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 inwOΔμ,−.
LetPμbe the parabolic subgroup corresponding to the Lie algebrapμ. LetX =Xμ= 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 Xx = B−xeX = B−xPμ/Pμ. Note thatXx is a locally closedT-stable subscheme of X of codimension l(x) which is isomorphic toA∞. Consider theT-stable subschemes
Xx=Xx=
We call Xx afinite-codimensional affine Schubert variety. We call Xx 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 (Ωn) as in[23, lem. 4.4.3]. So, the projectionpn:Ω→Ωnis a good quotient by a congruence subgroup Bn− of B−. Letn be large enough. Then, we have ICT(Xx∩Ω) =p∗nICT(XΩ,nx ) with
→
i∈Z
HomDT(Ω)
kΩ, p∗nICT
XΩ,nx [i]
→IHT
Xx∩Ω
which yields an isomorphism IHT
Xx∩Ω
= lim−→nIHT XΩ,nx
.
ForΩ=Xwandxwwe abbreviateX[x,w] =Xx∩XwandXn[x,w] =pn(X[x,w]).
Since pn is a good quotient by Bn− and since Xx is Bn−-stable, we have an algebraic stratification Xnw =
xwXnx, where Xnx is an affine space whose Zariski closure is Xn[x,w].
Lemma A.2. (a) TheT-variety Xnw 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 inwGμ.
Proof. The T-variety Xnw is smooth by [23], because Xw is smooth and pn is a Bn−-torsor fornlarge enough. We claim that it is also quasi-projective.
LetX0be the stack ofG-bundles onP1. We may assume thatPμis 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 chooseBn− to be the kernel of the restrictionG(k[[t]])→G(k[t]/tn). Then, ak-point ofXn=X/Bn− 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 Xn×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 canonicalGLr(m+1)-action on Ym such that the morphismYm→ Um, (V, b)→V is a GLr(m+1)-bundle. Now, for n 0 the fiber productXn×X0 Um is representable by a quasi-projective variety, see e.g.,[41, thm. 5.0.14].
Next, note that Xnw is recovered by the open subsets Vnx = pn(Vx) 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
Xn[x,w]
Xny =
i
kXny
−l(y)
l(y)−l(x)−2i⊕Qμ,−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 graphwG∨. Proposition A.3.We have
(a)HT(Xw) =wZ¯∨S,μ,− andH(Xw) =wZ¯∨μ,− as gradedk-algebras, (b)IHT(X[x,w]) =wB¯∨S,μ,−(x)as a gradedwZ¯S,μ,−-module,
(c)IH(X[x,w]) =wB¯∨μ,−(x)as a gradedwZ¯μ,−∨ -module.
Proof. Assuming n to be large enough we may assume that HT(Xw) = HT(Xnw), IHT(X[x,w]) =IHT(Xn[x,w]), etc. ByLemma A.2theS-moduleHT(Xnw) 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 isomorphismIHT∗(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∈ZtidimkExtiDT(Xw)(ICT(X[x,w]), ICT(X[y,w])) =
zQμ(t)x,zQμ(t)y,z, (b)ExtDT(Xw)(ICT(X[x,w]), ICT(X[y,w])) = HomHT(Xw)(IHT(X[x,w]), IHT(X[y,w])), (c)ExtD(Xw)(IC(X[x,w]), IC(X[y,w])) = HomH(Xw)(IH(X[x,w]), IH(X[y,w])), (d)ExtD(Xw)(IC(X[x,w]), IC(X[y,w])) =kExtDT(Xw)(ICT(X[x,w]), ICT(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 wA¯∨
μ,−= EndwZ∨μ(wB¯∨μ,−)op. Proof. ApplyPropositions A.3, A.5. 2
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