MOTIVIC DECOMPOSITIONS OF TWISTED FLAG VARIETIES AND REPRESENTATIONS OF HECKE-TYPE ALGEBRAS

ALEXANDER NESHITOV, VICTOR PETROV, NIKITA SEMENOV, AND KIRILL ZAINOULLINE

Abstract. Motivated by the motivic Galois group and the Kostant-Kumar results on equivariant cohomology of flag varieties, we provide a uniform description of motivic (direct sum) decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil-Hecke algebra H.

More generally, we establish an equivalence between the h-motivic subcategory generated by the motive of E/B and the category of projective modules of the associated rational algebra D of push-pull operators, where E is a torsor for a split semisimple linear algebraic group G over a field k, B is a Borel subgroup of G, h is an algebraic oriented cohomology theory in the sense of Levine-Morel (e.g. Chow ring CH or an algebraic cobordism Ω). The algebra D can be think of as an integer-analogue of the ’Hopf-algebra of the h-motivic Galois group of E/B.

As an application, taking h = CH and specializing the coefficients to the finite field 𝔽_{p} we obtain that p-modular projective representations of D = H are generated by an irreducible H-module corresponding to the generalized Rost-Voevodsky motive for (G, p).

Contents

1. Introduction  2
2. Oriented (co-)homology  4
3. The equivariant Künneth isomorphism  6
4. The convolution product  9
5. The subring of push-pull operators  12
6. Self-duality of the algebra of push-pull operators  14
7. The rational algebra of push-pull operators  16
8. Applications to representation theory of Hecke rings  18
References  20

2010 Mathematics Subject Classification. 14F43, 14M15, 20C08, 14C15. Key words and phrases. linear algebraic group, torsor, flag variety, equivariant oriented coho-

mology, motivic decomposition, Hecke algebra.

The first author was supported by the Trillium Foundation (Ontario). The second author was supported by the Chebyshev Laboratory, St. Petersburg University via

RF Government grant 11.G34.31.0026, by JSC ‘Gazprom Neft’, and by RFBR grant 13-01-00429.

The third author was supported by the SPP 1786 ‘Homotopy theory and algebraic geometry’ (DFG).

The last author was supported by the NSERC Discovery grant and the Early Researcher Award (Ontario).

1. Introduction

Let G be a split semisimple linear algebraic group over a field k, let E be a G-torsor over K/k. Consider a twisted form E/B over K of the variety of Borel subgroups G/B of G, e.g., a variety of (complete) flags of ideals in a central division algebra over K. In general, such a variety neither have a K-rational point nor any (relative) cellular filtration over K.

Consider the pseudo-abelian tensor category of Grothendieck-Chow motives of smooth projective varieties over K with coefficients in a ring R. The main result of [25] says that the motive of E/B with finite coefficients (R = 𝔽_{p}) is always a direct sum of Tate twists of some indecomposable motive R_{E,p}, a generalization of the Rost-Voevodsky motive. In other words, the tensor subcategory ⟨E/B⟩_{𝔽_{p}} generated by all direct summands of E/B is, indeed, generated by R_{E,p}, i.e.,

⟨E/B⟩_{𝔽_{p}} = ⟨R_{E,p}⟩.

The motive R_{E,p} has several remarkable properties. For instance, it is closely related to various cohomological invariants of G-torsors. If p is not a torsion prime of G or if the coefficient ring R has characteristic 0, then R_{E,p} coincides with the motive of a point, so ⟨E/B⟩_{𝔽_{p}} gives the subcategory of Tate motives. While being indecomposable over k, the motive R_{E,p} becomes isomorphic to a direct sum of Tate motives over the splitting field k̄ of E (as k̄ one can always take the algebraic closure of k or the function field of E/B). Moreover, the generating function of R_{E,p} over k̄ (counting the number of Tate motives in each dimension) is given by an explicit cyclotomic polynomial involving the p-exceptional degrees of V.Kac [19]. For example, if E is a G-torsor, where G is an exceptional group of type F_{4} and E splits by a cubic field extension, then R_{E_{3}} corresponds to the Rost-Serre cohomological invariant and R_{E,3}|k̄ ≃ 𝔽_{3} ⊕ 𝔽_{3}(4) ⊕ 𝔽_{3}(8).

As for integer coefficients (R = ℤ) only very few facts are known concerning the category ⟨E/B⟩_{ℤ}. An integer version of the motive R_{E} was introduced and discussed in [26]; in [5], [10] it was shown that ⟨E/B⟩_{ℤ} is not Krull-Schmidt (the uniqueness of a direct sum decomposition fails).

In the present paper we cover all the mentioned cases (R = ℤ, 𝔽_{p}). More generally, we consider the category of h-correspondences with coefficients in R, where h is any algebraic oriented Borel-Moore homology of [23] (e.g. Chow groups, connective K-theory, elliptic cohomology, algebraic cobordism Ω of Levine-Morel) and R = h(K) is its coefficient ring. Let ⟨E/B⟩_{h} denote the respective tensor subcategory generated by indecomposable summands of the h-motive of E/B. Our main result (Theorem 8.1) establishes an equivalence between the motivic category ⟨E/B⟩_{h} and certain category of finitely generated projective \overline{D}_{F} -modules

(1)  ⟨E/B⟩_{h} ≃ P roj \overline{D}_{F} ,

where \overline{D}_{F} is the R-algebra defined using the formal push-pull operators for the group G and the theory h. So it provides a direct link between integer/modular h-motivic decompositions of twisted flag varieties and integer/modular representations of Hecke-type algebras \overline{D}_{F}.

If E is a versal (generic) torsor, then \overline{D}_{F} can be replaced by the formal affine Demazure algebra D_{F}. The theory of such algebras and formal push-pull operators has been recently developed in [6], [18], [7], [8], [9] motivated by Bernstein-Gelfand-Gelfand [2], Demazure [11], [12], Bressler-Evens [3], [4] and Kostant-Kumar [21],

[20] results. The key properties of D_{F} are

• it is a free module over the T -equivariant oriented cohomology ring S = h_{T} (K) of a point, where T is a split maximal torus in G;

• its S-dual D_{F}^{⋆} = Hom_{S}(D_{F} , S) is isomorphic to the T -equivariant oriented cohomology h_{T} (G/B) of G/B [9] and

• its structure (generators and relations) is very close to those of the affine Hecke algebra [18].

For example, if h = CH (Chow groups) and R = 𝔽_{p} as before, then D_{F} = H_{nil,p} is the affine nil-Hecke algebra (in the notation of Ginzburg [16, §12]) over 𝔽_{p} which is a free module of rank |W | over the polynomial ring S = 𝔽_{p}[x_{1}, …, x_{n}], where n is the rank of G and W is the Weyl group, and D_{F}^{⋆} ≃ CH_{T} (G/B; 𝔽_{p}) is the T -equivariant Chow groups. For a versal torsor E the equivalence (1) then turns into

⟨R_{E,p}⟩ ≃ P roj H_{nil,p}

meaning that all indecomposable projective H_{nil,p}-modules are isomorphic to each other (up to a shift). Moreover, their ranks over S equal to the p-part of the product of p-exceptional degrees of the group G.

Roughly speaking, the algebra \overline{D}_{F} can be viewed as an integral analogue of the Hopf-algebra of the motivic Galois group of E/B (see e.g. [1]). Indeed, if taken with ℚ-coeffcients (or if E is split), the algebra \overline{D}_{F} becomes isomorphic to End_{R} h(G/B) ≃ M_{|W |}(R) and, hence, the category P roj \overline{D}_{F} can be identified with the category of representations P roj ℚ[𝔾_{m}] = Rep 𝔾_{m} with 𝔾_{m} known to be the motivic Galois group of ⟨E/B⟩_{ℚ}. Observe that in general, \overline{D}_{F} is not a matrix algebra over R.

In the paper we restrict ourselves to varieties E/B of Borel subgroups only. However, by [5] we have ⟨E/B⟩_{h} = ⟨E/P ⟩_{h} for any special parabolic subgroup P. Hence, B can be replaced by any such P without affecting the equivalence (1). For instance, for G = P GL_{p^{n}}, h = CH, R = ℤ and E corresponding to a generic central division algebra A of degree p^{n} we get

⟨SB(A)⟩_{ℤ} ≃ P roj H_{nil,ℤ},

where SB(A) is the Severi-Brauer variety of A and H_{nil,ℤ} is the affine nil-Hecke algebra with integer coefficients.

The paper is organized as follows. In section 2 we recall definitions and basic facts concerning Borel-Moore homology h and the respective category of h-motives. We state a version of the Künneth isomorphism for cellular spaces. In the next section we generalize it to the equivariant setting. In section 4 we introduce the convolution product on the equivariant cohomology of group powers and study its properties. In the next section we identify this equivariant cohomology with the endomorphism ring on equivariant cohomology of G/B and then in section 6 with the formal affine Demazure algebra. In section 7 we introduce the notion of a rational algebra of push-pull operators \overline{D}_{F} and identify it with the subring of rational endomorhisms. In the last section we prove the equivalence (1) and provide some applications and examples.

