The connected case

In view of Proposition 5.4.8 we can restrict ourselves to the case of a connectedw.

Lemma 5.4.9. Letwbe connected and letK = Ker(Sym²(H_w²)→H_w⁴). Then the elements X_C :=P

s,t∈CcstPsPt, with C closed, generate K.

Proof. We know that dimK = #{(s, t) ∈S² |st 6≤w}+ 1 because Sym²(H_w²) → H_w⁴ is surjective. Since w is connected, if st6≤w then s and t are connected by an edge in the Dynkin diagram andts≤w.

Let (a, b) be any pair of elements of S such that ba≤ w and ab 6≤ w, i.e. such that in I_w there is an arrow a → b but not an arrow b → a. We can define a proper closed subset C_ab by taking the connected component of b inI_w after erasing the arrow a→ b.

Since there are no loops inI_w we have a6∈Cab. It is easy to see thatX_Cab together with X =X_S are linearly independent in Sym²(H_w²): in fact when we write them in the basis {P_sP_t}_s,t∈S we haveX_Cab ∈c_bbP_b²+R_ab, where

R_ab= spanhP_sPt|(s, t)6= (a, a),(b, b)i, while all the otherX_C

a0b0 are either inR_ab or in c_aaP_a²+c_bbP_b²+R_ab. Therefore when we quotient toSym²(H_w²)/R_ab, the termX_Cab is the only one which is not proportional to the image ofc_aaP_a²+c_bbP_b².

By the formula for the dimension of K given above, it remains to show that all the X_C, for C closed, lie in K. Let y denote the projection of an element y ∈ Sym²(H_w²) to H⁴(G/B,C). Let C be a closed subset and let

E :={a(i)→ⁱ b(i)|a(i)6∈C andb(i)∈C}

be the set of arrows starting outsideC and ending in C. Applying Lemma 5.3.1, on one hand we obtain:

X_C = X

s,t∈C

cstPsPt∈spanhP_st |s, t∈Ci ⊕spanhP_a(i)b(i)|i∈Ei ⊆H⁴(G/B,C). (5.4) On the other hand we have

X − X_C = X

s,t6∈C

c_stP_sP_t+X

i∈E

2c_a(i)b(i)P_a(i)P_b(i) ∈Sym²(H_w²).

SinceX = 0 inH⁴(G/B,C), projecting from R⁴ to H⁴(G/B,C) we obtain

X_C ∈spanhP_st |s, t6∈Ci ⊕spanhP_a(i)b(i)|i∈Ei ⊕spanhP_b(i)a(i)|i∈Ei. (5.5) Then (5.4) together with (5.5) implies that the projectionX_C ofX_C toH⁴(G/B,C)lies inspanhP_a(i)b(i)|i∈Ei. But, for anyi∈E,P_a(i)b(i)projects to0inH_w⁴ sincea(i)b(i)6≤w, whenceX_C ∈K.

For a closed C let N S(C) :=spanhP_s |s∈Ci ⊆ H_w². The proof of Proposition 5.4.2 applies also toN S(C)if we replaceX byX_C =P

s,t∈CcstPsPt. This means that whenever we have a decompositiong^C_{N S}(w) =g₁×g₂, thenN S(C) splits compatibly.

Remark 5.4.10. The elementX_C ∈Sym²(h^∗) should be thought as the restriction of the Killing form onspanhα_s |s∈Ci. This is non denegerate, so it means thatX_C 6∈Sym²(V) for any proper subspaceV ⊆spanhα_s|s∈Ci(cf. Remark 5.3.3).

Lemma 5.4.11. Let KC := K ∩Sym²(N S(C)). Then KC is generated by X_D, with D closed andD⊆C.

Proof. Assume thatP

ia_iX_Di ∈K∩Sym²(N S(C))withD_i closed anda_i∈C. Then it is easy to see thatP

iaiX_Di =P

iaiX_Di∩C ∈Sym²(N S(C)).

