PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL

BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG

Contents

1. Introduction  1
2. Formal Demazure and push-pull operators  3
3. Two bases of the formal twisted group algebra  6
4. The Weyl and the Hecke actions  8
5. Push-pull operators and elements  10
6. The push-pull operators on the dual  12
7. Another basis of the W_{Ξ}-invariant subring  14
8. The formal affine Demazure algebra  16
9. The algebraic restriction to the fixed locus  18
10. The push-pull operators on D_{F}^{⋆}  19
11. The non-degenerate pairing on the W_{Ξ}-invariant subring  21
References  24

1. Introduction

In a series of papers [KK86], [KK90] Kostant and Kumar introduced and successfully applied the techniques of nil (or 0-) Hecke algebras to study equivariant cohomology and K-theory of flag varieties. In particular, they showed that the dual of the nil Hecke algebra serves as an algebraic model for the T -equivariant singular cohomology of G/B (here G is a split semisimple linear algebrac group with a chosen split maximal torus T and G/B is the variety of Borel subgroups). In [HMSZ] and [CZZ], this formalism has been generalized using an arbitrary formal group law associated to an algebraic oriented cohomology theory in the sense of Levine-Morel [LM07], via the Quillen formula. Namely, given a formal group law F and a finite root system with a set of simple roots Π, one defines the formal affine Demazure algebra D_{F} and its dual D_{F}^{⋆} provides an algebraic model for the T -equivariant oriented cohomology h_{T} (G/B). Specializing to the additive and the multiplicative formal group laws, one recovers Chow groups (or singular cohomology) and K-theory respectively.

2010 Mathematics Subject Classification. 20C08, 14F43, 57T15.

The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005. The second author was supported by the NSERC Discovery grant 385795-2010, NSERC DAS grant 396100-2010 and the Early Researcher Award (Ontario). We appreciate the support of the Fields Institute; part of this work has been done while authors were attending the Thematic Program on Torsors, Nonassociative algebras and Cohomological Invariants at the Fields Institute.

Another motivation for studying the algebra D_{F} comes from its close relationship to Hecke algebras. Indeed, for the additive (resp. multiplicative) F it coincides with the completion of the nil (resp. 0-) affine Hecke algebra (see [HMSZ]). Moreover, in section 8, we show that for some elliptic formal group law F and a root system of Dynkin type A the non-affine part of D_{F} is isomorphic to the classical Iwahori-Hecke algebra, hence, relating it to equivariant elliptic cohomology.

In the present paper we pursue the ‘algebraization program’ for oriented cohomology theories started in [CPZ] and continued in [HMSZ] and [CZZ]; the general idea is to match cohomology rings of flag varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced simply and algebraically, in the spirit of [De73] or [KK86]. This approach is useful to study the structure of these rings, and to perform various computations. We focus here on algebraic constructions pertaining to T -equivariant oriented cohomology groups. The precise proofs and details of how our algebraic objects match cohomology groups will be treated in a forthcoming paper; however, for the convenience of the reader, we now give a brief description of the geometric setting.

Given an equivariant oriented cohomology theory h over a base field whose spectrum is denoted by pt, the formal group algebra S will correspond to h_{T} (pt). ^{1} It is an algebra over R = h(pt).

The T -fixed points of G/B are naturally in bijection with the Weyl group W. This gives a pull-back to the fixed locus map h_{T} (G/B) → h_{T} (W) ≃ ⨁_{w∈W} h_{T} (pt). This map happens to be injective. We do not know a direct geometric reason for that, but it follows from our algebraic description, in which it appears as the map D_{F}^{⋆} → S_{W}^{⋆} ≃ ⨁_{w∈W} S of Definition 9.1. It is then convenient to enlarge S to its localization Q at a multiplicative subset generated by Chern class of line bundles corresponding canonically to roots, which gives injections S ⊆ Q, S_{W} ⊆ Q_{W} and S_{W}^{⋆} ⊆ Q_{W}^{∗}. Although we do not know good geometric interpretations of Q, Q_{W} or Q_{W}^{∗}, all the formulas and operators we are interested in are easily defined at that localized level, because they involve denominators. The main technical difficulties then lie in proving that these operators actually restrict to S, S_{W}^{⋆}, D_{F}^{⋆} etc., or so to speak, that the denominators cancel out.

