Structure Constants in equivariant oriented cohomology of flag varieties

We obtain a formula for structure constants of certain variant form of Bott-Samelson classes for equivariant oriented cohomology of flag varieties. Specializing to singular cohomology/K-theory, we recover formulas of structure constants of Schubert classes of Goldin-Knutson, and that of structure constants of Segre-Schwartz-MacPherson classes of Su. We also obtain a formula for K-theoretic stable basis. Our method comes from the study of formal affine Demazure algebra, so is purely algebraic, while the above mentioned results are geometric.


Introduction
Flag varieties G/B are among the most studied varieties in topology and algebraic geometry.They have a cellular decomposition by Schubert cells, whose closures are called Schubert varieties.Schubert varieties are invariant under a torus action and, consequently, their torus-equivariant singular cohomology is spanned as a module by the Schubert classes.
Other classes associated to Schubert varieties in the equivariant singular cohomology H * T and equivariant K-theory K T of the flag variety G/B include Chern-Schwartz-MacPherson (CSM) classes and Motivic Chern (mC) classes, studied in [1,2,17,19,18,20,22].These classes coincide with the corresponding stable bases of Maulik-Okounkov [16] for H * T and K T , of the Springer resolutions.Due to this fact, we always refer to the CSM classes as the cohomological stable basis, and to the mC classes as the K-theoretic stable basis.These classes behave like Schubert classes in their corresponding theories.Roughly speaking, Schubert classes in H * T (G/B) and K T (G/B) are constructed by Demazure operators (also called divided difference operators), and elements of the stable bases are constructed by Demazure-Lusztig operators.All these operators generate various Hecke-type algebras.
Structure constants of Schubert classes are central objects in Schubert calculus, appearing in important questions of representation theory and combinatorics.In [11], the first author and Knutson obtain formulas for the structure constants in H * T (G/B) and K T (G/B) using geometric properties of Bott-Samelson resolutions of Schubert varieties.They pull-back the Schubert classes to the equivariant cohomology (or equivariant K-theory) of Bott-Samelson variety, apply the cup product in this variety, then push-forward back to G/B.In [21], Su generalized this method to the so-called Segre-Schwartz-MacPherson (SSM) classes, a variant form of CSM classes.
We are interested in generalized cohomology theories, called oriented cohomology theories, defined by Levine and Morel [15].These cohomologies are contravariant functors defined on the category of smooth projective varieties over a field k of characteristic 0 to the category of commutative rings, such that for proper maps, there is a push-forward map on cohomology groups.Examples include Chow rings (singular cohomology), K-theory and algebraic cobordism.Chern classes are defined for each oriented cohomology theory h, and there is an associated formal group law F defined over R = h(pt).The machinery works equivariantly as well, resulting in a cohomology theory h T with an associated formal group law F defined over R = h T (pt).
For flag varieties, generalizing work of Kostant and Kumar [13,14] on equivariant singular cohomology and equivariant K-theory of flag varieties, the ring h T (G/B) has a nice algebraic model, constructed in Hoffmann et al. in [12], and studied in [6,7,5] by Calmès, Zainoulline, and the second author.One can define the (formal) Demazure operators X α associated to each simple root α.These operators generate a non-commutative algebra, called the formal affine Demazure algebra D F .It is a free left h T (pt)-module with basis {X Iw | w ∈ W }, where X Iw is, roughly speaking, a product of the operators X α , with I w indicating a reduced word expression for w.
The algebra D F is also a co-commutative co-algebra, where the coproduct comes from the twisted Leibniz rule of the operator X α .Taking the h T (pt)-dual, one obtains a commutative ring D * F , a free h T (pt)-module isomorphic to h T (G/B), together with a dual basis {X * Iw | w ∈ W }. Indeed, for equivariant Chow group/singular cohomology/K-theory, X * Iw coincides, up to various normalizations, to the Schubert class associated with w.Then H * T (G/B) and K T (G/B) are achieved with the same module basis, and a restricted coefficient ring: a polynomial ring for H * T (G/B) and Laurent polynomial ring for K T (G/B).
We notice that the product structure on D * F is obtained by dualizing the coproduct structure of D F .It follows that the structure constants of the basis X * Iw may be deduced from the twisted Leibniz rule of the product X β1 X β2 • • • X β k for a reduced word s β1 • • • s β k of w ∈ W .This is the main idea of the proof of Theorem 3.7, which implies the main result, Theorem 4.1.Specializing h T to equivariant singular cohomology and equivariant K-theory, we recover the formulas of the first author and Knutson in [11].
In the case of H * T (G/B) and K T (G/B), replacing the Demazure operators X α by the Demazure-Lusztig operators T α and τ − α , one obtains the stable bases for H * T (G/B) and K T (G/B), respectively.Both the cohomology stable basis and the K-theory stable basis can be described in an analogous fashion to the story for Schubert classes.That is, the Demazure-Lusztig operators generate a degenerate affine Hecke algebra (for equivariant cohomology) and an affine Hecke algebra (for equivariant K-theory).The dual elements to products of these operators are essentially the cohomological/K-theoretic stable bases, so their respective twisted Leibniz rules result in a formula for the structure constants of stable bases.For instance, for cohomology, we recover the formula of Su [21] (see Remark 6.6).
To work with the Demazure operators X α and Demazure-Lusztig operators T α at the same time, we define a general operator Z α (see §3) in a ring containing D F , which can be specialized to X α and T α .Our main results are Theorems 4.1 and 6.3, which state a formula for structure constants of the basis determined by Z α and apply it to the cohomological stable basis.
The paper is organized as follows: In §2 we recall necessary notation introduced by the second author in [6,7,5].We recall the definition of a Demazure element, the formal affine Demazure algebra, its dual, and relation with h T (G/B).In §3 we prove the twisted Leibniz rule for the operator Z α , which is used to derive the structure constants of the basis Z * Iw in §4.In §5, we specialize our result to Demazure operators in singular cohomology and K-theory, and recover the formulas in [11].In §6 we specialize our result to Demazure-Lusztig operators in singular cohomology, which, as a by-product, recovers the formula due to Su in [21].In §7 we consider Demazure-Lusztig operators in K-theory and obtain a formula for the structure constants of the K-theoretic stable basis.In §8, for equivariant oriented cohomology, we generalize some results of Proposition 4.32], [14, Lemma 2.25]) by relating our formula for structure constants with a restriction formula of Schubert classes.Acknowledgments: The first author was partially supported by National Science Foundation grant DMS-2152312.