For any s∈S, let L_s=π₁(P_s)∈g₁ andR_s=π₂(P_s)∈g₂.

Lemma 5.4.12. Let C be a connected and closed subset of S. Assume that there exists a non-empty closed subset D ⊆ C such that N S(D) = π1(N S(C)). Then if D does not contain any sink we have D=C.

Proof. Let U = C \D and E := {a(i) →ⁱ b(i) | a(i) ∈ U andb(i) ∈ D} be the set of arrows starting in U and ending in D. The set {P_s}_s∈D = {L_s}_s∈D is a basis of N S(D) =π₁(N S(C)), therefore the set{R_u}_u∈U is a basis of π₂(N S(C)). We assume for contradiction thatU 6=∅. By writing the(2,2)-component of X_C − X_D we obtain

X

u∈U

X

s∈C

c_suL_s

!

⊗R_u = 0∈g₁⊗g₂

from which we getP

s∈CcsuLs= 0for any u∈U. Let Ue be a connected component ofU and let

Ee={a(i)→ⁱ b(i)|a(i)∈Ue and b(i)∈D} ⊆E.

SinceC is connected we haveEe 6=∅. Since Ue is connected and there are no loops in the Dynkin diagram, we haveb(i)6=b(j) for anyi6=j∈E, and moreover there are no arrowse betweenb(i) and b(j). Then for any u∈Ue we have

0 =X

s∈C

c_suL_s=X

s∈Ue

c_suL_s+X

i∈Ee

c_b(i)uL_b(i). (5.6)

Since the set{L_b(i)}_i∈

Eeis linearly independent, this can be thought as a non-degenerate system of linear equations inLs, withs∈Ue and it has a unique solution

L_s=X

i∈Ee

y(s, i)L_b(i) =X

i∈Ee

y(s, i)P_b(i) withy(s, i)∈R.

SubstitutingLs in (5.6) we get X

s∈Ue

y(s, i)c_su =

(0 if u6=a(i),

−c_a(i)b(i) if u=a(i), for all u∈Ue and i∈E.e (5.7) Claim 5.4.13. We havey(s, i)>0 for anys∈Ue and any i∈E.e

Proof of the claim. Let(−,−) be the Killing form on h^∗. From Equation (5.7) it is easy to see that



 X

s∈Ue

y(s, i) (αs, αs)αs, αu



=−δ_a(i),uc_a(i)b(i)(αu, αu) ∀u∈U ,e ∀i∈E.e HenceP

s∈Ue y(s,i)

(αs,αs)αs is (up to a positive scalar) equal to the fundamental weight ofa(i) in the root system generated by the simple roots inUe. Now the claim follows from the fact

that in any irreducible root system all the dominant weights have only positive coefficients when expressed in the basis of simple roots.

In fact, let 0 6= λ= P

s∈Ueλ_sα_s and assume (λ, α_s) ≥0 for all s ∈ Ue. If λ_s < 0 for some s, then (λs, αs) <0. Thus λs ≥0 for all s. Assume now λs = 0for some s. Then (λ, αs) ≥0 only if λt = 0for all t ∈S neighboring s in the Dynkin diagram. Since Ue is connected we obtainλ_s= 0 for alls, henceλ= 0 which is a contradiction.

For any s∈Ue we haveRs=Ps−P

i∈Eey(s, i)P_b(i)∈g₂. Now consider the element R^0,4 3 X

s,t∈Ue

cstRsRt= X

s,t∈Ue

cst



Ps−X

i∈Ee

y(s, i)P_b(i)







Pt−X

i∈Ee

y(t, i)P_b(i)



=

=



 X

s,t∈Ue

cstPsPt



−2X

i∈Ee



 X

s,t∈Ue

y(s, i)cstPt



P_b(i)+ X

i,j∈Ee



 X

s,t∈Ue

y(s, i)y(t, j)cst



P_b(i)P_b(j)

=



 X

s,t∈Ue

cstPsPt



+ 2X

i∈Ee

c_a(i)b(i)P_a(i)P_b(i)− X

i,j∈Ee

y(a(j), i)c_a(j)b(j)P_b(i)P_b(j) =

=X_D∪

Ue− X_D + Θ, whereΘ :=− X

i,j∈Ee

y(a(j), i)c_a(j)b(j)P_b(i)P_b(j). Letp:R⁴→H_w⁴ denote the projection. The previous equation implies that

p



 X

s,t∈Ue

c_stR_sR_t



=p(Θ).

