Equivariant Oriented Cohomology of Flag Varieties

Посвящается А.С. Меркурьеву, Ученому и Учителю

Baptiste Calmès^{1}, Kirill Zainoulline^{2}, Changlong Zhong^{3}

Received: September 30, 2014 Revised: March 25, 2015

Abstract. Given an equivariant oriented cohomology theory h, a split reductive group G, a maximal torus T in G, and a parabolic subgroup P containing T, we explain how the T -equivariant oriented cohomology ring h_{T} (G/P) can be identified with the dual of a coalgebra defined using exclusively the root datum of (G, T), a set of simple roots defining P and the formal group law of h. In two papers [CZZ,CZZ2] we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the T -fixed points of G/P which embeds the cohomology ring in question into a direct product of a finite number of copies of the T -equivariant oriented cohomology of a point.

2010 Mathematics Subject Classification: 14F43, 14M15, 19L41, 55N22, 57T15, 57R85

1Supported by the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005, as well as by the Alexander von Humboldt Foundation, through the special semester in homotopy theory, in Essen

2Supported by the NSERC Discovery grant 385795-2010, NSERC DAS grant 396100-2010 and the Early Researcher Award (Ontario)

3Supported by the research grants of the second author, V. Chernousov and S. Gille, by the Fields Institute and PIMS

Documenta Mathematica · Extra Volume Merkurjev (2015) 113–144

114 Baptiste Calmès, Kirill Zainoulline, Changlong Zhong

Contents

1. Introduction  114
2. Equivariant oriented cohomology theory  116
3. Torus-equivariant cohomology of a point  119
4. Equivariant cohomology of ℙ^{1}  121
5. Algebraic and combinatorial objects  124
6. Fixed points of the torus action  128
7. Bott-Samelson classes  130
8. Pull-back to T -fixed points  133
9. Invariant subrings and push-forward pairings  138
10. Borel-style presentation  139
11. Subgroups of T  141
References  141

1. Introduction

Equivariant Oriented Cohomology of Flag Varieties 115

Given an equivariant algebraic oriented cohomology theory h over a base field k, a split reductive group G over k, a maximal torus T in G and a parabolic subgroup P containing T, we explain how, as a ring, h_{T} (G/P) can naturally be identified with an algebraic object D_{Ξ}^{⋆} introduced in [CZZ2]. This D_{Ξ}^{⋆} is the dual of a coalgebra defined using exclusively the root datum of (G, T), a set of simple roots Ξ defining P and the formal group law F of h. In [CZZ2], we studied the properties of this object and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper is a companion paper to [HMSZ, CZZ, CZZ2] that justifies the definitions of D_{Ξ}^{⋆} and of other related algebraic objects and operators by explaining how to match them to equivariant cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. The starting point of our approach are celebrated papers by Bernstein-Gelfand-Gelfand and Demazure [BGG, D74] dedicated to (non-equivariant) Chow groups and K-theory, which then were extended to the respective T -equivariant setting by Arabia [Ar86, Ar89], Brion [Br97], Kostant, Kumar [KK86, KK90] and others. While the equivariant case looks more difficult, its big advantage is that the T -fixed points embedding injects h_{T} (G/P) into a very simple ring: a direct product of a finite number of copies of h_{T} (pt), where pt is Spec(k). This important property was already apparent in [Q71, Thm. 4.4] in the topological context (see also [CS74, AB84]). With this observation in hands, the study of the multiplication of Schubert classes (one of the major goals of Schubert calculus) turns into the study of the image of this injection, and then finding a good description of classes of geometric interest in this image, i.e. classes of Schubert varieties, or rather their Bott-Samelson desingularisations. We would like to point out several places where the case of an oriented cohomology theory with an arbitrary formal group law is significantly more complicated than the two classical cases of the additive law (Chow groups) and the multiplicative one (K-theory). First of all, in these two classical cases, the formal group law is given by very simple polynomials; it is easy to conceive that the computations increase in complexity with other formal group laws given by powers series with an infinite number of nonzero coefficients. Secondly, in both of these classical cases, the (non-equivariant) cohomology ring of a point is ℤ, which is a regular ring, while in general, this base ring can be arbitrary. In the work of Kostant and Kumar, the fraction field of the T -equivariant cohomology ring of the point is used as a crucial tool, but we are forced to invert fewer elements and use a more subtle localization process, for fear of killing everything in some cases (see the definition of Q from S in section 5). Thirdly, an important result by Bressler and Evens [BE90] shows that the additive and the multiplicative formal group laws are the only formal group laws for which the elements X_{I_{w}} and Y_{I_{w}} (see after Def. 5.2) are independent of the choice of a reduced decomposition I_{w} of w. Geometrically, this translates as the fact that for Chow groups or K-theory, the class of a Bott-Samelson desingularization corresponding to the reduced decomposition I_{w} only depends on w, and actually is the class of the (possibly singular) Schubert variety corresponding to w in Chow groups and the class of its structural sheaf in K-theory. This combinatorial/geometric independence plays a crucial role in the arguments dealing with Chow groups or K-theory: see [D73, Thm. 1] and how it is used in [D74, §4]; see also [KK86, Prop. 4.2] and its corollary Prop. 4.3. For an arbitrary oriented cohomology theory, for example for algebraic cobordism, this is simply not true: different desingularizations of the same Schubert variety give different classes. Let us mention some of the literature on cohomology theories that go beyond Chow groups or K-theory. Using the Bernstein-Gelfand-Gelfand approach, Bressler and Evens [BE90, BE92] described bases of the (non equivariant) topological complex cobordism ring using Bott-Samelson classes and depending on choosing a reduced decomposition for each Weyl group element. These results were extended later to the algebro-geometric setting independently in [HK] and [CPZ]; in the latter, the approach is algebraic as in [D73, D74] and an efficient algorithm for multiplying Bott-Samelson classes [CPZ, §15] is provided. In [HHH], Harada, Henriques and Holm prove the injectivity of the pull-back to fixed points map and the characterization of its image in the topological context of generalized cohomology theories, under an assumption that certain characteristic classes are prime to each other. Our Theorem 9.2 gives the precise cases when this happens; as all of our statements and proofs, it only relies on algebro-geometric methods, with no input from topology. In [KiKr, Thm. 5.1], a Borel-style presentation of equivariant algebraic cobordism is obtained after inverting the torsion index. The improvement of our Theorem 10.2 is that it applies to any oriented cohomology theory, and that, even over a field of characteristic zero, over which algebraic cobordism is the universal oriented cohomology theory, it gives a finer result than what one would get by specializing from cobordism, as one can see in the case of K-theory for which the Borel-style presentation always holds in the simply connected case, without inverting the torsion index. The techniques developed in the present paper (together with [HMSZ], [CZZ] and [CZZ2]) have been successfully applied to elliptic cohomology: see [LZ14], where the Billey-Graham-Willems formulas for the localization of Schubert classes at torus fixed points were extended to degenerate elliptic cohomology case. In [ZZ14], the authors establish a residue interpretation of the formal affine Hecke algebra H_{F} (a deformation of D_{Ξ}), which coincides with the residue construction of elliptic affine Hecke algebra of Ginzburg, Kapranov, and Vasserot [GKV97] for an arbitrary elliptic formal group law. They also constructed an isomorphism between H_{F} and the equivariant oriented cohomology of the Steinberg variety. Our main results (Theorems 8.11 and 9.1) identify the ring D_{Ξ}^{⋆} with the equivariant cohomology h_{T} (G/P), within the fixed points ring S_{W/W_{Ξ}}^{⋆} that is a direct product of copies of h_{T} (pt) and the image of the injective pull-back map h_{T} (G/P) → h_{T} (G/B) (B is a Borel subgroup) as the subring h_{T} (G/B)^{W_{Ξ}} of fixed elements under the parabolic Weyl group W_{Ξ} corresponding to P. In Theorem 10.2 we provide a Borel-style presentation h_{T} (pt) ⊗_{h_{T} (pt)^{W}} h_{T} (pt) ≃ h_{T} (G/B) under certain conditions. Other results are proved along the way: Theorem 9.2 gives an intrinsic characterization of the above mentioned image in the Borel case. Diagram (8.3) describes the push-forward map h_{T} (G/P′) → h_{T} (G/P), induced by the projections G/P′ → G/P for parabolic subgroups P′ ⊆ P of G. Lemma 7.6 describes the algebraic elements corresponding to Bott-Samelson classes, i.e. fundamental classes of desingularized Schubert varieties. Theorem 9.3 proves that the pairing defined by product and push-forward to h_{T} (pt) is non-degenerate. The paper is organized as follows. First, we state the properties that we use from equivariant oriented cohomology theories, in section 2. Then, in section 3, we describe h_{T} (pt) as the formal group ring of [CPZ, Def. 2.4]. In section 4, we compute the case of h_{T} (ℙ^{1}) when the action of T on the projective line ℙ^{1} = (𝔸^{2} \ {0})/𝔾_{m} is induced by a linear action of T on 𝔸^{2}. It enables us to compute the pull-back of Bott-Samelson classes ζ_{I} to h_{T} ((G/B)^{T}) in Lemma 7.6. By localization, some of these classes generate h_{T} (G/B) and this lets us prove the Borel case of Theorem 8.11. The parabolic cases are then obtained in the remaining sections, as well as the Borel-style presentation. In the last section, we explain how equivariant groups under subgroups of T (and in particular the trivial group which gives the non-equivariant case) can be recovered out of the equivariant one.

2. Equivariant oriented cohomology theory

In the present section we recall the notion of an equivariant algebraic oriented cohomology theory, essentially by compiling definitions and results of [Des09], [EG98], [HM13], [KiKr], [Kr12], [LM07], [Pa09] and [To99]. We present it here in a way convenient for future reference.

In this paper, k is always a fixed base field, and pt denotes Spec(k). By a variety we mean a reduced separated scheme of finite type over k. Let G be a smooth linear algebraic group over k, abbreviated as algebraic group. In this paper we are mostly interested in the case G = T. Let G-Var be the category of smooth quasi-projective varieties over k endowed with an action of G, and with morphisms respecting this action (i.e. G-equivariant morphisms). The tangent sheaf T_{X} of any X ∈ G-Var is locally free and has a natural G-equivariant structure. The same holds for the (co)normal sheaf of any equivariant regular embedding of a closed subscheme.

An equivariant oriented cohomology theory over k is an additive contravariant functor h_{G} from the category G-Var to the category of commutative rings with unit for any algebraic group G (for an equivariant morphism f, the map h_{G}(f) is denoted by f^{∗} and is called pull-back) together with

• a morphism f_{∗}: h_{G}(X) → h_{G}(Y) of h_{G}(Y)-modules (called push-forward) for any projective morphism f: X → Y in G-Var (here h_{G}(X) is an h_{G}(Y)- module through f^{∗}). That is, we have the projection formula

(2.1)  f_{∗}(f^{∗}(y)x) = yf_{∗}(x), x ∈ h_{G}(X), y ∈ h_{G}(Y).

• a natural transformation of functors res_{ϕ}: h_{H} → h_{G} ∘ Res_{ϕ} (called restriction) for any morphism of algebraic groups ϕ: G → H (here Res_{ϕ}: H -Var → G-Var simply restricts the action of H to an action of G through ϕ)

• a natural transformation of functors c^{G}: K_{G} → h̃_{G} (called the total equivariant characteristic class), where K_{G}(X) is the K-group of G-equivariant locally free sheaves over X and h̃_{G}(X) is the multiplicative group of the polynomial ring h_{G}(X)[t] (the coefficient at t^{i} is called the i-th equivariant characteristic class in the theory h and is denoted by c_{i}^{G})

that satisfy the following properties:

A 1 (Compatibility for push-forwards). The push-forwards respect composition and commute with pull-backs for transversal squares (a transversal square is a fiber product diagram with a nullity condition on Tor-sheaves, stated in [LM07, Def. 1.1.1]; in particular, this condition holds for any fiber product with a flat map).

A 2 (Compatibility for restriction). The restriction respects composition of morphisms of groups and commutes with push-forwards.

A 3 (Localization). For any smooth closed subvariety i: Z → X in G-Var with open complement u: U ↪ X, the sequence

h_{G}(Z) ⟶^{i_{∗}} h_{G}(X) ⟶^{u^{∗}} h_{G}(U) → 0

is exact.

A 4 (Homotopy Invariance). Let p: X ×𝔸^{n} → X be a G-equivariant projection with G acting linearly on 𝔸^{n}. Then the induced pull-back h_{G}(X) → h_{G}(X ×𝔸^{n}) is an isomorphism.

A 5 (Normalization). For any regular embedding i: D ⊂ X of codimension 1 in G-Var we have c_{1}^{G}(O(D)) = i_{∗}(1) in h_{G}(X), where O(D) is the line bundle dual to the kernel of the map of G-equivariant sheaves O → O_{D}.