Preliminary
We follow notation used in [6,7,5].Let Σ ֒→ Λ ∨ , α → α ∨ be a semi-simple root datum of rank n.That is, Σ is the finite set of roots, Λ is the lattice and Λ ∨ is its dual.Let {α 1 , ..., α n } be the set of simple roots, Σ + and Σ − be the set of positive and negative roots, respectively.
Let W be the Weyl group generated by the associated simple reflections s i := s αi .Denote by ≤ the Bruhat order, and let ℓ(v) be the length of an element v ∈ W .Note that W acts on Λ since it preserves the root system.For each sequence I = (i 1 , ..., i k ) with i j ∈ [n], denote the product s i1 • • • s i k ∈ W by I, in which we keep track both of the concatenated sequence of simple reflections and the resulting element of W .If I is a reduced word expression for the resulting Weyl group element, we say that I is a reduced sequence.Following [11, §1], define the Demazure product subject to the braid relations and s 2 i = s i for all i.Observe that I = I when I is a reduced sequence.When I is a reduced sequence for w, we may denote it by I w and abuse notation by calling it a reduced word for w.Finally, let I rev denote the sequence obtained by reversing the sequence I.
Let F be a formal group law over the coefficient ring R. Examples of formal group laws include the additive formal group law F a = x + y and the multiplicative formal group law F m = x+y−xy.Suppose the root datum together with the formal group law satisfy the regularity condition of [5,Lemma 2.7].This guarantees that all the properties that we use from [6,7,5] hold.Indeed, the regularity condition guarantees that the elements x α , α ∈ Λ defined in S below are non-zero-divisors.In particular, the Demazure operators X α for simple roots α are well defined.
Let G be a split semi-simple linear algebraic group with maximal torus T and a Borel subgroup B. Let the associated root datum of G be Σ ֒→ Λ ∨ , so Λ is the group of characters of T .
Let h be an oriented cohomology theory of Levine and Morel.Roughly speaking, it is a contravariant functor from the category of smooth projective varieties to the category of commutative rings such that there is a push-forward map for any proper map.The Chern classes of vector bundles are defined.Associated to h, there is a formal group law is F defined over R = h(pt).That is, the first Chern class of line bundles over a smooth projective variety X satisfies ).For example, F a (resp.F m ) is associated to the Chow group (or singular cohomology) (resp.K-theory).Both can be extended to the torus equivariant setting.We assume the equivariant cohomology theory h T is Chern-complete over the point for T , that is, the ring h T (pt) is separated and complete with respect to the topology induced by the γ-filtration [5, Definition 2.2].In particular, this includes the completed equivariant Chow ring, the completed equivariant K-theory and equivariant algebraic cobordism.
We will frequently need the special element of Q given by κ λ := 1 x λ + 1 x −λ .Note that κ λ actually belongs to S. Note also that the action of W on Λ induces an action of W on S.
Example 2.1.Two cases of the formal product appear widely in the literature [4, §2].
(1) If F = F a with R = Z, then h is the singular cohomology/Chow groups, and S ∼ = Sym Z (Λ) ∧ ( with x λ → λ) is the completion of the polynomial ring at the augmentation ideal.In this case is the completion of the Laurent polynomial ring at the augmentation ideal.
In this case x −λ = x λ x λ −1 , and κ λ = 1.To obtain equivariant cohomology H * T (X) and equivariant K-theory K T (X), we restrict the coefficient ring to S a = Sym[Λ] and S m = Z[Λ], respectively.
2.1.The operator algebras Q W and D F .This paper is concerned with various divided difference operators acting on h T (G/B), the equivariant oriented cohomology of G/B.To create an algebraic framework for these operators, following [6,7] we localize S at {x α } to create an algebra out of this localization and the Weyl group, as follows.
Let S be the ring described in (1), and let We shall see that Q W acts on its dual space Q * W , which is identified with Q ⊗ S h T (G/B), the cohomlogy of G/B with inverted Chern classes.
We impose a product on Q W by (pδ w )(p ′ δ w ′ ) = pw(p ′ )δ ww ′ , for all p, p ′ ∈ Q, and w, w ′ ∈ W, using the natural W action on Q induced from that on Λ and extending linearly.Note that Q identified with Qδ e is a subring of Q W under this product, where e ∈ W denotes the identity element of W .We routinely abuse notation and write δ α for δ sα , and use 1 = δ e to denote the identity element of This action factors through the coproduct ∆ : In other words, the coproduct structure on Q W is induced from the Q W -action on Q.
For any simple root α we, define the Demazure element X α and the push-pull element Y α in Q W : We observe the relationship Y α = κ α −X α .In particular, if The way Q W acts on Q implies that X α acts in a fashion similar to the Demazure operator defined in [8] (and there denoted D α ).In particular, X α • S ⊂ S and, for any r ∈ R, X α • r = 0 and In particular, X (i) = X αi and Y (i) = Y αi , though we eliminate parentheses when there is no confusion.We write X e := 1 ∈ Q W to indicate X I when I is the empty sequence.
The Demazure and push-pull elements have the following properties: Lemma 2.2.[24, Proposition 3.2] Let α and β be simple roots.The following identities hold in Q W : (1) Furthermore, κ αβ ∈ S by [12, Lemma 6.7].( 5) Suppose s α s β has order m with m = 4 or 6, and I w is a choice of reduced word for w ∈ W . Then Lemma 2.2.( 4)-( 5) imply that the operators X α (and similarly Y α ) do not satisfy braid relations for general F .For F = F a or F = F m , they do; in these cases, the coefficients κ αβ and c Iv all vanish.In general, X Iw and Y Iw depend on the choice of I w due to this failure of braid relations.
For the purposes of this paper, we fix a reduced sequence I w of w for each w ∈ W .While the specific coefficients and calculations regarding X Iw and Y Iw depend on this choice, statements regarding bases and ring phenomena do not.
By construction, {δ v : v ∈ W } form a basis of Q W as a module over Q.In [6], and extended in [7], the second author proves that {X Iv : v ∈ W } and {Y Iv : v ∈ W } also form bases of Q W as a module over Q, and that the change of basis matrix from {X Iv } (or from {Y Iv }) to {δ v } consists of elements of S. In particular, {δ v } are elements of D F .The lower-triangularity of the change of bases matrices is expressed in the following lemma.
Lemma 2.3.[7, Lemma 3.2, Lemma 3.3] For each v ∈ W , choose a reduced decomposition of v and let I v be its corresponding sequence.There exist elements Notice that nonzero coefficients b X w,Iv are elements of S with v ≤ w.Example 2.4.Consider the root datum A 2 , with where w 0 is the longest element and s i is the reflection corresponding to α i for i = 1, 2. We fix the reduced sequence I w0 = (1, 2, 1) for w 0 .For simplicity, let 2.2.The dual operator algebras.The dual Q-module One may think of Q * W as the T -equivariant oriented cohomology of W with the trivial T action, tensored with Q.In particular, and 0 otherwise, extended linearly to all elements of Q * W , and unity 1 = w∈W f w .This product structure is equivalent to the one induced from the coproduct structure (see §4 below).
The ring Iv for each v ∈ W are determined by duality.Under the dual pairing, and thus Fix the reduced sequence w 0 = s 1 s 2 s 1 .The calculations from Example 2.4 imply The following proposition explains the relationship between the algebraic construction above and equivariant oriented cohomology of G/B.
For each reduced sequence where the top horizontal map is the embedding of the S-module into the Q-module Q * W .By specializing the formal group law to F a or F m , respectively, and restricting ) to the equivariant cohomology or equivariant K-theory.The map remains an isomorphism over the corresponding module.From now on we will not distinguish between D * F and h T (G/B).