2. Oriented (co-)homology

We recall definitions of an algebraic oriented Borel-Moore homology and of the respective category of correspondences. We also recall a version of the Künneth isomorphism for cellular spaces (Lemmas 2.4 and 2.5).

Fix a smooth scheme S over a field k. Let Sch_{S} denote the category of finite type quasi-projective separated S-schemes and let Sm_{S} denote its full subcategory consisting of smooth quasi-projective S-schemes.

Following [23, Def. 5.1.3] consider an oriented graded Borel-Moore homology theory h_{•} defined on some admissible [23, (1.1)] subcategory V of Sch_{S}. So that there are pull-backs f^{∗}: h_{•}(X) → h_{•+d}(Y) for l.c.i. morphisms f: Y → X in V of relative dimension d and push-forwards f_{∗}: h_{•}(Y) → h_{•}(X) for projective morphisms f: X → Y in V. According to [23, Prop. 5.2.1] the Borel-Moore homology h_{•} restricted to Sm_{S} defines an algebraic oriented cohomology theory h^{•} (with values in the category of graded commutative rings with unit) in the sense of [23, Def. 1.1.2] by

h^{dim_{S} X−•}(X) := h_{•}(X), X ∈ Sm_{S}.

If the (co-)dimension is clear from the context we will write simply h(X).

Following [26, §2] (see also [14, §63]) we define the category of h-correspondences h-Corr_{S} over S. The objects are pairs ([X → S], i), where [X → S] is an isomorphism class of a smooth projective map X → S and i ∈ ℤ. The morphisms are defined by

Hom_{h-Corr_{S}}(([Y → S], i), ([X → S], j)) := ⨁_{l} Hom_{i−j}([Y_{l} → S], [X → S]),

taken over all connected components Y_{l} of Y, where

Hom_{•}([Y_{l} → S], [X → S]) := h_{dim_{S} Y_{l}+•}(Y_{l} ×_{S} X).

The composition of morphisms is given by the correspondence product. Namely, if p_{i}: X_{1} ×_{S} X_{2} ×_{S} X_{3} → X_{j} ×_{S} X_{j′} denotes the projection obtained by removing the i-th coordinate, then given α ∈ h(X_{1} ×_{S} X_{2}) and β ∈ h(X_{2} ×_{S} X_{3}) we set

(2)  β ∘ α := (p_{2})_{∗}(p_{1}^{∗}(β) · p_{3}^{∗}(α)) ∈ h(X_{1} ×_{S} X_{3}).

Let h-Corr_{S}^{+} denote the additive completion of h-Corr_{S}. We simply write X for the respective class in h-Corr_{S}^{+}.

Definition 2.1. (cf. [23, (CD’)]) Let X be smooth projective over S. Suppose that there is a filtration by proper closed subschemes

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} = X

such that

• each irreducible component X_{ij} of X_{i} \ X_{i−1} is a locally trivial affine fibration over S of rank d_{ij}, and

• the closure of X_{ij} in X admits a resolution of singularities X̃_{ij} → \overline{X}_{ij} over S; we set g_{ij}: X̃_{ij} → \overline{X}_{ij} ↪ X and, therefore, (g_{ij})_{∗}(1X̃_{ij}) ∈ h_{d_{ij}}(X).

We call such X (together with the filtration) a cellular space over S.

Definition 2.2. We say that the theory h satisfies the cellular decomposition (CD) property if given a cellular space X over S the respective elements (g_{ij})_{∗}(1X̃_{ij}) form a h(S)-basis of h(X).

Example 2.3. The property (CD) holds for any oriented Borel-Moore homology h over a field k of characteristic 0.

Indeed, the same reasoning as in [14, Thm. 66.2] shows that for every Z ∈ Sm_{S} there is an isomorphism

∑(g_{ij})_{∗}(1) × id_{Z}: ⨁_{ij} CH_{•−d_{ij}}(Z) → CH_{•}(Z ×_{S} X).

By the Yoneda lemma (cf. [14, Lemma 63.9]) the latter induces an isomorphism in the category CH-Corr_{S}^{+} (cf. [14, Cor. 66.4]).

Following [29, §2] consider the specialization functor Ω-Corr_{S}^{+} → CH-Corr_{S}^{+}, [f: Y → X] ↦ f_{∗}(1_{Y}). It is surjective on the classes of objects and morphisms. Moreover, for every X the kernel of

Ω_{dim_{S} X}(X ×_{S} X) ⟶ CH_{dim_{S} X}(X ×_{S} X)

is Ω_{⩾1}(k) · Ω_{•}(X ×_{S} X) by [23, Rem.4.5.6]. Hence for every y in this kernel

y^{∘(dim_{S} X+1)} ∈ Ω_{dim_{S} X}(X ×_{S} X) ∩ (Ω_{⩾(dim_{S}X+1)}(k) · Ω_{•}(X ×_{S} X)).

So y = 0 since Ω_{<0}(Y) = 0. Therefore, the kernel of

End_{Ω-Corr_{S}^{+}}(X, i) → End_{CH-Corr_{S}^{+}}(X, i)

consists of nilpotents.

Finally, by [29, Lemma 2.1] the isomorphism ∑_{ij}(g_{ij})_{∗}(1) in CH-Corr_{S}^{+} can be lifted to an isomorphism in the category Ω-Corr_{S}^{+}. Specializing it via Ω → h we obtain the desired isomorphism.

From this point on we assume that h satisfies the property (CD).

Lemma 2.4. Let X be a cellular space over S. Then there is an isomorphism in h-Corr_{S}^{+}

∑_{ij}(g_{ij})_{∗}(1X̃_{ij}): ⨁_{ij}(S, d_{ij}) → X,

where (g_{ij})_{∗}(1X̃_{ij}) ∈ h_{d_{ij}}(X) = Hom_{h-Corr_{S}^{+}}((S, d_{ij}), X).

Proof. Transversal base change implies that there is an isomorphism

∑(g_{ij})_{∗}(1) × id_{Z}: ⨁_{ij} h_{•−d_{ij}}(Z ×_{S} S) → h_{•}(Z ×_{S} X)

for any Z smooth projective over S. So by the Yoneda lemma (cf. [14, Lemma 63.9]) it induces an isomorphism in h-Corr_{S}^{+} (cf. [14, Cor. 66.4]). □

Lemma 2.5. The pairing (·, ·): h(X) ⊗_{h(S)} h(X) → h(S) given by (a, b) = p_{∗}(ab) is non-degenerate and the map

f: (h(X ×_{S} X), ∘) → End_{h(S)} h(X) given by a ↦ f_{a}, f_{a}(x) = (p_{2})_{∗}(p_{1}^{∗}(x) · a)

is an h(S)-linear isomorphism of graded rings. In particular, it gives an h(S)-linear isomorphism

(h_{dim_{S} X}(X ×_{S} X), ∘) ≃ End_{h-Corr_{S}^{+}}(X).

Observe that the endomorphism ring of h(S)-linear operators End_{h(S)}(h(X)) is a graded ring. Its n-th graded component consists of operators increasing the codimension by n. By definition the subring of degree-0 operators (preserving the codimension) coincides with End_{h-Corr_{S}^{+}}(X).

Proof. By the previous lemma there is an isomorphism

⨁_{ij} h(S) = ⨁_{k=−∞}^{∞} Hom((S, k), ⊕_{ij}(S, d_{ij})) →^{≃} ⨁_{k=−∞}^{∞} Hom((S, k), X) = h(X),

where each component is given by x ↦ x · (g_{ij})_{∗}(1). Let ∑_{ij} a_{ij}: X → ⊕_{ij}(S, d_{ij}) be the inverse isomorphism in h-Corr_{S}^{+}. Observe that

a_{ij} ∈ Hom(X, (S, d_{ij})) = h_{dim(X/S)−d_{ij}}(X).

Since a_{ij} ∘ (g_{ij})_{∗}(1) = p_{∗}(a_{ij} · (g_{ij})_{∗}(1)) = δ_{i,j}, the pairing (·, ·) is non-degenerate.

The pairing (·, ·) gives an isomorphism h(X) → Hom_{h(S)}(h(X), h(S)) and, hence, an isomorphism End_{h(S)} h(X) →^{≃} h(X) ⊗_{h(S)} h(X). Consider the composition

ρ: h(X ×_{S} X) →^{f} End_{h(S)} h(X) →^{≃} h(X) ⊗_{h(S)} h(X)

and a map π: h(X) ⊗ h(X) → h(X ×_{S} X) given by π(a ⊗ b) = p_{1}^{∗}(a) · p_{2}^{∗}(b). By definition, we have

f_{p_{1}^{∗}(a)p_{2}^{∗}(b)}(x) = (p_{2})_{∗}(p_{1}^{∗}(x)p_{1}^{∗}(a)p_{2}^{∗}(b)) = (x, a)b.