Our central object of study is a push-pull operator on D_{F}^{⋆}, which is an algebraic version of the composition

h_{T} (G/P) →^{p_{∗}} h_{T} (G/Q) →^{p^{∗}} h_{T} (G/P)

of the push-forward followed by the pull-back along the quotient map p: G/P → G/Q, where P ⊆ Q are two parabolic subgroups of G. Again p^{∗} happens to be injective, and it identifies h_{T} (G/Q) to a subring of h_{T} (G/P), namely the subring of invariants under the action of the parabolic subgroup W_{Q} of the Weyl group W. This does not seem to be straightforward from the geometry either, and it once more follows from our algebraic description: given subsets Ξ′ ⊆ Ξ of a given set of simple roots Π (each giving rise to a parabolic subgroup), we define an element Y_{Ξ/Ξ′} in Q_{W} (see 5.3). We define an action of the Demazure algebra D_{F} on its S-dual D_{F}^{⋆}, by precomposition by multiplication on the right. The action of Y_{Ξ/Ξ′} thus defines the desired push-pull operator A_{Ξ/Ξ′} : (D_{F}^{⋆})^{W_{Ξ′}} → (D_{F}^{⋆})^{W_{Ξ}}. The formula for the element Y_{Ξ/Ξ′} with Ξ′ = ∅ had already appeared in related contexts, namely, in discussions around the Becker-Gottlieb transfer for topological complex-oriented theories (see [BE90, (2.1)] and [GR12, §4.1]).

1We will require that the cohomology rings are ‘complete’ in some precise sense, but this is a technical point, that we prefer to hide here for simplicity.

Finally, we define the algebraic counterpart of the natural pairing h_{T} (G/B) ⊗ h_{T} (G/B) → h_{T} (pt) obtained by multiplication and push-forward to the point. It is a pairing D_{F}^{⋆} ⊗ D_{F}^{⋆} → S. We show that it is non-degenerate, and that algebraic classes corresponding to (chosen) desingularization of Schubert varieties form a basis of D_{F}^{⋆}, with a very simple dual basis with respect to the pairing. We provide the same kind of description for h_{T} (G/P). This generalizes (to parabolic subgroups and to equivariant cohomology groups) and simplifies several statements from [CPZ, §14], as well as results from [KK86] and [KK90] (to arbitrary oriented cohomology theories).

The paper is organized as follows. In sections 2 and 3, we recall definitions and basic properties from [CPZ, §2,3], [HMSZ, §6] and [CZZ, §4,5]: the formal group algebra S, the Demazure and push-pull operators Δ_{α} and C_{α} for every root α, the formal twisted group algebra Q_{W} and its Demazure and push-pull elements X_{α} and Y_{α}. In section 4, we introduce a left Q_{W} -action ‘•’ on the dual Q_{W}^{∗}. It induces both an action of the Weyl group W on Q_{W}^{∗} (the Weyl-action) and an action of X_{α} and Y_{α} on Q_{W}^{∗} (the Hecke-action). In sections 5 and 6, we introduce and study the more general push-pull elements in Q_{W} and operators on Q_{W}^{∗} with respect to given coset representatives of parabolic quotients of the Weyl group. In section 7, we construct a basis of the subring of invariants of Q_{W}^{∗}, which generalizes [KK90, Lemma 2.27].

In section 8, we recall the definition and basic properties of the formal (affine) Demazure algebra D_{F} following [HMSZ, §6], [CZZ, §5] and [Zh13]. We show that for the special elliptic formal group law, the formal Demazure algebra is related to the classical Iwahori-Hecke algebra. In section 9, we define the algebraic restriction to the fixed locus map which is used in section 10 to restrict all our push-pull operators and elements to D_{F} and its dual D_{F}^{⋆} as well as to restrict the non-degenerate pairing on D_{F}^{⋆}. At last, in section 11, we define and discuss the non-degenerate pairing on the subring of invariants of D_{F}^{⋆} under a parabolic subgroup of the Weyl group.

Acknowledgments We would like to mention that one of the ingredients of this paper, the push-pull formulas in the context of Weyl group actions, arose in discussions between the first author and Victor Petrov, whose unapparent contribution we therefore gratefully acknowledge.

2. Formal Demazure and push-pull operators

In this section we recall definitions of the formal group algebra and of the formal Demazure and push-pull operators, following [CPZ] and [CZZ].

Let R be a commutative ring with unit, and let F be a one-dimensional commutative formal group law (FGL) over R, i.e. F (x, y) ∈ R[[x, y]] satisfies

F (x, 0) = 0, F (x, y) = F (y, x), F (x, F (y, z)) = F (F (x, y), z).

Example 2.1. The additive FGL is defined by F_{a}(x, y) = x+y, and a multiplicative FGL is defined by F_{m}(x, y) = x + y − βxy with β ∈ R. The coefficient ring of the universal FGL is generated by the coefficients a_{ij} modulo relations induced by the above properties and is called the Lazard ring.

