We can continue this procedure to obtain a (maximal) Zr+1-filtration

86(2010) 303-354

FUNCTORIAL RELATIONSHIPS BETWEEN QH^∗(G/B) AND QH^∗(G/P)

Naichung Conan Leung & Changzheng Li

Abstract

We give a natural filtrationF onQH^∗(G/B), which respects the quantum product structure. Its associated graded algebra Gr^F(QH^∗(G/B)) is isomorphic to the tensor product ofQH^∗(G/P) and a corresponding graded algebra ofQH^∗(P/B) after localiza- tion. When the quantum parameter goes to zero, this specializes to the filtration on H^∗(G/B) from the Leray spectral sequence associated to the fibrationP/B→G/B−→G/P.

1. Introduction

LetGbe a simply connected complex simple Lie group,B be a Borel subgroup, and P ⊃ B be a parabolic subgroup of G. The natural fibration P/B→G/B−→G/P of homogeneous varieties gives rise to a Z²-filtrationF onH^∗(G/B) overQ(orC) such thatGr^F(H^∗(G/B))∼= H^∗(P/B)⊗H^∗(G/P) as graded algebras by the Leray-Serre spectral se- quence. Given another parabolic subgroupP^′ withB⊂P^′ ⊂P, we ob- tain the corresponding natural fibrationP^′/B→P/B −→P/P^′. Com- bining it with the former one, we obtain a Z³-filtration on H^∗(G/B).

We can continue this procedure to obtain a (maximal) Z^r+1-filtration.

In the present paper, we study the small quantum cohomology rings QH^∗(G/P)’s of homogeneous varietiesG/P’s, which are deformations of the ring structures onH^∗(G/P)’s by incorporating genus zero 3-pointed Gromov-Witten invariants ofG/P’s into the cup product. We show the

“functorial relationships” between QH^∗(G/B) and QH^∗(G/P) in the sense that theZ^r+1-filtration onH^∗(G/B) can be generalized to give a Z^r+1-filtration on QH^∗(G/B) and there exist canonical maps between quantum cohomologies, in analog with the classical ones. We begin with a toy example to illustrate our results.

Example 1.1. WhenG=SL(3,C),G/B ={V1 6V₂6C³| dim_CV_i

= 1, i= 1,2} =:F ℓ₃ is a complete flag variety. Given a maximal para- bolic subgroupP ⊃B, we haveP/B =P¹ andG/P =P²together with a natural fibration P¹ ֒→ⁱ F ℓ₃ −→^π P². The quantum cohomology ring

Received 3/1/2010.

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QH^∗(G/B) has a basis consisting of Schubert classesσ^w’s overQ[q₁, q₂], indexed by the Weyl group W =S₃ ={1, s1, s₂, s₁s₂, s₂s₁, s₁s₂s₁}. To obtain the Z²-filtration F on QH^∗(G/B), we need a deformation gr of the classical grading map which satisfies gr(q₁^aq^b₂σ^w) = agr(q₁) + bgr(q₂) +gr(σ^w). In this example, gris given explicitly by Table 1.

Table 1. gr(q^a₁q₂^bσ^w) = (i, j) with −2≤i≤4,0≤j≤6 4 q₁² q₁²σ^s2 q₁²σ^s1s2 q₁²q2σ^s1 q₁²q2σ^s2s1 q₁²q2σ^s1s2s1 q₁³q₂² 3 q1σ^s1 q1σ^s2s1 q1σ^s1s2s1 q₁²q2 q₁²q2σ^s2 q₁²q2σ^s1s2 q₁²q₂²σ^s1 2 q1 q1σ^s2 q1σ^s1s2 q1q2σ^s1 q1q2σ^s2s1 q1q2σ^s1s2s1 q₁²q₂² 1 σ^s1 σ^s2s1 σ^s1s2s1 q1q2 q1q2σ^s2 q1q2σ^s1s2 q1q₂²σ^s1 0 1 σ^s2 σ^s1s2 q2σ^s1 q2σ^s2s1 q2σ^s1s2s1 q1q²₂

−1 0 0 0 q2 q2σ^s2 q2σ^s1s2 q₂²σ^s1

−2 0 0 0 0 0 0 q₂²

i

j 0 1 2 3 4 5 6

This determines a Z²-filtration F = {Fc}_c∈Z² on QH^∗(G/B). The main point is that this filtration respects the quantum multiplication, i.e., F_cF_d⊂F_c+d. Indeed, this can be easily checked with the following well known quantum products for F ℓ₃:

