SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND WEYL GROUP INVARIANTS

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1 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND WEYL GROUP INVARIANTS HARRY TAMVAKIS In memory of Alain Lascoux Abstract. We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds. 1. Introduction The theory of Schubert polynomials due to Lascoux and Schützenberger [LS1] provides a canonical set of polynomial representatives for the Schubert classes on complete type A flag manifolds. The classical Schur polynomials are identified with the Schubert polynomials representing the classes which pull back from Grassmannians. There are natural analogues of these objects for the symplectic and orthogonal groups: the Schubert polynomials of Billey and Haiman [BH], and the theta and eta polynomials of Buch, Kresch, and the author [BKT1, BKT2], respectively. We also have double versions of the aforementioned polynomials, which represent the Schubert classes in the torus-equivariant cohomology ring, and in the setting of degeneracy loci of vector bundles (see [L2, F1, IMN1] and [KL, L1, W, IM, TW, AF2, T4], respectively). The goal of this work is to study the relation between these two families of polynomials from the point of view of Weyl group symmetries, following the program set out in [LS1, LS2, M2] in Lie type A. The key observation is that the theta and eta polynomials of a fixed level n form a basis of the Weyl group invariants in the associated ring of Schubert polynomials (Propositions 6 and 14). In this introduction, for simplicity, we review the story in type A, and describe its analogue in type C, in the case of single polynomials, leaving the extensions to the double case and the orthogonal Lie types B and D to the main body of the paper. Let S := k S k be the group of permutations of the positive integers which leave all but a finite number of them fixed. For any n 1, let S (n) denote the set of those permutations = ( 1, 2,...) in S such that n+1 < n+2 <. If X n := (x 1,...,x n )isafamilyofncommutingindependentvariables,thenthesingle Schubert polynomials S of Lascoux and Schützenberger [LS1], as ranges over Date: September 22, Mathematics Subject Classification. Primary 14M15; Secondary 05E05, 13A50, 14N15. Key words and phrases. Schubert polynomials, theta and eta polynomials, Weyl group invariants, flag manifolds, equivariant cohomology. The author was supported in part by NSF Grant DMS

2 2 HARRY TAMVAKIS S (n), form a Z-basis of the polynomial ring Z[X n ]. If M n := GL n /B denotes the complete type A flag manifold over C, then there is a surjective ring homomorphism ρ n : Z[X n ] H (M n ) which maps the polynomial S to the cohomology class [X ] of a codimension l( ) Schubert variety X in M n, if S n, and to zero, otherwise. The Weyl group S n acts on Z[X n ] by permuting the variables, and the subring Z[X n ] Sn of S n -invariants is the ring Λ n of symmetric polynomials in x 1,...,x n. We have Λ n = Z[e 1 (X n ),...,e n (X n )], where e i (X n ) denotes the i-th elementary symmetric polynomial. The kernel of ρ n is the ideal IΛ n of Z[X n ] generated by the homogeneous elements of positive degree in Λ n. We therefore have IΛ n = ZS = e 1 (X n ),...,e n (X n ) S (n) S n and recover the Borel presentation [Bo] of the cohomology ring H (GL n /B) = Z[X n ]/IΛ n. Any Schubert polynomial S which lies in Λ n is equal to a Schur polynomial s λ (X n ) indexed by a partition λ = (λ 1,...,λ n ) associated to the Grassmannian permutation, and these elements form a Z-basis of Λ n. One knows that (1) s λ (X n ) = A(x λ+δn 1 ) / A(x δn 1 ) where A := S n ( 1) l( ) is the alternating operator on Z[X n ], x α denotes x α1 1 xαn n for any integer vector exponent α, and δ k := (k,...,1,0) for every k 0. Equation (1) may be identified with the Weyl character formula for GL n. Alternatively, one has the (dual) Jacobi-Trudi identity (2) s λ (X n ) = det(e λ i +j i(x n )) i,j = i<j(1 R ij )e λ (X n ), where λ is the conjugate partition of λ, e ν := e ν1 e ν2 for any integer vector ν, and the R ij are Young s raising operators, with R ij e ν := e Rijν (see [M1, I.3]). If 0 denotes the longest permutation in S n, then the divided difference operator gives a Λ 0 n -linear map Z[X n ] Λ n, and the equation f,g = 0(fg) defines a scalar product on Z[X n ], with values in Λ n. The Schubert polynomials S } Sn form a basis for Z[X n ] as a Λ n -module, and satisfy an orthogonality property under this product, which corresponds to the natural duality pairing on H (M n ). The above narrative admits an exact analogue for the symplectic group. Let c := (c 1,c 2,...) be a sequence of commuting variables, and set c 0 := 1 and c p = 0 for p < 0. Consider the graded ring Γ which is the quotient of the polynomial ring Z[c] modulo the ideal generated by the relations p (3) c 2 p +2 ( 1) i c p+i c p i = 0, for all p 1. i=1 The ring Γ is isomorphic to the ring of Schur Q-functions [M1, III.8] and to the stable cohomology ring of the Lagrangian Grassmannian, following [P, J]. Let W k denote the hyperoctahedral group of signed permutations on the set 1,...,k}. For each k 1, we embed W k in W k+1 by adjoining the fixed point k + 1, and set W := k W k. For any n 0, let W (n) denote the set of those elements w = (w 1,w 2,...) in W such that w n+1 < w n+2 <. The type C single Schubert polynomials C w of Billey and Haiman [BH], as w ranges over W (n), form a

3 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 3 Z-basis of the ring Γ[X n ] (Proposition 1). If M n := Sp 2n /B denotes the complete symplectic flag manifold over C, then there is a surjective ring homomorphism π n : Γ[X n ] H (M n ) which maps the polynomial C w to the class [X w ] of a codimension l(w) Schubert variety X w in M n, if w W n, and to zero, otherwise. There is a natural action of the Weyl group W n on Γ[X n ] which extends the S n action on Z[X n ] (see 2.1). The subring Γ[X n ] Wn of W n -invariants is the ring Γ (n) of theta polynomials of level n (Proposition 6). The ring Γ (n) was defined in [BKT1, 5.1] as Γ (n) := Z[ n c 1, n c 2,...], where p n c p := c p j e j (X n ), for p 1. j=0 The kernel of π n is the ideal IΓ (n) of Γ[X n ] generated by the homogeneous elements of positive degree in Γ (n). We therefore have (4) IΓ (n) = ZC w = n c 1, n c 2,... w W (n) W n and obtain (Corollary 3) a canonical isomorphism H (Sp 2n /B) = Γ[X n ]/IΓ (n). Following [BKT1], any Schubert polynomial C w which lies in Γ (n) is equal to a theta polynomial n Θ λ indexed by an n-strict partition λ = λ(w) associated to the n-grassmannian element w, and these polynomials form a Z-basis of Γ (n). The polynomial n Θ λ was defined in [BKT1] using the raising operator formula (5) (1 R ij ) (1+R ij ) 1 ( n c) λ, n Θ λ := i<j (i,j) C(λ) where ( n c) ν := n c ν1 n c ν2 and C(λ) denotes the set of pairs (i,j) with i < j and λ i +λ j > 2n+j i. This is the symplectic version of equation (2). There is also a symplectic analogue of formula (1). Let w 0 denote the longest element of W n, define ŵ := ww 0 and consider the multi-schur Pfaffian (6) ν(ŵ) Q λ(ŵ) := i<j 1 R ij 1+R ij ν(ŵ) c λ(ŵ) where ν(ŵ) and λ(ŵ) are certain partitions associated to ŵ (see (13) and Definition 2). We then have (Theorem 2) ( (7) n Θ λ(w) = ( 1) n(n+1)/2 A ν(ŵ) Q λ(ŵ) )/A ( x δn+δn 1) where A := w W n ( 1) l(w) w is the alternating operator on Γ[X n ]. In the special case when w S, with λ = λ(w), equation (7) becomes n Θ λ = ( 1) n(n+1)/2 A ( δ n 1 Q δn+δ n 1+λ )/ A ( x δ n+δ n 1 ). The maximal divided difference operator w0 gives a Γ (n) -linear map Γ[X n ] Γ (n), and the equation f,g = w0 (fg) defines a scalar product on Γ[X n ], with values in Γ (n). The Schubert polynomials C w } w Wn form a basis for Γ[X n ] as a Γ (n) -module(corollary4), andsatisfyanorthogonalitypropertyunderthisproduct (Proposition 9), which corresponds to the duality pairing on H (M n ). A similar scalar product in the finite case was introduced and studied in [LP1].