Example 2.2. Consider an elliptic curve given in Tate coordinates by

(1 − μ_{1}t − μ_{2}t^{2})s = t^{3}.

The corresponding FGL over the coefficient ring R = ℤ[μ_{1}, μ_{2}] is given by, e.g. see [BB10, Example 63],

F_{e}(x, y) := \frac{x+y−μ_{1}xy}{1+μ_{2}xy}

and will be called a special elliptic FGL. Observe that

F_{e}(x, y) = x + y − xy(μ_{1} + μ_{2}F_{e}(x, y)),

and thus that the formal inverse of F_{e} is identical to the one of F_{m}, i.e. \frac{x}{μ_{1}x−1}, and

F_{e}(x, x) = \frac{2x−μ_{1}x^{2}}{1+μ_{2}x^{2}} .

Let Λ be an Abelian group and let R[[x_{Λ}]] be the ring of formal power series with variables x_{λ} for all λ ∈ Λ. Define the formal group algebra R[[Λ]]_{F} to be the quotient of R[[x_{Λ}]] by the closure of the ideal generated by elements x_{0} and x_{λ_{1}+λ_{2}} − F (x_{λ_{1}}, x_{λ_{2}}) for any λ_{1}, λ_{2} ∈ Λ. Here 0 is the identity element in Λ. Let I_{F} denote the kernel of the augmentation map ϵ: R[[Λ]]_{F} → R, x_{α} ↦ 0.

Assume that Λ is a free Abelian group of finite rank and let Σ be a finite subset of Λ. A root datum is an embedding Σ ↪ Λ^{∨}, α ↦ α^{∨} into the dual of Λ satisfying certain conditions [SGA, Exp. XXI, Def. 1.1.1]. The rank of the root datum is the ℚ-rank of Λ ⊗_{ℤ} ℚ. The root lattice Λ_{r} is the subgroup of Λ generated by Σ, and the weight lattice Λ_{w} is the Abelian group defined by

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

We always assume that the root datum is reduced and semisimple (ℚ-ranks of Λ_{r}, Λ_{w} and Λ are the same and no root is twice another one). We say that a root datum is simply connected (resp. adjoint) if Λ = Λ_{w} (resp. Λ = Λ_{r}), and then use the notation D_{n}^{sc} (resp. D_{n}^{ad}) for irreducible root data where D = A, B, C, D, E, F, G is one of the Dynkin types and n is the rank.

The Weyl group W of a root datum (Λ, Σ) is a subgroup of Aut_{ℤ}(Λ) generated by simple reflections s_{α} for all α ∈ Σ defined by

s_{α}(λ) := λ − α^{∨}(λ)α, λ ∈ Λ.

We fix a set of simple roots Π = {α_{1}, …, α_{n}} ⊂ Σ that is a basis of the root datum: each element of Σ is an integral linear combination of simple roots with either all positive coefficients or negative. This partitions Σ into Σ^{+} and Σ^{−} the subsets of positive roots, resp. negative roots. Let ℓ denote the length function on W with respect to the set of simple roots Π. Let w_{0} be the longest element of W with respect to ℓ and let N := ℓ(w_{0}).

Following [CZZ, Def. 4.4] we say that the formal group algebra R[[Λ]]_{F} is Σregular if x_{α} is not a zero-divisor in R[[Λ]]_{F} for all roots α ∈ Σ. We will always assume that:

The formal group algebra R[[Λ]]_{F} is Σ-regular. By [CZZ, Lemma 2.2] this holds if 2 is not a zero-divisor in R, or if the root datum does not contain any C^{sc} as an irreducible component.

Following [CPZ, Def. 3.5 and 3.12] for each α ∈ Σ we define two R-linear operators Δ_{α} and C_{α} on R[[Λ]]_{F} as follows:

(2.1) Δ_{α}(y) := \frac{y−s_{α}(y)}{x_{α}}, C_{α}(y) := κ_{α}y − Δ_{α}(y) = \frac{y}{x_{−α}} + \frac{s_{α}(y)}{x_{α}}, y ∈ R[[Λ]]_{F} ,

where κ_{α} := \frac{1}{x_{α}} + \frac{1}{x_{−α}} (note that κ_{α} ∈ R[[Λ]]_{F}). The operator Δ_{α} is called the Demazure operator and the operator C_{α} is called the push-pull operator or the BGG operator.