Generalized Demazure operators and the generalized Leibniz rule
In this section, we generalize the operators X Iv and Y Iv on h T (G/B) to a more general class of elements of Q W , and prove the generalized Leibniz rule for D F acting on Q.We use this result to compute the coproduct structure in Q W , and then the product structure in Q * W .Let {a α , b α ∈ Q : α ∈ Σ} be a set of elements with the property that, for all w ∈ W , , and b α are all invertible in Q.
For any simple root α, define operators Clearly X α and Y α result from Z α as special cases of a α and b α .For any sequence We call Z I generalized Demazure operators.
As before, we choose a reduced word expression I v for each v ∈ W .
Lemma 3.1.The set of generalized Demazure operators {Z Iv } forms a basis of Q W as a module over Q.
Proof.This follows from the fact that b α ∈ Q is invertible for all simple roots α (hence, for all roots α).bα ∈ S for all α.For example, this holds for X α , Y α , but fails for T α considered in Section 6 and 7.This is precisely why the stable basis is only a basis after localization.Lemma 3.3.For any sequence J, define coefficients c J,Iw ∈ Q by (4) Then c J,Iw = 0 unless w ≤ J.
Proof.Clearly Z α = a α + b α δ α has support on {w : w ≤ s α }.An immediate observation of the product in Q W shows inductively that Z J may be expressed as a Q-linear combination of δ v for v ≤ J.
For any v ∈ W and reduced sequence In particular, since b γj is invertible, so is w(b γj ) for any Weyl group element w, and thus the coefficient of δ v in Z Iv is nonzero.
Let A = {w ∈ W : c J,Iw = 0 and w ≤ J}, and assume A is nonempty.Pick v ∈ A to be a maximal element of A in the Bruhat order.By support considerations, the only terms contributing to the coefficient of δ v in ( 4) is c J,Iv Z Iv .Since the coefficient of δ v in Z Iv is a unit, we conclude c J,Iv = 0, contrary to assumption.
The structure constants c J,Iw reflect geometric properties in some special cases (see Section 5).When Z α = X α for all α or Z α = Y α for all α, and F = F a , the coefficients in the sum (4) vanish unless J is a reduced word for w, in which case c J,Iw = 1; this reflects the property that the pushforward map in homology sends the orientation class [BS J ] to the Schubert variety X(w) when J is a reduced word for w.When Z α = X α for all α or Z α = Y α for all α, and F = F m , coefficients vanish except when the Demazure product of J is w, which occurs exactly once and results in c J,Iw = 1.In this case, the K-theoretic pushforward of [O BSJ ] is the structure sheaf of X(w) when w = J.More generally, Z J is an (equivariant) operator whose dual has support only on those fixed points in the Schubert variety X(w), where w = J.
We have the following lemma describing the action of Z α on a product.
Lemma 3.4.For a simple root α, and p, q ∈ Q, we have Proof.One just has to plug in Z α = a α + b α δ sα and use the definition of the action δ sα • p = s α (p).A comparison of both sides yields the identity.
The coefficients occurring in Lemma 3.4 may be generalized to the case of the action of Z I on a product pq.
Example 3.6.Let γ j = α ij indicate the jth root listed in the sequence I.If Z = X, then using the specific choice of coefficients for the Demazure operator yields Similarly, if Z = Y indicate the push-pull operators, Theorem 3.7 (Generalized Leibniz Rule).Let Z I be a generalized Demazure operator for I = (i 1 , ..., i k ), and let γ j = α ij denote the jth simple root in the list.Then for any p, q ∈ Q, where C I E,F are the Leibniz coefficients defined in (5) Proof.For any simple root α, observe the following two identities: We prove the theorem by induction on k.If k = 1, the theorem holds by Lemma 3.4.Now assume it holds for all I with ℓ(I) < k, and let I = (i 1 , ..., i k ).Let J = (i 2 , ..., i k ) and let α = α i1 .We have by Equation (7).
Comparing the coefficients with B Z 1 • (C J E,F ) from ( 6), we see that they coincide.The proof then follows by induction.
The following corollary follows immediately.We see in Section 8 that the Leibniz coefficients C I [k],E arise as factors in summands of specific structure constants in Schubert calculus, justifying the name.Here [k] = {1, 2, ..., k}.
Corollary 3.8.[Generalized Billey's Formula] Let I = (i 1 , ..., i k ) be a sequence of indices of simple roots, and denote As a consequence of Theorem 3.7, [6, Proposition 9.5] and the coproduct defined in Equation (2), we obtain the following theorem.Theorem 3.9.Let Z α = a α + b α δ α ∈ Q W with b α invertible, then for any I = (i 1 , ..., i k ), we have where C I E,F are defined in Definition 3.5.We specialize Theorem 3.7 to the elements X I and Y I .For any index j, the operators B X j and B Y j preserve S under the action of Q W on Q, and thus B X j , B Y j ∈ D F (see [6,Remark 7.8]).The first statement in the next corollary is the result [6, Proposition 9.5].