4 4 HARRY TAMVAKIS As mentioned earlier, we provide analogues of most of the above facts for the double Schubert, theta, and eta polynomials. Our main new results (Theorems 1 and 3) are the double versions of equation (4), which exhibit natural generators for the kernel of the geometrization map of [IMN1, 10] from the (stable) ring of double Schubert polynomials to the equivariant cohomology ring of the corresponding (finite dimensional) symplectic or orthogonal flag manifold. This is done by using an idea from [T1, Lemma 1] together with the transition equations of [B, IMN1] to write the Schubert polynomials in this kernel as an explicit linear combination of these generators, which is important in applications. The double versions of formula (7) rely on the equality of the multi-schur Pfaffian (6) and its orthogonal analogue with certain double Schubert polynomials (Propositions 4 and 12). This latter fact is an extension of [IMN1, Thm. 1.2], which may be deduced from the (even more general) Pfaffian formulas of Anderson and Fulton [AF1]. We give an independent treatment here, using the right divided difference operators. This paper is organized as follows. In Section 2 we recall the type C double Schubert polynomials and the geometrization map π n from Γ[X n,y n ] to the equivariant cohomology ring of Sp 2n /B, and obtain canonical generators for the kernel of π n. In Section 3 we define the statistics ν(w) and λ(w) of a signed permutation w and prove the analogue of formula (7) for double theta polynomials. Section 4 examines some related facts about single type C Schubert polynomials, including the scalar product with values in the ring Γ (n) of W n -invariants. Sections 5, 6 and 7 study the corresponding questions in the orthogonal Lie types B and D. I dedicate this article to the memory of Alain Lascoux, whose warm personality and vision about symmetric functions and Schubert polynomials initially assisted, and subsequently inspired my research, from its beginning to the present day. 2. Double Schubert polynomials of type C 2.1. Preliminaries. We recall the type C double Schubert polynomials of Ikeda, Mihalcea, and Naruse [IMN1], employing the notational conventions of the introduction, which are similar to those used in [AF1]. These differ from the conventions for Schubert polynomials used in [BH, IMN1] and our papers [T2, T3, T5], in that the ring Γ is realized using the generators c p and relations (3) among them, instead of the formal power series known as Schur Q-functions, which are not required in the present work. Let X := (x 1,x 2,...) and Y := (y 1,y 2,...) be two lists of commuting independent variables, and set X n := (x 1,...,x n ) and Y n := (y 1,...,y n ) for each n 1. The Weyl group for the root system of type C n is the group of signed permutations on the set 1,...,n}, denoted W n. The group W = k W k is generated by the simple transpositions s i = (i,i + 1) for i 1 together with the sign change s 0, which fixes all j 2 and sends 1 to 1 (a bar over an integer here means a negative sign). For w W, we denote by l(w) the length of w, which is the least integer l such that we can write w = s i1 s il for some indices i j 0. There is an action of W on Γ[X,Y] by ring automorphisms, defined as follows. The simple reflections s i for i 1 act by interchanging x i and x i+1 while leaving all the remaining variables fixed. The reflection s 0 maps x 1 to x 1, fixes the x j for

5 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 5 j 2 and all the y j, and satisfies (8) s 0 (c p ) := c p +2 p x j 1 c p j for all p 1. j=1 For each i 0, define the divided difference operator x i on Γ[X,Y] by x 0f := f s 0f 2x 1, x i f := f s if x i x i+1 for i 1. Consider the ring involution ω : Γ[X,Y] Γ[X,Y] determined by ω(x j ) = y j, ω(y j ) = x j, ω(c p ) = c p and set y i := ω x i ω for each i 0. The double Schubert polynomials C w = C w (X,Y) for w W are the unique family of elements of Γ[X,Y] such that (9) i x C wsi if l(ws i ) < l(w), C w = y i 0 otherwise, C C siw if l(s i w) < l(w), w = 0 otherwise, for all i 0, together with the condition that the constant term of C w is 1 if w = 1, and 0 otherwise. For any w W, the corresponding (single) Billey-Haiman Schubert polynomial of type C is C w (X) := C w (X,0). It is known that the C w (X) for w W form a Z-basis of Γ[X] = Γ[x 1,x 2,...], and the C w (X,Y) for w W form a Z[Y]-basis of Γ[X,Y] = Γ[x 1,y 1,x 2,y 2,...]. See [IMN1] for further details, noting that the polynomial called C w (z,t; x) in op. cit., which is a formal power series in the x variables, would be the polynomial denoted by C w (z,t) here. In the sequel, for every i 0, we set i := i x. For any w W, we define a divided difference operator w := i1 il, for any choice of indices (i 1,...,i l ) such that w = s i1 s il and l = l(w). According to [IMN1, Prop. 8.4], for any u,w W, we have C (10) u C w (X,Y) = wu 1(X,Y) if l(wu 1 ) = l(w) l(u), 0 otherwise The set W (n) and the ring Γ[X n,y n ]. For every n 1, let W (n) := w W w n+1 < w n+2 < }. Proposition 1. The C w (X) for w W (n) form a Z-basis of Γ[X n ]. Proof. We have that C w (X) Γ[X n ] if and only if m C w (X) = 0 for all m > n if and only if w W (n). Suppose that f Γ[X n ] is a polynomial which is not in the Z-span of the C w (X), w W (n). Then f can be written as an integer linear combination of Schubert polynomials (11) f(x) = w e w C w (X) where there is at least one w with e w 0 and w / W (n). Hence for some m > n we have m C w = C wsm, and since m f = 0, we obtain from (11) a nontrivial linear dependence relation among the Schubert polynomials, which is a contradiction. This proves that the C w (X) for w W (n) span Γ[X n ], and therefore the result. Proposition 2. The C w (X,Y) for w W (n) form a Z[Y]-basis of Γ[X n,y].