σ^s1⋆ σ^s1=σ^s2s1 +q₁, σ^s1 ⋆ σ^s1s2 =σ^s1s2s1, σ^s1⋆ σ^s2s1=q₁σ^s2, σ^s2⋆ σ^s2=σ^s1s2 +q₂, σ^s2⋆ σ^s2s1 =σ^s1s2s1, σ^s2 ⋆ σ^s1s2=q₂σ^s1, σ^s1s2 ⋆ σ^s1s2s1=q1q2σ^s2, σ^s1s2⋆ σ^s1s2 =q2σ^s2s1, σ^s1⋆ σ^s1s2s1=q₁σ^s1s2+q₁q₂, σ^s2s1 ⋆ σ^s1s2s1=q₁q₂σ^s1,

σ^s2s1 ⋆ σ^s2s1 =q₁σ^s1s2, σ^s2 ⋆ σ^s1s2s1=q₂σ^s2s1 +q₁q₂, σ^s1 ⋆ σ^s2=σ^s1s2+σ^s2s1, σ^s1s2 ⋆ σ^s2s1 =q1q2, σ^s1s2s1 ⋆ σ^s1s2s1=q₁q₂σ^s1⋆ σ^s2.

This Z²-filtration of the algebra structure on QH^∗(G/B) is a q- deformation of the classical one on H^∗(G/B), which comes from the Leray spectral sequence. Due to the existence of such a filtration, we can easily check that there arealgebraisomorphisms ¯ϕ:QH^∗(G/B)/I −→

QH^∗(P/B) and ¯ψ : QH^∗(G/P) −→ A/J, where A := S

j≥0F_(0,j) is a subalgebra of QH^∗(G/B), J := F_(0,−1) is an ideal of A, and I is the ideal in QH^∗(G/B) spanned by thoseq^a₁q₂^bσ^w’s with their gradings (d₁, d₂) satisfying d₂ >0. Here ¯ϕsendsq^a₁q₂^bσ^w+I to y^j whereq₁^aq^b₂σ^w is the (unique) one among such expressions with its grading equal to (j,0), and ¯ψsendsx^j to the (unique)q₁^aq₂^bσ^w ∈QH^∗(G/B) whose grad- ing equals (0, j), in which we have taken the well known isomorphisms QH^∗(P/B) ∼=Q[y] and QH^∗(G/P) ∼=Q[x]. In particular, QH^∗(G/P) is the quotient of the subalgebra A generated by {q2σ^s1, σ^s2, q1q₂², q2} by the ideal J = q₂A. (We remark that in this case QH^∗(G/B) itself is generated by {σ^s1, σ^s2, q₁, q₂}.)

These algebra isomorphisms generalize the classical ones in an obvious way, namely, A, J, and I are q-deformations of A := π^∗(H^∗(G/P)), J = 0, and I =Q{σ^s2, σ^s1s2, σ^s2s1, σ^s1s2s1}, respectively.

All the above descriptions forG=SL(3,C) will be generalized to ar- bitrary complex semi-simple Lie groups. For simplicity, we assumeP/B is irreducible. (Note that any homogeneous variety splits into a direct product of irreducible ones.) All the results can be easily generalized for reducibleP/B’s, and we will describe such generalizations in section 5. Note that P/B is again a complete flag variety, isomorphic toG^′/B^′ for some other complex simple Lie group G^′. Then we denote by r the rank of G^′, which depends only on P/B. Since we exclude the trivial cases, namely,P equalsB orG, we always haver= 1 forG=SL(3,C).

In general, we consider a special iterated fibration{Pj−1/P₀ →P_j/P₀

−→P_j/P_j−1}^r+1_j=2 in which P_j’s are parabolic subgroups withB =P₀ ( P₁ ( · · ·(P_r =P (G= P_r+1. Consequently, we obtain a canonical Z^r+1-filtration on H^∗(G/B). Note that QH^∗(G/B) also has a natural basis of Schubert classes σ^w’s over Q[q]. As we will see in section 2.2, there exists a grading map gr giving gradings gr(q_λσ^w)∈Z^r+1 for the (Q-)basis q_λσ^w’s. The Peterson-Woodward comparison formula in [32]

plays a key role in defininggr. It is the only known formula that charac- terizes the relations of genus zero 3-pointed Gromov-Witten invariants betweenG/B and a generalG/P explicitly. grdefines aZ^r+1-filtration F ={Fa}_a∈Z^r+1 of subspaces inQH^∗(G/B), generalizing Example 1.1.

The next theorem says thatF respects the quantum product structure.

Theorem 1.2.QH^∗(G/B)is aZ^r+1-filtered algebra with filtrationF.

We can obtain several important consequences as below.

Theorem 1.3. The vector subspace I, spanned by those q_λσ^w’s with their gradings (d₁,· · ·, d_r+1) satisfying d_r+1 > 0, is an ideal of QH^∗(G/B). Furthermore, there is a canonical algebra isomorphism

QH^∗(G/B)/I −→^≃ QH^∗(P/B).

