2) such thatφ=α2◦φ. We must check thatφ◦γ1= 0.

To do that, set d= dimXx. Since Xx k^d, the fundamental classω of Xx belongs to H_c^2d(Xx). The cup product with ω restricts to 0 onX<x, because there is an exact sequence 0→Hc(Xx)→H(Xx)→H(X<x)→0. Thus, it yields a mapgs:H(Fs)→ Hc(j^∗Fs)[2d] which vanishes on Im(αs). We deduce that g2◦φ = 0. Since Hc(Xx) ⊂ Hc(Xx) and φ is Hc(Xx)-linear, we have also φ◦g1 = 0. Since g1 is surjective, we deduce thatφ◦γ1= 0. Hence, we haveφ◦γ1= 0, becauseα2 is injective.

Part (c) is proved as part (d), using equivariant cohomology.

Finally, we prove (e). By parity vanishing, the spectral sequence E₁^p,q=H^p+q

j_p^!IC( ¯Xx)

⇒ IH^p+q( ¯Xx)

degenerates at E₁. This yields the first claim, as in part (a). The second one follows from the first one, because the spectral sequence E^p,q₂ =S^p⊗IH^q( ¯X_x)⇒IH^p+q( ¯X_x) degenerates at E₂. 2

A.2. Equivariant perverse sheaves on infinite dimensional varieties

This section is a reminder on equivariant perverse sheaves on infinite dimensional varieties, to be used in the next section.

LetX be anessentially smoothT-scheme, in the sense of[23, sec. 1.6]. In[23, sec. 2]

the derived category of constructible complexes on X and perverse sheaves on X (for the analytic topology) are defined. Note that the convention for perverse sheaves here differs from the convention for perverse sheaves on varieties (as in Section A.1): indeed k_Y[−codimY] is perverse if Y is an essentially smooth T-scheme, while k_Y[dimY] is perverse for a smooth variety Y. We’ll use the same terminology as in [23, sec. 2]and we formulate our results in the T-equivariant setting. The equivariant version of the constructions in [23]is left to the reader.

LetDT(X) be theT-equivariant derived category onX. IfX, Y are essentially smooth and Y → X is a T-equivariant embedding of finite presentation, let ICT(Y) be the minimal extension of the equivariant constant shifted sheaf kY[−codimY] on Y. It is a perverse sheaf on X supported on the Zariski closure ¯Y in X. If F ∈ DT(X), we set H(F) =

i∈ZHom_DT(X)(kX,F[i]), a gradedS-module. We abbreviateIHT( ¯Y) = H(IC_T( ¯Y)) and H_T(Y) =H(k_Y). We call H_T(Y) theequivariant cohomology of Y. It is anH_T(X)-module.

Recall that for a morphism f : Z → X of essentially smooth T-schemes there is a functor f^∗ : D_T(X) → D_T(Z), see [23, sec. 3.7]. If f is the inclusion of a T-stable subscheme we writeFZ=i^∗F andIH_T(Y)_Z =H(IC_T(Y)_Z).

Now, assume thatX is the limit of a projective system of smooth schemes

X = lim_←−nX_n (A.4)

as in[23, sec. 1.3]. Letpn:X→Xn be the projection and setYn=pn(Y). Assume that Yn is locally closed in Xn. Since kY =p^∗_nkYn we have an obvious map

HT(Yn) =

A.3. Moment graphs and the Kashiwara flag manifold

In this section we consider the localization on Kashiwara’s flag manifold. The main motivation is Corollary A.6 below that we used in the rest of text, in particular in Corollary 4.50to prove that the functorVk is fully-faithful on projectives in^wO^Δ_μ,−.

LetP_μbe the parabolic subgroup corresponding to the Lie algebrap_μ. LetX =X_μ= G/Pμbe the Kashiwara partial flag manifold associated withgandpμ, see[22]. HereGis the schematic analogue ofG(k((t))) defined in [22], which has a locally free right action of the group-k-scheme Pμ and a locally free left action of the group-k-scheme B⁻, the Borel subgroup opposite toB. Recall thatXis an essentially smooth, not quasi-compact, T-scheme, which is covered byT-stable quasi-compact open subsets isomorphic toA^∞= Speck[xk; k∈N].

Let eX = Pμ/Pμ be the origin of X. For each x ∈ Iμ,+, we set X^x = B⁻xeX = B⁻xP_μ/P_μ. Note thatX^x is a locally closedT-stable subscheme of X of codimension l(x) which is isomorphic toA^∞. Consider theT-stable subschemes

X^x=X^x=

We call X^x afinite-codimensional affine Schubert variety. We call X^x an admissible open set.

IfΩis an admissible open set, there are canonical isomorphisms ICT

We can view Ω as the limit of a projective system of smooth schemes (Ω_n) as in[23, lem. 4.4.3]. So, the projectionpn:Ω→Ωnis a good quotient by a congruence subgroup B_n⁻ of B⁻. Letn be large enough. Then, we have ICT(X^x∩Ω) =p^∗_nICT(X_Ω,n^x ) with

→

i∈Z

HomDT(Ω)

kΩ, p^∗_nICT

X_Ω,n^x [i]

→IHT

X^x∩Ω

which yields an isomorphism IH_T

X^x∩Ω

= lim_−→nIH_T X_Ω,n^x

.

ForΩ=X^wandxwwe abbreviateX^[x,w] =X^x∩X^wandXn^[x,w] =pn(X^[x,w]).

Since p_n is a good quotient by B_n⁻ and since X^x is B_n⁻-stable, we have an algebraic stratification X_n^w =

xwX_n^x, where X_n^x is an affine space whose Zariski closure is Xn^[x,w].