6 6 HARRY TAMVAKIS Proof. The C w (X,Y) for w W are linearly independent over Z[Y]. By Proposition 1 we know that the C w (X) for w W (n) form a Z-basis of Γ[X n ]. According to [IMN1, Cor. 8.10], we have C w (X,Y) = S u 1( Y)C v (X) uv=w summed over all factorizations uv = w with l(u)+l(v) = l(w) and u S. Since the term of lowest y-degree in the sum is C w (X), the proposition follows. Let C w = C w(x n,y n ) be the polynomial obtained from C w (X,Y) by setting x j = y j = 0 for all j > n. Corollary 1. The C w for w W (n) form a Z[Y n ]-basis of Γ[X n,y n ] The geometrization map π n. The double Schubert polynomials C w(x,y) for w W n represent the equivariant Schubert classes on the symplectic flag manifold. Let e 1,...,e 2n } denote the standard symplectic basis of E := C 2n and let F i be the subspace spanned by the first i vectors of this basis, so that F n i = F n+i for 0 i n. Let B denote the stabilizer of the flag F in the symplectic group Sp 2n = Sp 2n (C), and let T be the associated maximal torus in the Borel subgroup B. The symplectic flag manifold given by M n := Sp 2n /B parametrizes complete flags E in E with E n i = E n+i for 0 i n. The T-equivariant cohomology ring H T(M n ) is defined as the cohomology ring of the Borel mixing space ET T M n. The ring H T(M n ) is a Z[Y n ]-algebra, where y i is identified with the equivariant Chern class c T 1(F n+1 i /F n i ), for 1 i n. The Schubert varieties in M n are the closures of the B-orbits, and are indexed by the elements of W n. Concretely, any w W n corresponds to a Schubert variety X w = X w (F ) of codimension l(w) by X w := E M n dim(e r F s ) d w (r,s) r,s} where d w (r,s) is the rank function specified in [T3, 6.2]. Since X w is stable under the action of T, we obtain an equivariant Schubert class [X w ] T := [ET T X w ] in H T(M n ). Following [IMN1], there is a surjective homomorphism of graded Z[Y n ]-algebras π n : Γ[X n,y n ] H T(M n ) such that π n (C w) = [X w ] T if w W n, 0 if w W (n) W n. WeletE i denotethei-thtautological vectorvectorbundleoverm n, for0 i 2n. The geometrization map π n is defined by the equations π n (x i ) = c T 1(E n+1 i /E n i ) and π n (c p ) = c T p(e E n F n ) for 1 i n and p 1. Here c T p(e E n F n ) denotes the degree p component of the total Chern class c T (E E n F n ) := c T (E)c T (E n ) 1 c T (F n ) 1.

7 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS The kernel of the map π n. For any integer j 0 and sequence of variables Z = (z 1,z 2,...), define the elementary and complete symmetric functions e j (Z) and h j (Z) by the generating series (1+z i t) = e j (Z)t j i=1 j=0 and (1 z i t) 1 = h j (Z)t j, respectively. Ifr 1thenwe let e r j (Z) := e j(z 1,...,z r )and h r j (Z) := h j(z 1,...,z r ) denote the polynomials obtained from e j (Z) and h j (Z) by setting z j = 0 for all j > r. Let e 0 j (Z) = h0 j (Z) = δ 0j, where δ 0j denotes the Kronecker delta, and for r < 0, define h r j (Z) := e r j (Z) and e r j (Z) := h r j (Z). For any k,k Z, define the polynomial k c k p = k c k p (X,Y) by Definition 1. Let k c k p := p p i=0 j=0 i=1 c p j i h k i (X)h k j ( Y). Γ (n) := Z[ n c n 1, n c n 2,...] and let IΓ (n) be the ideal of Γ[X n,y n ] generated by the homogeneous elements in Γ (n) of positive degree. For any p Z, define ê p Z[X n,y n ] by ê p = ê p (X n /Y n ) := p=0 p=0 j=0 i+j=p e i (X n )h j ( Y n ). We then have the generating function equation ( ) ( n ) n (12) n c n p t p = c p t p ê j t j = c p t p 1+x j t 1+y j t. Lemma 1. We have IΓ (n) Kerπ n. Proof. It suffices to show that n c n p Kerπ n for each p 1. We give two proofs of this result. A straightforward calculation using Chern roots shows that p=0 π n ( k c k p ) = c T p(e E n k F n+k ) for all p,k,k Z. Since E = F 2n, we deduce the lemma from this and the properties of Chern classes. Our second proof proceeds as follows. There is a canonical isomorphism of Z[Y n ]- algebras H T(M n ) = Z[A n,y n ]/K n, where A n := (a 1,...,a n ) and K n is the ideal of Z[A n,y n ] generated by the differences e i (A 2 n) e i (Yn) 2 for 1 i n (see for example [F2, 3]). The geometrization map π n satisfies π n (x j ) = a j for 1 j n, while π n (c p ) := e i (A n )h j (Y n ), p 0. i+j=p j=0 j=1

8 8 HARRY TAMVAKIS A straightforward calculation using (12) gives ( ) n π n n c n p t p 1 a 2 j = t2 1 yj 2t2. p=0 On the other hand, we have n 1 a 2 n n j t2 1 y 2 = 1+ (1 a 2 jt 2 ) (1 yjt 2 2 ) h p (Yn)t 2 2p j=1 jt2 j=1 j=1 p=0 ( n ) = 1+ ( 1) r (e r (A 2 n) e r (Yn))t 2 2r h p (Yn)t 2 2p. r=0 The result follows immediately. For any three integer vectors α,β,ρ Z l, which we view as integer sequences with finite support, define ρ c β α := ρ1 c β1 ρ 2 α 1 c β2 α 2. Given any raising operator R = i<j Rnij, let R ρ c β α := ρ c β Rα. Finally, define the multi-schur Pfaffian ρ Q β α by (13) j=1 ρ Q β α := R ρ c β α, where the raising operator expression R is given by R := i<j 1 R ij 1+R ij. The name multi-schur Pfaffian is justified because ρ Q β α is equal to the Pfaffian of the r r skew-symmetric matrix with } } ρi,ρ 1 R12 j Q βi,βj ρ α i,α j = i,ρ j c βi,βj α 1 i<j r 1+R i,α j 12 p=0 1 i<j r above the main diagonal, following Kazarian [K]; here r = 2 l/2. We adopt the convention that when some superscript(s) are omitted, the corresponding indices are equal to zero. Thus k c p := k c 0 p, c k p := 0 c k p, ρ c α := i ρi c 0 α i, ρ Q α := R ρ c α, Q α := R c α, etc. If λ = (λ 1 > λ 2 > > λ l ) is a strict partition of length l, let w λ be the corresponding increasing Weyl group element, so that the negative components of w λ are exactly ( λ 1,..., λ l ). Lemma 2. If λ is a strict partition with λ 1 > n, then C w λ (X n,y n ) IΓ (n). Proof. For p 0, recall that c n p := 0 c n p Γ[Y n ], so that we have ( ) n c n p t p = c p t p (1 y j t). p=0 p=0 One has the generating function equation ( )( ) n (14) n c n p t p c n p ( t) p = e j (X n )t j. It follows from (14) that p=0 p=0 j=1 j=0 n c n p n c n p 1c n 1 + +( 1) p c n p = e p (X n )