Since QH^∗(G/B) has a Z^r+1-filtration F, we obtain an associated Z^r+1-graded algebraGr^F(QH^∗(G/B)) =L

a∈Z^r+1Gr^F_a, whereGr_a^F :=

Fa

∪b<aF_b. For each j, we denote Gr_(j)^F (QH^∗(G/B)) :=L

i∈ZGr^F_iej. Theorem 1.4. For each 1≤j ≤r, there exists a canonical algebra isomorphism

Ψ_j :QH^∗(P_j/P_j−1)−→^≃ Gr^F_(j)(QH^∗(G/B)).

Furthermore, if P/B ∼= F ℓ_r+1, then there exists a canonical algebra isomorphism

Ψ_r+1:QH^∗(G/P) −→^≃ Gr_(r+1)^F (QH^∗(G/B)).

As a consequence, we have the following results for any G.

Theorem 1.5. Suppose P/B ∼=F ℓ_r+1. Then there exists a subalge- braAofQH^∗(G/B)together with an idealJ ofA, such thatQH^∗(G/P) is canonically isomorphic to A/J as algebras.

Theorem 1.6. Suppose P/B ∼= F ℓ_r+1. Then as graded algebras Gr^F(QH^∗(G/B))is isomorphic toQH^∗(P¹)⊗· · ·⊗QH^∗(P^r)⊗QH^∗(G/P) after localization.

We should point out that the requirement “P/B ∼=F ℓ_r+1” in Theo- rem 1.5 and Theorem 1.6 is not a strong assumption, because both of theorems can be easily generalized to the case “P/B is isomorphic to a product of F ℓ_k’s” (see section 5). As a consequence, all G/P’s for G being of A-type or G2-type satisfy this assumption. Furthermore, for each remaining type, more than half of the homogeneous varieties G/P’s also satisfy this. (We could also show Theorem 1.5 holds for any G/P with G=Sp(2n,C).)

As we saw in Example 1.1, the gradings of elements inQH^∗(F ℓ₃) only form a proper sub-semigroupSofZ², which looks like stairs, so that the Z²-filtration comes from an S-filtration. In general, the Z^r+1-filtration comes from a similar filtration. For this reason, we need localization to obtain the analog of graded-algebra isomorphism (in Theorem 1.6). In section 4, we will restate theorems 1.4, 1.5, and 1.6 more concretely. As we will see later, all the relevant maps generalize the classical ones in an obvious way, as in Example 1.1.

Our results relate the quantum cohomologies of the total space and the base space of the fibration P/B → G/B −→ G/P. Similar struc- tures occur when one studies the relationships ofJ-functions between an abelian quotient and a nonabelian quotient. Such relations were studied by Bertram, Ciocan-Fontanine, and Kim in [3] and [4]. (See also [33].) There were also relevant studies by Liu-Liu-Yau [23] and Paksoy [27]

by using mirror principle [22] .

Let us mention two more important problems on the study of QH^∗(G/P), for which our theorems may also be helpful. One can see

the excellent survey [9] and references therein for more details on the developments. As mentioned before, the (small) quantum cohomology ringQH^∗(G/P) has a basis of Schubert classesσ^w’s overQ[q]. In order to understand QH^∗(G/P), one would like to have (i) a (good) presen- tation of the ring structure onQH^∗(G/P) and (ii) a (nice) formula (or algorithm) for the quantum Schubert structure constants Nu,v^w,λP’s in the quantum product σ^u ⋆ σ^v = P

w,λPNu,v^w,λPq_λPσ^w. For classical co- homology H^∗(G/P), these natural and important problems have been solved in [5] for (i) and in [17] and [7] for (ii). However, for quan- tum cohomologyQH^∗(G/P), the answer to (i) is only known in certain cases—for instance, when G is of A-type (see [15], [1]) or P = B is a Borel subgroup [16]. For problem (ii), there were early studies for a few cases, including complex Grassmannians (see the survey [9]) and complete flag varieties ofA-type [8], besides the quantum Chevalley for- mula [11], which works for all cases. Recently, Mihalcea [26] has given an algorithm, and the authors ([20], [21]) have given a combinatorial formula for these structure constants.

All these problems were discussed in the unpublished work [28] by Dale Peterson. In [32], Woodward proves a comparison formula of Pe- terson. The Peterson-Woodward comparison formula explicitly charac- terizes the relations of the quantum Schubert structure constants be- tween QH^∗(G/P) and QH^∗(G/B). However, it does not tell us the relations of the algebra structures between them. Along Peterson’s ap- proach, Lam and Shimozono [18] show that the torus-equivariant exten- sion of QH^∗(G/P) is isomorphic to a quotient of the torus-equivariant homology of a based loop group after localization. In [29], K. Rietsch discusses the relationships between Peterson’s work and mirror symme- try. In [28], Peterson had also claimed there was an analogous isomor- phism for the (un-iterated) fibrationP/B →G/B −→G/P in terms of torus-equivariant homology of based loop groups after localization. We were motivated by his claim and the results by Woodward and Lam- Shimozono. We succeeded in obtaining natural generalizations of the classical isomorphisms. It is interesting to compare our results with Peterson’s claim. It is also interesting to compare our Theorem 1.5 with Theorem 10.16 of [18] by Lam and Shimozono. As commented by Thomas Lam, our results should be related to the discussions in section 10.4 of [18].

