Quantum double Schubert polynomials represent Schubert classes

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.


Introduction
Let H * (Fl), H T (Fl), QH * (Fl), and QH T (Fl) be the ordinary, T -equivariant, quantum, and T -equivariant quantum cohomology rings of the variety Fl = Fl n of complete flags in C n where T is the maximal torus of GL n . All cohomologies are with Z coefficients. The flag variety Fl n has a stratification by Schubert varieties X w , labeled by permutations w ∈ S n , which gives rise to Schubert bases for each of these rings.
This paper is concerned with the problem of finding polynomial representatives for the Schubert bases in a ring presentation of these (quantum) cohomology rings. These ring presentations are due to Borel [Bo] in the classical case, and Ciocan-Fontanine [Cio], Givental and Kim [GK] and Kim [Kim] in the quantum case. This is a basic problem in classical and quantum Schubert calculus.
This problem has been solved in the first three cases: the Schubert polynomials are known to represent Schubert classes in H * (Fl n ) by work of Bernstein, Gelfand, and Gelfand [BGG] and Lascoux and Schützenberger [LS]; the double Schubert polynomials, also due to Lascoux and Schützenberger, represent Schubert classes in H T (Fl n ) (see for example [Bi]); and the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov [FGP] represent Schubert classes in QH * (Fl). These polynomials are the subject of much research by combinatorialists and geometers and we refer the reader to these references for a complete discussion of these ideas. Our first main result (Theorem 5) is that the quantum double Schubert polynomials of [KM, CF] represent equivariant quantum Schubert classes in QH T (Fl). Anderson and Chen [AC] have announced a proof of this result using the geometry of Quot schemes. Now let SL n /P be a partial flag variety, where P denotes a parabolic subgroup of SL n . In non-quantum Schubert calculus, the functorality of (equivariant) cohomology implies that the (double) Schubert polynomials labeled by minimal length coset representatives again represent Schubert classes in H * (SL n /P ) or H T (SL n /P ). This is not the case in quantum cohomology. Ciocan-Fontanine [Cio2] solved the corresponding problem in QH * (SL n /P ), extending Fomin, Gelfand, and Postnikov's work to the parabolic case. Here we introduce the parabolic quantum double Schubert polynomials. We show that these polynomials represent Schubert classes in the torus-equivariant quantum cohomology QH T (SL n /P ) of a partial flag variety. Earlier, Mihalcea T.L. was supported by NSF grant DMS-0901111, and by a Sloan Fellowship. M.S. was supported by NSF DMS-0652641 and DMS-0652648. [Mi2] had found polynomial representatives for the Schubert basis in the special case of the equivariant quantum cohomology of the Grassmannian.
as algebras over Z, S, Z[q], and S[q] respectively. The presentation of H * (Fl) is a classical result due to Borel. The presentations of QH * (Fl) and QH T (Fl) are due to Ciocan-Fontanine [Cio], Givental and Kim [GK] and Kim [Kim].
2.2. Schubert bases. Let X w = B − wB/B ⊂ Fl be the opposite Schubert variety, where w ∈ W = S n is a permutation, B ⊂ SL n is the upper triangular Borel and B − the opposite Borel. The ring H * (Fl) (resp. H T (Fl)) has a basis over Z (resp. S) denoted [X w ] (resp. [X w ] T ) associated with the Schubert varieties.
Given three elements u, v, w ∈ W and an element of the coroot lattice β ∈ Q ∨ one may define a genus zero Gromov-Witten invariant c w uv (β) ∈ Z ≥0 (see [GK, Kim]) and an associative ring QH * (Fl) with Z[q]-basis {σ w | w ∈ W } (called the quantum Schubert basis) such that Similarly there is a basis of a ring QH T (Fl) with S[q]-basis given by the equivariant quantum Schubert classes σ w T , defined using equivariant Gromov-Witten invariants, which are elements of S.