Hence, ρ(π(a ⊗ b)) = a ⊗ b and the map ρ is surjective. By the property (CD) for X ×_{S} X → X, h(X ×_{S} X) is a free h(X)-module of rank rk_{h(S)}h(X). Thus, ρ is a surjective homomorphism between free modules of the same rank, hence, it is an isomorphism. □

3. The equivariant Künneth isomorphism

In the present section we introduce an equivariant Borel-Moore homology following [7, §2] and [17]. We provide an equivariant analogue of the Künneth isomorphism (Lemma 3.7).

Let G be a smooth group scheme over S. Consider an admissible subcategory V^{G} of the category of G-varieties X ∈ Sch_{S} with G-equivariant morphisms. By a G-equivariant oriented (graded) Borel-Moore homology theory we will call an additive functor h_{•}^{G} from V^{G} to graded abelian groups such that

1. There are pull-backs for l.c.i. maps and push-forwards for projective maps that satisfy

(TS) (l.c.i. base change) For a Cartesian square \xymatrix{X′ \ar[r]^{f′} \ar[d]_{g′} & Y′ \ar[d]^{g} \\ X \ar[r]^{f} & Y} where f (hence

f′) is l.c.i. and g (hence g′) is projective, we have f^{∗}g_{∗} = g′_{∗}(f′)^{∗}.

(Loc) (localization) If U ⊂ X is an open G-equivariant embedding with Z = X\U, then there is a right exact sequence:

h_{•}^{G}(Z) → h_{•}^{G}(X) → h_{•}^{G}(U) → 0.

2. The functor h_{•}^{G} restricted to Sm_{S} defines a graded G-equivariant oriented cohomology theory h_{G}^{•} in the sense of [9] (we refer to [9, §2, A1-9] for the precise definition) by

h_{G}^{dim_{S} X−•}(X) := h_{•}^{G}(X), X ∈ Sm_{S}.

In addition to the axioms of [9, §2] we require that h_{G} satisfies the following stronger version of the homotopy invariance axiom:

(HI) (extended homotopy invariance) Let p: Y → X be a G-equivariant torsor of a vector bundle of rank r over X, then the pull-back induced by projection

p^{∗}: h_{G}^{•}(X) → h_{G}^{•}(Y)

is an isomorphism. If a variety is smooth we will always use the cohomology notation.

Example 3.1. Given a linear algebraic group G over a field k of characteristic zero an example of such G-equivariant Borel-Moore homology theory h_{•}^{G} was constructed in [17] as follows.

Consider a system of G-representations V_{i} and its open subsets U_{i} ⊆ V_{i} such that

• G acts freely on U_{i} and the quotient U_{i}/G exists as a scheme over k,

• V_{i+1} = V_{i} ⊕ W_{i} for some representation W_{i},

• U_{i} ⊆ U_{i} ⊕ W_{i} ⊆ U_{i+1}, and U_{i} ⊕ W_{i} → U_{i+1} is an open inclusion, and

• codim(V_{i} \ U_{i}) strictly increases. Such a system is called a good system of representations of G.

Let X ∈ Sch_{k} be a G-variety. Following [17, §3 and §5] the inverse limit induced by pull-backs

\underleftarrow{lim}_{i} h_{•−dim G+dim U_{i}}(X ×^{G} U_{i}), X ×^{G} U_{i} = (X ×_{k} U_{i})/G,

does not depend on the choice of the system (V_{i}, U_{i}) and, hence, defines the Gequivariant oriented homology group h_{•}^{G}(X).

In the present paper we will extensively use the following property (cf. [9, §2, A6]) of an equivariant theory (Tor) Let X → X/G be a G-torsor over S and a G′-equivariant map for some

group scheme G′ over S. Then there is an isomorphism

h_{G×G′}^{•}(X) ⟶^{≃} h_{G′}^{•}(X/G).

that is natural with respect to the maps of pairs

(ϕ, γ): (X, G × G′) → (X_{1}, G_{1} × G′_{1}), ϕ(x · (g, g′)) = ϕ(x) · γ(g, g′).

Observe that the theory of Example 3.1 satisfies this property by [17, Prop. 27].

We have the following equivariant analogues of Definitions 2.1 and 2.2

Definition 3.2. Let X ∈ V^{G}. Suppose that there is a filtration by G-equivariant proper closed subschemes

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} = X

such that

• each irreducible component X_{ij} of X_{i} \ X_{i−1} is a G-equivariant (locally trivial) affine fibration over S of rank d_{ij}, and • the closure of X_{ij} in X admits a G-equivariant resolution of singularities

g_{ij}: X̃_{ij} → \overline{X}_{ij} over S. We call such X (together with the filtration) a G-equivariant cellular space over S.

Definition 3.3. We say that the equivariant theory h^{G} satisfies the cellular decomposition (CD) property if given a G-equivariant cellular space X over S the respective elements (g_{ij})_{∗}(1X̃_{ij}) form a h^{G}(S)-basis of h^{G}(X).

Lemma 3.4. Suppose a morphism f: X → Y in Sm_{k} factors as f: X →^{z} L →^{j} Y where p: L → X is a vector bundle, z: X → L is a zero section and j is an open embedding.

Then for every projective map a: Y′ → Y and X′ = X ×_{Y} Y′ the following diagram of pull-back and push-forward maps commutes (we omit the grading)

\xymatrix{h(X′) \ar[r]^{a′_{∗}} & h(X) \\ h(Y′) \ar[u]^{f^{′∗}} \ar[r]^{a_{∗}} & h(Y) \ar[u]_{f^{∗}}}

Proof. Observe that the map f′: X′ → Y′ factors as X′ →^{z′} L ×_{Y} Y′ →^{j′} Y′ where z′ is the zero section of the vector bundle p′: L′ = L ×_{Y} Y′ → X′ and j′ is an open embedding. Let b denote the canonical map L′ → L. Since j and j′ are flat, we have j^{∗}a_{∗} = b_{∗}j^{′∗} by the l.c.i. base change for oriented theories. Note that by the homotopy invariance z^{∗} = (p^{∗})^{−1} and z^{′∗} = (p^{′∗})^{−1}. Since p and p′ are flat, p^{∗}a′_{∗} = b_{∗}p^{′∗}. Then z^{∗}b_{∗} = a′_{∗}z^{′∗} and

f^{∗}a_{∗} = z^{∗}j^{∗}a_{∗} = z^{∗}b_{∗}j^{′∗} = a′_{∗}z^{′∗}j^{′∗} = a′_{∗}f^{′∗}.  □

Remark 3.5. If (V_{i}, U_{i}) is a good system of representations of Example 3.1, then for any G-variety X the connecting maps X×^{G}U_{i} → X×^{G}U_{i+1} factor as in Lemma 3.4, i.e., we have X ×^{G} U_{i} → X ×^{G} (U_{i} ⊕ W_{i}) → X ×^{G} U_{i+1}.

Example 3.6. Let h^{G} be the equivariant theory of Example 3.1. Then the property (CD) holds for h^{G}.

Indeed, consider a good system of representations {(V_{j}, U_{j})}_{j} for X. The subvarieties X_{i} ×^{G} U_{j}, i = 0 … n form a cellular filtration on X ×^{G} U_{j} over S ×^{G} U_{j}. Note that X̃_{i} ×^{G} U_{j} is a resolution of singularities of X_{i} ×^{G} U_{j}. By (CD) for h the set {(f_{i} ×^{G} id_{U_{j}})_{∗}(1)}_{i} forms a basis of h(X ×^{G} U_{j}) as a h(S ×^{G} U_{j})-module. By Lemma 3.4 the following diagram commutes:

\xymatrix{h(X̃_{i} ×^{G} U_{j+1}) \ar[r]^{(g_{i,j+1})_{∗}} \ar[d]^{ĩ_{j}^{∗}} & h(X ×^{G} U_{j+1}) \ar[d]^{i_{j}^{∗}} \\ h(X̃_{i} ×^{G} U_{j}) \ar[r]^{(g_{i,j})_{∗}} & h(X ×^{G} U_{j})}

So i_{m}^{∗}((f_{i} ×^{G} id_{U_{j+1}})_{∗}(1)) = (f_{i} ×^{G} id_{U_{j}})_{∗}(1), which implies that the elements f_{i∗}(1) = lim_{j}((f_{i} ×^{G} id_{U_{j}})_{∗}(1)) form a basis of h^{G}(X) over h^{G}(S).