We hope our results could be used to solve problem (i) by combining with Kim’s early work [16], where a nice presentation of the ring struc- ture on the complexified quantum cohomology QH^∗(G/B) was given.

This paper is organized as follows. In section 2, we define a grading map and prove our main result, Theorem 1.2, assuming the Key Lemma.

Then we devote the whole section 3 to the proof of the Key Lemma.

In section 4, we prove the remaining theorems discussed as above. In

section 5, we show how to generalize our results to the general case when P/B is reducible. Finally in section 6, we give an appendix which deals with exceptional cases in the proof of the Key Lemma. Our proofs are combinatorial in nature. We hope to find nice geometrical explanations of them later.

Acknowledgements.The authors thank Baohua Fu, Bumsig Kim, Thomas Lam, Augustin-Liviu Mare, and Leonardo Constantin Mihalcea for useful discussions. We also thank the referee for valuable suggestions.

The first author is supported in part by a RGC research grant from the Hong Kong government. The second author is supported in part by KRF-2007-341-C00006.

2. A filtration on QH^∗(G/B)

2.1. Preliminaries.We recall some basic notions and fix the notation.

See, for example, [12,13] for more details on Lie theory.

Let G be a simply-connected complex simple Lie group of rank n, B ⊂ G be a Borel subgroup and P ⊃ B be a proper parabolic sub- group of G. Fix a basis of simple roots ∆ ={α1,· · ·, α_n}(with respect to (G, B)). Then P corresponds canonically to a proper subset ∆_P of

∆. (In particular, B corresponds to the empty subset ∅.) Let h de- note the corresponding Cartan subalgebra; then h^∗ = Ln

i=1Cαi. Let {α^∨₁,· · · , α^∨_n} ⊂hbe the fundamental coroots and{χ1,· · · , χ_n} ⊂h^∗be the fundamental weights. For any 1 ≤i, j≤n, we have hχi, α^∨_ji=δi,j

with respect to the natural pairing h·,·i : h^∗ ×h → C. Furthermore, we have ρ := ¹₂P

γ∈R⁺γ = P_n

i=1χ_i. For each 1 ≤ i ≤ n, the simple reflection s_i:=s_αi acts on handh^∗ by

s_i(λ) =λ− hα_i, λiα^∨_i, forλ∈h; s_i(β) =β− hβ, α^∨_i iα_i, forβ ∈h^∗. The Weyl group W, which is generated by{s₁,· · · , s_n}, acts on h and h^∗ and preserves the natural pairing. The root system is given by R= W·∆ =R⁺⊔(−R⁺), whereR⁺=R∩Ln

i=1Z_≥0αi is the set of positive roots. Thus each rootγ ∈R is given byγ =w(α_i) for somew∈W and 1 ≤i≤n. Then we defineγ^∨ =w(α^∨_i) and s_γ =ws_iw⁻¹ ∈W, which is independent of the expressions of γ.

The length ℓ(w) of w∈ W (with respect to ∆) is defined by ℓ(1), 0 and ℓ(w) , min{k | w = s_i1· · ·s_ik} for w 6= 1. An expression w = s_i1· · ·s_iℓ is called reduced if ℓ = ℓ(w). Let ˜P = P_∆˜ denote the (standard) parabolic subgroup corresponding to a subset ˜∆ ⊂ ∆, WP˜ denote the subgroup generated by {s_j | α_j ∈ ∆}, and˜ ωP˜ denote the longest element in WP˜. For ¯∆ ⊂ ∆ with ¯˜ P := P∆¯, we denote W^P_˜^¯

P := {w∈W_P˜|ℓ(w) ≤ℓ(v), ∀v∈wWP¯}. Each coset in W_P˜/WP¯ has

a unique (minimal length) representative in W^P_˜^¯

P ⊂ W_P˜ ⊂ W. In par- ticular, we haveP_∆=Gand W_G=W, and simply denote W^P¯ :=W_G^P¯ and ω:=ω_G.