We shall use the following characterization of QH T (Fl) and its Schubert basis {σ w T | w ∈ W } due to Mihalcea [Mi]. Let Φ + be the set of positive roots and ρ the half sum of positive roots. For w ∈ W define Let ω i (a) = a 1 + · · · + a i ∈ S be the fundamental weight. We write A n w and B n w to emphasize that the computation pertains to Fl = SL n /B. Theorem 1. [Mi,Corollary 8.2] For w ∈ S n and 1 ≤ i ≤ n − 1 a Dynkin node, the equivariant quantum Schubert classes σ w T satisfy the equivariant quantum Chevalley-Monk formula Moreover these structure constants determine the Schubert basis {σ w T | w ∈ S n } and the ring QH T (SL n /B) up to isomorphism as Z[q 1 , . . . , q n−1 ; a 1 , . . . , a n ]-algebras.
and s a i is the operator that exchanges a i and a i+1 . Since the operators ∂ i = ∂ a i satisfy the braid relations one may define ∂ w = ∂ i1 · · · ∂ i ℓ where w = s i1 · · · s i ℓ is a reduced decomposition. For w ∈ S n define the double Schubert polynomial S w (x; a) [LS] and the quantum double Schubert polynomialS w (x; a) ∈ S[x; q] [KM, CF] by where w (n) 0 ∈ S n is the longest element. 1 Note that it is equivalent to define S w (x; a) by setting the q i variables to zero inS w (x; a).
Let S ∞ = n≥1 S n be the infinite symmetric group under the embeddings i n : S n → S n+1 that add a fixed point at the end of a permutation. Due to the stability property [KM]S in(w) (x; a) = S w (x; a) for w ∈ S n , the quantum double Schubert polynomialsS w (x; a) are well-defined for w ∈ S ∞ . Similarly, S w (x; a) is well-defined for w ∈ S ∞ .
For w ∈ S ∞ , define the (resp. quantum) Schubert polynomial S w (x) = S w (x; 0) (resp. S w (x) =S w (x; 0)) by setting the a i variables to zero in the (resp. quantum) double Schubert polynomial. Note that S w (x), S w (x; a),S w (x), andS w (x; a) are all homogeneous of degree ℓ(w). The original definition of quantum Schubert polynomial in [FGP] is different. However their definition and the one used here, are easily seen to be equivalent [KM], due to the commutation of the divided differences in the a variables and the quantization map θ of [FGP], which we review in §3.3. Lemma 2. [Mac] For w ∈ S ∞ , the term that is of highest degree in the x variables and then is the reverse lex leading such term, in any of Lemma 3. [Mac] There is a bijection from S ∞ to the set of tuples (c 1 , c 2 , . . . ) of nonnegative integers, almost all zero, given by w → code(w). Moreover it restricts to a bijection from S n to the set of tuples (c 1 , . . . , c n ) such that 0 ≤ c i ≤ n − i for all 0 ≤ i ≤ n.
. . , x n ; q 1 , . . . , q n−1 ; a 1 , . . . , a n ]/J q,a n . Proof. Since in each case the highest degree part of the j-th ideal generator in the x variables is e j (x 1 , . . . , x n ), any polynomial may be reduced modulo the ideal until its leading term in the x variables is x γ where γ = (γ 1 , . . . , γ n ) ∈ Z n ≥0 with γ i ≤ n − i for 1 ≤ i ≤ n. But these are the leading terms of the various kinds of Schubert polynomials.
given by By [FGP, CF, KM] we have (this is the definition of quantum Schubert polynomial in [FGP]) 3.4. Cauchy formulae. The double Schubert polynomials satisfy [Mac] where v ≤ L w denotes the left weak order, defined by ℓ(wv −1 ) + ℓ(v) = ℓ(w). For a geometric explanation of this identity see [And]. We have [KM, CF] The first equality follows from the divided difference definitions ofS w (x; a) and S w (x; a) and the commutation of the divided differences in the a-variables with quantization. The second equality follows from quantizing (9).
We require the explicit formulae for Schubert polynomials indexed by simple reflections.
Lemma 6. We have Proof. Since s i is an i-Grassmannian permutation with associated partition consisting of a single box, its Schubert polynomial is the Schur polynomial [Mac]

Chevalley-Monk rules for Schubert polynomials
The Chevalley-Monk formula describes the product of a divisor class and an arbitrary Schubert class in the cohomology ring H * (Fl). The goal of this section is to establish the Chevalley-Monk rule for quantum double Schubert polynomials. The Chevalley-Monk rules for (double, quantum, quantum double) Schubert polynomials should be viewed as product rules for the cohomologies of an infinite-dimensional flag ind-variety Fl ∞ of type A ∞ with Dynkin node set Z >0 and simple bonds between i and i + 1 for all i ∈ Z >0 .