But p(P

s,t∈Uec_stR_sR_t) ∈Hw^0,4 while p(Θ) ∈ Hw^4,0, because b(i) ∈ D and P_b(i) ∈ Hw^2,0 for anyi∈E. It follows thate p(Θ)∈Hw^4,0∩Hw^0,4={0}.

We can writeΘ = Θ₁+ Θ₂ with Θ₁ = X

i,j∈Ee

i6=j

y(a(j), i)c_a(j)b(j)P_b(i)P_b(j) and Θ₂ =X

i∈Ee

y(a(i), i)c_a(i)b(i)P_b(i)² .

Since there are no edges between b(i) and b(j), we have that p(P_b(i)P_b(j)) = P_b(i)b(j) for anyi, j∈Ee such thati6=j. Thus, by Lemma 5.3.1, we have

p(Θ1) = X

i,j∈Ee

i6=j

y(a(j), i)c_a(j)b(j)P_b(i)b(j)

p(Θ2) =−2X

i∈Ee

y(a(i), i)c_a(i)b(i)



 X

j∈Ei

(α_b(i), α_βi(j))

(α_βi(j), α_βi(j))P_βi(j)b(i)





whereE_i={b(i)→^j β_i(j)} is the set of arrows inI_w starting inb(i). It is easy to see that all the terms in p(Θ1) and p(Θ2) are linearly independent, whence p(Θ1) +p(Θ2) = 0 if and only if all their terms vanish. Recall that y(a(i), i)c_a(i)b(i) < 0 for all i ∈ E. Hencee p(Θ1) +p(Θ2) = 0 forces Ei =∅ for any i∈E. But this is a contradiction because theree are no sinks inD, whence U =∅ and C=D.

Lemma 5.4.14. Let C be a closed and connected subset of S. Assume that there are no sinks inC. Then N S(C)⊆g₁ or N S(C)⊆g₂.

Proof. We work by induction on the number of vertices inC. There is nothing to prove if C=∅.

LetD⊆C be a maximal proper closed subset. The kernel K_C :=K∩Sym²(N S(C)) is generated byX_C andX_D⁰ withD⁰ ⊆D. In fact, ifDe ⊆C is a proper closed subset and De 6⊆D, then by maximality De ∪D=C and X

De =X_C − X_D +X_D∩

De. In particular, we havedimK_C = dimK_D+ 1.

By induction on the number of vertices we can subdivide D into two subsets DL and D_R, each consisting of the union of connected components of D, such thatN S(D_L)⊆g₁ andN S(D_R)⊆g₂.

SinceN S(C)splits, then KC also splits asK_C^4,0⊕K_C^2,2⊕K_C^0,4 whereK_C^i,j =KC∩R^i,j. However, K_C^2,2 ⊆ K^2,2 = 0 since R^2,0⊗R^0,2 is mapped isomorphically to Hw^2,2. Using dimK_C = dimK_D + 1we getK_C^4,0 =K_D^4,0 or K_C^0,4 =K_D^0,4. Without loss of generality we can assumeK_C^4,0=K_D^4,0 =KDL.

This implies that X_C ∈K_C^4,0⊕K_C^0,4 =K_DL⊕K_C^0,4. It follows that X_C ∈Sym²(N S(D_L)⊕π₂(N S(C))).

SinceX_C is non-degenerate onN S(C), we getN S(DL) =π1(N S(C)). Now we can apply Lemma 5.4.12: ifDL6=∅, thenDL=C, otherwise π1(N S(C)) = 0and N S(C)⊆g₂. Theorem 5.4.15. For w ∈ W, if the graph I_w is connected and without sinks, then g_{N S}(w) =aut(IH_w, φ).