A 6 (Torsors). Let p: X → Y be in G-Var and let H be a closed normal subgroup of G acting trivially on Y such that p: X → Y is a H-torsor. Consider the quotient map ı: G → G/H. Then the composite p^{∗} ∘ res_{ı}: h_{G/H}(Y) → h_{G}(X) is an isomorphism. In particular, if H = G we obtain an isomorphism h_{{1}}(Y) ≃ h_{G}(X) for a G-torsor X over Y.

A 7. If G = {1} is trivial, then h_{{1}} = h defines an algebraic oriented cohomology in the sense of [LM07, Def. 1.1.2] (except that h takes values in rings, not in graded rings) with push-forwards and characteristic classes being as in [LM07].

A 8 (Self-intersection formula). Let i : Y ⊂ X be a regular embedding of codimension d in G-Var. Then the normal bundle to Y in X, denoted by N_{Y/X} is naturally G-equivariant and there is an equality i^{∗}i_{∗}(1) = c_{d}^{G}(N_{Y/X}) in h_{G}(Y).

A 9 (Quillen’s formula). If L_{1} and L_{2} are locally free sheaves of rank one, then

c_{1}(L_{1} ⊗ L_{2}) = c_{1}(L_{1}) +_{F} c_{1}(L_{2}),

where F is the formal group law of h (here G = {1}).

As consequences of the projection formula (2.1), we have:

Lemma 2.1. Let p : X → Y be a morphism in G-Var, with a section s : Y → X. Then for any u ∈ h_{G}(Y), one has

(a) s^{∗}s_{∗}(u · v) = u · s^{∗}s_{∗}(v) if s is projective.

(b) p_{∗}(s_{∗}(u)^{n}) = u·s^{∗}s_{∗}(u)^{n−1} for any n ≥ 1 if furthermore p is projective.

Proof. Part (a) follows from

s^{∗}s_{∗}(u·v) = s^{∗}s_{∗}(s^{∗}p^{∗}(u)·v) = s^{∗}(p^{∗}(u)·s_{∗}(v)) = s^{∗}p^{∗}(u)·s^{∗}s_{∗}(v) = u·s^{∗}s_{∗}(v)

and part (b) from

p_{∗}(s_{∗}(u)^{n}) = p_{∗}(s_{∗}(u)·s_{∗}(u)^{n−1}) = p_{∗}(s_{∗}(u·s^{∗}(s_{∗}(u)^{n−1}))) = u·s^{∗}s_{∗}(u)^{n−1}. □

This lemma applies in particular when p : X → pt is the structural morphism of X and s is therefore a G-fixed point of X.

For any X ∈ G-Var consider the γ-filtration on h_{G}(X), whose i-th term γ^{i} h_{G}(X) is the ideal of h_{G}(X) generated by products of equivariant characteristic classes of total degree at least i. In particular, a G-equivariant locally free sheaf of rank n over pt is the same thing as an n-dimensional k-linear representation of G, so γ^{i} h_{G}(pt) is generated by characteristic classes of such representations. This can lead to concrete computations when the representations of G are well described.

We introduce the following important notion

Definition 2.2. An equivariant oriented algebraic cohomology theory is called Chern-complete over the point for G, if the ring h_{G}(pt) is separated and complete with respect to the topology induced by the γ-filtration.

Remark 2.3. Assume that the ring h_{G}(pt) is separated for all G, and let h_{G}(pt)^{∧} be its completion with respect to the γ-filtration. We can Cherncomplete the equivariant cohomology theory by tensoring with − ⊗_{h_{G}(pt)} h_{G}(pt)^{∧}. In this way, we obtain a completed version of the cohomology theory, still satisfying the axioms. Note that this completion has no effect on the nonequivariant groups, since in h(pt), the characteristic classes are automatically nilpotent by [LM07, Lemma 1.1.3].

Here are three well-known examples of equivariant oriented cohomology theories.

Example 2.4. The equivariant Chow ring functor h_{G} = CH_{G} was constructed by Edidin and Graham in [EG98], using an inverse limit process of Totaro [To99]. In this case the formal group law is the additive one F (x, y) = x + y, the base ring CH(pt) is ℤ, and the theory is Chern-complete over the point for any group G by construction.

Example 2.5. Equivariant algebraic K-theory and, in particular, K_{0} was constructed by Thomason [Th87] (see also [Me05] for a good survey). The formal group law is multiplicative F (x, y) = x + y − xy, the base ring K_{0}(pt) is ℤ, and the theory is not Chern complete: for example, (K_{0})_{𝔾_{m}}(pt) ≃ ℤ[t, t^{−1}] with the γ^{i} generated by (1 − t)^{i}. Observe that (K_{0})_{G}(pt) consists of classes of k-linear finite dimensional representations of G.

Example 2.6 (Algebraic cobordism). Equivariant algebraic cobordism was defined by Deshpande [Des09], Malgón-López and Heller [HM13] and Krishna [Kr12]. The formal group law is the universal one over Ω(pt) = 𝕃 the Lazard ring. The equivariant theory is Chern complete over the point for any group G by construction.

By Totaro’s process one can construct many examples of equivariant theories, such as equivariant connective K-theory, equivariant Morava K-theories, etc. Moreover, in this way one automatically obtains Chern-complete theories.

3. Torus-equivariant cohomology of a point

From now on, T is always a split torus. In the present section we show that the completed T -equivariant oriented cohomology ring of a point can be identified with the formal group algebra of the respective group of characters (see Theorem 3.3).

Let Λ be the group of characters of T, which is therefore the Cartier dual of Λ. Let X be a smooth variety over k endowed with a trivial T -action. Consider the pull-back p^{∗}: h_{T} (pt) → h_{T} (X) induced by the structure map. Let γ_{pt}^{i} h_{T} (X) denote the ideal in h_{T} (X) generated by elements from the image of γ^{i} h_{T} (pt) under the pull-back. Since any representation of T decomposes as a direct sum of one dimensional representations, γ^{i} h_{T} (pt) is generated by products of first characteristic classes c_{1}^{T} (L_{λ}), λ ∈ Λ. Since characteristic classes commute with pull-backs, γ_{pt}^{i} h_{T} (X) is also generated by products of first characteristic classes (of pull-backs p^{∗}L_{λ}).

Let F be a one-dimensional commutative formal group law over a ring R. We often write x +_{F} y (formal addition) for the power series F (x, y) defining F. Following [CPZ, §2] consider the formal group algebra R[[Λ]]_{F}. It is an Ralgebra together with an augmentation map R[[Λ]]_{F} → R with kernel denoted by I_{F}, and it is complete with respect to the I_{F} -adic topology. Thus

R[[Λ]]_{F} = \underleftarrow{lim}_{i} R[[Λ]]_{F} /I_{F}^{i} ,

and it is topologically generated by elements of the form x_{λ}, λ ∈ Λ, which satisfy x_{λ+μ} = x_{λ} +_{F} x_{μ}. By definition (see [CPZ, 2.8]) the algebra R[[Λ]]_{F} is universal among R-algebras with an augmentation ideal I and a morphism of groups Λ → (I, +_{F}) that are complete with respect to the I-adic topology. The choice of a basis of Λ defines an isomorphism

R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]],

where n is the rank of Λ.

Set R = h(X). Then h_{T} (X) is an R-algebra together with an augmentation map h_{T} (X) → R via the restrictions induced by {1} → T → {1}. The assignment λ ∈ Λ ↦ c_{1}^{T} (L_{λ}) induces a group homomorphism Λ → (I, +_{F}), where I is the augmentation ideal. Therefore, by the universal property of R[[Λ]]_{F}, there is a morphism of R-algebras

ϕ: R[[Λ]]_{F} /I_{F}^{i} → h_{T} (X)/γ_{pt}^{i} h_{T} (X).

We claim that

Lemma 3.1. The morphism ϕ is an isomorphism.

Proof. We proceed by induction on the rank n of Λ. For n = 0, we have T = {1}, R = h_{T} (X), I_{F}^{i} = γ_{pt}^{i} h_{T} (X) = {0} and the map ϕ turns into an identity on R. For rank n > 0 we choose a basis {λ_{1}, …, λ_{n}} of Λ. Let {L_{1}, …, L_{n}} be the respective one-dimensional representations of T. This gives isomorphisms Λ ≃ ℤ^{n} and T ≃ 𝔾_{m}^{n} and 𝔾_{m}^{n} acts on L_{i} by multiplication by the i-th coordinate. Let 𝔾_{m}^{n} act on 𝔸^{i} by multiplication by the last coordinate. Consider the localization sequence (A3)

h_{𝔾_{m}^{n}}(X) ⟶ h_{𝔾_{m}^{n}}(X × 𝔸^{i}) ⟶ h_{𝔾_{m}^{n}}(X × (𝔸^{i} \ {0})) ⟶ 0.

After identifying

h_{𝔾_{m}^{n}}(X) →^{∼} h_{𝔾_{m}^{n}}(X × 𝔸^{i}) and h_{𝔾_{m}^{n−1}}(X × ℙ^{i−1}) →^{∼} h_{𝔾_{m}^{n}}(X × (𝔸^{i} \ {0}))

via (A8) and (A6), we obtain an exact sequence

h_{𝔾_{m}^{n}}(X) ⟶^{c_{1}(L_{n})^{i}} h_{𝔾_{m}^{n}}(X) ⟶ h_{𝔾_{m}^{n−1}}(X × ℙ^{i−1}) ⟶ 0.

where the first map is obtained by applying self-intersection (A5) and homotopy invariance (A4) properties. By definition, all these maps are R-linear, and the action of 𝔾_{m}^{n−1} on X × ℙ^{i−1} is the trivial one. Since the last map is given by pull-back maps and restrictions (although not all in the same direction), and since equivariant characteristic classes commute with these, one checks that it sends c_{1}(L_{i}) to c_{1}(L_{i}) for any i ≤ n − 1 and c_{1}(L_{n}) to c_{1}(O(1)); this last case holds because O(1) on ℙ^{i−1} goes (by restriction and pull-back) to the equivariant line bundle on 𝔸^{i} \ {0} with trivial underlying line bundle, but where 𝔾_{m}^{n} acts by λ_{n} on fibers. By the projective bundle theorem, we have R′ := h(X × ℙ^{i−1}) ≃ R[y]/y^{i} with c_{1}(O(1)) = y. By induction, we obtain for any i an isomorphism

h_{𝔾_{m}^{n−1}}(X × ℙ^{i−1})/γ_{pt}^{i} ≃ R′[[Λ′]]_{F} /(I′_{F})^{i},

where Λ′ = ℤ^{n−1} and I′_{F} is the augmentation ideal of R′[[Λ′]]_{F}. Using the isomorphisms R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]] and R′[[Λ′]]_{F} ≃ R′[[x_{1}, …, x_{n−1}]] induced by the basis of Λ, we are reduced to checking that

\begin{matrix}R[[x_{1}, …, x_{n}]]/I_{F}^{i} & ⟶ & (R[y]/y^{i})[[x_{1}, …, x_{n−1}]]/J \\ x_{i} & ⟼ & \begin{cases}x_{i} & if i ≤ n − 1 \\ y & if i = n. \end{cases} \end{matrix}

is an isomorphism, when J = (I′_{F})^{i} + y · (I′_{F})^{i−1} + ⋯ + y^{i}. The latter then follows by definition. □

Remark 3.2. Similar statements can be found in [HM13, 3.2.1] or [Kr12, 6.7], but we gave a full proof for the sake of completeness.

We obtain a natural map of R-algebras

h_{T} (pt) → \underleftarrow{lim}_{i} h_{T} (pt)/γ^{i} h_{T} (pt) ≃ \underleftarrow{lim}_{i} R[[Λ]]_{F} /I_{F}^{i} = R[[Λ]]_{F}

and, therefore, by Lemma 3.1, we have:

Theorem 3.3. If h is (separated and) Chern-complete over the point for T, then the natural map h_{T} (pt) → R[[Λ]]_{F} is an isomorphism. It sends the characteristic class c_{1}^{T} (L_{λ}) ∈ h_{T} (pt) to x_{λ} ∈ R[[Λ]]_{F}.

4. Equivariant cohomology of ℙ^{1}

In the present section we compute equivariant cohomology h_{T} (ℙ(V_{1} ⊕ V_{2})) of a projective line, where a split torus T acts on one-dimensional representations V_{1} and V_{2} by means of characters λ_{1} and λ_{2}.

Assumption 4.1. For the rest of the paper we assume that the equivariant cohomology of the point h_{T} (pt) is (separated and) complete for the γ-filtration in the sense of Definition 2.2.

Let X be a smooth T -variety. By section 3, the ring h_{T} (X) can be considered as a ring over S := R[[Λ]]_{F} via the identification S ≃ h_{T} (pt) of Theorem 3.3 and the pull-back map h_{T} (pt) → h_{T} (X). By convention, we’ll use the same notation for an element u of S and the element u · 1 ∈ h_{T} (X), where 1 is the unit of h_{T} (X). Thus, for example, x_{λ} = c_{1}^{T} (L_{λ}) in h_{T} (X).