For w ∈ S ∞ let A w and B w be defined as before Theorem 1 but using the infinite set of positive roots Φ + and letting ρ = (0, −1, −2, . . . ). To distinguish between the finite and limiting infinite cases, we denote by A n w the set A w for w ∈ S n which uses the positive roots of SL n . 4.1. Schubert polynomials.
Proposition 7. [Che, Mo]. For w ∈ S n and 1 ≤ Proposition 8. [Mac] For w ∈ S ∞ and i ∈ Z >0 the Schubert polynomials satisfy the identity in Z[x] given by Example 1. It is necessary to take a large-rank limit (n ≫ 0) to compare Propositions 7 and 8. Let n = 2. We have [X s1 ] 2 = 0 in H * (Fl 2 ) since A s1 = ∅ for SL 2 . Lifting to polynomials we have S 2 s1 = x 2 1 = S s2s1 since A s1 = {α 13 }, which is not a positive root for SL 2 . Note that S s2s1 ∈ J 2 and s 2 s 1 ∈ S 3 \ S 2 . In H * (Fl n ) for n ≥ 3 we have [X s1 ] 2 = [X s2s1 ].

Quantum Schubert polynomials.
Proposition 9. [FGP] For w ∈ S n and 1 ≤ i ≤ n − 1, in QH * (Fl n ) we have Proposition 10. [FGP] For i ∈ Z >0 and w ∈ S ∞ the quantum Schubert polynomials satisfy the identity in Z[x; q] given bỹ 4.3. Double Schubert polynomials.
Proposition 11. [KK] [Rob]. For w ∈ S n and 1 ≤ The following is surely known but we include a proof for lack of a known reference.
Proposition 12. For w ∈ S ∞ and i ∈ Z >0 , the double Schubert polynomials satisfy the identity in Z[x, a] given by Proof. Fix w ∈ S ∞ . We observe that the set A w is finite. Let N be large enough so that all appearing terms make sense for S N . By [Bi] under the isomorphism (2) (17) holds modulo an element f ∈ J a N . We and only finitely many are nonzero. Choose n ≥ N large enough so that v ∈ S n and b v ∈ Z[a 1 , . . . , a n ] for all v with b v = 0. Applying Proposition 11 again for H T (Fl n ) we deduce that f ∈ J a n . By Lemma 4 it follows that f = 0 as required.
4.4. Quantum double Schubert polynomials. Theorem 1 gives the equivariant quantum Chevalley-Monk rule for QH T (Fl n ). We cannot use the multiplication rule in Theorem 1 directly because we are trying to prove that the quantum double Schubert polynomials represent Schubert classes. We deduce the following product formula by cancelling down to the equivariant case which was proven above.
Proposition 13. The quantum double Schubert polynomials satisfy the equivariant quantum Chevalley-Monk rule in Z[x, q, a]: for all w ∈ S ∞ and i ≥ 1 we havẽ Proof. Starting with (17) and using Lemma 6 and (9) we have Quantizing and rearranging, we have Therefore to prove (18) it suffices to show that To prove (19) it suffices to show that there is a bijection A → B given by (v, α) → (vs α , α).

Proof of Theorem 5
Let I a be the ideal in Z[x, a] generated by e p i (x) − e p i (a) for p ≥ n and i ≥ 1, and a i for i > n. Let J a ⊂ Z[x, a] be the Z[a]-submodule spanned by S w (x; a) for w ∈ S ∞ \ S n and a i S u (x; a) for i > n and any u ∈ S ∞ . We shall show that I a = J a . Let c i,p = s p+1−i · · · s p−2 s p−1 s p ∈ S ∞ \ S p be the cycle of length i. We note that the family {e p i (x) − e p i (a) | 1 ≤ i ≤ p} is unitriangular over Z[a] with the family {S ci,p (x; a) | 1 ≤ i ≤ p}. Since Z[x, a] = u∈S∞ Z[a]S u (x; a), to show that I a ⊂ J a it suffices to show that S u (x; a)S ci,p (x, a) ∈ J a for all p ≥ n, i ≥ 1, and u ∈ S ∞ . But this follows from the fact that the product of S u (x, a)S v (x; a) is a Z[a]-linear combination of S w (x; a) where w ≥ u and w ≥ v.