Proof. Applying Lemma 5.4.14 to C =S we see that any decomposition of g^C_{N S}(w) must be trivial, hence by Proposition 5.3.10 we getg_{N S}(w) =aut(IHw, φ).

Example 5.4.16. It is in general false thatg_{N S}(w) is simple for any connected w.

LetW be the Weyl group of type A₃ (i.e. W =S₄) whereS ={s, t, u}. We consider the elementusts∈W whose graph I_usts is

t u s

The closed subsets inI_usts areS,{u}and ∅. Theng_{N S}(usts)∼=g_{N S}(u)×g_{N S}(sts)∼= sp₂(R)×sp₆(R)∼=sl₂(R)×sp₆(R). The splitting induced on H_w² is

H_w² =π₁(H_w²)⊕π₂(H_w²) =CP_u⊕

C(P_t−2

3P_u) +C(P_s−1 3P_u)

.

As we explain in the next section, we have a similar behavior more generally: for any w∈ Sn+1, with S ={s₁, . . . , sn}, such that w= s1w⁰ wherew⁰ is the longest element in W_{s2,...,sn} the Lie algebrag_{N S}(w) is isomorphic tosl₂(R)×g_{N S}(w⁰).

Example 5.4.17. The following example demonstrates that having no sinks in I_w is not a necessary condition for the algebrag_{N S}(w)to be simple.

LetW be the Weyl group of type B3, where we label the simple reflections as follows:

s t u

Then forw₁=ustswe get againg_{N S}(w₁)∼=g_{N S}(u)×g_{N S}(sts)∼=sl₂(R)×sp₆(R), but for w2 =stut the Lie algebra g_{N S}(w2) is simple (hence it is isomorphic to so_6,6(R)). Notice that the graphsI_w1 andI_w2 are isomorphic.

Remark 5.4.18. We have chosen to restrict ourselves to the case of finite Weyl groups in order to be able to state the results using only “classical” Schubert calculus. However, the results given in this section work in the same way for any irreducible finite Coxeter groups using a realization of Type I, i.e. the geometric representation. Note that this includes the groupsH3 and H4. We briefly explain how.

We replace everywhere the intersection cohomology of Schubert variety IH_w by the indecomposable Soergel modulesBwand the Killing form by the positive definite symmetric formBdefined in [Hum78, §5.2]. Assumeuis a simple reflection such thatwu < u. Because of Chapter 4 we can define the Néron-Severi Lie algebrag_{N S}(B_wu^u ) of the singular Soergel moduleB_wu^u . This Lie algebra is semisimple and its action on B_wu^u is irreducible.

We need an argument to replace the recourse to the relative hard Lefschetz in the proof of Theorem 5.3.6. We have

B^u_wu⊗_R^uR[1]∼=Bw,

therefore any decompositionR∼=R^u⊕R^u[−2]asR^u-modules induces a decomposition Bw ∼=B_wu^u [1]⊕B_wu^u [−1]

of (R, R^u) bimodules. We choose this decomposition as in the proof of [EW14, Theorem 6.19] (cf. Theorem 4.5.4). With respect to this decomposition multiplication byρ induces the map

∂u(ρ) :B_wu^u [1]→B_wu^u [−1]

which is clearly an isomorphism ifρ is ample.

The rest of arguments go through using the Schubert basis from Chapter 3. We obtain thus the same criterion: if w is connected and there are not sinks in the graph I_w then g_{N S}(w) is maximal, i.e. it coincides withaut(Bw, φ).

For infinite Coxeter groupsW our methods do not apply directly. In fact, in general a reflection faithful representation ofW is not irreducible, thus Lemma 5.3.2 does not hold and the kernel of the mapR→Bw seems harder to compute.

Im Dokument Hodge theoretic aspects of Soergel bimodules and representation theory (Seite 86-90)