From this point on we assume that h_{•}^{G} satisfies the property (CD). As for usual oriented theories we then obtain Lemma 3.7. The pairing (·, ·): h^{G}(X) ⊗_{h^{G}(S)} h^{G}(X) → h^{G}(S) given by (a, b) = p_{∗}(ab) is non-degenerate and the map

f: (h^{G}(X ×_{S} X), ∘) → End_{h^{G}(S)} h^{G}(X) given by a ↦ f_{a}, f_{a}(x) = (p_{2})_{∗}(p_{1}^{∗}(x)·a)

is an h^{G}(S)-linear isomorphism of rings. In particular, there is an h^{G}(S)-linear isomorphism

(h_{dim_{S} X}^{G}(X ×_{S} X), ∘) → End_{h^{G}-Corr_{S}^{+}}(h^{G}(X)),

where h^{G}-Corr_{S}^{+} is the respective category of G-equivariant correspondences.

4. The convolution product

In the present section we introduce the convolution product on the equivariant Borel-Moore homology (Definition 4.3) of group power. We relate this product to the usual correspondence product for the associated torsors (Lemma 4.6) and study its behaviour under the base change (diagram (6)).

Let G be a smooth algebraic group over k and let E be a G-torsor over k (G acts on the right). By definition there is an isomorphism ρ: E ×_{k} G →^{≃} E ×_{k} E given on points by (e, g) ↦ (e, eg). For each i ≥ 0 it induces an isomorphism

ρ_{i}: E ×_{k} G^{i} ⟶ E^{i+1}, (e, g_{1}, g_{2}, …, g_{i}) ↦ (e, eg_{1}, eg_{2}, …, eg_{i}).

Consider the composition

γ_{i}: E^{i+1} ⟶^{ρ_{i}^{−1}} E ×_{k} G^{i} = E ×_{k} G^{i} ⟶^{pr} G^{i}.

The coordinate-wise right G^{i+1}-action on E^{i+1} induces an action on E ×_{k} G^{i} and, hence, on G^{i}. For instance, on points it is given by

(3)  (e, g_{1}, …, g_{i}) · (h_{1}, …, h_{i+1}) = (eh_{1}, h_{1}^{−1}g_{1}h_{2}, …, h_{1}^{−1}g_{i}h_{i+1}).

Consider projections p_{j}: E^{i+1} → E^{i} obtained by removing the j-th coordinate and the respective G^{i}-action on E^{i}. For each i ≥ 1, 1 ≤ j ≤ i + 1 there is a commutative diagram of G^{i}-equivariant maps

(4)  \xymatrix{E^{i+1} \ar[r]^{γ_{i}} \ar[d]_{p_{j}} & G^{i} \ar[d]^{π_{j}} \\ E^{i} \ar[r]^{γ_{i−1}} & G^{i−1}}

where π_{1}(g_{1}, …, g_{i}) = (g_{1}^{−1}g_{2}, …, g_{1}^{−1}g_{i}) and π_{j}(g_{1}, …, g_{i}) = (g_{1}, …, ĝ_{j−1}, …, g_{i}) for j > 1.

Example 4.1. For i = 1 it gives a commutative diagram of G-equivariant maps

\xymatrix{E ×_{k} E \ar[r]^{γ_{1}} \ar[d]_{p_{j}} & G \ar[d]^{π_{j}} \\ E \ar[r]^{γ_{0}} & Spec k}

where γ_{0}, π_{1}, π_{2} are the structure maps, p_{1}, p_{2} are the corresponding projections and γ_{1}(e, eg) = g. Moreover, if E is trivial, then γ_{1} = π_{1} : G×_{k}G → G, (g_{1}, g_{2}) ↦ g_{1}^{−1}g_{2}.

Let H be an algebraic subgroup of G such that G/H is a smooth variety over k. We can view G^{i} as an H-torsor over G^{i}/H, where H acts on G^{i} via the jth coordinate of G^{i+1}. By definition, the H^{i}-equivariant map π_{j} factors as

π_{j}: G^{i} ⟶^{q} G^{i}/H ⟶^{π̄_{j}} G^{i−1},

where the second map π̄_{j} is a fibration with a fibre G/H.

Example 4.2. The map π_{1} factors through the quotient maps modulo the diagonal action

π_{1} : G^{i} ⟶^{q} G^{i}/Δ(H) ⟶^{π̄_{1}} G^{i}/Δ(G) = G^{i−1}.

which are equivariant with respect to the usual coordinate-wise H^{i}-action.

Consider an equivariant Borel-Moore homology theory h. For every 1 ⩽ j ⩽ i+1 consider the action of the j-th copy of H on G^{i}. The property (Tor) gives an isomorphism

(5)  h_{H^{i}}(G^{i}/H) ⟶^{≃} h_{H^{i+1}}(G^{i}),

where H^{i+1} acts on G^{i} as in (3). Unless explicitly mentioned we will always identify these two rings.

Set S = h_{H}(G^{0}) = h_{H}(k) and set the convolution product on S to be the usual intersection product.

Definition 4.3. Assume that G/H is a smooth projective variety over k. We define the S-linear convolution product ′∘′ on h_{H^{i}}(G^{i−1}), i ≥ 2 to be the composite

h_{H^{i}}(G^{i−1}) ⊗ h_{H^{i}}(G^{i−1}) ⟶^{π̄_{i−1}^{∗}⊗π̄_{i+1}^{∗}} h_{H^{i+1}}(G^{i}) ⊗ h_{H^{i+1}}(G^{i}) ⟶^{′·′}

h_{H^{i+1}}(G^{i}) ⟶^{(π̄_{i})_{∗}} h_{H^{i}}(G^{i−1}),

where h_{H^{i+1}}(G^{i}) is identified with h_{H^{i}}(G^{i}/H) via (5) and π̄_{i} is projective because so is G/H.

The central object of the present paper is the convolution ring (h_{H^{2}}(G), ∘), i.e., the case i = 2. In the next sections we will show that (h_{B^{2}}(G), ∘) (where B is a Borel subgroup of a semisimple split G) can be identified with the formal affine Demazure algebra.

Example 4.4. In the case i = 3 the convolution ring (h_{H^{3}}(G^{2}), ∘) is isomorphic to h_{Δ(H)}((G/H)^{2}) with respect to the usual correspondence product. Indeed, the maps π_{i}: G^{3} → G^{2}, i = 2, 3, 4 induce Δ(H)-equivariant projections (G/H)^{3} → (G/H)^{2}. The isomorphism then follows by (Tor).

Observe that if G/H is an H-equivariant cellular space and h_{H} satisfies (CD), then by Lemma 3.7 there is an S-linear ring isomorphism

(h_{H^{3}}(G^{2}), ∘) ≃ End_{S} h_{H}(G/H).

Lemma 4.5. For i ≥ 1 the map π_{1} induces an injective ring homomorphism with respect to the convolution products

(h_{H^{i}}(G^{i−1}), ∘) ⟶^{π̄_{1}^{∗}} (h_{H^{i+1}}(G^{i}), ∘).

Proof. For i = 1 it follows from the fact that the convolution product on h_{H^{2}}(G) is S-linear.

For i ≥ 2 for each i − 1 ≤ j ≤ i + 1 we have π_{j} ∘ π_{1} = π_{1} ∘ π_{j+1}. Since pushforwards commute with flat pull-backs by (TS), there are commutative diagrams in equivariant cohomology

\xymatrix{h_{H^{i+1}}(G^{i}) \ar[r]^{π̄_{1}^{∗}} \ar[d]_{(π̄_{i})_{∗}} & h_{H^{i+2}}(G^{i+1}) \ar[d]_{(π̄_{i+1})_{∗}} \\ h_{H^{i}}(G^{i−1}) \ar[r]^{π̄_{1}^{∗}} \ar[u]_{π̄_{i−1}^{∗},π̄_{i+1}^{∗}} & h_{H^{i+1}}(G^{i}) \ar[u]_{π̄_{i}^{∗},π̄_{i+2}^{∗}}}

Finally, there is a H^{i}-equivariant section of the map π̄_{1} : G^{i}/Δ(H) → G^{i−1} given by (g_{1}, …, g_{i−1}) ↦ (1, g_{1}, …, g_{i−1}), so π̄_{1}^{∗} is injective. □

Lemma 4.6. The map γ_{1} induces a ring homomorphism

(h_{H^{2}}(G), ∘) ⟶^{γ_{1}^{∗}} (h_{H^{2}}(E^{2}), ∘) ⟶^{≃} (h((E/H)^{2}), ∘),

where the last ring is viewed with respect to the correspondence product (2).