9 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 9 for each p 0. We deduce that c n p IΓ (n) when p n+1. According to [IMN1, Thm. 6.6], we have (15) C wλ (X,Y) = Q β(λ) λ = R c β(λ) λ in Γ[X,Y], where β(λ) is equal to the integer vector (1 λ 1,...,1 λ l ). It follows that (16) C w λ (X n,y n ) = Q β(λ) λ where Q β(λ) λ is obtained from Q β(λ) λ by setting y j = 0 for all j > n. The conclusion of the lemma now follows immediately by expanding the raising operator formula (15) for the double Schur Q-polynomial Q β(λ) λ and noting that in each monomial of the result, the first factor is equal to c n p for some p > n. Lemma 3. For any w W W n, we have C w IΓ (n). Proof. For any positive integers i < j we define reflections t ij S and t ij,t ii W by their right actions (...,w i,...,w j,...)t ij = (...,w j,...,w i,...), (...,w i,...,w j,...)t ij = (...,w j,...,w i,...), and (...,w i,...)t ii = (...,w i,...), and let t ji := t ij. Let w be an element of W. According to [B, Lemma 2], if i j, then l(wt ij ) = l(w) + 1 if and only if (i) w i < w j, (ii) in case i < j, either w i < 0 or w j < 0, and (iii) there is no p < i such that w j < w p < w i, and no p < j such that w i < w p < w j. The group W acts on the polynomial ring Z[y 1,y 2,...] in the usual way, with s i for i 1 interchanging y i and y i+1 and leaving all the remaining variables fixed, and s 0 mapping y 1 to y 1 and fixing the y j with j 2. Let w W be nonincreasing, let r be the last positive descent of w, let s := max(i > r w i < w r ), and let v := wt rs. Following [IMN1, Prop. 6.12], the double Schubert polynomials C u = C u (X,Y) obey the transition equations (17) C w = (x r v(y r ))C v + C vtir + 1 i<r l(vt ir )=l(w) i 1 l(vt ir )=l(w) C vtir in Γ[X,Y]. The recursion (17) terminates in a Z[X,Y]-linear combination of elements C wν (X,Y) for strict partitions ν. For any w W, let µ(w) denote the strict partition whose parts are the elements of the set w i : w i < 0}. Clearly we have µ(w) = µ(wu) for any u S. In equation (17), we therefore have µ(v) = µ(vt ir ) = µ(w). Moreover, condition (i) above shows that the parts of µ(vt ir ) are greater than or equal to the parts of µ(w). In particular, if µ(w) 1 > n, then µ(vt ir ) 1 > n. Assume first that w W n+1 W n. If w i = n 1 for some i n+1, we use the transition recursion (17) to write C w as a Z[X n,y n ]-linear combination of elements C w ν for strict partitions ν with ν 1 > n. Lemma 2 now implies that C w IΓ (n). Next, we consider the case when w i = n+1 for some i n. Let v 2,...,v n } := w 1,...,ŵ i,...,w n }

10 10 HARRY TAMVAKIS with v 2 > > v n, and define and u := (n+1,v 2,...,v n,w n+1 ) W n+1 u := us 0 = (n+1,v 2,...,v n,w n+1 ). We have C u IΓ (n) from the previous case, and, using (9), that 0 (C u ) = C u. For any integer i [0,n 1], it is easy to check that s i ( n c n p) = n c n p, and therefore that i ( n c n p) = 0. It follows that i ( IΓ (n) ) IΓ (n) for all indices i [0,n 1]. Since C u IΓ (n), we deduce that C u IΓ (n). There exists a permutation σ S n such that u = wσ and l(σ) = l(u) l(w). Using (10), we have C w = σ (C u), and hence conclude that C w lies in IΓ (n). Finally assume w / W n+1 and let m be minimal such that w W m. Then w W m W m 1, so the above argument applies with m 1 in place of n. The result now follows by setting x j = y j = 0 for all j > n. Theorem 1. Let J n := w W (n) W n Z[Y n ]C w. Then we have (n) (18) IΓ = Jn = Z[Y n ]C w = Kerπ n. w W W n We have a canonical isomorphism of Z[Y n ]-algebras H T(Sp 2n /B) = Γ[X n,y n ]/ IΓ (n). Proof. Lemmas 1 and 3 imply that (19) J n Z[Y n ]C w IΓ (n) Kerπ n. w W W n We claim that Kerπ n J n. Indeed, if f Kerπ n then by Corollary 1 we have a unique expression (20) f = w W (n) f w C w for some coefficients f w Z[Y n ]. Applying the map π n to (20) and using (19) gives w W n f w [X w ] T = 0. Since the equivariant Schubert classes [X w ] T are a Z[Y n ]-basis of H T(Sp 2n /B), we deduce that f w = 0 for all w W n. It follows that f J n. Remark 1. (a)itiseasytoshowthat n c n p liesin w W W n Z[Y n ]C w foralln,p 1. This follows from the fact that n c n p = n+p 1 c (n+p 1)+1 p p (X n,y n ) is equal to the (restricted) double Schubert polynomial C w (p), where w (p) := s n s n+1 s n+p 1. In fact, C w(p) (X,Y) is equal to the double theta polynomial n+p 1 Θ p (X,Y) of level n+p 1, for every p 1 (the definition of n+p 1 Θ p (X,Y) is recalled in (30)). (b) The equality w W W n Z[Y n ]C w = Kerπ n in (18) was proved earlier in [IMN1, Prop. 7.7] using different methods.

11 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 11 For any elements f,g Γ[X n,y n ], we define the congruence f g to mean f g IΓ (n). We claim that any element of Γ[X n,y n ] is equivalent under to a polynomial in Z[X n,y n ]. Indeed, we have that ( )( ) (21) n c n p t p c p ( t) p = ê j t j. It follows from (21) that (22) p=0 p=0 j=0 n c n p n c n p 1c 1 + +( 1) r c p = ê p for each p 0. The relation (22) implies that c p ( 1) p ê p (X n /Y n ), for all p 0, proving the claim. We deduce that c α ( 1) α e α (X n /Y n ) for each integer sequence α, and that Q λ = Q λ (c) ( 1) λ Qλ (X n /Y n ) for any partition λ. Here Q λ (X n /Y n ) denotes a supersymmetric Q-polynomial, namely Q λ (X n /Y n ) := R ê λ (X n /Y n ). The reader can compare this with the remarks in [T3, 7.3] Partial symplectic flag manifolds. Following [Bo, KK], there is a standard way to generalize the presentation in Theorem 1 to partial flag manifolds Sp 2n /P, where P is a parabolic subgroup of Sp 2n. The parabolic subgroups P containing B correspond to sequences a : a 1 < < a p of nonnegative integers with a p < n. The manifold Sp 2n /P parametrizes partial flags of subspaces 0 E 1 E p E = C 2n with dim(e j ) = n a p+1 j for each j [1,p] and E p isotropic. AsequenceaasabovealsoparametrizestheparabolicsubgroupW P ofw n,which is generated by the simple reflections s i for i / a 1,...,a p }. Let Γ[X n,y n ] WP be the subring of elements in Γ[X n,y n ] which are fixed by the action of W P, that is, Γ[X n,y n ] WP = f Γ[X n,y n ] s i (f) = f, i / a 1,...,a p }, i < n}. Since the action of W n on Γ[X n,y n ] is Z[Y n ]-linear, we see that Γ[X n,y n ] WP is a Z[Y n ]-subalgebra of Γ[X n,y n ]. Let W P W (n) denote the set W P := w W (n) l(ws i ) = l(w)+1, i / a 1,...,a p }, i < n}. Proposition 3. We have (23) Γ[X n,y n ] WP = w W P Z[Y n ]C w. Proof. If f is any element in Γ[X n,y n ] WP, Corollary 1 implies that we have an expansion f = w W (n) d wc w for some coefficients d w in Z[Y n ]. If u / W P, there is an index i < n with i / a 1,...,a p } and l(us i ) = l(u) 1. We have i f = 0, and on the other hand, using (9), we see that i f = u d u C us i summed over all u such that l(us i ) = l(u) 1. It follows that d u = 0 for all such u, and thus that Γ[X n,y n ] WP is contained in the sum on the right hand side of (23).