Example 2.3. For the special elliptic formal group law F_{e} we have κ_{α} = μ_{1} + μ_{2}F_{e}(x_{−α}, x_{α}) = μ_{1} for each α ∈ Σ. If the root datum is of type A_{1}^{sc}, we have Σ = {±α}, Λ = ⟨ω⟩ with simple root α = 2ω and

C_{α}(x_{α}) = \frac{x_{α}}{x_{−α}} + \frac{x_{−α}}{x_{α}} = μ_{1}x_{α}−1+ \frac{1}{μ_{1}x_{α}−1}, C_{α}(x_{ω}) = \frac{x_{ω}}{x_{−α}} + \frac{x_{−ω}}{x_{α}} = μ_{1}x_{ω} − \frac{1+μ_{2}x_{ω}^{2}}{1−μ_{1}x_{ω}} .

If it is of type A_{2}^{sc} we have Σ = {±α_{1}, ±α_{2}, ±(α_{1} + α_{2})}, Λ = ⟨ω_{1}, ω_{2}⟩ with simple roots α_{1} = 2ω_{1} − ω_{2}, α_{2} = 2ω_{2} − ω_{1} and x_{α_{1}} = \frac{2x_{1}−μ_{1}x_{1}^{2}−x_{2}−μ_{2}x_{1}^{2}x_{2}}{1+μ_{2}x_{1}^{2}−μ_{1}x_{2}−2μ_{2}x_{1}x_{2}} ,

C_{α_{2}}(x_{1}) = μ_{1}x_{1}, C_{α_{1}}(x_{1}) = μ_{1}x_{1} − \frac{1+μ_{2}x_{1}^{2}−μ_{1}x_{2}−2μ_{2}x_{1}x_{2}}{1−μ_{1}x_{1}−μ_{2}x_{1}x_{2}} ,

where x_{1} := x_{ω_{1}} and x_{2} := x_{ω_{2}}.

According to [CPZ, §3] the operators Δ_{α} satisfy the twisted Leibniz rule

(2.2)  Δ_{α}(xy) = Δ_{α}(x)y + s_{α}(x)Δ_{α}(y), x, y ∈ R[[Λ]]_{F} ,

i.e. Δ_{α} is a twisted derivation. Moreover, they are R[[Λ]]_{F}^{W_{α}}-linear, where W_{α} = {e, s_{α}}, and

(2.3)  s_{α}(x) = x if and only if Δ_{α}(x) = 0.

Remark 2.4. Properties (2.2) and (2.3) suggest that the Demazure operators can be effectively studied using the theory of twisted derivations and the invariant theory of W. On the other hand, push-pull operators do not satisfy properties (2.2) and (2.3) but according to [CPZ, Theorem 12.4] they correspond to the push-pull maps between flag varieties and, hence, are of geometric origin.

For the i-th simple root α_{i}, let Δ_{i} := Δ_{α_{i}} and s_{i} := s_{α_{i}}. Given a sequence I = (i_{1}, …, i_{m}) with i_{j} ∈ {1, …, n}, denote |I| = m and define

Δ_{I} := Δ_{i_{1}} ∘ ⋯ ∘ Δ_{i_{m}}, C_{I} := C_{i_{1}} ∘ ⋯ ∘ C_{i_{m}} and set Δ_{∅} = C_{∅} = id.

We say that a sequence I is reduced in W if s_{i_{1}}s_{i_{2}} … s_{i_{m}} is a reduced expression in W. In this case we also say that I = I_{w} is a reduced sequence of the element w(I) := s_{i_{1}}s_{i_{2}} … s_{i_{m}}. We set I_{e} = ∅ (here e is the identity element of W).

Remark 2.5. It is well-known that for a nontrivial root datum, all w ∈ W, Δ_{I_{w}} and C_{I_{w}} are independent of the choice of a reduced sequence I_{w} of w ∈ W if and only if F is of the form F (x, y) = x+y+βxy, β ∈ R. The “if” part of the statement is due to Demazure [De73, Th. 1] and the “only if” part to Bressler and Evens [BE90, Theorem 3.7]. So for such F we can define Δ_{w} := Δ_{I_{w}} and C_{w} := C_{I_{w}} for each w ∈ W.

The operators Δ_{w} and C_{w} play a crucial role in Schubert calculus and computations of the singular cohomology (F = F_{a}) and the K-theory (F = F_{m}) rings of flag varieties.

For a general F, e.g. for F = F_{e}, the situation becomes much more intricate as we have to rely on choices of reduced decomposition I_{w}.