The (co)homology of a homogeneous variety X =G/P has an addi- tive basis of Schubert (co)homology classes indexed byW^P: H_∗(X,Z) = L

v∈W^P Zσ_v, H^∗(X,Z) = L

u∈W^P Zσ^u with hσ^u, σ_vi = δ_u,v for any u, v ∈ W^P [2]. In particular, H₂(X,Z) = L

αi∈∆\∆P Zσ_si. Set Q^∨ = L_n

i=1Zα^∨_i andQ^∨_P =L

αi∈∆PZα^∨_i. Then we can identifyH₂(X,Z) with Q^∨/Q^∨_P canonically, by mapping P

αj∈∆\∆P

a_jσ_sj toλ_P = P

αj∈∆\∆P

a_jα^∨_j+ Q^∨_P. For eachα_j ∈∆\∆_P, we introduce a formal variableq_α^∨

j+Q^∨_P. For such λ_P, we denote q_λP =Q

αj∈∆\∆P q^a_α^j∨ j+Q^∨_P.

LetM_0,m(X, λ_P) be the moduli space of stable maps of degreeλ_P ∈ H₂(X,Z) ofm-pointed genus zero curves intoX[10], and ev_i denote the ith canonical evaluation map ev_i :M0,m(X, λ_P) →X given by ev_i([f : C →X;p₁,· · · , p_m]) =f(p_i). The genus zero Gromov-Witten invariant forγ₁,· · ·, γ_m ∈H^∗(X) =H^∗(X,Q) is defined asI_0,m,λP(γ₁,· · ·, γ_m) = R

M0,m(X,λP)ev^∗₁(γ₁)∪ · · · ∪ev^∗_m(γ_m).The (small) quantum product for a, b∈H^∗(X) is a deformation of the cup product, defined by

a ⋆ b, X

u∈W^P,λP∈H2(X,Z)

I_0,3,λP(a, b,(σ^u)^♯)σ^uq_λP, where{(σ^u)^♯|u∈W^P}are the elements inH^∗(X) satisfyingR

X(σ^u)^♯∪ σ^v = δ_u,v for any u, v ∈ W^P. The quantum product ⋆ is associative, making (H^∗(X)⊗Q[q], ⋆) a commutative ring. This ring is denoted as QH^∗(X) and called the (small) quantum cohomology ring of X.

The same Schubert classes σ^u =σ^u⊗1 form a basis for QH^∗(X) over Q[q], and we write

σ^u⋆ σ^v = X

w∈W^P,λP∈Q^∨/Q^∨_P

N_u,v^w,λPq_λPσ^w.

The coefficientsN_u,v^w,λP’s are called thequantum Schubert structure con- stants. They generalize the well known Littlewood-Richardson coeffi- cients when X = Gr(k, n+ 1) is a complex Grassmannian. It is also well known that the quantum Schubert structure constants are non- negative.

When P = B, we have Q^∨_P = 0, WP = {1} and W^P = W. In this case, we simply denote λ = λ_P and q_j = q_α^∨

j. A combinatorial formula for Nu,v^w,λ’s has been given by the authors recently [20]. As a consequence, we can obtain the combinatorial formula for N_u,v^w,λP’s for general G/P, due to the following comparison formula.

Proposition 2.1 (Peterson-Woodward comparison formula [32]; see also [18]).

1) Let λ_P ∈ Q^∨/Q^∨_P. Then there is a unique λ_B ∈ Q^∨ such that λ_P = λ_B +Q^∨_P and hα, λ_Bi ∈ {0,−1} for all α ∈ R⁺_P (= R⁺∩ L

αj∈∆P Zα_j).

2) For every u, v, w∈W^P, we have

N_u,v^w,λP =N_u,v^{wωPω′,λB},

where ω^′ =ω_P^′ with∆_P^′ ={α_i ∈∆_P | hα_i, λ_Bi= 0}.

Thanks to Proposition 2.1, we have canonical representatives ofW/WP

×Q^∨/Q^∨_P inW ×Q^∨ with respect to the pair (∆,∆_P), which is a gen- eralization of the case W/W_P →^≃ W^P ⊂ W. We will discuss them in more details in the next subsection.

When v is a simple reflection s_i, we have the following (Peterson’s) quantum Chevalley formula for σ^u⋆ σ^si, which has been proved earlier in [11].

Proposition 2.2 (Quantum Chevalley Formula forG/B). For u ∈ W,1≤i≤n,

σ^u⋆ σ^si =X

γ

hχi, γ^∨iσ^usγ +X

γ

hχi, γ^∨iq_γ^∨σ^usγ,

where the first sum is over roots γ in R⁺ for which ℓ(us_γ) = ℓ(u) + 1, and the second sum is over roots γ in R⁺ for whichℓ(us_γ) =ℓ(u) + 1− h2ρ, γ^∨i.