12 12 HARRY TAMVAKIS For the reverse inclusion, is suffices to show that C w Γ[X n,y n ] WP for all w W P. The definition of W P implies that we have i C w = 0, or equivalently s i C w = C w, for all i < n with i / a 1,...,a p }. The result follows. Corollary 2. There is a canonical isomorphism of Z[Y n ]-algebras H T(Sp 2n /P) = Γ[X n,y n ] WP / IΓ (n) P where P is the ideal of Γ[X n,y n ] WP generated by the homogeneous elements in Γ (n) of positive degree. IΓ (n) Proof. It is well known that the canonical projection map h : G/B G/P induces an injection h : H T(Sp 2n /P) H T(Sp 2n /B) on equivariant cohomology rings, with the image of h equal to the W P -invariants in H T(Sp 2n /B) (see for example [KK, Cor. (3.20)]). In fact, since the C w for w W P W n represent the equivariant Schubert classes coming from H T(Sp 2n /P), we deduce from Proposition 3 that the restriction of the geometrization map π n : Γ[X n,y n ] H T(Sp 2n /B) to the W P - invariants induces a surjection Γ[X n,y n ] WP H T(Sp 2n /P). The result follows easily from this and Theorem Divided differences and double theta polynomials 3.1. Preliminaries. For every i 0, the divided difference operator i = x i on Γ[X,Y] satisfies the Leibnitz rule (24) i (fg) = ( i f)g +(s i f) i g. Observe that ω( k c r p) = r c k p, for all k,r,p Z. By applying this, it is easy to prove the following dual versions of [IM, Lemmas 5.4 and 8.2]. Lemma 4. Suppose that k,p,r Z. For all i 0, we have k 1 i ( k c r c r p 1 if k = ±i, p) = 0 otherwise. Lemma 5. Suppose that k 0 and r 1. Then we have We also require the following lemma. k c r p = k+1 cp r+1 (x k+1 +y r ) k cp 1 r+1. Lemma 6 ([IM], Prop. 5.4). Suppose that k,r 0 and p > k +r. Then we have (k 1,...,k,k,...,k l ) Q (r1,..., r, r,...,r l) (p 1,...,p,p,...,p l ) = The shape of a signed permutation. We proceed to define certain statistics of an element of W. Definition 2. Let w W be a signed permutation. The strict partition µ = µ(w) is the one whose parts are the absolute values of the negative entries of w, arranged in decreasing order. The A-code of w is the sequence γ = γ(w) with γ i := #j > i w j < w i }. We define a partition δ = δ(w) whose parts are the non-zero entries γ i arranged in weakly decreasing order, and let ν(w) := δ(w) be the conjugate of δ. Finally, the shape of w is the partition λ(w) := µ(w)+ν(w).

13 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 13 It is easy to see that w is uniquely determined by µ(w) and γ(w), and that λ(w) = l(w). The shape λ(w) of an element w W is a natural generalization of the shape of a permutation, as defined in [M2, Chp. 1]. Example 1. (a) For the signed permutation w = [3,2,7,1,5,4,6] in W 7, we obtain µ = (7,6,3,1), γ = (2,3,0,1,2,1,0), δ = (3,2,2,1,1), ν = (5,3,1), and λ = (12,9,4,1). (b) An element w W is n-grassmannian if l(ws i ) > l(w) for all i n, while a partition λ is called n-strict if all its parts λ i greater than n are distinct. Following [BKT1, 6.1], these two objects are in one-to-one correspondence with each other. If w is an n-grassmannian element of W, then λ(w) is the n-strict partition associated to w, in the sense of op. cit. Lemma 7. If i 1, w W, and γ = γ(w), then γ i > γ i+1 w i > w i+1 l(ws i ) = l(w) 1. If any of the above conditions hold, then γ(ws i ) = (γ 1,...,γ i 1,γ i+1,γ i 1,γ i+2,γ i+3,...). Proof. These follow immediately from [M2, (1.23) and (1.24)]. Let β(w) be the sequence defined by β(w) i = min(1 µ(w) i,0) for each i 1. For each n 1, let w (n) 0 := [1,...,n] denote the longest element in W n. Proposition 4. Suppose that m > n 0 and w W m is an n-grassmannian element. Set ŵ := ww (n) 0. Then we have Cŵ(X,Y) = ν(ŵ) Q β(ŵ) λ(ŵ) in the ring Γ[X n,y m 1 ]. In particular, if w S m, then we have Cŵ(X,Y) = δn 1 Q (1 wn,...,1 w1) δ n+δ n 1+λ(w). Proof. We first consider the case when w S m. We have w = (a 1,...,a n,d 1,...,d r ) where r = m n, 0 < a 1 < < a n and 0 < d 1 < < d r. If λ := λ(w) then λ j = n+j d j = m d j (r j) for 1 j r. Let w (m) 0 := [1,...,m] be the longest element in W m. Then we have w (m) 0 = ŵv 1 v r, where l(w (m) 0 ) = l(ŵ)+ r j=1 l(v j) and v j = s n+j 1 s 1 s 0 s 1 s dj 1, 1 j r. One knows from [IMN1, Thm. 1.2] that the equation C (m) w (X,Y) = δm 1 Q δm 1 δ 0 m+δ m 1 holds in Γ[X,Y]. It follows from this and (10) that ) ( ) (25) Cŵ = v1 vr (C (m) w = v1 δm 1 vr Q δm 1 δ 0 m+δ m 1. Using Lemmas 4 and 5, for any p,q Z with p 1, we obtain (26) p ( p c p q ) = p 1 c p q 1 = p c 1 p q 1 (x p +y p ) p 1 c 1 p q 2.

14 14 HARRY TAMVAKIS Let ǫ j denote the j-th standard basis vector in Z m. The Leibnitz rule and (26) imply that for any integer vector α = (α 1,...,α m ), we have ( δm 1 ) p c δm 1 α = δ m 1 c δm 1+ǫm p α ǫ m p (x p +y p ) δm 1 ǫm p c δm 1+ǫm p α 2ǫ m p. We deduce from this and Lemma 6 that p δ m 1 Q δm 1 δ m+δ m 1 = δm 1 Q δm 1+ǫm p δ m+δ m 1 ǫ m p = (m 1,...,1,0) Q (1 m,..., 1 p,1 p,1 p,2 p,..., 1,0) (2m 1,...,2p+3,2p,2p 1,2p 3,...,1). Iterating this calculation for p = d r 1,...,1 gives ( 1 dr 1)C w (m) 0 Since 0 ( 0 c 0 1) = 1 c 0 0 = 1, it follows that = (m 1,...,1,0) Q (1 m,..., dr,2 dr,3 dr,..., 1,0,0) (2m 1,...,2d r+1,2d r 2,2d r 4,...,2,1). ( 0 1 dr 1)C (m) w = (m 1,...,1) Q (1 m,..., dr,2 dr,3 dr,..., 1,0) (2m 1,...,2d. 0 r+1,2d r 2,2d r 4,...,2) Applying Lemma 4 alone m 1 times now gives vr (C (m) w ) = δm 2 Q (1 m,..., dr,2 dr,3 dr,..., 1,0) (2m 2,...,2d 0 r,2d r 3,...,1) = δm 2 Q (1 m,..., 1 d r,...,0) δ m 1+δ m 2+1 m dr. Finally, we use (25) and repeat the above calculation r 1 more times to get where and Cŵ = δn 1 Q ρ δ n+δ n 1+ξ ρ = (1 m,..., 1 d r,..., 1 d 1,..., 1,0) = (1 w n,...,1 w 1 ) ξ = r 1 m dj (r j) = j=1 r 1 n+j dj = j=1 r 1 λj = λ(w). j=1 Next consider the general case. Let p > n and suppose that w = (a 1,...,â i1,...,â ip n,...,a p, a ip n,..., a i1,d 1,...,d r ) where r = m p, 0 < a 1 < < a p and 0 < d 1 < < d r. If and û := uw (p) 0, then u := (a 1,...,a p,d 1,...,d r ) (27) û = ŵv p n v 1 where v j = s p j s ij j+2s ij j+1 for 1 j p n. Now Cû is known by the previous case, and Cŵ = v p n v 1 (Cû). The proof is now completed by induction, using Lemma 7. The key observation is the following: Suppose that u 0 = û > u 1 > > u d = ŵ is the sequence of coverings in the right weak Bruhat order corresponding to the factorization(27), sothatu i+1 = u i s ri withl(u i+1 ) = l(u i ) 1foreachi [0,d 1]. Thenifγ := γ(u i ), wehaveγ ri+1 = γ ri 1. ThereforeLemma7impliesthatγ(u i+1 ) has two equal entries in positions r i and r i +1. Moreover, γ(u j ) is a partition for all j [0,d], and hence ν(u j ) is the conjugate of γ(u j ).