Lemma A.2. (a) TheT-variety X_n^w is smooth and good.

(b) It is covered by T-stable open affine subsets with an attractive fixed point. The fixed points subset is naturally identified with ^wIμ,+.

(c) There is a finite number of one-dimensional orbits. The closure of each of them is smooth. Two fixed points are joined by a one-dimensional orbit if and only if the corresponding points in ^wIμ,+ are joined by an edge in^wGμ.

Proof. The T-variety X_n^w is smooth by [23], because X^w is smooth and pn is a B_n⁻-torsor fornlarge enough. We claim that it is also quasi-projective.

LetX₀be the stack ofG-bundles onP¹. We may assume thatP_μis maximal parabolic.

Then, by the Drinfeld–Simpson theorem, ak-point of X is the same as ak-point ofX₀ with a trivialization of its pullback to Spec(k[[t]]). Heretis regarded as a local coordinate at∞ ∈P¹and we identifyB⁻with the Iwahori subgroup inG(k[[t]]). We may chooseB_n⁻ to be the kernel of the restrictionG(k[[t]])→G(k[t]/tⁿ). Then, ak-point ofX_n=X/B_n⁻ is the same as ak-point ofX0with a trivialization of its pullback to Spec(k[t]/tⁿ). We’ll prove that there is an increasing system of open subsets Um ⊂ X0 such that for each m and for n 0 the fiber product X_n×X0Um is representable by a quasi-projective variety. This implies our claim.

Choosing a faithful representationG⊂SLrwe can assume thatG=SLr. So ak-point of X₀ is the same as a rank r vector bundle on P¹ of degree 0. For an integerm > 0 let Um(k) be the set of V in X0(k) with H¹(P¹, V ⊗ O(m)) = 0 which are generated by global sections. It is the set of k-points of an open substack Um of X0. Note that Um⊂ Um+1 andX₀=

mUm. Now, the setYm(k) of pairs (V, b) whereV ∈ Um(k) and bis a basis ofH⁰(P¹, V⊗O(m)) is the set ofk-points of a quasi-projective varietyYmby the Grothendieck theory of Quot-schemes. Further, there is a canonicalGL_r(m+1)-action on Ym such that the morphismYm→ Um, (V, b)→V is a GL_r(m+1)-bundle. Now, for n 0 the fiber productX_n×X0 Um is representable by a quasi-projective variety, see e.g.,[41, thm. 5.0.14].

Next, note that X_n^w is recovered by the open subsets V_n^x = pn(V^x) with x w.

Each of them contains a unique fixed point under theT-action and finitely many one-dimensional orbits.

Finally, the parity vanishing holds: sinceIC(X^[x,w]) =p^∗_nIC(Xn^[x,w]) we have IC

X_n^[x,w]

Xn^y =

i

k_Xn^y

−l(y)

l(y)−l(x)−2i⊕Q^μ,−1x,y,i

by[26, thm. 1.3]. The change in the degrees with respect to Section A.1 is due to the change of convention for perverse sheaves mentioned above. 2

Now we setV =t^∗ and we consider the moment graph^wG^∨. Proposition A.3.We have

(a)HT(X^w) =^wZ¯^∨_S,μ,− andH(X^w) =^wZ¯^∨_μ,− as gradedk-algebras, (b)IH_T(X^[x,w]) =^wB¯^∨_S,μ,−(x)as a graded^wZ¯_S,μ,−-module,

(c)IH(X^[x,w]) =^wB¯^∨_μ,−(x)as a graded^wZ¯_μ,−^∨ -module.

Proof. Assuming n to be large enough we may assume that H_T(X^w) = H_T(X_n^w), IHT(X^[x,w]) =IHT(Xn^[x,w]), etc. ByLemma A.2theS-moduleHT(X_n^w) is free. Thus we can apply the localization theorem[19, thm. 6.3], which proves (a).

Now, we concentrate on (b). The gradedk-moduleIH(Xn^[x,w]) vanishes in odd degree by Proposition A.1 and Lemma A.2. Thus, applying [7] to Xn^[x,w], we get a graded

wZ¯_S,μ,−-module isomorphismIH_T^∗(Xn^[x,w]) =^wB¯^∨_S,μ,−(x).

Part (c) follows from (b),Proposition A.1andLemma A.2. 2 Corollary A.4. We have a gradedS-module isomorphism

wB¯^∨_S,μ,−(x)y=

i0

S

−l(x)−2i ^⊕Q

μ,−1 x,y,i

.

Proof. ApplyProposition A.3and[26, thm. 1.3(i)]. 2 Proposition A.5.For eachx, ywwe have

(a)

i∈ZtⁱdimkExtⁱ_DT(Xw)(IC_T(X^[x,w]), IC_T(X^[y,w])) =

zQ_μ(t)_x,zQ_μ(t)_y,z, (b)Ext_DT(Xw)(ICT(X^[x,w]), ICT(X^[y,w])) = Hom_HT(Xw)(IHT(X^[x,w]), IHT(X^[y,w])), (c)Ext_D(Xw)(IC(X^[x,w]), IC(X^[y,w])) = Hom_H(Xw)(IH(X^[x,w]), IH(X^[y,w])), (d)Ext_D(Xw)(IC(X^[x,w]), IC(X^[y,w])) =kExt_DT(Xw)(IC_T(X^[x,w]), IC_T(X^[y,w])).

Proof. ApplyProposition A.1andLemma A.2. 2 Finally, we obtain the following.

Corollary A.6. We have a graded k-algebra isomorphism ^w_A¯∨

μ,−= End^wZ^∨_μ(^wB¯^∨_μ,−)^op. Proof. ApplyPropositions A.3, A.5. 2

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