Note that we have fixed a base ∆ = {α1,· · · , α_n}. As a subset of

∆, we can write ∆_P = {α_i1,· · · , α_ir}. Then giving an order on ∆_P is equivalent to giving a permutation of ∆_P. Once such a permutation Υ is given, we denote α^′_j = Υ(αij) for each 1 ≤ j ≤ r and then nat- urally rewrite the remaining simple roots so that ∆ = {α^′₁,· · · , α^′_n}.

In the present paper, we always keep the information on the order, whenever referring to (∆P,Υ) or (an ordered set) ∆P = (α^′₁,· · ·, α^′_r).

Furthermore for convenience, we simply denote α^′_j’s by α_j’s. (In other words, we take Υ = id∆P under the assumption in the beginning that

∆_P ={α1,· · · , α_r} satisfies certain properties on its associated Dynkin diagram.)

Notation 2.3. Let (∆_P,Υ) be given with ∆_P = (α₁,· · · , α_r).

For any integers k, m with 1 ≤ k ≤ m ≤ r, we denote u^∆_[k,m]^P,(r) :=

s_αks_αk+1· · ·s_αm.If k > m, then we just denote u^∆_[k,m]^P,(r) := 1. Further- more, we defineu^∆_[k,m]^P,(m)=u^∆_[k,m]^P,(r) and denoteu^∆_i ^P,(m):=u^∆_[m−i+1,m]^P,(m) for i= 0,1,· · · , m. Whenever there is no confusion, we simply denote

s_j :=s_αj, u^(r)_[k,m]:=u^∆_[k,m]^P,(r), and u^(m)_i :=u^∆_i ^P,(m).

Let ∆_j := {α1,· · ·, α_j} and P_j := P_∆j for each 1≤ j ≤ r. Denote P₀ = B and P_r+1 = G. A decomposition of w ∈W associated to (∆_P,Υ) is an expression w=v_r+1· · ·v₁ with v_i ∈W_P^Pi−1

i for each 1≤ i≤r+ 1, whereW_P^P0

1 =W_P1. By the iterated fibration associated to (∆_P,Υ), we mean the family of fibrations of homogeneous varieties, given by {Pj−1/P₀ →P_j/P₀ −→P_j/P_j−1}^r+1_j=2.

We denote byDyn(∆^′) the Dynkin diagram associated to a base ∆^′. Example 2.4. Suppose Dyn(∆P) is given by ◦−−◦ · · · ◦−−◦

α1 α2 αr . Con- sider the iterated fibration {Pj−1/P₀ → P_j/P₀ −→ P_j/P_j−1}^r+1_j=2 as- sociated to ∆_P = (α₁,· · · , α_r). Then we have P_r+1/P_r = G/P and P_j/P_j−1 = P^j for each 1 ≤ j ≤ r. Furthermore, the natural inclu- sion {α₁,· · · , α_r−1} ֒→ ∆_P (or SL(r,C) ֒→ SL(r + 1,C)) induces a canonical embedding P_r−1/B = F ℓ_r−1 ֒→ F ℓ_r = P/B of complete flag varieties, which maps a flag V1 6 · · · 6 Vr−1 in C^r to the flag V₁6· · ·6V_r−1 6C^r inC^r+1.

Due to the following well known lemma (see e.g., [14]), we obtain Corollary 2.6.

Lemma 2.5. Let γ ∈R⁺ and w=s_i1· · ·s_iℓ be a reduced expression of w∈W.

1) w∈W^P if and only if w(α) ∈R⁺ for any α ∈∆_P.

2) If ℓ(ws_γ) < ℓ(w), thenw(γ)∈−R⁺ and there is a unique 1≤k≤ℓ such that

s_ik· · ·s_iℓs_γ=s_ik+1· · ·s_iℓ and γ =s_iℓs_iℓ−1· · ·s_ik+1(α_ik).

Furthermore for 1 ≤ j ≤ n, ℓ(wsj) = ℓ(w)−1 if and only if w(α_j)∈ −R⁺.

Corollary 2.6. For each w ∈ W, there exists a unique decompo- sition w = v_r+1· · ·v₁ associated to ∆_P = (α₁,· · ·, α_r). Furthermore, we assume that Dyn({α1,· · · , α_m}) is given by ◦−−◦ · · · ◦−−◦

α1 α2 αm, where m ≤ r. Then for each 1 ≤j ≤m, ℓ(v_j) = i_j if and only if v_j = u^(j)_i

j . (In particular, the expression u^(j)_i

j itself is reduced.)

Proof. The former is also well known (see e.g., [14]). The latter statement is a direct consequence of Lemma 2.5, by noting |W_P^P_j^j−1|= j + 1 and u^(j)₀ ,· · ·, u^(j)_j are distinct elements of W_Pj for which (1) of

Lemma 2.5 can be applied. q.e.d.

The following lemma should also be well known.