15 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 15 Remark 2. The work of Anderson and Fulton [AF1] associates a partition λ to certain triples of l-tuples of integers which define a class of symplectic degeneracy loci. Theshapeλ(w)ofanelementw W indefinition2(anditsevenorthogonal counterpart in Definition 5) is consistent with op. cit. In particular, Propositions 4 and 12 follow from the more general formulas for double Schubert polynomials which are established in [AF1]. We give here an alternative proof, using [IMN1, Thm. 1.2] and the right divided difference operators Double theta polynomials and alternating sums. Let n 0 and w W be an n-grassmannian element. Let λ = λ(w) be the n-strict partition which corresponds to w, define a sequence β(λ) = β i (λ)} i 1 by w n+i +1 if w n+i < 0, (28) β i (λ) := w n+i if w n+i > 0, and a set of pairs C(λ) by (29) C(λ) := (i,j) N N 1 i < j and w n+i +w n+j < 0} (this agrees with the set C(λ) given in the introduction). The double theta polynomial n Θ λ (X,Y) of [TW, W] is defined by (30) (1 R ij ) (1+R ij ) 1 ( n c) β(λ) n Θ λ (X,Y) := i<j (i,j) C(λ) λ. In the above formula, for any integer sequence α = (α 1,α 2,...), we let ( n c) α β(λ) := i n c βi(λ) α i, and the raising operators R ij act by R ij ( n c) α β(λ) := ( n c) β(λ) R. Note ijα that n Θ λ (X,Y) lies in Γ[X n,y] for any n-strict partition λ. To be precise, the polynomial n Θ λ (X,Y) is the image of the double theta polynomial Θ λ (c t) of [TW] (with k = n) in the ring Γ[X n,y]. Let A : Γ[X n,y] Γ[X n,y] be the operator given by A(f) := ( 1) l(w) w(f). w W n Let w 0 = w (n) 0 denote the longest element in W n and set ŵ := ww 0. Theorem 2. For any n-strict partition λ, we have ( ) (31) n Θ λ (X,Y) = ν(ŵ) w0 Q β(ŵ) λ(ŵ) ( )/ (32) = ( 1) n(n+1)/2 A ν(ŵ) Q β(ŵ) λ(ŵ) A ( x δn+δn 1). Proof. We deduce from (9) that the double Schubert polynomial C w (X,Y) satisfies (33) C w (X,Y) = w0 (Cŵ(X,Y)). The equality (31) follows from (33), Proposition 4, and the fact, proved in [IM, Thm. 1.2], that C w (X,Y) = n Θ λ (X,Y) in the ring Γ[X n,y]. To establish the equality (32), recall from [D, Lemma 4] and [PR, Prop. 5.5] that we have 1 w0 (f) = ( 1) n(n+1)/2 2 n x 1 x n (x 2 i x 2 j) A(f). 1 i<j n

16 16 HARRY TAMVAKIS On the other hand, it follows from [PR, Cor. 5.6(ii)] that w0 (x δn+δn 1 ) = ( 1) n(n+1)/2 and hence that A(x δn+δn 1 ) = 2 n x 1 x n (x 2 i x 2 j). 1 i<j n The proof of (32) is completed by using these two equations in (31). 4. Single Schubert polynomials of type C In this section, we work with the single type C Schubert polynomials C w (X). The entire section is inspired by [LS1, LS2, M2] and [PR, LP1] Theta polynomials as Weyl group invariants. Let χ : Γ[X n ] Z be the homomorphism defined by χ(c p ) = χ(x j ) = 0 for all p,j. In other words, χ(f) is the constant term of f, for each polynomial f Γ[X n ]. Proposition 5. For any f Γ[X n ], we have f = w W (n) χ( wf)c w (X n ). Proof. By Proposition 1 and linearity, it is only necessary to verify this when f is a Schubert polynomial C v (X n ), v W (n). In this case, it follows from the properties of Schubert polynomials in 2.1 that χ( w (C v (X n ))) is equal to 1 when w = v and equal to zero, otherwise. Following [BKT1, 5.1], let Γ (n) := Z[ n c 1, n c 2,...] be the ring of theta polynomials of level n. Notice that the elements denoted by ϑ r (x;y) in loc. cit., with k replaced by n, correspond to the generators n c r here. According to [BKT1, Thm. 2], the single theta polynomials n Θ λ = n Θ λ (X) for all n-strict partitions λ form a Z-basis of Γ (n). In the next result, the Weyl group W n acts on the ring Γ[X n ] in the usual way. Proposition 6. The ring Γ (n) is equal to the subring Γ[X n ] Wn of W n -invariants in Γ[X n ]. Proof. We have g Γ[X n ] Wn if and only if s i g = g for all i [0,n 1] if and only if i g = 0 for 0 i n 1. Suppose that f Γ[X n ] Wn and employ Proposition 1 to write (34) f(x n ) = w W (n) a w C w (X n ). Applying the divided differences i for i [0,n 1] to (34) and using (9), we deduce that a w = 0 for all w W (n) such that l(ws i ) < l(w) for some i [0,n 1]. Therefore, f is in the Z-span of those C w (X n ) for w W (n) with l(ws i ) > l(w) for all i [0,n 1]. These are exactly the n-grassmannian elements w in W. According to [BKT1, Prop. 6.2], for any such w, we have C w (X n ) = n Θ λ(w) (X) in Γ[X n ]. It follows that f is a Z-linear combination of theta polynomials of level n, and hence that f Γ (n). The converse is clear, since i h = 0 for all i [0,n 1] and h Γ (n).