Let K be the ideal in Z[x, a] generated by a i for i > n. Then I a /K has Z[a 1 , . . . , a n ]-basis given by standard monomials e I with i r > 0 for some r ≥ n, while J a /K has Z[a 1 , . . . , a n ]-basis given by S w (x; a) for w ∈ S ∞ \ S n . The quotient ring Z[x, a]/K has Z[a 1 , . . . , a n ]-basis given by all standard monomials e I for I = (i 1 , i 2 , . . . ) with 0 ≤ i p ≤ p for all p ≥ 1 and almost all i p zero, and also by all double Schubert polynomials S w (x; a) for w ∈ S ∞ . But the standard monomials e I with i n = i n+1 = · · · = 0 are in graded bijection with the S w (x; a) for w ∈ S n . It follows that I a = J a by graded dimension counting.
Let J qa ∞ be the ideal of Z[x, q, a] generated by E p i − e p i (a) for all i ≥ 1 and p ≥ n, together with q i for i ≥ n and a i for i > n. We wish to show that For this it suffices to show that To prove (21) it suffices to show that θ(e I (e p i (x) − e p i (a))) ∈ J qa ∞ for standard monomials e I , i ≥ 1 and p ≥ n.
To apply θ to this element we must express e I e p i in standard monomials. The only nonstandardness that can occur is if i p > 0. In that case one may use [FGP,(3.2)]: e p i e p j = e p i−1 e p j+1 + e p j e p+1 i − e p i−1 e p+1 j+1 .
Note that ultimately the straightening of e I e p i into standard monomials, only changes factors of the form e q k for k ≥ 1 and q ≥ p. Let E I = r≥1 E r ir for I = (i 1 , i 2 , . . . ). If we consider E I (E p i − e p i (a)) and use [FGP,(3.6)] . to rewrite it into quantized standard monomials, we see that the two straightening processes differ only by multiples of q p , q p+1 , etc. Therefore (22) holds and (20) follows. The ring Z[x, q, a]/J qa ∞ has a Z[q 1 , . . . , q n−1 ; a 1 , . . . , a n ]-basis given byS w (x; a) for w ∈ S n . This follows from Lemmata 2 and 3. Moreover this basis satisfies the equivariant quantum Chevalley-Monk rule for SL n by Proposition 13. By Theorem 1 there is an isomorphism of Z[q 1 , . . . , q n−1 ; a 1 , . . . , a n ]-algebras QH T (SL n /B) → Z[x, q, a]/J qa ∞ . Moreover, σ w T andS w (x; a) (or rather, its preimage in QH T (Fl)) are related by an automorphism of QH T (Fl). But the Schubert divisor class σ si T is (by definition) represented by a usual double Schubert polynomial S si (x; a) =S si (x; a) (Lemma 6) in Kim's presentation, and these divisor classes generate QH T (Fl) over Z[q 1 , . . . , q n−1 ; a 1 , . . . , a n ]. Thus the automorphism must be the identity, completing the proof.
6. Parabolic case 6.1. Notation. Fix a composition (n 1 , n 2 , . . . , n k ) ∈ Z k >0 with n 1 + n 2 + · · · + n k = n. Let P ⊂ SL n (C) be the parabolic subgroup consisting of block upper triangular matrices with block sizes n 1 , n 2 , . . . , n k . Then SL n /P is isomorphic to the variety of partial flags in C n with subspaces of dimensions N j := n 1 + n 2 + · · · + n j for 0 ≤ j ≤ k. Denote by W P the Weyl group for the Levi factor of P and W P the set of minimum length coset representatives in W/W P . For every w ∈ W there exists unique elements w P ∈ W P and w P ∈ W P such that w = w P w P ; moreover this factorization is length-additive. Let w 0 ∈ W be the longest element and let w 0 = w P 0 w 0,P so that w 0,P ∈ W P is the longest element.
Following [AS] [Cio2] let D = D P be the n × n matrix with entries x i on the diagonal, −1 on the superdiagonal, and entry (N j+1 , N j−1 + 1) given by − The polynomial G j i is homogeneous of degree i. For w ∈ W P , we define the parabolic quantum double Schubert polynomialS P w (x; a) bỹ Example 3. Continuing Example 2 we havẽ The parabolic quantum double Schubert polynomialsS P w (x; a) have specializations similar to the quantum double Schubert polynomials. Let w ∈ W P .