Proof. By (TS) the diagram (4) gives rise to commutative diagrams in cohomology

\xymatrix{h_{H^{3}}(G^{2}) \ar[r]^{γ_{2}^{∗}} \ar[d]_{(π̄_{2})_{∗}} & h_{H^{3}}(E^{3}) \ar[d]_{(p_{2})_{∗}} \\ h_{H^{2}}(G) \ar[r]^{γ_{1}^{∗}} \ar[u]_{π̄_{1}^{∗},π̄_{3}^{∗}} & h_{H^{2}}(E^{2}) \ar[u]_{p_{1}^{∗},p_{3}^{∗}}}

The last isomorphism follows by (Tor). □

Let k̄ denote the splitting field of a G-torsor E so that Gk̄ = Ek̄. Since the base change preserves the convolution product, combining Lemmas 4.5 and 4.6 we obtain two commutative diagrams of convolution (correspondence) rings

\xymatrix{γ_{1}^{∗}: h_{H^{2}}(G) \ar[r]^{pr^{∗}} \ar[d]_{res_{k̄/k}} & h_{H^{2}}(E ×_{k} G) \ar[r]^{ρ_{1}^{∗}}_{≃} & h_{H^{2}}(E^{2}) \ar[d]^{res_{k̄/k}} \\ γ_{1}^{∗}: h_{H^{2}}(Gk̄) \ar[r]^{π̄_{1}^{∗}} & h_{H^{2}}(G2 ; k̄/Δ(H)) \ar[r]^{q^{∗}} & h_{H^{2}}(G2 ; k̄)}

and

\xymatrix{γ_{0}^{∗}: h_{H}(k) \ar[r]^{ρ_{0}^{∗}∘pr^{∗}} \ar[d]_{res_{k̄/k}}^{≃} & h_{H}(E) \ar[d]^{res_{k̄/k}} \\ γ_{0}^{∗}: h_{H}(k̄) \ar[r]^{q^{∗}∘π̄_{1}^{∗}} & h_{H}(Gk̄)}

where res_{k̄/k} is the base change map. Combining these two diagrams we obtain a commutative diagram of convolution rings

(6)  \xymatrix{h_{H}(E) ⊗_{S} h_{H^{2}}(G) \ar[r]^{(p_{1}^{∗},γ_{1}^{∗})} \ar[d]_{res_{k̄/k}} & h_{H^{2}}(E^{2}) \ar[d]^{res_{k̄/k}} \\ h_{H}(Gk̄) ⊗_{S} h_{H^{2}}(Gk̄) \ar[r]^{(p_{1}^{∗},γ_{1}^{∗})} & h_{H^{2}}(G2 ; k̄),}

where the left convolution rings are h_{H}(E)- and h_{H}(Gk̄)-linear.

5. The subring of push-pull operators

In the present section we prove that if H is the Borel subgroup of a split semisimple linear algebraic group, then the convolution ring h_{H^{2}}(G) of Definition 4.3 can be identified with the subring of push-pull operators (Corollary 5.3). Our arguments are essentially based on the Bruhat decomposition of G stated using the G-orbits on the product G/H ×_{k} G/H and the resolution of singularities (8).

As before assume that G/H is a smooth projective variety over k. In the notation of the previous section consider the H^{2}-equivariant maps of Example 4.2.

π_{1} : G^{2} ⟶^{q} G^{2}/Δ(H) ⟶^{π̄_{1}} G^{2}/Δ(G) = G, (g_{1}, g_{2}) ↦ g_{1}^{−1}g_{2}.

Since G^{2} is a Δ(G)-torsor over G (Δ(H)-torsor over G^{2}/Δ(H)), by the property (Tor) the induced Δ(G) × H^{2}-equivariant pull-backs on cohomology coincide with the forgetful maps

(7) \xymatrix{γ_{1}^{∗}: h_{H^{2}}(G) ≃ h_{Δ(G)×H^{2}}(G^{2}) \ar@{^{(}->}[r]^{π̄_{1}^{∗}} \ar[d]_{≃} & h_{Δ(H)×H^{2}}(G^{2}) \ar[r]^{q^{∗}} \ar[d]^{≃} & h_{H^{2}}(G^{2}) \ar[d]^{≃} \\ h_{G}((G/H)^{2}) \ar@{^{(}->}[r] & h_{H}((G/H)^{2}) \ar[r] & h((G/H)^{2})}

Moreover, by Lemma 4.5 it is a commutative diagram of convolution rings.

Let G be a split semisimple linear algebraic group over k and let h be an equivariant theory that satisfies property (CD). We fix a Borel subgroup B of G containing a split maximal torus T. By Bruhat decomposition (e.g. [27])

G = ⨿_{w∈W} BẇB, ẇ ∈ N_{T} ,

is the disjoint union of B^{2}-orbits of G, where W = N_{T} /T is the Weyl group and N_{T} is the normalizator of T in G. Projecting this decomposition onto X = G/B gives a B-equivariant cellular filtration on X by closures \overline{X}_{w} of affine spaces X_{w} = BẇB/B of dimension l(w) (the length of w).

The preimage π_{1}^{−1}(BẇB) is a Δ(G)-orbit in G^{2} (here H = B). Let O_{w} denote its image via G^{2} → X^{2} and let \overline{O}_{w} denote its closure. Observe that both O_{w} and \overline{O}_{w} are Δ(G)-invariant in X^{2}. By properties of the Bruhat decomposition (see [27, §1]) it follows that the projection O_{w} → X^{2} → X is a torsor of a vector bundle over X with fibre X_{w}. Indeed, the transition functions are affine since they are given by the action of B on the left on BẇB/B that is by T acting on the product of the respective root subgroups ∏_{α∈Φ^{+}∩w(Φ^{−})} U_{α} via the conjugation and, hence, by T acting on the product of the respective 𝔾_{a}’s via the multiplication t · x = α(t)x, t ∈ T, x ∈ 𝔾_{a}. So X^{2} is a G-equivariant (G acts diagonally) cellular space over X with filtration given by the closures \overline{O}_{w}.

Assume that for each w ∈ W we are given a G-equivariant resolution of singularities Õ_{w} → \overline{O}_{w}. Let [Õ_{w}]_{G} denote the respective class in h_{G}^{dim_{k} X−l(w)}(X^{2}). Then by the property (CD) the cohomology h_{G}(X^{2}) (resp. h_{B}(X^{2}) and h(X^{2})) is a free module over h_{G}(X) (resp. over h_{B}(X) and h(X)) with basis {[Õ_{w}]_{G}}_{w∈W} (resp. {[Õ_{w}]_{B}}_{w∈W} and {[Õ_{w}]}_{w∈W}). Hence, the forgetful maps of (7) send [Õ_{w}]_{G} ↦ [Õ_{w}]_{B} ↦ [Õ_{w}] and change the coefficients by − ⊗_{h_{G}(X)} h_{B}(X) and − ⊗_{h_{B}(X)} h(X) respectively, where the map S = h_{G}(X) ↪ h_{B}(X) → h(X) is the classical characteristic map.

We now construct such G-equivariant resolutions as follows. For the i-th simple reflection s_{i} we denote X_{s_{i}} (resp. O_{s_{i}}) simply by X_{i} (resp. by O_{i}). Let P_{i} be the minimal parabolic subgroup corresponding to a simple root α_{i} and let q_{i}: X → G/P_{i} denote the respective quotient map.

Lemma 5.1. We have \overline{O}_{i} = X ×_{G/P_{i}} X and, in particular, \overline{O}_{i} is smooth.

Proof. We have (g_{1}B, g_{2}B) ∈ X ×_{G/P_{i}} X, g_{1}, g_{2} ∈ G if and only if g_{1}P_{i} = g_{2}P_{i}, so g_{2} = g_{1}h for some h ∈ P_{i}. Since P_{i} = B ∪ Bs_{i}B, it means that either g_{2}B = g_{1}B or g_{2}B = g_{1}Bs_{i}B, so (g_{1}B, g_{2}B) ∈ O_{s_{i}} ∪ Δ_{X} = \overline{O}_{i}. □

For any w ∈ W we choose a reduced decomposition w = s_{i_{1}}s_{i_{2}} … s_{i_{l}} and set I_{w} = (i_{1}, i_{2}, …, i_{l}). Consider a variety

(8)  Õ_{I_{w}} = X ×_{G/P_{i_{1}}} X ×_{G/P_{i_{2}}} … ×_{G/P_{i_{l}}} X.

The projection on the first and the last factor pr: Õ_{I_{w}} → X ×_{k} X gives a Gequivariant resolution of singularities of \overline{O}_{w}.

Theorem 5.2. For H = B or 1, the image of [Õ_{I_{w}}]_{H} ∈ h_{H}(X ×_{k} X) under the Künneth isomorphism

(h_{H}(X ×_{k} X), ∘) ⟶^{≃} End_{h_{H}(k)}(h_{H}(X))

is the composition of push-pull operators q_{i_{1}}^{∗}q_{i_{1}∗} ∘ … ∘ q_{i_{l}}^{∗}q_{i_{l}∗}.

Proof. By definition the image of [Õ_{I_{w}}]_{H} is the h_{H}(k)-linear operator

h_{H}^{•}(X) ⟶^{p_{1}^{∗}} h_{H}(X ×_{k} X) ⟶^{·[Õ_{Iw}]} h_{H}(X ×_{k} X) ⟶^{p_{2∗}} h_{H}^{•−l(w)}(X).