17 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 17 Example 2. It follows from Proposition 6 that Γ (n) Z[X n ] = Z[X n ] Wn = Z[e 1 (X 2 n),...,e n (X 2 n)] where Xn 2 := (x 2 1,...,x 2 n). This can also be seen directly, using the identities p ( n c p ) 2 +2 ( 1) i ( n c p+i )( n c p i ) = e p (Xn) 2 i=1 for all p 0 (compare with [BKT1, Eqn. (19)]). Let IΓ (n) = n c 1, n c 2,... be the ideal of Γ[X n ] generated by the homogeneous elements in Γ (n) of positive degree, and let IΓ (n) P be the corresponding ideal of Γ[X n ] WP. The following result about the cohomology ring of Sp 2n /P is an immediate consequence of Theorem 1, Corollary 2 and the discussion in 2. Corollary 3. There is a canonical ring isomorphism H (Sp 2n /B) = Γ[X n ]/IΓ (n) which maps the cohomology class of the codimension l(w) Schubert variety X w to the Schubert polynomial C w (X), for any w W n. Moreover, for any parabolic subgroup P of Sp 2n, there is a canonical ring isomorphism H (Sp 2n /P) = Γ[X n ] WP /IΓ (n) P. Example 3. The version of Lemma 2 for single polynomials states that if λ is a strict partition of length l and p > max(n,λ 1 ), then Q (p,λ) IΓ (n). We can exhibit this containment more explicitly as follows. For any integer m > n, we have c m = ( 1) j 1 c n m j c j. j=1 This implies that for any integer vector α = (α 1,...,α l ), the equality c (m,α) = ( 1) j 1 c n (m j,α) c j holds, and therefore, by applying the Pfaffian operator R, that (35) Q (p,λ) = ( 1) j 1 Q n (p j,λ) c j. j=1 j=1 It is important to notice that the terms Q (p j,λ) in (35) can be non-zero even when p j < 0. The straightening law for such terms was found by Hoffman and Humphreys. For any integer k, let n(k) := #i λ i > k }, and define the sets A λ := r [0,p 1] r λ i for all i l} and B λ := λ 1,...,λ l }. It follows from [HH, Thm. 9.2] that for any integer k < p, we have ( 1) n(k) Q λ k if k A λ, Q (k,λ) = ( 1) k+n(k) 2Q λ k if k B λ, 0 otherwise,

18 18 HARRY TAMVAKIS where λ k and λ k denote the partitions obtained by adding (resp. removing) a part equal to k (resp. k ) from λ. Applying this in (35), we obtain (36) Q (p,λ) = ( 1) p 1 r+n(r) Q n λ r c p r +2 ( 1) p 1+n(r) Q n λ r c p+r. r A λ r B λ The particular case of (36) when (p,λ) = δ n+1 reads n Q δn+1 = Q n δn c n+1 +2 ( 1) r Q n δn r c n+1+r. r= The ring Γ[X n ] as a Γ (n) -module. Set e p := e p (X n ) for each p Z, and recall that e α := i e α i for any integer sequence α. Let P n denote the set of all strict partitions λ with λ 1 n. Proposition 7. Γ[X n ] is a free Γ (n) -module of rank 2 n n! with basis e λ ( X n )x α λ P n, 0 α i n i, i [1,n]}. Proof. Itiswellknown(seee.g[M2, (5.1 )])thatγ[x n ]isafreeγ[e 1,...,e n ]-module with basis given by the monomials x α with 0 α i n i for i [1,n]. It will therefore suffice to show that Γ[e 1,...,e n ] is a free Γ (n) -module with basis e λ ( X n ) for λ P n. Setting y j = 0 for 1 j n in equation (21) gives ( )( ) E(X n,t) := e p t p = n c p t p c p ( t) p. p=0 Using this and the relations (3), we obtain ( )( ) E(X n,t)e(x n, t) = n c p t r n c p ( t) p and therefore that e 2 p( X n )+2 p=0 p=0 p=0 p=0 p ( 1) i e p+i ( X n )e p i ( X n ) Γ (n) i=1 for each p 1. It follows that the monomials e λ ( X n ) for λ P n generate Γ[e 1,...,e n ] as a Γ (n) -module. It remains to prove that these monomials e λ ( X n ) are linearly independent over Γ (n). We claim that the Schubert polynomials C w (X n ) for w W n are linearly independent over Γ (n). Indeed, suppose that w W n f w C w (X n ) = 0 for some coefficients f w Γ (n), and that v W n is an element of maximal length such that f v 0. Then, by applying (10), we have ( ) 0 = v f w C w (X n ) = f v v (C v (X n )) = f v, w W n which is a contradiction, proving the claim. We have used here the fact that the divided differences i are Γ (n) -linear for each i [0,n 1]. It follows that the Schur Q-polynomials Q λ = Q λ (c) for λ P n are linearly independent over Γ (n) (since these are exactly the Schubert polynomials C w (X n )

19 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 19 which lie in Γ, with w = w λ W n ). But the elements Q λ } and c λ } for λ P n are related by an unitriangular change of basis matrix, and so are the elements c λ } and e λ ( X n )}. It follows that the Q λ for λ P n generate Γ[e 1,...,e n ] as a Γ (n) -module, and hence that the three aforementioned sets each form a basis. Following [PR], for any partition λ, the Q-polynomial is defined by (37) Qλ (X n ) := R e λ (X n ). Corollary 4. The ring Γ[X n ] is a free Γ[X n ] Sn -module with basis S (X)} for S n. The ring Γ[X n ] Sn is a free Γ (n) -module with basis Q λ ( X n )} for λ P n. The ring Γ[X n ] is a free Γ (n) -module on the basis C w (X n )} of single type C Schubert polynomials for w W n, and is also free on the product basis Q λ ( X n )S (X)} for λ P n and S n. Proof. Since Γ[X n ] Sn = Γ[e 1,...,e n ], the first statement follows from Proposition 7 and [M2, (4.11)]. The assertions involving the polynomials Q λ ( X n ) are justified using Proposition 7 and equation (37), and the fact that the Schubert polynomials C w (X n )} for w W n form a basis is also clear A scalar product on Γ[X n ]. Recall that w 0 = [1,...,n] denotes the element of longest length in W n. If f Γ[X n ], then i ( w0 f) = 0 for all i with 0 i n 1. Proposition 6 implies that w0 (f) Γ (n), for each f Γ[X n ]. Definition 3. We define a scalar product, on Γ[X n ], with values in Γ (n), by the rule f,g := w0 (fg), f,g Γ[X n ]. Proposition 8. The scalar product, : Γ[X n ] Γ[X n ] Γ (n) is Γ (n) -linear. For any f,g Γ[X n ] and w W n, we have w f,g = f, w 1g. Proof. The scalar product is Γ (n) -linear, since the same is true for the operator w0. For the second statement, given f,g Γ[X n ], it suffices to show that i f,g = f, i g for 0 i n 1. We have i f,g = w0 (( i f)g) = w0s i i (( i f)g) = w0s i (( i f)( i g)) because s i ( i f) = i f. The expression on the right is symmetric in f and g, hence i f,g = i g,f = f, i g, as required. Proposition 9. Let u,v W n be such that l(u)+l(v) = n 2. Then we have 1 if v = w 0 u, C u (X n ),C v (X n ) = 0 otherwise. Proof. Using (10) and Proposition 8, we obtain C u (X n ),C v (X n ) = u 1 w 0 C w0 (X n ),C v (X n ) = C w0 (X n ), w0uc v (X n ).