3. Two bases of the formal twisted group algebra

We now recall definitions and basic properties of the formal twisted group algebra Q_{W}, Demazure elements X_{α} and push-pull elements Y_{α}, following [HMSZ] and [CZZ]. For a chosen set of reduced sequences {I_{w}}_{w∈W} we introduce two bases {X_{I_{w}}} and {Y_{I_{w}}} of Q_{W} and describe the matrices (a_{v,w}^{X}) and (a_{v,w}^{Y}) by expressing them on the canonical basis {δ_{w}} of Q_{W}. We also relate the coefficients a_{v,w}^{X} and the corresponding coefficients a_{v,w}^{′X} of the reversed elements X_{I_{w}^{rev}}.

For simplicity we write S := R[[Λ]]_{F}. Since the formal group algebra S is Σregular, it embeds into the localization Q = S[\frac{1}{x_{α}} | α ∈ Σ]. Let Q_{W} be the twisted group algebra of Q and of the group ring R[W] over R, i.e. Q_{W} = Q ⊗_{R} R[W] as an R-module and the product in Q_{W} is given by

(q ⊗ δ_{w})(q′ ⊗ δ_{w′}) = qw(q′) ⊗ δ_{ww′}, q, q′ ∈ Q, w, w′ ∈ W.

where δ_{w} is the canonical element corresponding to w in R[W]. Note that Q_{W} is not a Q-algebra since the embedding Q ↪ Q_{W} , q ↦ q ⊗ δ_{e} is not central.

Inside Q_{W}, we use the notation q := q ⊗ δ_{e} and δ_{w} := 1 ⊗ δ_{w}, 1 := δ_{e} and δ_{α} := δ_{s_{α}} for a root α ∈ Σ. Thus qδ_{w} = q ⊗ δ_{w} and δ_{w}q = w(q) ⊗ δ_{w}. Clearly, {δ_{w}}_{w∈W} is a basis of Q_{W} as a left Q-module.

For each α ∈ Σ we define the following elements of Q_{W} (corresponding to the operators Δ_{α} and C_{α}, respectively, by the action of (4.3)):

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

called the Demazure elements and the push-pull elements, respectively.

By straightforward computations, for each α ∈ Σ we have

(3.1)  X_{α}^{2} = κ_{α}X_{α} = X_{α}κ_{α} and Y_{α}^{2} = κ_{α}Y_{α} = Y_{α}κ_{α},
X_{α}q = s_{α}(q)X_{α} + Δ_{α}(q) and Y_{α}q = s_{α}(q)Y_{α} + Δ_{−α}(q), q ∈ Q,
X_{α}Y_{α} = Y_{α}X_{α} = 0.

Let δ_{i} := δ_{s_{i}}, X_{i} := X_{α_{i}} and Y_{i} := Y_{α_{i}} be the i-th simple root α_{i}. Given a sequence I = (i_{1}, i_{2}, …, i_{m}) with i_{j} ∈ {1, …, n}, let the product X_{i_{1}}X_{i_{2}} … X_{i_{m}} be denoted by X_{I} and Y_{i_{1}}Y_{i_{2}} … Y_{i_{m}} by Y_{I}. Set X_{∅} = Y_{∅} = 1.

By [Bo68, Ch. VI, §1, No 6, Cor. 2] if v ∈ W has a reduced decomposition v = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{m}}, then

(3.2)  vΣ^{−} ∩ Σ^{+} = {α_{i_{1}}, s_{i_{1}}(α_{i_{2}}), …, s_{i_{1}}(s_{i_{2}}(⋯ s_{i_{m−1}}(α_{i_{m}}) ⋯))}.

In particular, s_{i}Σ^{−} ∩ Σ^{+} = {α_{i}}.

Lemma 3.1. Let I_{v} be a reduced sequence of an element v ∈ W.

Then X_{I_{v}} = ∑_{w≤v} a_{v,w}^{X}δ_{w} for some a_{v,w}^{X} ∈ Q, where the sum is taken over all elements of W less or equal to v with respect to the Bruhat order and a_{v,v}^{X} = (−1)^{ℓ(v)} ∏_{α∈vΣ^{−}∩Σ^{+}} x_{α}^{−1}.

Moreover, we have δ_{v} = ∑_{w≤v} b_{v,w}^{X}X_{I_{w}} for some b_{v,w}^{X} ∈ S such that b_{v,e}^{X} = 1 and b_{v,v}^{X} = (−1)^{ℓ(v)} ∏_{α∈vΣ^{−}∩Σ^{+}} x_{α}.

Proof. It follows from [CZZ, Lemma 5.4, Corollary 5.6] and the fact that δ_{α} = 1 − x_{α}X_{α}. □

Similar for Y_{I}’s we have Lemma 3.2. Let I_{v} be a reduced sequence for an element v ∈ W.