By the projection formula and (TS) it can be also written as

h_{H}^{•}(X) ⟶^{pr_{l+1}^{∗}} h_{H}(Õ_{I_{w}}) ⟶^{pr_{1}∗} h_{H}^{•−l(w)}(X),

where pr_{j} denotes the projection on the j-th coordinate (recall that p_{j} denotes the projection obtained by removing the j-th coordinate).

By the property (TS) we obtain a commutative diagram

\xymatrix{h_{H}(X) \ar[r]^{pr_{2}^{∗}} \ar[d]_{q_{i_{l}∗}} & h_{H}(Õ_{i_{l}}) \ar[d]_{pr_{1∗}} \ar[r]^{pr_{23}^{∗}} & h_{H}(Õ_{(i_{l−1},i_{l})}) \ar[d]_{pr_{12∗}} \ar[r]^{pr_{234}^{∗}} & … \ar[r] & h_{H}(Õ_{I_{w}}) \ar[d] \\ h_{H}(G/P_{i_{l}}) \ar[r]^{q_{i_{l}}^{∗}} & h_{H}(X) \ar[d]_{q_{i_{l−1}∗}} \ar[r]^{pr_{2}^{∗}} & h_{H}(Õ_{i_{l−1}}) \ar[d]_{pr_{1∗}} &  & … \ar[d] \\  & h_{H}(G/P_{i_{l−1}}) \ar[r]^{q_{i_{l−1}}^{∗}} & h_{H}(X) \ar[d]_{q_{i_{l−2}∗}} &  & … \ar[d] \\  &  & … \ar[r] & … \ar[r] & h_{H}(X)}

where pr_{ijk...} denote the projection on the i-th, j-th, k-th, …, coordinates. The result then follows since the top horizontal row gives pr_{l+1}^{∗} and the right vertical column gives pr_{1∗}. □

Combining Diagram (7) and Theorem 5.2 we obtain

Corollary 5.3. There is a commutative diagram of convolution rings

\xymatrix{h_{B^{2}}(G) \ar@{^{(}->}[r]^{π̄_{1}^{∗}} & h_{Δ(B)×B^{2}}(G^{2}) \ar[r]^{≃} \ar[d]_{q^{∗}} & h_{B}(X^{2}) \ar[r]^{≃} \ar[d] & End_{S}(h_{B}(X)) \ar[d] \\  & h_{B^{2}}(G^{2}) \ar[r]^{≃} & h(X^{2}) \ar[r]^{≃} & End_{R}(h(X))}

where the image of (h_{B^{2}}(G), ∘) in End_{S}(h_{B}(X)) is the subring generated by the push-pull operators q_{i}^{∗}q_{i_{∗}} (of degree (−1)) and the image of the forgetful map S = h_{G}^{•}(X) → h_{B}^{•}(X) (of degrees ’•’) and the last vertical arrow is induced by the augmentation map S → R = h(k).

6. Self-duality of the algebra of push-pull operators

In the present section we identify the convolution ring h_{B^{2}}(G) with the formal affine Demazure algebra D_{F} of [18] and show that it is self-dual with respect to the convolution product (Theorem 6.2). Our arguments are based on the results of [18], [7], [8] and, especially, [9]. We use the notation of [9].

Recall that algebraic oriented cohomology theories h correspond (up to universality) to one-dimensional commutative formal group laws F (u, v): the formal group law corresponds to h by means of the Quillen formula expressing the first characteristic classes

c_{1}^{h}(L_{1} ⊗ L_{2}) = F (c_{1}^{h}(L_{1}), c_{1}^{h}(L_{2}))

and the respective cohomology theory h is defined from F by tensoring with the algebraic cobordism

h(−) = Ω(−) ⊗_{Ω(k)} R,

where Ω(k) → R defines F by specializing the coefficients in the Lazard ring (see [9, §2] for details). For example, the additive formal group law correspond to Chow groups and the periodic multiplicative law corresponds to K-theory.

By [9, Thm. 3.3] the completed B-equivariant coefficient ring S = h_{B}(k) can be identified with the formal group algebra R[[T^{∗}]]_{F}, where T^{∗} is the group of characters of a split maximal torus T ⊂ B and F is the respective formal group law.

Following [9, §5] consider the localized algebra Q = R[[T^{∗}]]_{F} [\frac{1}{x_{α}}]_{α} (where α runs through all simple roots) and the smash products Q_{W} = Q#_{R}R[W] and S_{W} = S#_{R}R[W] with the multiplication given by

qδ_{w} · q′δ_{w′} = q(wq′)δ_{ww′}

for q, q′ ∈ Q (respectively S) and w, w′ ∈ W (the Weyl group). Consider the duals Q_{W}^{⋆} = Hom_{Q}(Q_{W} , Q) and S_{W}^{⋆} = Hom_{S}(S_{W} , S). By definition Q_{W}^{⋆} and S_{W}^{⋆} can be identified with the ring of functions Hom(W, Q) and Hom(W, S) respectively

As in [18, Def. 6.2, 6.3] for each simple root α_{i} of the root system for G define the push-pull element

Y_{i} = (1 + δ_{i})\frac{1}{x_{−i}} ∈ Q_{W} .

Define the formal affine Demazure algebra D_{F} as the subalgebra of Q_{W} generated by multiplications by S and the elements Y_{i}.

By [7, Thm. 7.9] (see also [18, Thm. 5.14]) the R-algebra D_{F} satisfies the following (complete) set of relations: for i, j = 1 … rk(G) and u ∈ S

• Y_{i}^{2} = κ_{i}Y_{i}, where κ_{i} = \frac{1}{x_{i}} + \frac{1}{x_{−i}} and x_{i} = x_{α_{i}},

• Y_{i}u = s_{i}(u)Y_{i} + Δ_{−i}(u), where Δ_{−i}(u) = \frac{u−s_{i}(u)}{x_{−i}},

• (Y_{i}Y_{j})^{m_{ij}} − (Y_{j}Y_{i})^{m_{ij}} = ∑_{I_{w}} c_{I_{w}}Y_{I_{w}}, where the sum is taken over all reduced expressions I_{w} of elements w of the subgroup ⟨s_{i}, s_{j}⟩ ⊆ W, and the coefficients c_{I_{w}} are given by the formulas of [18, Prop. 5.8]

Example 6.1. If F corresponds to Chow groups, then D_{F} = H_{nil} is the affine nil-Hecke algebra over ℤ in the notation of [16]. If F corresponds to K-theory, then D_{F} is the 0-affine Hecke algebra over ℤ (q → 0 in the affine Hecke algebra). If F corresponds to the generic hyperbolic formal group law of [8, §9], then by [8, Prop. 9.2] the constant part of D_{F} is isomorphic to the localized classical Iwahori-Hecke algebra.

Let D_{F}^{⋆} = Hom_{S}(D_{F} , S) denote its dual. Observe that the main result of [9] (Thm. 8.2 loc.cit.) says that D_{F}^{⋆} is isomorphic to the R-algebra h_{T} (X).

Theorem 6.2. Let G be a split semisimple linear algebraic group over a field k and let h be an equivariant theory that satisfies property (CD).

Then the convolution algebra (h_{B^{2}}(G), ∘) is isomorphic (as an R-algebra) to the formal affine Demazure algebra D_{F}. So there is an R-algebra isomorphism

(D_{F}^{⋆} , ∘) ≃ (D_{F} , ·)

Proof. By Corollary 5.3 the ring (h_{B^{2}}(G), ∘) ≃ (h_{B}(X), ∘) is isomorphic to the subalgebra of End_{S}(h_{B}(X)) generated by the image of the forgetful map h_{G}(X) → h_{B}(X) and push-pull operators q_{i}^{∗}q_{i∗}. Since the map B → B/T is an affine fibration, the natural map h_{B}(X) → h_{T} (X) is an isomorphism. Hence we may identify S with h_{T} (k) and End_{S}(h_{B}(X)) with End_{S}(h_{T} (X)). Observe that these identifications preserve push-pull operators. The inclusion of T -fixed point set W → X gives an embedding h_{T} (X) → h_{T} (W) = S_{W}^{⋆} ⊆ Q_{W}^{⋆}. By [9, Corollary 8.7] there is the following commutative diagram

(9)  \xymatrix{h_{T} (X) \ar[r] \ar[d]_{q_{i}^{∗}q_{i∗}} & S_{W}^{⋆} \ar@{^{(}->}[r] & Q_{W}^{⋆} \ar[d]^{A_{i}} \\ h_{T} (X) \ar[r] & S_{W}^{⋆} \ar@{^{(}->}[r] & Q_{W}^{⋆}}

where the Hecke operator A_{i} is given by

A_{i}(f)(x) = f(x · Y_{i}) for x ∈ Q_{W} , f ∈ Q_{W}^{⋆} .