Given a morphism f : X → Y in T -Var, the pull-back map f^{∗} is a morphism of rings over S and the push-forward map f_{∗} (when it exists) is a morphism of S-modules by the projection formula.

Remark 4.2. Note that we are not claiming that S injects in h_{T} (X) for all X ∈ T -Var; it will nevertheless hold when X has a k-point that is fixed by T, as most of the schemes considered in this paper have.

We now concentrate on the following setting. Let λ_{1} and λ_{2} be characters of T, and let V_{1} and V_{2} be the corresponding one dimensional representations of T, i.e. t ∈ T acts on v ∈ V_{i} by t · v = λ_{i}(t)v. Thus, the projective space ℙ(V_{1} ⊕ V_{2}) is endowed with a natural T -action, induced by the action of T on the direct sum of representations V_{1} ⊕ V_{2}. Furthermore, the line bundle O(−1) has a natural T -equivariant structure, that can be described in the following way: The geometric points of the total space of O(−1) are pairs (W, w) where W is a rank one sub-vector space of V_{1} ⊕ V_{2} and w ∈ W. The torus T acts by t · (W, w) = (t(W), t(w)). Two obvious embeddings V_{i} ⊆ V_{1} ⊕ V_{2} induce two T -fixed points closed embeddings σ_{1}, σ_{2} : pt ↪ ℙ(V_{1} ⊕ V_{2}). The open complement to σ_{1} is an affine space isomorphic to V_{1} ⊗ V_{2}^{∨}, with T -action by the character λ_{1} − λ_{2}. We set α := λ_{2}−λ_{1}. By homotopy invariance (A4) applied to the pull-back induced by the structural morphism of V_{1}, we have h_{T} (pt) →^{∼} h_{T} (V_{1}) with inverse given by the pull-back σ_{2}^{∗} (which actually lands in V_{1}). The exact localization sequence (A3) can therefore be rewritten as

\xymatrix{h_{T} (pt) \ar[r]^{(σ_{1})_{∗}} & h_{T} (ℙ(V_{1} ⊕ V_{2})) \ar[r]^{σ_{2}^{∗}} & h_{T} (pt) \ar[r] & 0}

Using the structural map p : ℙ(V_{1} ⊕ V_{2}) → pt, we get a splitting p^{∗} of σ_{2}^{∗} and a retract p_{∗} of (σ_{1})_{∗}. Thus, the exact sequence is in fact injective on the left, and we can decompose h_{T} (ℙ(V_{1} ⊕ V_{2})) using mutually inverse isomorphisms

(4.1)  \xymatrix{h_{T} (pt) ⊕ h_{T} (pt) \ar[r]_{((σ_{1})_{∗}, p^{∗}−(σ_{1})_{∗}p_{∗}p^{∗})} & h_{T} (ℙ(V_{1} ⊕ V_{2})) \ar[l]_{^{(\begin{matrix}p_{∗} \\ σ_{2}^{∗} \end{matrix})}}}

Lemma 4.3. (a) As T -equivariant bundles, we have σ_{i}^{∗}(O(−1)) = V_{i}.

(b) We have (σ_{1})_{∗}(1) = c_{1}(O(1)⊗p^{∗}(V_{2})) and (σ_{2})_{∗}(1) = c_{1}(O(1)⊗p^{∗}(V_{1})) in h_{T} (ℙ(V_{1} ⊕ V_{2})).

(c) For any u ∈ h_{T} (pt), we have σ_{1}^{∗}(σ_{1})_{∗}(u) = x_{α}u, σ_{2}^{∗}(σ_{2})_{∗}(u) = x_{−α}u and σ_{1}^{∗}(σ_{2})_{∗}(u) = σ_{2}^{∗}(σ_{1})_{∗}(u) = 0.

Proof. The first part is easily checked on the geometric points of total spaces and is left to the reader. The second part follows from (A5), given the exact sequence of T -equivariant sheaves

0 → O(−1) ⊗ p^{∗}(V_{2})^{∨} → O → O_{σ_{1}} → 0,

where O_{σ_{1}} is the structural sheaf of the closed subscheme given by σ_{1}. Again this exact sequence is easy to check and we leave it to the reader. In the third part, the last equality holds by transverse base change through the empty scheme, while the first two follow from Lemma 2.1 and

σ_{1}^{∗}(σ_{1})_{∗}(1) = σ_{1}^{∗}c_{1}(O(1) ⊗ p^{∗}(V_{2})) =

= c_{1}(σ_{1}^{∗}(O(1) ⊗ p^{∗}(V_{2}))) = c_{1}(V_{1}^{∨} ⊗ V_{2}) = x_{λ_{2}−λ_{1}}.

or a symmetric computation for σ_{2}^{∗}(σ_{2})_{∗}(1). □

Lemma 4.4. If x_{α} is not a zero divisor in S, then the push-forward

p_{∗}: h_{T} (ℙ(V_{1} ⊕ V_{2})) → h_{T} (pt) satisfies p_{∗}(1) = \frac{1}{x_{α}} + \frac{1}{x_{−α}} .

(Observe that p_{∗}(1) ∈ S by [CPZ, 3.12], where it is denoted by e_{α}.)

Proof. By Lemma 4.3, we have

\begin{aligned}x_{α} & = c_{1}(p^{∗}(V_{2} ⊗ V_{1}^{∨})) = c_{1}(O(1) ⊗ p^{∗}(V_{2}) ⊗ (O(1) ⊗ p^{∗}(V_{1}))^{∨}) \\ & = c_{1}(O(1) ⊗ p^{∗}(V_{2})) −_{F} c_{1}(O(1) ⊗ p^{∗}(V_{1})) = (σ_{1})_{∗}(1) −_{F} (σ_{2})_{∗}(1). \end{aligned}

By transverse base change, we have (σ_{1})_{∗}(1) · (σ_{2})_{∗}(1) = 0, and therefore

(σ_{1})_{∗}(1) −_{F} (σ_{2})_{∗}(1) = (σ_{1})_{∗}(1) + ( −_{F} (σ_{2})_{∗}(1)).

Since x_{α} is not a zero divisor in S, it suffices to prove that

x_{α} · p_{∗}(1) = 1 + \frac{x_{α}}{x_{−α}} ,

where \frac{x_{α}}{x_{−α}} ∈ S^{×} is the power series \frac{−_{F} (x)}{x} applied to x = x_{−α}. Now,

\begin{aligned}x_{α}p_{∗}(1) & = p_{∗}(x_{α}) = p_{∗}((σ_{1})_{∗}(1) + (−_{F} (σ_{2})_{∗}(1))) \\ & = 1 + p_{∗}(−_{F} (σ_{2})_{∗}(1)) = 1 + \frac{x_{α}}{x_{−α}} . \end{aligned}

where the last equality follows from Lemma 2.1, part (b). □

Let σ = σ_{1} ⊔ σ_{2} : pt ⊔ pt → ℙ(V_{1} ⊕ V_{2}) be the inclusion of both T -fixed points.

Lemma 4.5. If x_{α} is not a zero divisor in S, the pull-back σ^{∗} is injective, and

im σ^{∗} = {(u, v) ∈ h_{T} (pt) ⊕ h_{T} (pt) | x_{−α}u + x_{α}v ∈ x_{α}x_{−α} · h_{T} (pt)}.

Proof. Since h_{T} (pt⊔pt) = h_{T} (pt)⊕h_{T} (pt) identifies σ^{∗} with (σ_{1}^{∗}, σ_{2}^{∗}), it suffices to check that the composition

\xymatrix{h_{T} (pt) ⊕ h_{T} (pt) \ar[r]^{((σ_{1})_{∗}, p^{∗}−(σ_{1})_{∗}p_{∗}p^{∗})}_{≃} & h_{T} (ℙ(V_{1} ⊕ V_{2})) \ar[r]^{^{(\begin{matrix}σ_{1}^{∗} \\ σ_{2}^{∗} \end{matrix})}} & h_{T} (pt) ⊕ h_{T} (pt)}

is injective. Indeed, it is given by the matrix

(\begin{matrix}σ_{1}^{∗}(σ_{1})_{∗} & σ_{1}^{∗}p^{∗} − σ_{1}^{∗}(σ_{1})_{∗}p_{∗}p^{∗} \\ σ_{2}^{∗}(σ_{1})_{∗} & σ_{2}^{∗}p^{∗} − σ_{2}^{∗}(σ_{1})_{∗}p_{∗}p^{∗} \end{matrix}) = (\begin{matrix}x_{α} & 1 − x_{α} · p_{∗}(1) \\ 0 & 1 \end{matrix}) = (\begin{matrix}x_{α} & − \frac{x_{α}}{x_{−α}} \\ 0 & 1 \end{matrix} )

where in the first equality, we have used p ∘ σ_{i} = id, Lemma 4.3 part (c), to get the 1’s and the 0, and then the projection formula p_{∗}p^{∗}(u) = u · p_{∗}(1) and Lemma 2.1 to get σ_{1}^{∗}(σ_{1})_{∗}p_{∗}p^{∗}(u) = x_{α}p_{∗}(1) · u. The last equality holds by Lemma 4.4. Finally, the image of this matrix is of the expected form. □

Let S[\frac{1}{x_{α}}] be the localization of S at the multiplicative subset generated by x_{α}. Since \frac{x_{α}}{x_{−α}} is invertible, there is a canonical isomorphism S[\frac{1}{x_{α}}] ≃ S[\frac{1}{x_{−α}}]. We consider the S[\frac{1}{x_{α}}]-linear operator

A: S[\frac{1}{x_{α}}] ⊕ S[\frac{1}{x_{α}}] ⟶ S[\frac{1}{x_{α}}] given by (u, v) ↦ \frac{u}{x_{α}} + \frac{v}{x_{−α}} .

Note that by the previous lemma, it sends the image of σ^{∗} to S inside S[\frac{1}{x_{α}}].

Lemma 4.6. If x_{α} is not a zero divisor in S, the following diagram commutes.

[figure: h_{T} (ℙ(V_{1} ⊕ V_{2})); p_{∗}; σ^{∗}; h_{T} (pt) ⊕ h_{T} (pt); S ⊕ S; ≃; ⊆; S[\frac{1}{x_{α}}] ⊕ S[\frac{1}{x_{α}}]; A; h_{T} (pt); S; ≃; ⊆; S[\frac{1}{x_{α}}]]

Proof. It suffices to check the equality of the two maps after precomposition by the isomorphism h_{T} (pt) ⊕ h_{T} (pt) → h_{T} (ℙ(V_{1} ⊕ V_{2})) given in (4.1). Using the matrix already computed in the proof of Lemma 4.5, one obtains that the upper right composition sends (u, v) to u. The lower left composition sends (u, v) to

p_{∗}((σ_{1})_{∗}(u) + p^{∗}(v) − (σ_{1})_{∗}p_{∗}p^{∗}(v)) = u + p_{∗}p^{∗}(v) − p_{∗}p^{∗}(v) = u. □

5. Algebraic and combinatorial objects

Let us now introduce the main algebraic objects D^{⋆}, D_{Ξ}^{⋆}, S_{W}^{⋆} and S_{W/W_{Ξ}}^{⋆} that play the role of algebraic replacements for some equivariant cohomology groups in the remaining of this paper. These objects were discussed in detail in [CZZ] and [CZZ2], and we only give a brief overview here. Their geometric interpretation will be explained in the next sections.

Let Σ ↪ Λ^{∨}, α ↦ α^{∨} be a root datum in the sense of [SGA3, Exp. XXI, §1.1]. Thus, Λ is a lattice and Σ is a non-empty finite subset of Λ, called the set of roots. The rank of the root datum is the dimension of ℚ ⊗_{ℤ} Λ, and elements in Σ are called roots. The root lattice Λ_{r} is the subgroup of Λ generated by elements in Σ, and the weight lattice is defined as

Λ_{w} = {ω ∈ ℚ ⊗_{ℤ} Λ | α^{∨}(ω) ∈ ℤ for all α ∈ Σ}.

We have Λ_{r} ⊆ Λ ⊆ Λ_{w}. We always assume that the root datum is semisimple (the ranks of Λ, Λ_{r}, Λ_{w} are equal and no root is twice any other root). The root datum is called simply connected (resp. adjoint) if Λ = Λ_{w} (resp. Λ = Λ_{r}) and if it is furthermore irreducible of rank n, we use the notation D_{n}^{sc} (resp. D_{n}^{ad}) for its Dynkin type, with D among A, B, C, D, G, F, E.

The Weyl group W of the root datum is the subgroup of Aut_{ℤ}(Λ) generated by simple reflections

s_{α}(λ) = λ − α^{∨}(λ)α, λ ∈ Λ.

Fixing a set of simple roots Π = {α_{1}, ..., α_{n}} induces a partition Σ = Σ^{+} ∪ Σ^{−}, where Σ^{+} is the set of positive roots and Σ^{−} = −Σ^{+} is the set of negative roots. The Weyl group W is actually generated by s_{i} := s_{α_{i}}, i = 1, ..., n.