20 20 HARRY TAMVAKIS Also l(w 0 u) = l(w 0 ) l(u) = l(v), and we deduce that 1 if v = w 0 u, w0uc v (X n ) = 0 otherwise. Since C w0 (X n ),1 = w0 (C w0 (X n )) = 1, the result follows. Although the elements of the Γ (n) -basis Q λ ( X n )S (X)} of Γ[X n ] do not represent the Schubert classes on the symplectic flag manifold, this product basis is convenient for computational purposes. Indeed, following Lascoux and Pragacz [LP1] (in the finite case), one can identify the dual Γ (n) -basis of Γ[X n ] relative to the scalar product,, by working as shown below. Let 0 = (n,n 1,...,1) denote the permutation of longest length in S n, and define v 0 := w 0 0 = 0w 0. We have (38) w0 = v0 0 = 0 v0. We define a Γ[X n ] Sn -valued scalar product (, ) on Γ[X n ] by the rule (f,g) := 0(fg), f,g Γ[X n ]. According to [M2, (5.12)], the Schubert polynomials S u (X) for u S n satisfy the orthogonality relation (S u (X), 0S u 0( X)) = δ u,u for any u,u S n. Furthermore, define a Γ (n) -valued scalar product, } on Γ[X n ] Sn by the rule f,g} := v0 (fg), f,g Γ[X n ] Sn. According to [PR, Thm. 5.23], for any two partitions λ,µ P n, we have Qλ ( X n ), Q δn µ( X n )} = δ λ,µ, where δ n µ is the strict partition whose parts complement the parts of µ in the set n,n 1,...,1}, and δ λ,µ denotes the Kronecker delta. Observe that (, ) is Γ[X n ] Sn -linear and, } is Γ (n) -linear. Then (38) gives f,g = (f,g)}, for any f,g Γ[X n ], and moreover the orthogonality relation Qλ ( X n )S u (X), Q δn µ( X n )( 0S u 0( X)) = δ u,u δ λ,µ holds, for any u,u S n and λ,µ P n. The reader should compare this to the discussion in [LP1, 1]. 5. Double Schubert polynomials of types B and D 5.1. Preliminaries. Let b := (b 1,b 2,...) be a sequence of commuting variables, and set b 0 := 1 and b p = 0 for p < 0. Consider the graded ring Γ which is the quotient of the polynomial ring Z[b] modulo the ideal generated by the relations p 1 b 2 p +2 ( 1) i b p+i b p i +( 1) p b 2p = 0, for all p 1. i=1

21 SCHUBERT POLYNOMIALS, THETA AND ETA POLYNOMIALS, AND W-INVARIANTS 21 The ring Γ is isomorphic to the ring of Schur P-functions. Following [P], the P-functions map naturally to the Schubert classes on maximal (odd or even) orthogonal Grassmannians. We regard Γ as a subring of Γ via the injective ring homomorphism which sends c p to 2b p for every p 1. The Weyl group for the root system of type B n is the same group W n as the one for type C n. The Ikeda-Mihalcea-Naruse type B double Schubert polynomials B w (X,Y) for w W form a natural Z[Y]-basis of Γ [X,Y]. For any Weyl group element w, the polynomial B w (X,Y) satisfies B w (X,Y) = 2 s(w) C w (X,Y), where s(w) denotes the number of indices i such that w i < 0. The algebraic theory of these polynomials is thus nearly identical to that in type C, provided that one uses coefficients in the ring Z[ 1 2 ]. If B w = B w(x,y) is the polynomial obtained from B w (X,Y) by setting x j = y j = 0 for all j > n, then the B w for w W (n) form a Z[Y n ]-basis of Γ [X n,y n ]. The polynomials B w for w W n represent the equivariant Schubert classes on the odd orthogonal flag manifold SO 2n+1 /B, whose equivariant cohomology ring (with Z[ 1 2 ]-coefficients) is isomorphic to that of Sp 2n/B. For further details, the reader may consult the references [IMN1] and [T3, 6.3.1]. In the rest of this paper we discuss the corresponding theory for the even orthogonal group, that is, in Lie type D, and assume that n 2. The Weyl group W n for the root system D n is the subgroup of W n consisting of all signed permutations with an even number of sign changes. The group W n is an extension of S n by the element s = s 0 s 1 s 0, which acts on the right by (w 1,w 2,...,w n )s = (w 2,w 1,w 3,...,w n ). There is a natural embedding W k W k+1 of Weyl groups defined by adjoining the fixedpointk+1, andwelet W := k Wk. TheelementsofthesetN :=,1,...} index the simple reflections in W. The length l(w) of an element w W is defined as in type C. We define an action of W on Γ [X,Y] by ring automorphisms as follows. The simple reflections s i for i > 0 act by interchanging x i and x i+1 and leaving all the remaining variables fixed. The reflection s maps (x 1,x 2 ) to ( x 2, x 1 ), fixes the x j for j 3 and all the y j, and satisfies, for any p 1, p 1 s (b p ) := b p +(x 1 +x 2 ) x a 1x b 2 c p 1 j. j=0 a+b=j For each i N, define the divided difference operator x i on Γ [X,Y] by x f := f s f x 1 x 2, x i f := f s if x i x i+1 for i 1. Consider the ring involution ω : Γ [X,Y] Γ [X,Y] determined by ω(x j ) = y j, ω(y j ) = x j, ω(b p ) = b p and set y i := ω x i ω for each i N.

22 22 HARRY TAMVAKIS The Ikeda-Mihalcea-Naruse double Schubert polynomials D w = D w (X,Y) for w W are the unique family of elements of Γ [X,Y] satisfying the equations (39) i x D wsi if l(ws i ) < l(w), D w = y i 0 otherwise, D D siw if l(s i w) < l(w), w = 0 otherwise, for all i N, together with the condition that the constant term of D w is 1 if w = 1, and 0 otherwise. The operators i := i x for i N satisfy the same Leibnitz rule (24) as in the type C case, and for any w W, the divided difference operator w is defined as before. For any u,w W, we have D u D w (X,Y) = wu 1(X,Y) if l(wu 1 ) = l(w) l(u), 0 otherwise The set W (n) and the ring Γ [X n,y n ]. It is known that the D w for w W form a Z[Y]-basis of Γ [X,Y]. Let D w = D w(x n,y n ) be the polynomial obtained from D w (X,Y) by setting x j = y j = 0 for all j > n. For every n 1, let W (n) := w W w n+1 < w n+2 < }. Let D w (X) := D w (X,0) denote the single Schubert polynomial. Proposition 10. The D w (X) for w W (n) form a Z-basis of Γ [X n ], and a Z[Y]- basis of Γ [X n,y]. The D w for w W (n) form a Z[Y n ]-basis of Γ [X n,y n ]. Proof. The argument is the same as for the proofs of Propositions 1, 2, and Corollary 1 in The geometrization map π n. The double Schubert polynomials D w(x,y) for w W n represent the equivariant Schubert classes on the even orthogonal flag manifold. Let e 1,...,e 2n } denote the standard orthogonal basis of E := C 2n and let F i be the subspace spanned by the first i vectors of this basis, so that F n i = F n+i for 0 i n. We say that two maximal isotropic subspaces L and L of E are in the same family if dim(l L ) n (mod 2). The orthogonal flag manifold M n parametrizes complete flags E in E with E n i = E n+i for 0 i n, and E n in the same family as e n+1,...,e 2n. Equivalently, E n is in the same family as F n, if n is even, and in the opposite family, if n is odd. We have that M n = SO 2n /B for a Borel subgroup B of the orthogonal group SO 2n = SO 2n (C). If T denotes the associated maximal torus in B, then the T-equivariant cohomology ring H T(M n) is a Z[Y n ]-algebra, where y i is identified with the equivariant Chern class c T 1(F n+1 i /F n i ), for 1 i n. The Schubert varieties in M n are the closures of the B-orbits, and are indexed by the elements of W n. Concretely, any w W n corresponds to a Schubert variety X w = X w (F ) of codimension l(w), which is the closure of the B-orbit X w := E M n dim(e r F s ) = d w(r,s) r,s}, where d w(r,s) denotes the rank function specified in [T3, 6.3.2]. Since X w is stable undertheactionoft, weobtainanequivariant Schubert class[x w ] T := [ET T X w ] in H T(M n). Following [IMN1], there is a surjective homomorphism of graded Z[Y n ]-algebras π n : Γ [X n,y n ] H T(M n)