Then Y_{I_{v}} = ∑_{w≤v} a_{v,w}^{Y}δ_{w} for some a_{v,w}^{Y} ∈ Q and a_{v,v}^{Y} = (−1)^{ℓ(w)}a_{v,v}^{X}. Moreover, we have δ_{v} = ∑_{w≤v} b_{v,w}^{Y}Y_{I_{w}} for some b_{v,w}^{Y} ∈ S and b_{v,v}^{Y} = (−1)^{ℓ(v)}b_{v,v}^{X}.

Proof. We follow the proof of [CZZ, Lemma 5.4] replacing X by Y. By induction we have

Y_{I_{v}} = (x_{−β}^{−1} + x_{β}^{−1}δ_{β}) ∑_{w≤v′} a_{v′,w}^{Y}δ_{w} = x_{β}^{−1}s_{β}(a_{v′,v′}^{Y})δ_{v} + ∑_{w<v} a_{v,w}^{Y}δ_{w},

where I_{v} = (i_{1}, …, i_{m}) is a reduced sequence of v, β = α_{i_{1}} and v′ = s_{β}v. This implies the formulas for Y_{I_{v}} and for a_{v,v}^{Y}. Remaining statements involving b_{v,w}^{Y} follow by the same arguments as in the proof of [CZZ, Corollary 5.6] using the fact that δ_{α} = x_{α}Y_{α} − \frac{x_{α}}{x_{−α}} and \frac{x_{α}}{x_{−α}} ∈ S. □

As in the proof of [CZZ, Corollary 5.6], Lemmas 3.1 and 3.2 immediately imply:

Corollary 3.3. The elements {X_{I_{v}}}_{v∈W} (resp. {Y_{I_{v}}}_{v∈W}) form a basis of Q_{W} as a left Q-module.

Example 3.4. For the root data A_{1}^{ad} or A_{1}^{sc} and the formal group law F_{e} we have x_{Π} = x_{−α} and

(a_{v,w}^{Y})_{v,w∈W} = (\begin{matrix}1 & 0 \\ μ_{1} − \frac{1}{x_{α}} & \frac{1}{x_{α}} \end{matrix} )

where the first row and column correspond to e ∈ W and the second to s_{α} ∈ W.

Given a sequence I = (i_{1}, …, i_{m}), let I^{rev} := (i_{m}, …, i_{1}). We set

x_{Π} := ∏_{α∈Σ^{−}} x_{α} ∈ S.

Observe that by (3.2) we have s_{α}(x_{Π})x_{−α} = x_{Π}x_{α} for α ∈ Σ^{+}.

Lemma 3.5. Let I = (i_{1}, …, i_{m}) be a sequence in {1, …, n}. Let

X_{I} = ∑_{v∈W} a_{I,v}^{X}δ_{v} and X_{I^{rev}} = ∑_{v∈W} a_{I,v}^{′X}δ_{v} for some a_{I,v}^{X}, a_{I,v}^{′X} ∈ Q,

then v(x_{Π}) a_{I,v}^{′X} = v(a_{I,v^{−1}}^{X}) x_{Π}. Similarly, let

Y_{I} = ∑_{v∈W} a_{I,v}^{Y}δ_{v} and Y_{I^{rev}} = ∑_{v∈W} a_{I,v}^{′Y}δ_{v} for some a_{I,v}^{Y}, a_{I,v}^{′Y} ∈ Q,

then v(x_{Π}) a_{I,v}^{′Y} = v(a_{I,v^{−1}}^{Y}) x_{Π}.

Proof. If I = (i), then Y_{I} = Y_{I^{rev}}. So a′_{I,v} = a_{I,v} for all v ∈ W (we omit the superscripts ‘Y ’). For v ∉ {e, s_{i}} we have a′_{I,v} = a_{I,v} = 0. For v = e we have x_{Π}a′_{I,e} = x_{Π}a_{I,e}. For v = s_{i} we have s_{i}(x_{Π})a′_{I,s_{i}} = x_{Π}s_{i}(a_{I,s_{i}}), since a_{I,s_{i}} = \frac{1}{x_{α_{i}}}. The proof is analogous for X_{i}.

Now the conclusion immediately follows by induction on the length |I| using Lemma 3.6 below with x = x_{Π}, f = Y_{J}, f′ = Y_{J^{rev}}, g = Y_{K} and g′ = Y_{K^{rev}} for any splitting of I in smaller sequences J followed by K. □

Lemma 3.6. Given x ∈ Q, assume that elements

f = ∑_{v∈W} a_{v}δ_{v} and f′ = ∑_{v∈W} a′_{v}δ_{v} with a_{v}, a′_{v} ∈ Q

g = ∑_{v∈W} b_{v}δ_{v} and g′ = ∑_{v∈W} b′_{v}δ_{v} with b_{v}, b′_{v} ∈ Q

of Q_{W} satisfy v(x)a′_{v} = v(a_{v^{−1}}) x and v(x)b′_{v} = v(b_{v^{−1}}) x for all v ∈ W. Then the product fg = ∑ c_{v}δ_{v} and the element (fg)′ := g′f′ = ∑ c′_{v}δ_{v} bear the same relation: v(x)c′_{v} = v(c_{v^{−1}}) x.