The structure constants of equivariant oriented cohomology of flag varieties
In this section we prove the main result, i.e., the formulas of structure constants of Z * Iw in h T (G/B), with resulting formulas for the structure constants of X * Iw and of Y * Iw .Let {Z * Iw } be the basis of Q * W (as a module over Q) dual to the basis {Z Iw } of Q W introduced in Section 3.
Theorem 4.1.For any u, v ∈ W , the product Z * Iu Z * Iv is given by and c {2,3,4,5,6},Iu = c {1,2,3,5,6},Iv = 1.Therefore, Proof of Theorem 4.1.The coproduct structure ∆ on Q W (Equation ( 2)) naturally induces a product on Note that this product corresponds to the product on Q * W introduced at the beginning of Section 2.2 since From Theorem 3.9 we have Finally we obtain the coefficient by calculating the pairing: In §5 we show that these coefficients simplify in the case that F = F a or F = F m , resulting in Theorem 1 from [11].It is worth noting that the formula (10) can be used to prove the Leray-Hirsch Theorem for flag varieties (see [9]).
Example 4.3.Assume the root datum is of type A 1 , then W = {e, s 1 }.We calculate the basis change explicitly: and then we may obtain the products directly: , and note that it agrees with Theorem 4.1 with Z = X.
We use the calculation in Example 2.5, and the product structure on Q * W to obtain the multiplication table for {X Iv }.Recall that f u f v = 1 if u = v and 0 otherwise, and that The other products are as follows: Here y was defined in Example 2.5.
One can check that the above coefficients a Iw Iu,Iv agree with the formula (10).Note that when computing a Iw 0 1,1 , one needs to compute the following coefficients: As an application, we consider the case of a partial flag variety.Let K be a subset of [n].Let P K be the standard parabolic subgroup, W K < W the corresponding subgroup, and W K ⊂ W be the set of minimal length representatives of W/W K .We say a set of reduced sequences Proof.It follows from [7,Corollary 8.4] that X * Iu , u ∈ W K is a basis of (Q * W ) WK .Moreover, from Lemma 4.3 of loc.it., we know Geometrically, under the assumption of this theorem, it follows from [7, Corollary 8.4] that {X * Iw } w∈W K is a basis of (D * F ) WK ∼ = h T (G/P K ).So the product X * Iu X * Iv , u, v ∈ W K is a linear combination of X * Iw , w ∈ W K .Corollary 4.6.Let F = F a or F m , and suppose u ∈ W satisfies that u ∈ W K for some K and u is the longest element in W K .Then for any v ∈ W K , a w u,v = 0 for any w ∈ W , unless w = u.
Proof.In these cases, the braid relations are satisfied, so the structure constants do not depend on the choice of reduced sequences.In other words, fixing u and K, we can assume we have chosen K-compatible reduced sequences.Then Theorem 4.5 applies, which implies that for any v ∈ W K , w ∈ W , we have a Iw u,v = 0 unless w ∈ W K and w ≥ u.Since u is maximal in W K , so w = u.