Lemma 2.7. Let ∆¯ ⊂ ∆˜ ⊂ ∆, P˜ =P_∆˜ and P¯ =P∆¯. Let w = vu with u∈WP¯ and v=s_i1· · ·s_im being a reduced expression of v∈W^P_˜^¯

P. For any1≤j≤m, we havev^′ :=s_ij+1· · ·s_im∈W^P_˜^¯

P and(v^′u)⁻¹(α_ij)∈ R⁺_˜

P \RP¯.

Proof. Assume that the set{a|s_ia+1s_ia+2· · ·s_im ∈/ W^P_˜^¯

P,1≤a≤m}is non-empty. Then we can take the minimumkof this set. Consequently, w^′ := s_ik+1· · ·s_im ∈/ W^P_˜^¯

P and s_ikw^′ ∈ W^P_˜^¯

P. Hence, there exists α ∈

∆ such that¯ w^′(α) ∈ −R⁺ and s_ikw^′(α) ∈ R⁺. Since s_ik preserves

−R⁺\ {−αik}, we have w^′(α) =−αik. Thusw^′sαw^′−1=s−α_ik =sik so thatℓ(w^′s_α) =ℓ(s_ikw^′) =ℓ(w^′)+1. This impliesw^′(α)∈R⁺by Lemma 2.5 and therefore deduces a contradiction. Hence, for any 1 ≤j ≤ m, we have v^′ :=si_j+1· · ·sim ∈W^P_˜^¯

P. Note that (v^′u)⁻¹(α_ij)∈R⁺_˜

P andγ :=v^′−1(α_ij)∈R⁺_˜

P. We claim γ /∈ RP¯; otherwise we would conclude v^′s_γ(γ) = −v^′(γ) ∈ −R_P⁺_¯, contrary to v^′sγ(γ) = sijv^′(γ) ∈ R⁺. Since u ∈ WP¯, we have (v^′u)⁻¹(αij) =

u⁻¹(γ)∈/ RP¯. q.e.d.

2.2. Definition of gradings.In this subsection, we define a grading map gr with respect to an ordered set (∆_P,Υ), which is used for con- structing a filtration on QH^∗(G/B). In order to obtain gr, we first define “PW-lifting” (Peterson-Woodward lifting) as follows.

Definition 2.8. Given (∆_P,Υ) with ∆_P = (α₁,· · · , α_r), we de- note ∆_j = {α1,· · · , α_j}, P_j = P_∆j, and Q^∨_j = Lj

i=1Zα^∨_i for each j ≤ r. By the PW-lifting associated to (∆_P,Υ), we mean the fam- ily {ψ∆j+1,∆j}^r_j=1 of injective maps defined as follows. (We denote Q^∨_r+1 = Q^∨,∆_r+1 = ∆ and P_r+1 = G.) For each 1 ≤ j ≤ r, the map

ψ_∆j+1,∆j :W_P^Pj

j+1×Q^∨_j+1/Q^∨_j −→W ×Q^∨

is defined by sending (v,λ) to its associated elements (vω¯ _Pjω_P^′

j, λ^′) as described by the Peterson-Woodward comparison formula (see Proposi- tion 2.1) with respect to (∆_j+1,∆_j). That is, λ^′ is the unique element in Q^∨_j+1 ⊂ Q^∨ satisfying ¯λ = λ^′ +Q^∨_j and hα, λ^′i ∈ {0,−1} for all α∈R⁺∩Lj

i=1Zα_i; ∆_P^′

j ={α∈∆_j | hα, λ^′i= 0}.

Remark 2.9. Each ψ_∆j+1,∆j also defines an injective map in the canonical way:

QH^∗(P_j+1/P_j)−→QH^∗(P_j+1/B);q¯λσ^v 7→q_λ^′σ^vωPjωPj′.

Recall that a natural basis of QH^∗(G/B)[q₁⁻¹,· · · , q_n⁻¹] is given by q_λσ^w’s labeled by (w, λ)∈W ×Q^∨. We simply denote both of them as

q_λw(orwq_λ) by abuse of notation. Note that q_λw∈QH^∗(G/B) if and only if q_λ∈Q[q] is a polynomial.

Definition 2.8 (continued). Let {e1,· · · ,e_r+1} be the standard basis of Z^r+1. We define a grading map gr:W ×Q^∨ −→ Z^r+1 associated to (∆_P,Υ) as follows.

1) Forw∈W, we take its (unique) decompositionw=v_r+1· · ·v₁ as- sociated to(∆_P,Υ). Then we definegr(w) :=gr(w,0) =^r+1P

j=1

ℓ(v_j)e_j. 2) For all α ∈ ∆, we simply denote gr(q_α^∨) := gr(1, q_α^∨). Using the PW-lifing associated to (∆_P,Υ), we can define all gr(q_j)’s recursively in the following way. Define gr(q1) = 2e1; for any α∈∆_j+1\∆_j, we define

gr(q_α^∨) = ℓ(ω_Pjω_P^′

j)+2+Xj i=12a_i

e_j+1−gr(ωPjω_P^′

j)−Xj

i=1a_igr(q_i), where ω_Pjω_P^′

j and a_i’s satisfy (ω_Pjω_P^′

j, α^∨+

j

P

i=1

a_iα^∨_i)=ψ_∆j+1,∆j(1, α^∨+Q^∨_j).