Moreover, the forgetful map

S ≅ h_{G}(X) → h_{T} (X) = ⊕_{w∈W} S

is given by the formula s ↦ (w · s)_{w∈W} for any s ∈ S. Then the multiplication in h_{T} (X) = S_{W}^{∗} by the image of any element in s ∈ h_{G}(X) induces a right multiplication by s in Q_{W}^{∗}. Since Q_{W} is a free Q-module of finite rank, the natural map ı: Q_{W} → End_{Q}(Q_{W}^{⋆}) given by ı(x)(f)(y) = f(yx) is an inclusion. Note that every A_{i} lies in the image of ı. Then by diagram (9) the image of h_{B^{2}}(G) is isomorphic to a subalgebra of Q_{W} generated by S and Y_{i} which is D_{F}. □

7. The rational algebra of push-pull operators

In the present section we introduce the rational algebra of push-pull operators \overline{D}_{F} and show that it can be identified with the subring of rational endomorphisms of G/B (Theorem 7.5).

The B^{2}-equivariant isomorphism E ×_{k} G → E ×_{k} E, (e, g) ↦ (e, eg) induces an isomorphism E×^{B}G/B → E/B×_{k}E/B. For all w ∈ W fix a reduced decomposition I_{w} = (i_{1}, …, i_{l}) and the corresponding Bott-Samelson resolution X_{I_{w}} → G/B of the Schubert cell. This map is B-equivariant, so it descends to a map Y_{I_{w}} = E ×^{B} X_{I_{w}} → E ×^{B} G/B.

Lemma 7.1. The classes [Y_{I_{w}}] form a basis of h(E/B ×_{k} E/B) over h(E/B), where the module structure is given by the pullback of the projection pr_{1}^{∗}: h(E/B) → h(E/B ×_{k} E/B).

Proof. Since B is special, G-torsor E splits over the function field of E/B. Then by [25, Lemma 3.3] projection pr_{1} : E/B ×_{k} E/B → E/B is a cellular fibration in the sense of [25, Definition 3.1] so that (E/B)^{2} is a cellular space over E/B. Let ξ be the generic point of E/B. The pullback of an open embedding j^{∗}: h(E/B ×_{k} E/B) → h(ξ ×_{k} E/B) ≃ h(G/B) is surjective and any preimage of R-basis of h(G/B) gives a basis of h(E/B ×_{k} E/B). Thus it is sufficient to check that j^{∗} sends [Y_{I_{w}}] to a basis of h(ξ ×_{k} E/B). Let p: E → E/B be the projection. Note that

E ×^{B} X_{I_{w}} ×_{(E/B×_{k}E/B)} ξ ×_{k} E/B = p^{−1}(ξ) ×^{B} X_{I_{w}} = ξ × X_{I_{w}},

since p^{−1}(ξ) → ξ is a trivial B-torsor. Thus j^{∗}([Y_{I_{w}}]) = [ξ ×X_{i_{w}}] that forms a basis of h(ξ × E/B) = h(ξ × G/B) over h(ξ) = R. □

Consider a B-equivariant map

f: E ×^{B} G → B\G, (e, g)B ↦ Bg.

Let X′_{I_{w}} = (P_{i_{1}} × … × P_{i_{l}})/B^{l} where B^{l}-action on P_{i_{1}} × … × P_{i_{l}} is given by (p_{1}, …, p_{l}) · (b_{1}, …, b_{l}) = (b_{1}^{−1}p_{1}b_{2}, …, b_{l}^{−1}p_{l}). Then X′_{I_{w}} gives the Bott-Samelson class for B\G.

Lemma 7.2. The composition h_{B}(B\G) →^{f^{∗}} h_{B}(E ×^{B} G) ≃ h(E/B ×_{k} E/B) maps [X′_{I_{w}}]_{B} to [Y_{I_{w}}].

Proof. Consider the map P_{i_{1}} ×^{B} P_{i_{2}} ×^{B} … ×^{B} P_{i_{l}} → G given by (p_{1}, …, p_{l}) → p_{1} … p_{l}. It is B-equivariant with respect to the left multiplication, so it descends to a map M_{I_{w}} = E ×^{B} P_{i_{1}} ×^{B} P_{i_{2}} ×^{B} … ×^{B} P_{i_{l}} → E ×^{B} G. By construction we have an isomorphism

M_{I_{w}} ≃ Y_{I_{w}} ×_{E×^{B}(G/B)} (E ×^{B} G).

Then [M_{I_{w}}]_{B} is mapped to [Y_{I_{w}}] via the isomorphism h(E ×^{B} G/B) → h_{B}(E ×^{B} G). Thus it is sufficient to check that f^{∗}[X′_{I_{w}}]_{B} = [M_{I_{w}}]_{B}, which follows from the fact that

M_{I_{w}} = E ×^{B} (P_{i_{1}} × … × P_{i_{l}}/B^{l−1}) ≃ (E ×^{B} G) ×_{B\G} X′_{I_{w}}.  □

Lemma 7.3. (cf. [25, Corollary 3.4]) The composition

(p_{1}^{∗}, γ_{1}^{∗}): h_{B}(E) ⊗_{S} h_{B^{2}}(G) ⟶ h_{B^{2}}(E^{2}) ≃ h((E/B)^{2})

of the diagram (6) (for H = B) is an isomorphism.

Proof. Consider the basis of h_{B^{2}}(G) over S given by the classes of Bott-Samelson resolutions ζ_{I_{w}}. Then by Lemma 7.2 γ_{1}^{∗}(ζ_{I_{w}}) forms a basis of h_{B^{2}}(E^{2}) over h_{B}(E) induced by the respective cellular filtration. □

Consider the restriction map h(E/B) → h(Ek̄/B) = h(Xk̄) on cohomology induced by the scalar extension k̄/k (here k̄ is a splitting field of E). Let \overline{h}(X) denote its image.

Corollary 7.4. The image of the ring homomorphism

res_{k̄/k}: (h(E/B ×_{k} E/B), ∘) ⟶ (h(Xk̄ ×k̄ Xk̄), ∘).

is the subalgebra generated by the multiplication by the elements of \overline{h}(X) and the push-pull operators q_{i}^{∗}q_{i∗}: h(X) → h(G/P_{i}) → h(X) for all simple roots α_{i}.

Proof. Follows by (6), Lemma 7.3 and Corollary 5.3. □

There is a natural action of W on \overline{h}(X) that comes from the W -action on E/T. So we can endow \overline{h}(X) ⊗_{S} Q_{W} with a structure of an R-algebra. Let \overline{D}_{F} denote its subalgebra \overline{h}(X) ⊗_{S} D_{F}. We call it the rational algebra of push-pull operators. We endow \overline{D}_{F} (and D_{F}) with a grading assuming that all Y_{i}’s have degree (−1) and elements of \overline{h}^{•}(X) (and of S = h_{B}^{•}(k)) have degree ’•’. By \overline{D}_{F}^{(m)} we denote its degree m homogeneous component. Let N = dim X.

Theorem 7.5. Consider the restriction

res_{k̄/k}: End_{h-Corr_{k}^{+}}(E/B) ⟶ End_{h-Corr+ ; k̄}(Xk̄)

on endomorphism rings of the respective motives (i.e., preserving the grading of h(X)). Its image can be identified with \overline{D}_{F}^{(0)} via the injective forgetful map

ϕ: ((\overline{h}(X) ⊗_{S} h_{G}(X2 ; k̄))^{(N)}, ∘) ⟶ (h^{N}(X2 ; k̄), ∘).

Proof. By (7) both h_{G}(X^{2}) and h(X^{2}) are free modules over h_{G}(X) and h(X) with basis given by the classes [Õ_{I_{w}}]_{G} and [Õ_{I_{w}}] respectively. The map ϕ sends [Õ_{I_{w}}]_{G} ↦ [Õ_{I_{w}}] and leaves the coefficients invariant. The result follows by Corollary 7.4, Corollary 5.3 and Theorem 6.2. □

We say that a (co-)homology theory h satisfies the Dimension Axiom if (Dim) For any smooth variety Y over k we have h^{n}(Y) = 0 for all n > dim Y.

Example 7.6. Any theory h over a field k of characteristic 0 obtained by specialization of coefficients of the Lazard ring (e.g. Chow groups, connective K-theory, algebraic cobordism Ω) satisfies (Dim).

The graded K-theory K_{0}(−)[β, β^{−1}] of [23, Example 1.1.5] does not satisfy (Dim).

Observe that the image of the characteristic map c: S → h(X) is contained in \overline{h}(X) (see [15, Thm. 4.5]). Consider both the induced map 𝔠: D_{F}^{(0)} → \overline{D}_{F}^{(0)} and the restriction map res_{k̄/k}: (h_{B}(E) ⊗_{S} D_{F})^{(0)} → \overline{D}_{F}^{(0)}.