Let F be a one-dimensional commutative formal group law over a commutative ring R. Let S = R[[Λ]]_{F}. From now on we always assume that

Assumption 5.1. The algebra S is Σ-regular, that is, x_{α} is regular in S for all α ∈ Σ (see [CZZ, Def. 4.4]).

This holds if 2 is regular in R, or if the root datum does not contain an irreducible component of type C_{k}^{sc} [CZZ, Rem. 4.5].

The action of W on Λ induces an action of W on S, and let S_{W} be the R-algebra defined as S ⊗_{R} R[W] as an R-module, and with product given by

qδ_{w}q′δ_{w′} = qw(q′)δ_{ww′}, q, q′ ∈ S, w, w′ ∈ W.

Let Q = S[\frac{1}{x_{α}} |α ∈ Σ] and Q_{W} = Q ⊗_{S} S_{W}, with ring structure given by the same formula with q, q′ ∈ Q. Then {δ_{w}}_{w∈W} is an S-basis of S_{W} and a Q-basis of Q_{W}. There is an action of Q_{W} on Q, restricting to an action of S_{W} on S, and given by

qδ_{w} · q′ = qw(q′), q, q′ ∈ Q, w ∈ W.

For each α ∈ Σ, we define κ_{α} = \frac{1}{x_{α}} + \frac{1}{x_{−α}} ∈ S.

Definition 5.2. For any α ∈ Σ, let

X_{α} = \frac{1}{x_{α}} − \frac{1}{x_{α}} δ_{s_{α}}, Y_{α} = κ_{α} − X_{α} = \frac{1}{x_{−α}} + \frac{1}{x_{α}} δ_{s_{α}},

in Q_{W}, respectively called a formal Demazure element and a formal push-pull element.

For each sequence (i_{1}, ..., i_{k}) with 1 ≤ i_{j} ≤ n, we define X_{I} = X_{α_{i_{1}}} ⋯ X_{α_{i_{k}}} and Y_{I} = Y_{α_{i_{1}}} ⋯ Y_{α_{i_{k}}}.

Definition 5.3. Let D be the R-subalgebra of Q_{W} generated by elements from S and the elements X_{α}, α ∈ Σ.

Since δ_{s_{i}} = 1 − x_{α_{i}}X_{α_{i}}, we have S_{W} ⊆ D. By [CZZ, Prop. 7.7], D is a free S-module and for any choice of reduced decompositions I_{w} for every element w ∈ W the family {X_{I_{w}}}_{w∈W} is an S-basis of D.

There is a coproduct structure on the Q-module Q_{W} defined by

Q_{W} → Q_{W} ⊗_{Q} Q_{W} , qδ_{w} ↦ qδ_{w} ⊗ δ_{w},

with counit Q_{W} → Q, qδ_{w} ↦ q. Here Q_{W} ⊗_{Q} Q_{W} is the tensor product of left Q-modules. By the same formula, one can define a coproduct structure on the S-module S_{W}. The coproduct on Q_{W} induces a coproduct structure on D as a left S-module.

On duals S_{W}^{⋆} = Hom_{S}(S_{W} , S), D^{⋆} = Hom_{S}(D, S) and Q_{W}^{∗} = Hom_{Q}(Q_{W} , Q) (notice the different stars ⋆ for S-duality and ∗ for Q-duality), the respective coproducts induce products. In S_{W}^{⋆} or Q_{W}^{∗}, this product is given by the simple formula

f_{v}f_{w} = δ_{v,w}^{Kr}f_{v}

on the dual basis {f_{v}}_{w∈W} to {δ_{w}}_{w∈W}, with δ_{v,w}^{Kr} the Kronecker delta. The multiplicative identity is 1 = ∑_{v∈W} f_{v}. Let η be the inclusion S_{W} ⊆ D. It induces an S-algebra map η^{⋆} : D^{⋆} → S_{W}^{⋆}, which happens to be injective [CZZ2, Lemma 10.2]. Furthermore, after localization, η_{Q} : Q_{W} → Q ⊗_{S} D^{⋆} is an isomorphism and by freeness, we have Q ⊗_{S} D^{⋆} ≃ Hom_{Q}(Q ⊗_{S} D, Q) and thus Q ⊗_{S} D^{⋆} ≃ Q_{W}^{∗}, as left Q-rings.

There is a Q-linear action of the R-algebra Q_{W} on Q_{W}^{∗} given by

(z • f)(z′) = f(z′z), z, z′ ∈ Q_{W} , f ∈ Q_{W}^{∗} .

as well as S-linear actions of S_{W} on S_{W}^{⋆} and of D on D^{⋆}, given by the same formula. With this action, it is proved in [CZZ2, Theorem 10.13] that D^{⋆} is a free D-module of rank 1 and any w ∈ W gives a one-element basis {x_{Π} • f_{w}} of it, where x_{Π} = ∏_{α∈Σ^{−}} x_{α}. The map c_{S} : S → D^{⋆} sending s to s • 1 is called the algebraic (equivariant) characteristic map, and it is of special importance (see section 10).

We now turn to the setting related to parabolic subgroups. Let Ξ ⊆ Π be a subset and let W_{Ξ} be the subgroup of W generated by the s_{i} with α_{i} ∈ Ξ. Let Σ_{Ξ} = {α ∈ Σ|s_{α} ∈ W_{Ξ}}, and define Σ_{Ξ}^{+} = Σ^{+} ∩ Σ_{Ξ} and Σ_{Ξ}^{−} = Σ^{−} ∩ Σ_{Ξ}. For Ξ′ ⊆ Ξ ⊆ Π, let Σ_{Ξ/Ξ′}^{+} = Σ_{Ξ}^{+}\Σ_{Ξ′}^{+} and Σ_{Ξ/Ξ′}^{−} = Σ_{Ξ}^{−}\Σ_{Ξ′}^{−}. In S, we set

x_{Ξ/Ξ′} = ∏_{α∈Σ_{Ξ/Ξ′}^{−}} x_{α} and x_{Ξ} = x_{Ξ/∅}.

Let S_{W/W_{Ξ}} be the free S-module with basis {δw̄}_{w̄∈W/W_{Ξ}} and let Q_{W/W_{Ξ}} = Q ⊗_{S} S_{W/W_{Ξ}} be its localization.

As on Q_{W}, one can define a coproduct structure on Q_{W/W_{Ξ}} and S_{W/W_{Ξ}}, by the same diagonal formula. Let

S_{W/W_{Ξ}}^{⋆} = Hom_{S}(S_{W/W_{Ξ}}, S) and Q_{W/W_{Ξ}}^{∗} = Hom_{Q}(Q_{W/W_{Ξ}}, Q)

be the respective dual rings of the corings S_{W/W_{Ξ}} and Q_{W/W_{Ξ}}. On the basis {fv̄}_{v̄∈W/W_{Ξ}} dual to the basis {δw̄}_{w̄∈W/W_{Ξ}}, the unit element is 1_{Ξ} = ∑_{v̄∈W/W_{Ξ}} fv̄, both in S_{W/W_{Ξ}}^{⋆} and in Q_{W/W_{Ξ}}^{∗}.

Assume Ξ′ ⊆ Ξ. Let w̄ ∈ W/W_{Ξ′} and let ŵ denote its class in W/W_{Ξ}. Consider the projection and the sum over orbits

\begin{aligned}p_{Ξ/Ξ′} : & S_{W/W_{Ξ′}} & → & S_{W/W_{Ξ}} \\ & δw̄ & ↦ & δŵ \end{aligned} and \begin{aligned}d_{Ξ/Ξ′} : & S_{W/W_{Ξ}} & → & S_{W/W_{Ξ′}} \\ & δŵ & ↦ & ∑_{\begin{matrix}v̄∈W/W_{Ξ′} \\ v̂=ŵ \end{matrix}} δv̄ \end{aligned}

with S-dual maps

\begin{aligned}p_{Ξ/Ξ′}^{⋆} : & S_{W/W_{Ξ}}^{⋆} & → & S_{W/W_{Ξ′}}^{⋆} \\ & fŵ & ↦ & ∑_{\begin{matrix}v̄∈W/W_{Ξ′} \\ v̂=ŵ \end{matrix}} fv̄ \end{aligned} and \begin{aligned}d_{Ξ/Ξ′}^{⋆} : & S_{W/W_{Ξ′}}^{⋆} & → & S_{W/W_{Ξ}}^{⋆} \\ & fw̄ & ↦ & fŵ \end{aligned} .

Note that p_{Ξ/Ξ′} respects coproducts, so p_{Ξ/Ξ′}^{⋆} is a ring map while d_{Ξ/Ξ′}^{⋆} isn’t.

We set p_{Ξ} = p_{Ξ/∅}. Let D_{Ξ} denote the image of D via p_{Ξ}. The coproduct structure on Q_{W/W_{Ξ}} induces an S-linear coproduct structure on D_{Ξ}, so its S-dual D_{Ξ}^{⋆} has a ring structure. In summary, we have the following diagram followed by its dualization

\xymatrix{S_{W/W_{Ξ′}} \ar@{^{(}->}[r]^{η_{Ξ′}} \ar@{->>}[d]^{p_{Ξ/Ξ′}} & D_{Ξ′} \ar@{^{(}->}[r] \ar@{->>}[d]^{p_{Ξ/Ξ′}} & Q_{W/W_{Ξ′}} \ar@{->>}[d]^{p_{Ξ/Ξ′}} & D_{Ξ′}^{⋆} \ar@{^{(}->}[r]^{η_{Ξ′}^{⋆}} & S_{W/W_{Ξ′}}^{⋆} \ar@{^{(}->}[r] & Q_{W/W_{Ξ′}}^{∗} \\ S_{W/W_{Ξ}} \ar@{^{(}->}[r]^{η_{Ξ}} & D_{Ξ} \ar@{^{(}->}[r] & Q_{W/W_{Ξ}} & D_{Ξ}^{⋆} \ar@{^{(}->}[u]^{p_{Ξ/Ξ′}^{⋆}} \ar@{^{(}->}[r]^{η_{Ξ}^{⋆}} & S_{W/W_{Ξ}}^{⋆} \ar@{^{(}->}[u]^{p_{Ξ/Ξ′}^{⋆}} \ar@{^{(}->}[r] & Q_{W/W_{Ξ}}^{∗} \ar@{^{(}->}[u]^{p_{Ξ/Ξ′}^{⋆}}}

in which all horizontal maps become isomorphisms after tensoring by Q on the left. It will receive a geometric interpretation as Diagram (8.2). Moreover, by [CZZ2, Lemma 11.7], the image of p_{Ξ}^{⋆} in D^{⋆} (or S_{W}^{⋆}, Q_{W}^{∗}) is the subset of W_{Ξ}-invariant elements.

There is no ‘•’-action of S_{W/W_{Ξ}} on S_{W/W_{Ξ}}^{⋆} because S_{W/W_{Ξ}} is not a ring. But since x_{Π/Ξ} ∈ S^{W_{Ξ}}, the element x_{Π/Ξ} • f is well-defined for any f ∈ S_{W/W_{Ξ}}^{⋆} and actually belongs to D_{Ξ}^{⋆} inside S_{W/W_{Ξ}}^{⋆}, by [CZZ2, Lemma 15.3]. This defines a map D_{Ξ}^{⋆} → S_{W/W_{Ξ}}, interpreted geometrically in Diagram (8.1).

For a given set of representatives of W_{Ξ}/W_{Ξ′} we define the push-pull element by

Y_{Ξ/Ξ′} = ∑_{w∈W_{Ξ/Ξ′}} δ_{w} \frac{1}{x_{Ξ/Ξ′}} ∈ Q_{W} .

We set Y_{Ξ} = Y_{Ξ/∅}. If Ξ = {α_{i}}, then Y_{Ξ} = Y_{α_{i}}. By [CZZ2, Lemma 10.12], Y_{Ξ} ∈ D.

Let

A_{Ξ/Ξ′} : (Q_{W}^{∗})^{W_{Ξ′}} → (Q_{W}^{∗})^{W_{Ξ}}_{f ↦ Y_{Ξ/Ξ′} • f} and A_{Ξ/Ξ′} : Q_{W/W_{Ξ′}}^{∗} →  Q_{W/W_{Ξ}}^{∗}

f ↦ d_{Ξ/Ξ′}^{⋆}(\frac{1}{x_{Ξ/Ξ′}} • f)

and respectively call them push-pull operator and push-forward operator. The operator A_{Ξ/Ξ′} is actually independent of the choice of representatives [CZZ2, Lem. 6.5]. We have A_{Ξ/Ξ′}((D^{⋆})^{W_{Ξ′}}) = (D^{⋆})^{W_{Ξ}} by [CZZ2, Cor. 14.6] and A_{Ξ/Ξ′} induces a map A_{Ξ/Ξ′} : D_{Ξ′}^{⋆} → D_{Ξ}^{⋆} by [CZZ2, Lemma 15.1]. These two operators are related by the commutative diagram on the left below, becoming the one on the right after tensoring by Q.