(1) We define the parabolic quantum Schubert polynomials by the specializationS P w (x) = S P w (x; 0) which sets a i = 0 for all i. In Lemma 18 it is shown that these polynomials coincide with those of Ciocan-Fontanine [Cio2], whose definition uses a parabolic analogue of the quantization map of [FGP].
(2) Setting q i = 0 for all i one obtains the double Schubert polynomial S w (x; a).
(3) Setting both a i and q i to zero one obtains the Schubert polynomial S w (x).
) generated by the elements e n i (x), (resp. e n i (x) − e n i (a), G k i , G k i − e n i (a)) for 1 ≤ i ≤ n. The aim of this section is to establish (4) of the following theorem.
(1) There is an isomorphism of Z-algebras [BGG, LS] (2) There is an isomorphism of S-algebras [Bi] Here [X w ], [X w ] T , σ P,w , and σ P,w T , denote the Schubert bases for their respective cohomology rings for w ∈ W P .
The isomorphism (24) is due to [Kim3]. We shall establish (25), namely, that under this isomorphism, the parabolic quantum double Schubert polynomials are the images of parabolic equivariant quantum Schubert classes.
6.3. Stability of parabolic quantum double Schubert polynomials. The following Lemma can be verified by direct computation and induction.
Suppose w ∈ W P is such that w(r) = r for N k−1 < r ≤ n. Let w (p,q) 0 be the minimum length coset representative of the longest element in S p+q /(S p × S q ) and let w P− 0 be the minimum length coset representative of the longest element in S N k−1 /(S n1 × · · · × S n k−1 ). We have the length- ). Using the above Lemma repeatedly we have . This means that if we append a block of size m to n • and append m fixed points to w ∈ W P , the parabolic quantum double Schubert polynomial remains the same. 6.4. Stable parabolic quantization. This section follows [Cio2]. Consider an infinite sequence of positive integers n • = (n 1 , n 2 , . . . ). Let N j = n 1 + · · · + n j for j ≥ 1. Let W = S ∞ = n≥1 S n be the infinite symmetric group (under the inclusion maps S n → S n+1 that add a fixed point at the end), W P the subgroup of W generated by s i for i / ∈ {N 1 , N 2 , . . . }, W P the set of minimum length coset representatives in W/W P , etc. Let Y P be the set of tuples of partitions λ • = (λ (1) , λ (2) , . . . ) such that λ (j) is contained in the rectangle with n j+1 rows and N j columns and almost all λ (j) are empty. Define the standard monomial by Nj r (x). The following is a consequence of [Cio2] by taking a limit.
We also observe that By Z[a]-linearity it defines a Z[q, a]-module automorphism of Z[x, q, a], also denoted θ P .
Proof. Let w ∈ W P . We may computeS P w (x; a) by working with respect to (n 1 , n 2 , . . . , n k ) for some finite k; the result is independent of k by the previous subsection. Then (27) is an immediate consequence of the commutation of θ P and the divided difference operators in the a variables. Equation (26) follows from (27) by setting the a variables to zero. 6.5. Cauchy formula. Keeping the notation of the previous subsection, we have Proposition 19. For all w ∈ W P we havẽ 6.6. Parabolic Chevalley-Monk rules. Fix n • = (n 1 , n 2 , . . . , n k ) with k j=1 n j = n and let P be the parabolic defined by n • and so on. Let Q ∨ P , Φ + P , and ρ P be respectively the coroot lattice, the positive roots, and the half sum of positive roots, for the Levi factor of P . Let η P : Q ∨ → Q ∨ /Q ∨ P be the natural projection. Let π P : W → W P be the map w → w P where w = w P w P and w P ∈ W P and w P ∈ W P . Define the sets of roots A n P,w = {α ∈ Φ + \ Φ + P | ws α ⋗ w and ws α ∈ W P } (29) The following is Mihalcea's characterization of QH T (SL n /P ) which extends Theorem 1.