Proof. By definition of the product, we have

c′_{v} = ∑_{v_{1}v_{2}=v} b′_{v_{1}}v_{1}(a′_{v_{2}}) and c_{v^{−1}} = ∑_{v_{1}v_{2}=v} a_{v_{2}^{−1}}v_{2}^{−1}(b_{v_{1}^{−1}}).

We then compute (all sums still being over pairs (v_{1}, v_{2}) ∈ W^{2} such that v_{1}v_{2} = v)

\begin{matrix}v(c_{v^{−1}}) x = ∑ v_{1}v_{2}(a_{v_{2}^{−1}}v_{2}^{−1}(b_{v_{1}^{−1}})) x = ∑ v_{1}v_{2}(a_{v_{2}^{−1}})v_{1}(b_{v_{1}^{−1}}) x \\ = ∑ v_{1}v_{2}(a_{v_{2}^{−1}})v_{1}(x) b′_{v_{1}} = ∑ v_{1}(v_{2}(a_{v_{2}^{−1}})x) b′_{v_{1}} = ∑ v_{1}(v_{2}(x)a′_{v_{2}}) b′_{v_{1}} \\ = v(x) ∑ b′_{v_{1}}v_{1}(a′_{v_{2}}) = v(x)c′_{v}. \end{matrix} □

4. The Weyl and the Hecke actions

In the present section we recall several basic facts concerning the Q-linear dual Q_{W}^{∗} following [HMSZ] and [CZZ]. We introduce a left Q_{W} -action ‘•’ on Q_{W}^{∗}. The latter induces an action of the Weyl group W on Q_{W}^{∗} (the Weyl-action) and the action by means of X_{α} and Y_{α} on Q_{W}^{∗} (the Hecke-action). These two actions will play an important role in the sequel.

Let Q_{W}^{∗} := Hom_{Q}(Q_{W} , Q) denote the Q-linear dual of the left Q-module Q_{W}. By definition, Q_{W}^{∗} is a left Q-module via (qf)(z) := qf(z) for any z ∈ Q_{W}, f ∈ Q_{W}^{∗} and q ∈ Q. Moreover, there is a Q-basis {f_{w}}_{w∈W} of Q_{W}^{∗} dual to the basis {δ_{w}}_{w∈W} defined by f_{w}(δ_{v}) := δ_{w,v} (the Kronecker symbol), w, v ∈ W.

Definition 4.1. We define a left action of Q_{W} on Q_{W}^{∗} as follows:

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

By definition, this action is left Q-linear, i.e. z • (qf) = q(z • f) and it induces a different left Q-module structure on Q_{W}^{∗} via the embedding q ↦ qδ_{e}, i.e.

(q • f)(z) := f(zq).

It also induces a Q-linear action of W on Q_{W}^{∗} via w(f) := δ_{w} • f.

Lemma 4.2. We have q • f_{w} = w(q)f_{w} and w(f_{v}) = f_{vw^{−1}} for any q ∈ Q and w, v ∈ W.

Proof. We have (q • f_{w})(δ_{v}) = f_{w}(v(q)δ_{v}) = v(q)δ_{w,v} which shows that q • f_{v} = v(q)f_{v}. As for the second we have [w(f_{v})](δ_{u}) = f_{v}(δ_{u}δ_{w}) = δ_{v,uw}, so w(f_{v}) = f_{vw^{−1}}. □

There is a coproduct on Q_{W} defined by [CZZ, Def. 8.9]:

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

Here ⊗_{Q} is the tensor product of left Q-modules. It is cocommutative with co-unit ε : Q_{W} → Q, qδ_{w} ↦ q [CZZ, Prop. 8.10]. The coproduct structure on Q_{W} induces a product structure on Q_{W}^{∗}, which is Q-bilinear for the natural action of Q on Q_{W}^{∗} (not the one using •). In terms of the basis {f_{w}}_{w∈W} the product is given by component-wise multiplication:

(4.1)  ( ∑_{v∈W} q_{v}f_{v})( ∑_{w∈W} q′_{w}f_{w}) = ∑_{w∈W} q_{w}q′_{w}f_{w}, q_{w}, q′_{w} ∈ Q.