\xymatrix{D_{Ξ′}^{⋆} \ar[r]^{p_{Ξ′}^{⋆}}_{≃} \ar[d]_{A_{Ξ/Ξ′}} & (D^{⋆})^{W_{Ξ′}} \ar[d]^{A_{Ξ/Ξ′}} & Q_{W/W_{Ξ′}}^{∗} \ar[r]^{p_{Ξ′}^{⋆}}_{≃} \ar[d]_{A_{Ξ/Ξ′}} & (Q_{W}^{∗})^{W_{Ξ′}} \ar[d]^{A_{Ξ/Ξ′}} \\ D_{Ξ}^{⋆} \ar[r]^{p_{Ξ}^{⋆}}_{≃} & (D^{⋆})^{W_{Ξ}} & Q_{W/W_{Ξ}}^{∗} \ar[r]^{p_{Ξ}^{⋆}}_{≃} & (Q_{W}^{∗})^{W_{Ξ}}}

Again, when Ξ′ = ∅, we set A_{Ξ} = A_{Ξ/∅} and A_{Ξ} = A_{Ξ/∅}.

6. Fixed points of the torus action

We now consider a split semi-simple algebraic group G over k containing T as a maximal torus, with character group Λ. Let W be the Weyl group associated to (G, T), with roots Σ ⊆ Λ. We choose a Borel subgroup B of G containing T. It defines a set Π of simple roots in W. Given a subset Ξ ⊆ Π, the subgroup generated by B and representatives in G(k) of reflections with respect to roots in Ξ is a parabolic subgroup, denoted by P_{Ξ}. The map sending Ξ to P_{Ξ} is a bijection between subsets of Π and parabolic subgroups of G containing B. Let W_{Ξ} be the subgroup of W generated by reflections with respect to roots in Ξ. We will abuse the notation by also writing W (or W_{Ξ}, etc.) when referring to the constant finite algebraic group over pt whose set of points over any field is W.

For any parabolic subgroup P, the quotient variety G/P is projective and we consider it in T -Var by letting T act on G by multiplication on the left. After identifying W ≃ N_{G}(T)/T, the Bruhat decomposition says that G/P = ⨆_{w∈W^{Ξ}} BwP_{Ξ}/P_{Ξ}, where the union is taken over the set W^{Ξ} of minimal left coset-representatives of W/W_{Ξ}. It induces a bijection between k-points of G/P_{Ξ} that are fixed by T and the set W^{Ξ} (or W/W_{Ξ}). In particular, fixed k-points of G/B are in bijection with elements of W.

Let (G/P_{Ξ})^{T} = ⨆_{w̄∈W/W_{Ξ}} ptw̄ denote the closed subvariety of T -fixed k-points, then by additivity there is an S = h_{T} (pt)-algebra isomorphism

Θ_{Ξ} : h_{T} ((G/P_{Ξ})^{T}) ⟶^{≃} ∏_{w̄∈W/W_{Ξ}} h_{T} (ptw̄) = ∏_{w̄∈W/W_{Ξ}} S ≅ S_{W/W_{Ξ}}^{⋆}.

If Π = ∅, we denote Θ : h_{T} ((G/B)^{T}) = h_{T} (W) → S_{W}^{⋆} .

Let ı_{Ξ} : (G/P_{Ξ})^{T} ↪ G/P_{Ξ} denote the (closed) embedding of the T -fixed locus, and let ıw̄ ; Ξ : ptw̄ ↪ G/P_{Ξ} denote the embedding corresponding to w̄. Given Ξ′ ⊆ Ξ ⊆ Π, we define projections

π_{Ξ/Ξ′}: G/P_{Ξ′} → G/P_{Ξ}  and  ρ_{Ξ/Ξ′}: W/W_{Ξ′} → W/W_{Ξ}

(here we view W/W_{Ξ} as a variety that is a disjoint union of copies of pt indexed by cosets). If Ξ = {α} consists of a single simple root α, we omit the brackets in the indices, i.e. we abbreviate W_{{α}} as W_{α}, P_{{α}} as P_{α}, etc. If Ξ′ = ∅, we omit the ∅ in the notation, i.e. π_{Ξ/∅} = π_{Ξ}, ρ_{Ξ/∅} = ρ_{Ξ}, etc. By definition, we have

(6.1) Θ_{Ξ} ∘ (ρ_{Ξ/Ξ′})_{∗} = d_{Ξ/Ξ′}^{⋆} ∘ Θ_{Ξ′} and Θ_{Ξ′} ∘ (ρ_{Ξ/Ξ′})^{∗} = p_{Ξ/Ξ′}^{⋆} ∘ Θ_{Ξ}.

The following lemma is easy and well-known. We include a proof for the sake of completeness.

Lemma 6.1. Let w ∈ W be a representative of w̄ ∈ W/W_{Ξ}. The pull-pack (ıw̄ ; Ξ)^{∗}T_{G/P_{Ξ}} of the tangent bundle T_{G/P_{Ξ}} of G/P_{Ξ} is the representation of T (the T -equivariant bundle over a point) with weights {w(α) | α ∈ Σ_{Π/Ξ}^{−}}. (This set is indeed independent of the choice of a representative w, e.g. by [CZZ2, Lemma 5.1].)

Proof. Consider the exact sequence of T -representations at the neutral element e ∈ G

0 → T_{P_{Ξ},e} → T_{G,e} → T_{G/P_{Ξ},e} → 0

(it is exact by local triviality of the right P_{Ξ}-torsor G → G/P_{Ξ}). By definition of the root system associated to (G, T), the roots Σ are the characters of T_{G,e}. By definition of the parabolic subgroup P_{Ξ}, the characters of T_{P_{Ξ},e} are Σ^{+} ⊔ Σ_{Ξ}^{−}. This proves the lemma when w = e. For an arbitrary w, we consider the diagram

\xymatrix{pt_{e} \ar[r]^{ı_{Ξ}^{e}} \ar[dr]_{ıē ; Ξ} & G \ar[r]^{w·} \ar[d] & G \ar[d] \\  & G/P_{Ξ} \ar[r]^{w·} & G/P_{Ξ}}

which is T -equivariant if T acts by multiplication on the left on the right column and through conjugation by w^{−1} and then by multiplication on the left on the left column. Since ıw̄ ; Ξ is the bottom composite from pt_{e} to G/P_{Ξ}, the fiber of T_{G/P_{Ξ}} at w̄ is isomorphic to its fiber at e, but for every character α, the action of T is now by t(v) = α(w̄^{−1}tw̄) · v = α(w^{−1}(t)) · v = w(α)(t) · v, in other words by the character w(α). □

Proposition 6.2. We have (ıw̄ ; Ξ)^{∗}(ı_{Ξ}^{w̄′})_{∗}(1) = 0 if w̄ ≠ w̄′ ∈ W/W_{Ξ} and

(ıw̄ ; Ξ)^{∗}(ıw̄ ; Ξ)_{∗}(1) = ∏_{α∈Σ_{Π/Ξ}^{−}} x_{w(α)} = w(x_{Π/Ξ}).

Proof. The case w̄ ≠ w̄′ holds by transverse base change through the empty scheme. Since the normal bundle to a point in G/P_{Ξ} is the tangent bundle of G/P_{Ξ} pulled back to that point, and since any T -representation splits into one-dimensional ones, the case w̄ = w̄′ follows from (A8) using Lemma 6.1 to identify the characters. □

Remark 6.3. Note that in the Borel case, the inclusion of an individual fixed point is local complete intersection as any other morphism between smooth varieties, but not “global” complete intersection, in the sense that it is not the zero locus of transverse sections of a globally defined vector bundle. Otherwise, for Chow groups, such a point would be in the image of the characteristic map as a product of first characteristic classes, and it isn’t for types for which the simply connected torsion index isn’t 1. Locally, on an open excluding other fixed points, it becomes such a product, as the previous proposition shows.

Corollary 6.4. We have Θ_{Ξ}(ı_{Ξ})^{∗}(ı_{Ξ})_{∗}(1) = x_{Π/Ξ} • 1_{Ξ}.

Proof. Since ı_{Ξ} = ⨆_{w̄∈W/W_{Ξ}} ıw̄ ; Ξ, we have

Θ_{Ξ}(ı_{Ξ})^{∗}(ı_{Ξ})_{∗}(1) = Θ_{Ξ}( ∑_{v̄, w̄∈W/W_{Ξ}} (ıv̄ ; Ξ)^{∗}(ıw̄ ; Ξ)_{∗}(1)) = Θ_{Ξ}( ∑_{w̄∈W/W_{Ξ}} w(x_{Π/Ξ})1_{ptw̄})
= ∑_{w̄∈W/W_{Ξ}} w(x_{Π/Ξ})fw̄ = x_{Π/Ξ} • 1_{Ξ}.  □

7. Bott-Samelson classes

In the present section we describe the Bott-Samelson classes in the T - equivariant cohomology of G/P_{Ξ}.

Let Ξ ⊆ Π as before. For each w̄ ∈ W/W_{Ξ} consider the B-orbit BwP_{Ξ}/P_{Ξ} of the point in G/P_{Ξ} corresponding to w̄. It is isomorphic to the affine space 𝔸^{l(v)} where v ∈ W^{Ξ} is the representative of w̄ of minimal length l(v). Its closure \overline{BwP_{Ξ}/P_{Ξ}} is called the Schubert variety at w̄ with respect to Ξ and is denoted by X Ξ ; w̄. If Ξ = ∅, we write X_{w} for X_{w}^{∅}. Moreover, by Bruhat decomposition the closed complement of BwP_{Ξ}/P_{Ξ} is the union of Schubert varieties X Ξ ; ū with ū < w̄ for the Bruhat order on W/W_{Ξ}. For any w ∈ W, the projection map G/B → G/P_{Ξ} induces a projective map X_{w} → X_{w}^{Ξ}. Moreover, if w ∈ W^{Ξ}, then the projection X_{w} → X_{w}^{Ξ} is (projective and) birational.

The variety X Ξ ; w̄ is not smooth in general, but it admits nice desingularizations, that we now recall, following [D74]. Given a sequence of simple reflections I = (s_{1}, …, s_{l}) corresponding to simple roots (α_{1}, …, α_{l}), the Bott-Samelson desingularization of X_{I} is defined as

X̂_{I} = P_{α_{1}} ×^{B} P_{α_{2}} ×^{B} ⋯ ×^{B} P_{α_{l}}/B

where ×^{B} means the quotient by the action of B given on points by b · (x, y) = (xb^{−1}, by). By definition, the multiplication of all factors induces a map q_{I}: X̂_{I} → G/B which factors through a map μ_{I}: X̂_{I} → X_{w(I)} where w(I) = s_{1} ⋯ s_{l}. It is easy to see that if I′ = (s_{1}, …, s_{l−1}), the diagram

(7.1)  \xymatrix{X̂_{I} \ar[r]^{q_{I}} \ar[d]_{p′} & G/B \ar[d]^{π_{α_{l}}} \\ X̂_{I′} \ar[r]^{π_{α_{l}} ∘q_{I′}} & G/P_{α_{l}}}

is cartesian, when p′ is projection on the first l − 1 factors. By induction on l, the variety X̂_{I} is smooth projective and the morphism μ_{I} is projective. When furthermore I is a reduced decomposition of w(I), meaning that it is of minimal length among the sequences J such that w(J) = w(I), the map μ_{I} is birational (still by Bruhat decomposition). We can compose this map with the projection to get a map X̂_{w} → X Ξ ; w̄ and thus when w ∈ W^{Ξ}, we obtain a (projective birational) desingularization X̂_{w} → X Ξ ; w̄. It shows that, G/P_{Ξ} has a cellular decomposition with desingularizations, as considered just before [CPZ, Thm. 8.8], with cells indexed by elements of W/W_{Ξ}.

Remark 7.1. The flag varieties, the Schubert varieties, their Bott-Samelson desingularizations and the various morphisms between them that we have just introduced are all B-equivariant when B acts on the left, and therefore are T -equivariant.

Definition 7.2. Let q_{I}^{Ξ} = π_{Ξ} ∘ q_{I}, let ζ_{I}^{Ξ} be the push-forward (q_{I}^{Ξ})_{∗}(1) in h_{T} (G/P_{Ξ}), and let ζ_{I} = ζ_{I}^{∅} in h_{T} (G/B).

Note that by definition, we have (π_{Ξ})_{∗}(ζ_{I}) = ζ_{I}^{Ξ}.

Lemma 7.3. For any choice of reduced sequences {I_{w}}_{w∈W^{Ξ}}, the classes ζ_{I_{w}}^{Ξ} generate h_{T} (G/P_{Ξ}) as an S-module.