Structure constants in singular cohomology and K-theory
We restrict our attention to H * T (G/B) and K T (G/B) to recover formulas in [11] of structure constants of Schubert classes for singular cohomology (F = F a ) and Ktheory (F = F m ).We first simplify the coefficients c X I,Iw and c Y I,Iw in these two cases.Recall that, when the formal group law is F = F a or F = F m , the braid relations are satisfied for Z α = X α and Z α = Y α .We consider the equivariant oriented cohomology together with either the additive or multiplicative formal group law, and restrict the coefficient ring to S a or S m .Lemma 5.1.Let J be a word in the Weyl group.As in Lemma 3.3, define coefficients c J,Iw by 1, if J is a reduced word for w; 0, else. ( Proof.When F = F a or F = F m , it is well-known that the braid relations are satisfied.We write c J,w for the coefficient c J,Iw .When When F = F m , we have Z 2 α = Z α and thus Z J = Z w where w := J.It follows that c J,w = 1 and c J,v = 0 for v = w.For each w ∈ W , fix a reduced sequence I w .From the specialization of Theorem 4.1, we have defining relations where the second sum is over E, F whose corresponding products of reflections are reduced and equal to u, v respectively.Recall that The coefficients b Iw u,v coincide with the structure constants c w uv in [11, Theorem 1]. The action of X α (resp.Y −α ) on K T (pt) corresponds to the action of the ordinary (resp.isobaric) Demazure operator in [11].
Fixing a reduced sequence I w for each w, we have where by Lemma 5.1(2), the second sum is over all E, F ⊂ [ℓ(w)] such that E = u and F = v.Here, we have where The classes {ξ w : w ∈ W } in [11] are defined as the dual basis to [O X(w) (−∂X(w))] under the pairing obtained by taking the equivariant cap product and pushing forward to a point.Each ξ w coincides with the Poincaré dual class to [O Y (w) ].In Example 2.7 we note that X * w = (−1) ℓ(w) [O Y (w) ], and thus ξ w = (−1) ℓ(w) X * w .Therefore, It follows that the coefficients (−1) ℓ(u)+ℓ(v)+ℓ(w) a Iw u,v coincide with a w uv in [11], as is clear from the formula.
With the observation that the classes { ξw : w ∈ W } defined in [11] satisfy ξw = Y * w , a similar argument implies that b w u,v coincide with the structure constants åw u,v defined in [11]. Similarly, For the A 3 case, one can also compute Example 5.5.Let F = F m .Consider the A 3 case, with