3) In general, x = wQn

k=1q^b_k^k; then we define gr(x) = gr(w) + Pn

k=1b_kgr(q_k).

Furthermore for 1≤k≤m≤ |∆P|, we define

gr_m :=gr_[1,m], with gr_[k,m]:W ×Q^∨ →Z^m−k+1

being the composition of the natural projection map and the grading map gr. Precisely, writegr(q_λw) =P_r+1

i=1d_ie_i; then we definegr_[k,m](q_λw) = Pm

i=kd_ie_i.

Recall that theinversion set of w∈W is defined to be Inv(w) ={γ∈R⁺ |w(γ)∈ −R⁺}.

It is well known that ℓ(w) = |Inv(w)| (see e.g., [14]). Take the de- composition w = v_r+1· · ·v₁ of w associated to (∆_P,Υ). For each k, we note v_r+1· · ·v_k+1 ∈ W^Pk and v_k· · ·v₁ ∈ W_Pk. Thus for γ ∈ RPk, v_k· · ·v1(γ) ∈ −R⁺ if and only if w(γ) ∈ −R⁺. Consequently, ℓ(v_k· · ·v₁) =|{γ ∈ R_P⁺

k | w(γ) ∈ −R⁺}|= |Inv(w)∩R⁺_P

k|. Note that ℓ(v_k· · ·v₁) =Pk

i=1ℓ(v_k). Hence, we have gr(w) =

r+1

X

k=1

|Inv(w)∩(R⁺_P

k\R_P⁺

k−1)|e_k.

Remark 2.10. We would like to thank the referee for reminding us of the above expression of gr(w). Following the suggestions of the referee, the proof of Proposition 3.1 has been simplified substantially in the present version. In type A, the vector gr(w) is essentially what

is known as an “inversion table” (see e.g., [31]). The referee has also made the following conjecture:

gr(q_γ^∨) =

r+1

X

k=1

h X

β∈R⁺_Pk\R⁺_Pk−1

β, γ^∨ie_k.

If it is true, the proofs of our main results might also be simplified substantially.

In Proposition 3.10, Proposition 3.12, Lemma 3.26, and (the proof of) Lemma 3.27, we will explicitly describe all the gradings gr(qj)’s with respect to a fixed (∆_P,Υ). In particular, we will see thatgr(q_j) = (1−j)e_j−1+ (1 +j)e_j for 2≤j ≤r−1 (which also holds for j =r if

∆_P is of A-type).

2.3. Proof of Theorem 1.2.Assuming Dyn(∆_P) is connected, we always consider (∆_P,Υ) with the fixed order ∆_P = (α₁,· · · , α_r) in a special way that will be explained in section 2.4. In this subsection, we construct a filtration on QH^∗(G/B) with respect to a totally ordered sub-semigroup S of Z^r+1 and prove Theorem 1.2, which is the most essential part of our main results.

Unless otherwise stated, we will always use the lexicographical or- der whenever referring to a partial order on (a sub-semigroup of) Z^m in the present paper. (Recall that a <b, wherea = (a₁,· · ·, a_m) and b = (b1,· · · , bm), if and only if there is 1 ≤ j ≤ m such that aj < bj

and a_k=b_k for each 1≤k < j.)

Definition 2.11. We define a subset S of Z^r+1 and a family F = {Fa}_a∈S of subspaces ofQH^∗(G/B) as follows:

S ,{gr(qλw)|q_λw∈QH^∗(G/B)}; F_a, M

gr(qλw)≤a

Qq_λw⊂QH^∗(G/B).

As will be shown in section 4, we have:

Lemma 2.12. S is a totally-ordered sub-semigroup of Z^r+1. Now we can state Theorem 1.2 more explicitly as follows.

Theorem 1.2. QH^∗(G/B) is an S-filtered algebra with filtration F. Furthermore, this S-filtered algebra structure is naturally extended to a Z^r+1-filtered algebra structure on QH^∗(G/B).

That is, we need to show FaFb⊂F_a+b for any a,b∈S. In order to prove it, we need to assume the following Key Lemma first.

Key Lemma. Let u∈W and γ ∈R⁺.

a) If ℓ(us_γ) =ℓ(u) + 1, then we havegr(us_γ)≤gr(u) +gr(s_i) when- ever the fundamental weight χ_i satisfies hχi, γ^∨i 6= 0.