Proof. The proof of [CPZ, Theorem 8.8] goes through when h is replaced by h_{T}, since all morphisms involved are T -equivariant; it only uses homotopy invariance and localization. □

Let V_{0} (resp. V_{α}) be the 1-dimensional representation of T corresponding to the 0 (resp. α) character. Let σ_{0} and σ_{α} be the inclusions of T -fixed points corresponding to V_{0} and V_{α} in ℙ(V_{0} ⊕ V_{α}) as in the setting of Section 4. Consider the projection π_{α}: G/B → G/P_{α}. Given an element w ∈ W, with image w̄ in W/W_{α} and any lifting w′ of w in G, the fiber over the fixed point

ıw̄ ; α: ptw̄ → G/P_{α} is w′P_{α}/B.

Lemma 7.4. There is a T -equivariant isomorphism w′P_{α}/B ≃ ℙ(V_{0} ⊕ V_{−w(α)}), such that the closed fixed point ı^{w}: pt_{w} → w′P_{α}/B ↪ G/B (resp. ı^{ws_{α}}) is sent to σ_{0} : pt → ℙ(V_{0} ⊕ V_{−w(α)}) (resp. to σ_{−w(α)}).

Proof. Multiplication on the left by w′ defines an isomorphism P_{α}/B → w′P_{α}/B and it is T -equivariant if T acts by multiplication on the left on w′P_{α}/B and through conjugation by (w′)^{−1} and then by multiplication on the left on P_{α}/B. Thus, we can reduce to the case where w′ = e: the general case follows by replacing the character α by w(α). First, let us observe that PGL_{2} acts on the projective space ℙ^{1} by projective transformations, i.e.

\overline{(\begin{matrix}t & b \\ c & d \end{matrix})}[x : y] = [tx + by : cx + dy]

with its Borel subgroup B_{PGL_{2}} of upper triangular matrices fixing the point [1 : 0], which therefore gives an identification PGL_{2}/B_{PGL_{2}} ≃ ℙ^{1}. So, its maximal torus 𝔾_{m} of matrices such that b = c = 0 and d = 1 acts by t[x : y] = [tx : y] = [x : t^{−1}y]. Thus, as a 𝔾_{m}-variety, this ℙ^{1} is actually ℙ(V_{1} ⊕ V_{0}) ≃ ℙ(V_{0} ⊕ V_{−1}). The adjoint semi-simple quotient of P_{α} is of rank one, so it is isomorphic to PGL_{2}. The maximal torus T maps to a maximal torus 𝔾_{m} and the Borel B to a Borel in this PGL_{2}. Up to modification of the isomorphism by a conjugation, we can assume that this Borel of PGL_{2} is indeed B_{PGL_{2}} as above. The map T → 𝔾_{m} is ±α (the sign depends on how the maximal torus of PGL_{2} is identified with 𝔾_{m}). Since P_{α}/B ≃ PGL_{2}/B_{PGL_{2}}, we are done by the PGL_{2} case. □

Recall the notation from section 5.

Lemma 7.5. The following diagram commutes.

[figure: h_{T} (G/B); π_{α}^{∗}(π_{α})_{∗}; ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}; A_{α}; h_{T} (G/B); ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}]

Proof. In view of Lemma 7.4, the strategy is to reduce to the case of Lemma 4.6 by restricting to the fiber over one fixed point of G/P_{α} at a time. We decompose Q_{W}^{∗} = ⨁_{w∈W^{α}}(Q · f_{w} ⊕ Q · f_{ws_{α}}) and note that A_{α} preserves this decomposition since

A_{α}(f_{w}) = \frac{1}{x_{−w(α)}} (f_{w} + f_{ws_{α}}), A_{α}(f_{ws_{α}}) = \frac{1}{x_{w(α)}} (f_{w} + f_{ws_{α}})

and A_{α} is Q-linear. It therefore suffices to check the commutativity of the diagram after extending both rows on the right by a projection Q_{W}^{∗} → Q · f_{w} ⊕ Q · f_{ws_{α}}, for all w ∈ W^{α}. But then, the composite horizontal maps h_{T} (G/B) → Q · f_{w} ⊕ Q · f_{ws_{α}} factor as

h_{T} (G/B) → h_{T} (P_{α}wB/B) → h_{T} (pt) ⊕ h_{T} (pt)≃ ≃ S ⊕ S ⊆ S[\frac{1}{x_{w(α)}} ] ⊕ S[\frac{1}{x_{w(α)}} ] ⊆ Q ⊕ Q.

Using proper base change on the diagram

\xymatrix{G/B \ar[d]_{π_{α}} & w′P_{α}/B \ar[d] \ar@{^{(}->}[l] \\ G/P_{α} & pt \ar@{^{(}->}[l]_{ıw̄ ; α}}

and identifying w′P_{α}/B with ℙ(V_{0} ⊕ V_{−w(α)}) by Lemma 7.4, we are reduced to proving the commutativity of

[figure: h_{T} (ℙ(V_{0} ⊕ V_{−w(α)})); p^{∗}p_{∗}; σ^{∗}; h_{T} (pt) ⊕ h_{T} (pt); S ⊕ S; ≃; ⊆; S[ \frac{1}{x_{w(α)}} ] ⊕ S[ \frac{1}{x_{w(α)}} ]; A_{α}; h_{T} (ℙ(V_{0} ⊕ V_{−w(α)})); σ^{∗}; h_{T} (pt) ⊕ h_{T} (pt); S ⊕ S; ≃; ⊆; S[ \frac{1}{x_{w(α)}} ] ⊕ S[ \frac{1}{x_{w(α)}} ]]

which immediately reduces to the diagram of Lemma 7.5 followed by an obvious commutative diagram involving pull-backs

[figure: h_{T} (ℙ(V_{0} ⊕ V_{−w(α)})); σ^{∗}; h_{T} (pt) ⊕ h_{T} (pt); S ⊕ S; ≃; ⊆; S[ \frac{1}{x_{w(α)}} ] ⊕ S[ \frac{1}{x_{w(α)}} ]; h_{T} (pt); p^{∗}; Δ; S; ≃; ⊆; Δ; S[ \frac{1}{x_{w(α)}} ]; Δ]

in which Δ is the diagonal morphism. □

Lemma 7.6. For any sequence I = (i_{1}, …, i_{l}), the Bott-Samelson class ζ_{I} ∈ h_{T} (G/B) maps to

Θ ∘ ı^{∗}(ζ_{I}) = A_{I^{rev}}(x_{Π} · f_{e})

in S_{W}^{⋆}.

Proof. By induction using diagram (7.1), we have

ζ_{I} = π_{α_{i_{l}}}^{∗} (π_{α_{i_{l}}})_{∗} ∘ ⋯ ∘ π_{α_{i_{1}}}^{∗} (π_{α_{i_{1}}} )_{∗} ∘ (ı^{e})_{∗}(1).

Since Θı^{∗}(ı^{e})_{∗}(1) = x_{Π} · f_{e} by Proposition 6.2, the conclusion follows from Lemma 7.5. □

8. Pull-back to T -fixed points

In the present section we describe the T -equivariant cohomology of an arbitrary split flag variety G/P_{Ξ} via the pull-back map to the cohomology of T -fixed points.

First, consider the complete flag variety G/B.

Proposition 8.1. For any choice of reduced decompositions (I_{w})_{w∈W}, the family (ζ_{I_{w}})_{w∈W} form a basis of h_{T} (G/B) over S = h_{T} (pt).

Proof. By Lemma 7.6, the element ζ_{I_{w}} pulls-back to A_{I^{rev}}(x_{Π} · f_{e}) in S_{W}^{⋆} and these are linearly independent over S by [CZZ2, Theorem 12.4]. They generate h_{T} (G/B) by Lemma 7.3. □

Theorem 8.2. The pull-back map to fixed points ı^{∗} : h_{T} (G/B) → h_{T} (W) is injective, and the isomorphism Θ : h_{T} (W) ≃ S_{W}^{⋆}, identifies its image to D^{⋆} ⊆ S_{W}^{⋆}.

Proof. This follows from Lemma 8.1 and the fact that the A_{I^{rev}}(x_{Π} · f_{e}) form a basis of D^{⋆} as a submodule of S_{W}^{⋆}, still by [CZZ2, Theorem 12.4]. □

Remark 8.3. We do not know a direct geometric proof that h_{T} (G/B) injects into h_{T} ((G/B)^{T}), which is of course well known for Chow groups or K-theory. To prove injectivity for Chow groups, one usually argues along the following lines:

(a) the composition ı^{∗} ∘ ı_{∗} becomes an isomorphism over Q (see Prop. 6.2);

(b) CH_{T} (G/B) is a free CH_{T} (pt)-module of rank |W | and so is CH_{T} ((G/B)^{T}).

(c) the pull-back Q ⊗_{S} CH_{T} (G/B) → Q ⊗_{S} CH_{T} ((G/B)^{T}) is an isomorphism as any surjection of free modules of the same rank over a noetherian ring (Q is a localization of CH_{T} (pt) = ℤ[[x_{1}, …, x_{n}]]).

However, in the general case, localization arguments only give generating families so the freeness part of (b) does not follow, and in (c), Q is not noetherian (e.g. the Lazard ring is not noetherian), so we need to look more carefully into the structure of the image as a submodule of h_{T} ((G/B)^{T}). This is done in the algebraic world: the Bott-Samelson classes considered are linearly independent when pulled to S_{W}^{⋆} and the map D^{⋆} → S_{W}^{⋆} is indeed an injection by [CZZ2, Lemma 10.2].

Corollary 8.4. The pull-back map ı^{∗} : h_{T} (G/B) → h_{T} (W) becomes an isomorphism after localization at the multiplicative subset generated by all x_{α} where α is a root.

Proof. After localization at this subset, the inclusion D^{⋆} ⊆ S_{W}^{⋆} becomes an isomorphism (see [CZZ2, Lemma 10.2]). □

Lemma 8.5. The following diagram commutes

\xymatrix{h_{T} (W) \ar[d]_{≃}^{Θ} \ar[r]^{ı_{∗}} & h_{T} (G/B) \ar@{^{(}->}[r]^{ı^{∗}} \ar[d]_{≃}^{Θ} & h_{T} (W) \ar[d]_{≃}^{Θ} \\ S_{W}^{⋆} \ar[r]^{x_{Π}•(−)} & D^{⋆} \ar@{^{(}->}[r]^{η^{∗}} & S_{W}^{⋆}}

Proof. This follows from Corollary 6.4 and Theorem 8.2. □

We now consider an arbitrary flag variety G/P_{Ξ}.

Lemma 8.6. The following diagram commutes.

[figure: h_{T} (G/B); (π_{Ξ})_{∗}; ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}; A_{Ξ}; h_{T} (G/P_{Ξ}); ı_{Ξ}^{∗}; h_{T} (W/W_{Ξ}); Θ_{Ξ}; ≃; S_{W/W_{Ξ}}^{⋆}; ⊆; Q_{W/W_{Ξ}}^{∗}]

Proof. After tensoring the whole diagram with Q over S, the morphism ı^{∗} becomes an isomorphism by Corollary 8.4. The family ((ı^{w})_{∗}(1))_{w∈W} is a Q-basis of Q ⊗_{S} h_{T} (G/B), since by Proposition 6.2, Θ ∘ ı^{∗} ∘ (ı^{w})_{∗}(1) is f_{w} multiplied by an element that is invertible (in Q). It therefore suffices to check the equality of both compositions in the diagram when applied to all (ı^{w})_{∗}(1) with w ∈ W:

A_{Ξ} ∘ Θ ∘ ı^{∗} ∘ (ı^{w})_{∗}(1) = A_{Ξ}(w(x_{Π})f_{w}) = w(x_{Π})A_{Ξ}(f_{w}) =^{(∗)}

=^{(∗)} w(x_{Π/Ξ})fw̄ = Θ_{Ξ}(ı_{Ξ})^{∗}(ıw̄ ; Ξ)_{∗}(1) = Θ_{Ξ}(ı_{Ξ})^{∗}(π_{Ξ})_{∗}(ı^{w})_{∗}(1)

where equality (∗) follows from the definition of A_{Ξ}. □

Corollary 8.7. The following diagram commutes.

[figure: h_{T} (G/B); (π_{Ξ})^{∗}(π_{Ξ})_{∗}; ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}; A_{Ξ}; h_{T} (G/B); ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}]

Proof. Using equation (6.1), one easily checks the commutativity of diagram involving pull-backs

[figure: h_{T} (G/B); ı^{∗}; h_{T} (W); Θ; ≃; S_{W}^{⋆}; ⊆; Q_{W}^{∗}; h_{T} (G/P_{Ξ}); π_{Ξ}^{∗}; ı^{∗}; h_{T} (W/W_{Ξ}); Θ_{Ξ}; ≃; S_{W/W_{Ξ}}^{⋆}; ⊆; Q_{W/W_{Ξ}}^{∗}; p_{Ξ}^{⋆}]

where p_{Ξ}^{⋆} is the sum over orbits: p_{Ξ}^{⋆}(fw̄) = ∑_{v̄=w̄} f_{v}. The result follows from the combination of this diagram and the one of Lemma 8.6. □

Lemma 8.8. For any sequence I = (i_{1}, …, i_{l}), the Bott-Samelson class ζ_{I}^{Ξ} ∈ h_{T} (G/P_{Ξ}) maps to

Θ ∘ (ı_{Ξ})^{∗}(ζ_{I}^{Ξ}) = A_{Ξ} ∘ A_{I^{rev}}(x_{Π}f_{e})

in S_{W}^{⋆}.