Structure constants of cohomological stable bases
In this section, we let F = F a and R = R a = Z[h].We recall the definition of the cohomological stable basis of Maulik-Okounkov, and generalize Su's formula of structure constants for Segre-Schwartz-MacPherson classes (Theorem 6.3).We use the twisted group algebra language for singular cohomology, whose K-theory version was given in [22].As the framework and proofs are very similar to earlier sections, we will only review essential properties.Some of the notation introduced below is restricted to this section only.
Let R a = Z[h], S a = Sym R a (Λ) and Q a = Frac(S a ).Define basis δ w , w ∈ W .For simplicity we introduce the following notation: Finally, for any simple root α, define an operator associated to this root by By direct computation, the set {T α } α∈{α1,...,αn} satisfies the braid relations, and T 2 α = 1.Indeed, the algebra generated by {T α } is called the degenerate (or graded) Hecke algebra.Note that T α is a special case of Z α , occurring over R = R a .
For any sequence I = (i 1 , . . ., i ℓ ) (not necessarily reduced), we define the Demazure-Lusztig operator T I = T αi 1 . . .T αi ℓ in cohomology to be the product of the operators indicated in the list I.It follows from the relations that, if I and I ′ are two sequences with w := I = I ′ , then T I = T I ′ , and we denote it T w .The set .
is the algebraic analogue of the composition of the map , where the last map is the equivariant pushforward of cohomology class on G/B to a point on the second term.The proofs in [7,Lemma 7.1] and [22,Lemma 5.1] easily extend to show that, for any f, g Definition 6.1.We define two bases of (Q a W ) * as a module over Q a .Let stab + w = T w −1 • (α w0 f e ), and stab − w = (−1) ℓ(w0) T w −1 w0 • (α w0 f w0 ).Then {stab + w : w ∈ W } and {stab − w : w ∈ W } each form a basis for (Q a W ) * as a module over Q a .We call these bases the cohomological stable bases.See [20] for more details.
It is immediate from the definition that stab + w has support on {f v : v ≤ w} and stab − w has support on {f v : v ≥ w}.The following lemma is the analogue of Theorem 5.7 and Lemma 5.6 in [22].The first identity was due to Maulik-Okounkov originally.Lemma 6.2.We have We now present the main result about the stable basis {stab − w }.Theorem 6.3.The classes stab − w and the coefficients t w u,v satisfy the following properties: (1) We have stab − w = (−1) ℓ(w0) α w0 T * w .
(2).For each w ∈ W , we fix a reduced decomposition.We have stab Therefore, it suffices to consider the structure constants for T * u .But the elements T u are an instantiation of Z Iu with the coefficient ring R a , with a αi j = −h/α ij and b αi j = α ij /α ij .Thus Theorem 4.1 indicates how to multiply the corresponding dual elements, resulting in B T j defined as above.When h = −1, the Demazure Lusztig operator T α specializes to the operator considered by Su in [21], allowing us to recover his formula for the structure constants from the SSM classes from Theorem 6.3.