Lemma 7.7. Assume that the theory h satisfies (Dim), then the kernels of 𝔠 and res_{k̄/k} are nilpotent.

In other words, there is a commutative diagram of maps of convolution rings

\xymatrix{D_{F}^{(0)} \ar[r]^{γ_{1}^{∗}} \ar[dr]_{𝔠} & h^{N}((E/B)^{2}) \ar@{->>}[d]^{res_{k̄/k}} \\  & _{\overline{D}{(0) \atop F}}}

with nilpotent kernels.

Proof. Let f = 𝔠 or res_{k̄/k}. Then x ∈ ker f means that x = ∑_{w} a_{w}[Õ_{I_{w}}]_{G} with f(a_{w}) = 0. By Theorem 5.2 each [Õ_{I_{w}}] corresponds to the composite of pushpull elements Y_{I_{w}} in \overline{D}_{F}, so that x corresponds to ∑_{w} a_{w}Y_{I_{w}} ∈ \overline{D}_{F}^{(0)} and x^{∘n} corresponds to

(∑_{w} a_{w}Y_{I_{w}})^{n} = ∑_{w} a_{w,n}Y_{I_{w}}, a_{w,n} ∈ (ker f)^{n},

Since ker f is contained in the augmentation ideal, (ker f)^{n} ⊂ S^{(≥n)}. Finally, observe that deg Y_{I_{w}} ≤ −N, hence, for n > 2N we get x^{∘n} = 0. □

Lemma 7.8. If E is a versal G-torsor, then the map γ_{1}^{∗} and, hence, 𝔠, of the lemma 7.7 is surjective.

Proof. Observe that if E is versal, then it admits an open G-equivariant embedding into 𝔸_{k}^{N}. So the projection E ×_{k} G^{i} → G^{i} in the definition of γ_{i} factors through 𝔸_{k}^{N} ×_{k} G^{i}. By (Loc) and (HI) the induced pullback γ_{i}^{∗} is surjective. □

8. Applications to representation theory of Hecke rings

Let G be a split semisimple linear algebraic group over a field k and let E be a G-torsor over k. Let k̄ be a splitting field of E and let E/B be the twisted form of G/B by means of E. By definition, we have

E/B ×_{k} k̄ ≃ G/B ×_{k} k̄.

Let h, h_{B} be an (B-equivariant) oriented theory over k that satisfies both (CD) and (Dim) axioms, e.g., Chow groups, connective K-theory or algebraic cobordism Ω. Let R = h(k) and S = h_{B}(k) be the respective coefficient rings.

Consider the endomorphism ring of the h-motive of E/B

C_{F} = (End_{h-Corr_{k}^{+}}(E/B), ∘).

By definition, any direct sum decomposition of the motive [E/B] is given by a complete set of primitive pairwise orthogonal idempotents on C_{F} so that ⟨E/B⟩_{h} ≃ P roj C_{F}. Our main result (Theorem 7.5) together with Lemma 7.7 says that the restriction map gives a surjective ring homomorphism with nilpotent kernel

res_{k̄/k}: C_{F} → \overline{D}_{F}^{(0)}.

By the standard idempotent lifting (see e.g. [25, §2 and Prop. 2.6]) we then obtain P roj C_{F} ≃ P roj \overline{D}_{F}^{(0)}, so that we get the following Theorem 8.1. There is a one-to-one correspondence between direct sum decompositions of the h-motive [E/B] and direct sum decompositions of \overline{D}_{F} -module \overline{D}_{F}. This correspondence, induces an equivalence between the category ⟨E/B⟩_{h} and the category P roj \overline{D}_{F} of finitely generated projective \overline{D}_{F} -modules.

If E is versal, then the algebra \overline{D}_{F} can be replaced by the algebra D_{F}.

Observe that in general the ring \overline{D}_{F} is not Krull-Schmidt (and not semi-simple).

Lemma 8.2. If the coefficient ring R is Artinian, then both C_{F} and \overline{D}_{F} and, hence, the categories ⟨E/B⟩_{h} and P roj \overline{D}_{F} satisfy the Krull-Schmidt property (uniqueness of a direct sum decomposition).

Proof. If R is Artinian, then both \overline{D}_{F} and C_{F} are Artinian (as \overline{D}_{F} is finite dimensional over R). So they are both Noetherian which implies that the respective tautological modules \overline{D}_{F} and C_{F} have finite length and, hence, the Krull-Schmidt property holds for both \overline{D}_{F} and C_{F}. □

As a direct application of the main result of [25] one obtains the following characterization of modular representations of the (affine) nil-Hecke algebra (F is additive and h = CH).

Corollary 8.3. Let G be a split semisimple linear algebraic group over a field k. Consider the affine nil-Hecke algebra H_{nil} for G with coefficients in R = 𝔽_{p}, p is a prime. Then

P roj H_{nil} ≃ ⟨R_{E,p}⟩,

where E is a versal G-torsor.

In particular, all indecomposable submodules of H_{nil} are free S-modules isomorphic to each other and their S-rank equals to the p-part of the product of pexceptional degrees of G.

Proof. The S-rank coincides with the number of Tate motives in the decomposition of R_{E,p} over a splitting field of E, that is ∏_{i=1}^{r} \frac{1−t^{d_{i}p^{k_{i}}}}{1−t^{d_{i}}} |_{t=1} (in the notation of [25]) which is equal to the p-part p^{∑_{i=1}^{r} k_{i}} of p-exceptional degrees of [19, p.73]. □

Example 8.4. Consider the root system of type A_{1}. In this case T^{∗} = ℤω (G = SL_{2}) or T^{∗} = ℤα (G = P GL_{2}), α = 2ω is the simple root and ω is the fundamental weight. The Weyl group W = {1, s} acts by s: ω ↦ −ω, where s is the simple reflection. By definition, S = R[[x]]_{F} (where x = x_{ω} or x = x_{α}), Q = S[\frac{1}{x}], Q_{W} = {q(x)δ_{w} | q(x) ∈ Q, w ∈ W } with

q(−_{F} x)δ_{s} = s(q(x))δ_{s} = δ_{s}q(x),

where −_{F} x is the formal inverse of x. Observe that x_{α} = x_{ω+ω} = F (x_{ω}, x_{ω}) in S. The R-algebra D_{F} is a free left S-submodule of rank 2 in Q_{W} with basis

\{1, Y = \frac{1}{−_{F} x_{α}} + \frac{1}{x_{α}} δ_{s}\}.

It satisfies the relations

Y^{2} = κY and Y q(x) = q(−_{F} x)Y + Δ(q(x)),

where κ = \frac{1}{−_{F} x_{α}} + \frac{1}{x_{α}} and Δ(q(x)) = \frac{q(x)−q(−_{F} x)}{x_{−α}}.

Let p = a + bY, a, b ∈ S be an idempotent in D_{F}, i.e., p^{2} = p. Since deg p = 0, we have deg a = 0 and deg b = 1 (the coefficient a_{ij} ∈ R at u^{i}v^{j} in F (u, v) has degree 1 − i − j). Then

(a+bY)^{2} = a^{2}+abY +bY a+bY bY = a^{2}+abY +b(s(a)Y +Δ(a))+b(s(b)Y +Δ(b))Y =

(a^{2} + bΔ(a)) + (ab + bs(a) + bs(b)κ + bΔ(b))Y.

So that

a^{2} + bΔ(a) = a and ab + bs(a) + bs(b)κ + bΔ(b) = b.

Assume b is a non-zero divisor, then we obtain (in S)

(10)  a^{2} + bΔ(a) = a and a + s(a) + s(b)κ + Δ(b) = 1.

In the case h = CH and R = ℤ (F (u, v) = u + v) we have κ = 0, −_{F} x = −x, x_{α} = 2x_{ω}, a ∈ ℤ, b = cx, c ∈ ℤ and, (10) becomes

a = 0 and Δ(b) = \frac{cx}{−x_{ω}} = 1

or

a = 1 and Δ(b) = \frac{cx}{−x_{ω}} = −1

which have solutions only if x = x_{ω} and c = ±1. Therefore, D_{F} has only two indecomposable submodules corresponding to the idempotents 1 − xY and xY.

The latter translates into the following well-known fact concerning motivic decompositions:

The Chow motive of a conic E/B (for a versal G-torsor E) decomposes as a direct sum of two indecomposable summands if and only if G = SL_{2}, i.e.. E is trivial and E/B = ℙ^{1}. In this case each summand correspond to the (shifted) Tate motive.

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Alexander Neshitov, University of Ottawa, Canada / Steklov Institute at St.Petersburg, Russia

Victor Petrov, Steklov Institute at St.Petersburg, Russia

Nikita Semenov, University of Mainz, Germany

Kirill Zainoulline, University of Ottawa, Canada