Proof. We have

Θ(ı_{Ξ})^{∗}(ζ_{I_{w}}^{Ξ}) = Θ(ı_{Ξ})^{∗}(π_{Ξ})_{∗}(ζ_{I_{w}}) = A_{Ξ} ∘ Θ ∘ ı^{∗}(ζ_{I_{w}}) = A_{Ξ} ∘ A_{I_{w}^{rev}}(x_{Π}f_{e})

using Lemma 8.6 and Lemma 7.6 for the last two equalities. □

Proposition 8.9. For any choice of reduced decompositions (I_{w})_{w∈W^{Ξ}} for elements minimal in their W_{Ξ}-cosets, the classes ζ_{I_{w}}^{Ξ} form an S-basis of h_{T} (G/P_{Ξ}).

Proof. By Lemma 7.3, the classes ζ_{I_{w}}^{Ξ} generate h_{T} (G/P_{Ξ}) as an S-module. We have

Θı^{∗}(π_{Ξ})^{∗}(ζ_{I_{w}}^{Ξ}) = Θı^{∗}(π_{Ξ})^{∗}(π_{Ξ})_{∗}(ζ_{I_{w}}) = A_{Ξ}Θı^{∗}(ζ_{I_{w}}) = A_{Ξ}A_{I_{w}^{rev}}(x_{Π}f_{e})

and these elements are linearly independent by [CZZ2, Theorem 14.3]. □

Let Ξ′ ⊆ Ξ ⊆ Π.

Corollary 8.10. The push-forward map (π_{Ξ/Ξ′})_{∗} : h_{T} (G/P_{Ξ′}) → h_{T} (G/P_{Ξ}) is surjective and the pull-back map (π_{Ξ/Ξ′})^{∗} : h_{T} (G/P_{Ξ}) → h_{T} (G/P_{Ξ′}) is injective.

Proof. Surjectivity is obvious from the fact that ζ_{I_{w}} maps to the basis element ζ_{Iw̄}^{Ξ} for any w ∈ W^{Ξ} and injectivity can be seen in the proof of Proposition 8.9: the elements ζ_{Iw̄}^{Ξ} stay independent when pulled back all the way to h_{T} (W) through h_{T} (G/B). □

Theorem 8.11. The pull-back map ı_{Ξ}^{∗} : h_{T} (G/P_{Ξ}) → h_{T} (W/W_{Ξ}) is injective and the isomorphism Θ_{Ξ} : h_{T} (W/W_{Ξ}) →^{∼} S_{W/W_{Ξ}}^{⋆} identifies its image to D_{Ξ}^{⋆} ⊆ S_{W/W_{Ξ}}^{⋆}.

Proof. As seen in the proof of Corollary 8.10, pulling back further to h_{T} (W) is injective, so injectivity of ı_{Ξ}^{∗} is clear. By Lemma 8.8, for any w ∈ W^{Ξ}, the Bott-Samelson class ζ_{I_{w}}^{Ξ} is sent to A_{Ξ}A_{I_{w}^{rev}}(x_{Π/Ξ}f_{e}). These elements form a basis of D_{Ξ}^{⋆} by [CZZ2, Theorem 14.3 and Lemma 15.1]. □

Corollary 8.12. The pull-back map ı_{Ξ}^{∗} : h_{T} (G/P_{Ξ}) → h_{T} (W/W_{Ξ}) becomes an isomorphism after localization at the multiplicative subset generated by all x_{α} where α is a root.

Proof. After localization at this subset, the inclusion D_{Ξ}^{⋆} ⊆ S_{W/W_{Ξ}}^{⋆} becomes an isomorphism (see [CZZ2, Lemma 11.5]). □

As for G/B, we have the following commutative diagram

(8.1)  \xymatrix{h_{T} (W/W_{Ξ}) \ar[r]^{(ı_{Ξ})_{∗}} \ar[d]_{≃}^{Θ_{Ξ}} & h_{T} (G/P_{Ξ}) \ar@{^{(}->}[r]^{(ı_{Ξ})^{∗}} \ar[d]_{≃}^{Θ_{Ξ}} & h_{T} (W/W_{Ξ}) \ar[d]_{≃}^{Θ_{Ξ}} \\ S_{W/W_{Ξ}}^{⋆} \ar[r]^{x_{Π/Ξ}•(−)} & D_{Ξ}^{⋆} \ar@{^{(}->}[r]^{η_{Ξ}^{∗}} & S_{W/W_{Ξ}}^{⋆}}

Lemma 8.13. The following diagram commutes.

[figure: h_{T} (G/P_{Ξ′}); (π_{Ξ/Ξ′})_{∗}; (ı_{Ξ′})^{∗}; h_{T} (W/W_{Ξ′}); Θ_{Ξ′}; ≃; S_{W/W_{Ξ′}}^{⋆}; ⊆; Q_{W/W_{Ξ′}}^{∗}; A_{Ξ/Ξ′}; h_{T} (G/P_{Ξ}); (ı_{Ξ})^{∗}; h_{T} (W/W_{Ξ}); Θ_{Ξ}; ≃; S_{W/W_{Ξ}}^{⋆}; ⊆; Q_{W/W_{Ξ}}^{∗}]

Proof. By the surjectivity claim in Corollary 8.10, we can precompose the diagram by π_{Ξ′}. Since A_{Ξ} = A_{Ξ/Ξ′} ∘ A_{Ξ′}, the result follows from Lemma 8.6 applied first to Ξ′ and then to Ξ. □

Summarizing, we have the following commutative diagrams describing the correspondence between the cohomology rings and their algebraic counterparts:

(8.2)  [figure: h_{T} (W/W_{Ξ′}); ≃; Θ_{Ξ′}; S_{W/W_{Ξ′}}^{⋆}; h_{T} (G/P_{Ξ′}); ≃; Θ_{Ξ′}; ı_{Ξ′}^{∗}; D_{Ξ′}^{⋆}; h_{T} (W/W_{Ξ}); ≃; Θ_{Ξ}; (ρ_{Ξ/Ξ′})^{∗}; S_{W/W_{Ξ}}^{⋆}; (p_{Ξ/Ξ′})^{⋆}; h_{T} (G/P_{Ξ}); ≃; Θ_{Ξ}; (π_{Ξ/Ξ′})^{∗}; ı_{Ξ}^{∗}; D_{Ξ}^{⋆}]

For push-forwards, instead, the morphism A_{Ξ/Ξ′} : Q_{W/W_{Ξ′}}^{∗} → Q_{W/W_{Ξ}}^{∗} induces a map A_{Ξ/Ξ′} : D_{Ξ′}^{⋆} → D_{Ξ}^{⋆} by [CZZ2, Lemma 15.1], and we have: (8.3)

[figure: h_{T} (W/W_{Ξ′}); ≃; Θ_{Ξ′}; S_{W/W_{Ξ′}}^{⋆}; Q_{W/W_{Ξ′}}^{∗}; A_{Ξ/Ξ′}; h_{T} (G/P_{Ξ′}); (π_{Ξ/Ξ′})_{∗}; ≃; Θ_{Ξ′}; ı_{Ξ′}^{∗}; D_{Ξ′}^{⋆}; A_{Ξ/Ξ′}; h_{T} (W/W_{Ξ}); ≃; Θ_{Ξ}; S_{W/W_{Ξ}}^{⋆}; Q_{W/W_{Ξ}}^{∗}; h_{T} (G/P_{Ξ}); ≃; Θ_{Ξ}; ı_{Ξ}^{∗}; D_{Ξ}^{⋆}]

Notice that on this diagram, there is no map from h_{T} (W/W_{Ξ′}) to h_{T} (W/W_{Ξ}), nor from S_{W/W_{Ξ′}}^{⋆} to S_{W/W_{Ξ}}^{⋆} because the operator A_{Ξ/Ξ′} is not defined at that level.

By (8.1) and the identity x_{Π/Ξ′} = x_{Π/Ξ}x_{Ξ/Ξ′}, we finally have the following. (8.4)

[figure: h_{T} (W/W_{Ξ′}); (ı_{Ξ′})_{∗}; (ρ_{Ξ/Ξ′})_{∗}; ≃; Θ_{Ξ′}; S_{W/W_{Ξ′}}^{⋆}; x_{Π/Ξ′} •; (d_{Ξ/Ξ′})^{⋆}; Q_{W/W_{Ξ′}}^{∗}; (d_{Ξ/Ξ′})^{⋆}; h_{T} (G/P_{Ξ′}); ≃; Θ_{Ξ′}; (π_{Ξ/Ξ′})_{∗}; D_{Ξ′}^{⋆}; A_{Ξ/Ξ′}; h_{T} (W/W_{Ξ}); ≃; Θ_{Ξ}; (ı_{Ξ})_{∗}; S_{W/W_{Ξ}}^{⋆}; x_{Π/Ξ′} •; Q_{W/W_{Ξ}}^{∗}; h_{T} (G/P_{Ξ}); ≃; Θ_{Ξ}; D_{Ξ}^{⋆}]

9. Invariant subrings and push-forward pairings

We now describe how the Weyl group W, as an abstract group, acts on h_{T} (G/B), and how W_{Ξ}-invariant elements of this action are related to h_{T} (G/P_{Ξ}).

Since the projection G/T → G/B is an affine bundle, by homotopy invariance the induced pull-back h_{T} (G/B) →^{∼} h_{T} (G/T) is an isomorphism. The Weyl group action is easier to describe geometrically on h_{T} (G/T). Since W ≃ N_{G}(T)/T, multiplication on the right by w ∈ W defines a right action of W on G/T, by T -equivariant morphisms. Action by induced pull-backs, therefore, defines a left action of W on h_{T} (G/T). Similarly, a right action of W on the T -fixed points (G/T)^{T} = W induces a left action of W on h_{T} (W), and the pullback h_{T} (G/T) → h_{T} (W) is W -equivariant. Identifying h_{T} (G/T) ≃ h_{T} (G/B), we obtain the Weyl group action on h_{T} (G/B) with ı^{∗}: h_{T} (G/B) → h_{T} (W) being W -equivariant.

One easily checks on S-basis elements f_{w} that through Θ, this W -action on h_{T} (W) corresponds to the W -action on S_{W}^{⋆} by the Hecke action w(z) = δ_{w} • z, as described in [CZZ2, §4] (by definition, we have δ_{w} • f_{v} = f_{vw^{−1}}).

Theorem 9.1. The image of the injective pull-back map h_{T} (G/P_{Ξ}) → h_{T} (G/B) is h_{T} (G/B)^{W_{Ξ}}.

Proof. In Diagram (8.2), the upper square is W -equviariant. Since ı^{∗} is both W -equivariant and injective, we are reduced to showing that p_{Ξ}^{⋆} identifies S_{W/W_{Ξ}}^{⋆} to (S_{W}^{⋆})^{W_{Ξ}}, which follows from [CZZ2, Lemma 11.7]. □

The following theorem generalizes [Br97, Proposition 6.5.(i)]. According to the irreducible Dynkin types of the group, regularity assumptions on elements of the base ring R (or weaker assumptions on elements in R[[x]]) are needed. They are carefully summarized in [CZZ2, Lemma 2.7], but as a first approximation, regularity in R of 2, 3 and divisors of |Λ_{w}/Λ_{r}| cover all types, except the C_{n}^{sc} case, in which one needs 2 to be invertible.^{4}

Theorem 9.2. Under the regularity assumptions of [CZZ2, Lemma 2.7], the image of the injective pull-back ı^{∗} : h_{T} (G/B) → h_{T} (W) ≅ S_{W}^{⋆} is the set of element ∑_{w∈W} q_{w}f_{w} such that x_{α}|(q_{w} − q_{s_{α}w}) for all roots α.

Proof. If follows from [CZZ2, Theorem 10.7]. □

We now describe the pairing given by multiplication and then push-forward to the point, that we call the push-forward pairing. Let

h_{T} (G/P_{Ξ}) ⊗_{S} h_{T} (G/P_{Ξ}) \underrightarrow{⟨−, −⟩_{Ξ}}  S
ξ ⊗ ξ′  ⟼ ⟨ξ, ξ′⟩_{Ξ} = (π_{Π/Ξ})_{∗}(ξ · ξ′)

It is clearly S-bilinear and symmetric. Through the isomorphism Θ, this pairing corresponds to

⟨ξ, ξ′⟩_{Ξ} = A_{Π/Ξ}(Θ_{Ξ}(ξ) · Θ_{Ξ}(ξ′))

by Diagram (8.3).

Theorem 9.3. The push-forward pairing h_{T} (G/P_{Ξ})⊗_{S} h_{T} (G/P_{Ξ}) → h_{T} (pt) ≃ S, sending (ξ, ξ′) to ⟨ξ, ξ′⟩_{Ξ} is non-degenerate.