Example 6.4. Consider the
Similarly, for v ′ = s 2 s 1 , we have Example 6.5.Consider the A 3 case.For This is equal to our −T α with = −1.

Structure constants for K-theoretic stable bases
In this section, we give a formula of the structure constants of the K-theory stable basis.Similar to our strategy in §6, we use the twisted group algebra method.This method was introduced by Su, Zhao and the second author in [22]; we only recall the definitions below.Here we use We use the following notation in this section: xα , q w = q ℓ(w) .
Let Q m = Frac(S m ) and apply the twisted group algebra construction to obtain the module Observe that τ − α is a special case of Z α when Q = Q m .A simple calculation shows that (τ − α ) 2 = (q − 1)τ − α + q, and that {τ α } satisfies the braid relations.It follows that the K-theoretic Demazure-Lusztig operator τ − w , given by the product τ For each not-necessarily reduced sequence I = (i 1 , . . ., i ℓ ), let τ − I be the concate- , and define the structure constants c τ − I,w ∈ R m by the equations Proof.Statement (1) follows from the quadratic relation (τ − α ) 2 = (q − 1)τ − α + q.Statement (2) follows from the braid relations satisfied by the τ − α .The analogous statement to Theorem 3.7 is the following proposition.P I E,F (τ I|E • p)(τ I|F • q), p, q ∈ Q.
where α ) 2 = (q − 1)τ − α + q, it is difficult to express the sum in terms of formulas in Section 5 and Section 6.Indeed, this is also the reason why it is difficult to express the restriction formula of stab − w in [22] in terms of an AJS-Billey-Graham-Willems type formula.