Proposition 21. For w ∈ W P and i such that i ∈ {N 1 , N 2 , . . . , } (that is, s i ∈ W P ), the parabolic quantum double Schubert polynomials satisfy the identitỹ Sketch of proof. The proof proceeds entirely analogously to that of Proposition 13, starting with the Chevalley-Monk formula for double Schubert polynomials (applied in the special case that w ∈ W P and s i ∈ W P ) and reducing to the following identity: For this it suffices to show that there is a bijection (v, α) → (π P (vs α ), α) and inverse bijection (u, α) → (π P (us α ), α) between the sets To establish this bijection we use [LamSh,Lemma 10.14], which asserts that elements α ∈ B P,w automatically satisfy the additional condition ℓ(ws α ) = ℓ(w) + 1 − α ∨ , 2ρ .
Given the finite sequence (n 1 , . . . , n k ) and parabolic subgroup P , consider the extension (n 1 , . . . , n k , 1, 1, 1, . . . ) by an infinite sequence of 1s. We use the notation P ∞ to label the corresponding objects. That is, W P ∼ = W P∞ is the subgroup of S ∞ , W P∞ is the set of minimum-length coset representatives in S ∞ /W P∞ , and so on.
Let R P = Z[x, a] WP ∞ and R q P = Z[x, q, a] WP ∞ , where x = (x 1 , x 2 , . . . ), q = (q 1 , q 2 , . . .), and q 1 , q 2 , . . . , q k−1 are identified with the quantum parameters in Theorem 14(4). Define J a P,∞ to be the ideal of R P generated by g p i (x) − e Np i (a) for i ≥ 1 and p ≥ k and by a i for i > n. Note that for p = k + j with j ≥ 0 we have g p i (x) − e Np i (a) = e n+j i (x) − e n+j i (a). Let I a P,∞ be the Z[a]-submodule of R P spanned by S w (x; a) for w ∈ W P∞ \ S n , and by a i S u (x; a) for i > n and any u ∈ W P∞ . We claim that I a P,∞ is an ideal. This follows easily from the fact that the only double Schubert polynomials occurring in the expansion of a product S w (x; a)S v (x; a) lie above w and v in Bruhat order. But then it follows from Theorem 14(2) and dimension counting that I a P,∞ = J a P,∞ . Define J qa P,∞ to be the ideal of R q P generated by G p i − e Np i (a) for p ≥ k and i ≥ 1, and a i for i > n, and q i for i ≥ k. We claim thatS P w (x; a) ∈ J qa P,∞ for w ∈ W P∞ \ S n . This would follow from the definitions if we could establish that θ P∞ (J a P,∞ ) ⊂ J qa P,∞ . Since θ P∞ is trivial on the equivariant variables a 1 , a 2 , . . ., it suffices to show that θ P∞ (g P λ • (g p i (x)−e Np i (a))) ∈ J qa P,∞ , for each i ≥ 1, p ≥ k, and each standard monomial g P λ • . To apply θ P∞ to g P λ • g p i we must first standardize this monomial. This can be achieved by using a parabolic version of the straightening algorithm of [FGP,Proposition 3.3]. The only nonstandard part of g P λ • g p i is the possible presence of a factor g p j g p i . This can be standardised using only the (non-parabolic) relation [FGP,(3.2)] -any factors g k ′ m for k ′ < k in g P λ • g p i are not modified. Similarly, the product G P λ • G p i can be standardized using the following variant of [FGP,Lemma 3.5] which can be deduced from [Cio2,(3.5)] : We note that when p ≥ k, this relation only involves quantum variables q k , q k+1 , . . . . Thus modulo q k , q k+1 , . . . , the straightening relation for g P λ • g p i and for G P λ • G p i coincide. It follows that θ P∞ (g P λ • (g p i (x) − e Np i (a)) = G P λ • (G p i − e Np i (a)) mod J qa P,∞ , and thusS P w (x; a) ∈ J qa P,∞ for w ∈ W P∞ \ S n . But R qa P,∞ /J qa P,∞ has rank |W P | over Z[q 1 , . . . , q k−1 , a 1 , . . . , a n ] and so it follows that J qa P,∞ is spanned byS P w (x; a) ∈ J qa P,∞ for w ∈ W P∞ \ S n together with q iS P u (x; a) for i ≥ k, a iS P u (x; a) for i > n and any u ∈ W P∞ . Theorem 14(4) now follows from Proposition 21 and the determination of QH T (SL n /P ) in Theorem 20.