Proof. This follows from [CZZ2, Theorem 15.6]. □

Remark 9.4. Note that in [CZZ2, Theorem 15.5], we describe a basis that is dual to the basis of Bott-Samelson classes for the push-forward pairing on G/B. That dual basis can be very useful for algorithmic computations. However, it is given in combinatorial terms, and we do not have a geometric interpretation of its elements. When the formal group law is additive, this problem disappears since the basis is auto-dual (up to a permutation), see [D74, Prop. 1, p. 69], but for general formal group laws, this is not the case.

10. Borel-style presentation

The geometric (equivariant) characteristic map c_{g} : h_{T} (pt) → h_{T} (G/B) is defined as the composition

h_{T} (pt) →^{∼} h_{T ×G}(G) ←^{∼} h_{G}(G/T) → h_{T} (G/T) ←^{∼} h_{T} (G/B)

where the first two maps are isomorphisms from Axiom (A6), the third is the restriction to the subgroup T of G and the fourth is the pull-back map, an isomorphism by Axiom (A4) of homotopy invariance. In h_{T ×G}(G), the action of T × G on G is by (t, g) · g′ = gg′t^{−1}, and the other non-trivial actions are by multiplication on the left. Note that c_{g} is R = h(pt)-linear, although not h_{T} (pt)-linear. By restricting further to h(G/B), one obtains the nonequivariant characteristic map c : h_{T} (pt) → h(G/B). Recall the algebraic characteristic map c_{S} : S → D^{⋆}, sending s → s • 1, defined in section 5.

4Regarding these assumptions, there is a slight omission in the statement of [Br97, Proposition 6.5.(i)]. One needs to add that no root is divisible in the lattice for the statement to hold integrally. Otherwise, for example, the product of all roots divided by 2 gives a counterexample in the C_{2}^{sc} case.

Documenta Mathematica · Extra Volume Merkurjev (2015) 113–144

Lemma 10.1. The algebraic and geometric characteristic maps coincide with each other, up to the identifications S ≃ h_{T} (pt) of Theorem 3.3 and Θ : h_{T} (G/B) ≃ D^{⋆} of Theorem 8.2.

Proof. It suffices to show the equality after embedding in S_{W}^{⋆} ≃ h_{T} (W), which decomposes as copies of S. In other words, it suffices to compare, for every w ∈ W, a map ϕ_{w} from S to itself, and a map ψ_{w} from h_{T} (pt) to itself. Both are continuous R-algebra maps, ψ_{w} for the topology induced by the γfiltration and ϕ_{w} for the I_{F} -adic topology, which correspond to each other through S ≃ h_{T} (pt). Since S is (topologically) generated by elements x_{λ}, corresponding to first characteristic classes of line bundles c_{1}^{T} (L_{λ}) in h_{T} (pt), it suffices to compare ϕ_{w}(x_{λ}) and ψ_{w}(c_{1}^{T} (L_{Λ})). By definition of c_{S}, we have ϕ(x_{λ}) = x_{w(λ)}. Since c_{g} is defined using only pull-back and restriction maps, both commuting with taking characteristic classes, it suffices to verify that when h = K, the Grothendieck group, we have ψ_{w}([L_{λ}]) = [L_{w(λ)}]. This is easily checked by using total spaces of bundles, and the formalism of points. For this purpose, let us consider the following equivariant bundles:

• M_{λ}, the T × G-equivariant line bundle over G, whose total space is L_{λ} × G mapping by the second projection to G, and with action given on points by (t, g) · (v, g′) = (λ(t)v, gg′t^{−1});

• N_{λ}, the G-equivariant line bundle over G/T, whose total space is G ×^{T} L_{λ}, the quotient of G × L_{λ} by the relation (gt, v) = (g, λ(t)v), mapping to G/T by the first projection, and with G action by g·(g′, v) = (gg′, v);

• M′_{λ}, the T × G-equivariant line bundle over G, whose total space is G ×_{G/T} G ×^{T} L_{λ}, mapping to G by the first projection, with action of T × G given by (t, g) · (g_{1}, g_{2}, v) = (gg_{1}t^{−1}, gg_{2}, v).

It is clear that L_{λ} restricts to T × G and pulls-back over G to M_{λ}. Similarly, N_{λ} restricts and pulls-back to M′_{λ}. But M_{λ} maps isomorphically to M′_{λ} by the map (v, g) ↦ (g, g, v). Therefore, [L_{λ}] maps to [N_{λ}] by the map K_{T} (pt) →^{∼} K_{T ×G}(G) ←^{∼} K_{G}(G/T). Furthermore, N_{λ} restricts and pulls-back as a T - equivariant bundle to the fixed point w in G/T (or G/B) as wT ×^{T} L_{λ} with T -action on the left, isomorphic to L_{w(λ)}. This completes the proof. □

Let 𝔱 be the torsion index of the root datum, as defined in [D73, §5]. See also [CPZ, 5.1] for a table giving the values of its prime divisors for each simply connected type. For other types, one just needs to add the prime divisors of |Λ_{w}/Λ| by [D73, §5, Prop. 6]. Together with the previous lemma, [CZZ, Thm. 11.4] immediately implies a Borel-style presentation of h_{T} (G/B). Let π : G/B → pt be the structural map.

Theorem 10.2. If 2𝔱 is regular in R, then the map h_{T} (pt) ⊗_{h_{T} (pt)^{W}} h_{T} (pt) → h_{T} (G/B) sending a ⊗ b to π^{∗}(a)c_{g}(b) is an h_{T} (pt)-linear ring isomorphism if and only if the (non-equivariant) characteristic map c : h_{T} (pt) → h(G/B) is surjective.

In particular, it will hold for K-theory, since the characteristic map is always surjective for K-theory. It will also hold for any cohomology theory if 𝔱 is invertible in R, as [CPZ, Cor. 13.9] shows that the non-equivariant characteristic map is then surjective. As mentioned in the introduction, this presentation was obtained in [KiKr] for algebraic cobordism, with the torsion index inverted, and by using comparisons with complex cobordism.

11. Subgroups of T

Let H be a subgroup of T given by the embedding h : H ↪ T. For example H could be the trivial group, a finite multiplicative group or a subtorus of T. For any X ∈ T -Var, and thus in H -Var by restriction, there is a restriction ring map res_{h} : h_{T} (X) → h_{H}(X), in particular if X = pt, which induces a canonical morphism h_{H}(pt)⊗_{h_{T} (pt)}h_{T} (X) → h_{H}(X) of rings over h_{H}(pt), sending a⊗b to a·res_{h}(b). This “change of coefficients” morphism is compatible with pull-backs and push-forwards.

Lemma 11.1. The morphism h_{H}(pt) ⊗_{h_{T} (pt)} h_{T} (X) → h_{H}(X) is an isomorphism when X = G/P_{Ξ} or X = W/W_{Ξ}.

Proof. The case of X = W/W_{Ξ} is obvious, since as as scheme, it is simply a disjoint union of copies of pt. If X = G/P_{Ξ}, the left-hand side is free, with a basis of Bott-Samelson classes. So is the right-hand side: it is still generated as an h_{H}(pt)-module by the corresponding Bott-Samelson classes because the proof of Lemma 7.3 works for H as well as for T. Thus, the change of coefficients is surjective. The push-forward pairing is perfect and commutes to the restriction map from T to H, so these classes stay independent in h_{H}(G/P_{Ξ}) (they have a dual family). Thus, the change of coefficients is injective. □

This shows that Diagram (8.2) for H is obtained by change of coefficients, as well as Diagram (8.3) and Diagram (8.4) except their rightmost columns involving Q. Theorem 9.3 on the bilinear pairing stays valid for H instead of T.

References

[Ar86] A. Arabia, Cycles de Schubert et cohomologie équivariante de K/T, Invent. Math. 85 (1986), no. 1, 39–52.

[Ar89] , Cohomologie T -équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France 117, no. 2, 129–165, 1989.

[AB84] M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23, no. 1, 1-–28, 1984.

[BGG] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys 28, 1–26, 1973.

[BE90] P. Bressler and S. Evens, The Schubert calculus, braid relations and generalized cohomology, Trans. Amer. Math. Soc. 317, no.2, 799–811, 1990.

[BE92] , Schubert calculus in complex cobordisms, Trans. Amer. Math. Soc. 331, no.2, 799–813, 1992.

[Br97] M. Brion, Equivariant Chow groups for torus actions, Transform. Groups 2 (3), 225—267, 1997.

[CPZ] B. Calmès, V. Petrov and K. Zainoulline, Invariants, torsion indices and oriented cohomology of complete flags, Ann. Sci. École Norm. Sup. (4) 46(3), 405–448, 2013.

[CZZ] B. Calmès, K. Zainoulline and C. Zhong, A coproduct structure on the formal affine Demazure algebra, to appear in Math. Zeitschrift, arXiv:1209.1676v2.

[CZZ2] , Push-pull operators on the formal affine Demazure algebra and its dual, preprint arXiv:1312.0019v2, 2013.

[CS74] T. Chang and T. Skjelbred, The topological Schur lemma and related results, Ann. of Math. (2) 100 (1974), 307-321.

[CG10] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Modern Birkhauser Classics. Birkhauser Boston Inc., Boston, MA, 2010. Reprint of the 1997 edition.

[D73] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21, 287–301, 1973.

[D74] , Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4), 7, 53–88, 1974.

[SGA3] M. Demazure, A. Grothendieck, Schémas en groupes, (SGA 3, vol. I and III), Séminaire de Géométrie Algébrique du Bois Marie 1962-1964, Documents mathématiques 7 and 8, 2011.

[Des09] D. Deshpande, Algebraic Cobordism of Classifying Spaces, preprint arXiv:0907.4437v1, 2009.

[EG98] D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math., 131(3), 595–634, 1998.

[GR12] N. Ganter and A. Ram, Generalized Schubert Calculus, J. Ramanujan Math. Soc. 28A (Special Issue-2013): 149–190, 2013.

[GKV97] V. Ginzburg, M. Kapranov, and E. Vasserot, Residue construction of Hecke algebras, Adv. Math. 128 (1997), no. 1, 1-19.

[HHH] M. Harada, A. Henriques and T. Holm, Computation of generalized equivariant cohomologies of Kac-Moody flag varieties, Adv. Math. 197, 198–221, 2005.

[HM13] J. Heller and J. Malagón-López, Equivariant algebraic cobordism, J. Reine Angew. Math., 684, 87–112, 2013.

[HMSZ] A. Hoffnung, J. Malagón-López, A. Savage, and K. Zainoulline, Formal Hecke algebras and algebraic oriented cohomology theories, to appear in Selecta Math.

[HK] J. Hornbostel and V. Kiritchenko. Schubert calculus for algebraic cobordism, J. Reine Angew. Math., 656, 59–85, 2011.

[KiKr] V. Kiritchenko and A. Krishna, Equivariant cobordism of flag varieties and of symmetric varieties, Transform. Groups, 18(2), 391–413, 2013.

[KK86] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G^{∗}, Advances in Math. 62, 187–237, 1986.

[KK90] , T -equivariant K-theory of generalized flag varieties, J. Differential geometry 32, 549–603, 1990.

[Kr12] A. Krishna, Equivariant cobordism of schemes, Doc. Math., 17, 95– 134, 2012.

[Ku02] S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser, Boston, MA, 2002.

[LZ14] C. Lenart and K. Zainoulline, Towards generalized cohomology Schubert calculus via formal root polynomials, preprint arXiv:1408.5952, 2014.

[LM07] M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2007.

[Me05] A. S. Merkurjev, Equivariant K-theory, in Handbook of K-theory, 925–954, Springer-Verlag, Berlin, 2005.

[Pa09] I. Panin, Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov), Homology, Homotopy Appl., 11, 349– 405, 2009.

[Q71] D. Quillen. The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2), 94, 549–572; ibid. (2) 94 573–602, 1971.

[To99] B. Totaro, The Chow ring of a classifying space, Algebraic K-theory (Seattle, WA, 1997), Proc. Sympos. Pure Math., 67, 249–281, 1999.

[Th87] R. W. Thomason, Algebraic K-theory of group scheme actions, in Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud. 113, 539–563, 1987.

[ZZ14] G. Zhao and C. Zhong, Geometric representations of the formal affine Hecke algebra, preprint arXiv:1406.1283, 2014.

[Zh13] C. Zhong, On the formal affine Hecke algebra, to appear in J. Inst. Math. Jussieu, http://dx.doi.org/10.1017/S1474748014000188.

Baptiste Calmès Université d’Artois Laboratoire de Mathématiques de Lens France baptiste.calmes@univ-artois.fr Kirill Zainoulline Department of Mathematics and Statistics University of Ottawa Canada kirill@uottawa.ca

Changlong Zhong

Department of Mathematical and Statistical Sciences University of Alberta Canada zhongusc@gmail.com