The restriction formula
In this section we relate the structure constants of Z * Iw with its restriction coefficients.This generalizes such relations in cohomology and K-theory due to Kostant and Kumar in [13,Proposition 4.32] and [14,Lemma 2.25].

Example 2 . 7 .
Let X(w) = BwB/B be the Schubert variety and Y (w) = B − wB/B be the opposite Schubert variety.For H * T (G/B) (with F = F a ) or K T (G/B) (with F = F m ), we write w for I w since X Iw and Y Iw are independent of the reduced sequence.(1) [11, §1.2]For H * T (G/B), ζ Y w = [X(w)], and ζ X w = (−1) ℓ(w) [X(w)], where each homology class is identified with its dual cohomology class.Then Y * w = [Y (w)] and similarly X * w

Remark 3 . 2 .
Note that Z α ∈ D F if and only it satisfies the residue condition [23, Definition 3.7].If this is satisfied, then Z Iv ∈ D and equivalently, Z * Iv ∈ D * F .Moreover, Z Iv forms a basis of D F if and only if 1

Definition 3 . 5 .
For each simple root α, let Z α = a α + b α δ α with a α , b α ∈ Q and b α invertible.Let I = (i 1 , . . ., i k ) be a sequence of indices of simple roots, with γ j := α ij corresponding to the jth entry of I.For E, F ⊂ {1, . . .k}, define the Leibniz coefficients C Now we prove the main technical result of this paper, generalizing[6, Lemma  4.8].

Example 4 . 2 .
Leibniz coefficients given in Definition 3.5.As before, the Q elements c E,Iu and c F,Iv are defined as constants appearing in the expansion (9) Z J = w∈W c J,Iw Z Iw .Consider the A 3 -case.Consider I u = (2, 3, 1, 2, 1), I v = (1, 2, 3, 2, 1), then c Iw Iu,Iv = 0 unless w = w 0 is the longest element.Fix I w0 = (1, 2, 3, 1, 2, 1), in which case we have C Iw 0 {2,3,4,5,6},{1,2,3,5,6} Iv .Let I w | E be the subsequence obtained from restricting I w to E. Since w = I w ≥ (I w | E ) for any E ⊂ [ℓ(w)], by Lemma 3.3, c Iw Iu,Iv = 0 unless u ≤ w and v ≤ w.The coproduct structure on the left Q-module Q W restricts to a coproduct structure on the left S-module D F [6, Theorem 9.2].Consequently, the embedding D * F ⊂ Q * W is an embedding of subrings.So the structure constants of the S-bases {X * Iw } and {Y * Iw } in D * F are precisely those of the Q-bases {X * Iw } and {Y * Iw } in Q * W . Specializing Theorem 4.1 to the X-operators, we have = E,F ⊂[ℓ(w)] A Iw E,F c Iw|E ,Iu c Iw|F ,Iv , (10) where c I,Iv are the coefficients that occur in the expansion X I = v c I,Iv X Iv .It follows from [6, Theorem 9.2 and Proposition 7.7] that A Iw E,F ∈ S, that c I,Iw ∈ S, so a Iw Iu,Iv ∈ S. Similarly, specializing to the Y -operators, the structure constants for Y * Iw are denoted by b Iw Iv ,Iu and can be expressed as b Iw Iu,Iv = E,F ⊂[ℓ(w)] B Iw E,F c Iw |E,Iu c Iw|F ,Iv , where now the coefficients c I,Iv are those appearing in the expansion of Y I .As before, B Iw E,F ∈ S and c I,Iw ∈ S, so b Iw Iu,Iv ∈ S.

Example 5 . 2 .
For H * (G/B) and F = F a , as described in Example 2.7 and Proposition 2.7, the element ζ X w in D * F corresponds under a natural isomorphism D * Fa −→ h T (G/B) to the equivariant cohomology class Poincaré dual to [X(w)], where [X(w)] is the homology class of the Schubert variety.Furthermore, the first Chern classes of the corresponding line bundles are x α = α for all simple roots α.

Lemma 7 . 1 . 1 )
The coefficients c τ − I,w ∈ R m in (11) satisfy the following: (For all w ∈ W and sequences I, c τ − I,w = 0 unless w ≤ I. (2) If I is reduced, then c τ − I,w = 0 if w = I 1 if w = I.