In other words, if we identify the dual Q_{W}^{∗} with the Q-module of maps Hom(W, Q) via

Q_{W}^{∗} → Hom(W, Q), f ↦ f′, f′(w) := f(δ_{w}),

then the product is the classical multiplication of functions with values in a ring.

The multiplicative identity 1 of this product corresponds to the counit ε and equals 1 = ∑_{w∈W} f_{w}. We also have

(4.2)  q • (ff′) = (q • f)f′ = f(q • f′) for q ∈ Q and f, f′ ∈ Q_{W}^{∗} .

Lemma 4.3. For any α ∈ Σ and f, f′ ∈ Q_{W}^{∗} we have s_{α}(ff′) = s_{α}(f)s_{α}(f′), i.e. W acts on the algebra Q_{W}^{∗} by Q-linear automorphisms.

Proof. By Q-linearity of the action of W and of the product, it suffices to check the formula on basis elements f = f_{w} and f′ = f_{v}, for which it is straightforward. □

Observe that the ring Q can be viewed as a left Q_{W} -module via the following action:

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

Then by definition we have

(4.4)  (q • 1)(z) = z · q, z ∈ Q_{W} .

Definition 4.4. For α ∈ Σ we define two Q-linear operators on Q_{W}^{∗} by

A_{α}(f) := Y_{α} • f and B_{α}(f) := X_{α} • f, f ∈ Q_{W}^{∗} .

An action by means of A_{α} or B_{α} will be called a Hecke-action on Q_{W}^{∗}.

Remark 4.5. If F = F_{m} (resp. F = F_{a}) one obtains actions introduced by Kostant–Kumar in [KK90, I_{18}] (resp. in [KK86, I_{51}]).

As in (2.2) and (2.3) we have

(4.5) B_{α}(ff′) = B_{α}(f)f′ + s_{α}(f)B_{α}(f′) and B_{α} ∘ s_{α} = −B_{α}, for f, f′ ∈ Q_{W}^{∗} ,
(4.6)  B_{α}(f) = 0 if and only if f ∈ (Q_{W}^{∗})^{W_{α}}.

Indeed, using (4.2) and Lemma 4.3 we obtain

B_{α}(f)f′ + s_{α}(f)B_{α}(f′) = [\frac{1}{x_{α}} (1 − δ_{α}) • f]f′ + s_{α}(f)[\frac{1}{x_{α}} (1 − δ_{α}) • f′]
= [\frac{1}{x_{α}} • (f − s_{α}(f))]f′ + s_{α}(f)[\frac{1}{x_{α}} • (f′ − s_{α}(f′))]
= \frac{1}{x_{α}} • (ff′ − s_{α}(f)s_{α}(f′)) = B_{α}(ff′)

and B_{α}(s_{α}(f)) = \frac{1}{x_{α}} (1 − δ_{α}) • s_{α}(f) = \frac{1}{x_{α}} • (s_{α}(f) − f) = −B_{α}(f). As for (4.6) we have 0 = B_{α}(f) = X_{α} • f = \frac{1}{x_{α}} • [(1 − δ_{α}) • f] which is equivalent to f = s_{α}(f).

And similarly to (3.1) we obtain

(4.7) A_{α}^{∘2}(f) = κ_{α} • A_{α}(f) = A_{α}(κ_{α} • f), B_{α}^{∘2}(f) = κ_{α} • B_{α}(f) = B_{α}(κ_{α} • f),
A_{α} ∘ B_{α} = B_{α} ∘ A_{α} = 0.

We set A_{i} = A_{α_{i}} and B_{i} := B_{α_{i}} for the i-th simple root α_{i}. We set A_{I} = A_{i_{1}} ∘ … ∘ A_{i_{m}} and B_{I} = B_{i_{1}} ∘ … ∘ B_{i_{m}} for a sequence I = (i_{1}, …, i_{m}) with i_{j} ∈ {1, …, n}. The operators A_{I} and B_{I} are key ingredients in the proof that the natural pairing of Theorem 10.7 on the dual of the formal affine Demazure algebra is non-degenerate.

Lemma 4.6. For any sequence I, we have

A_{I^{rev}}(x_{Π}f_{e}) = ∑_{v∈W} v(x_{Π})a_{I,v}^{Y}f_{v} and B_{I^{rev}}(x_{Π}f_{e}) = ∑_{v∈W} v(x_{Π})a_{I,v}^{X}f_{v}.

Proof. We prove the first formula only. The second one is obtained using similar arguments. Let Y_{I^{rev}} = ∑_{v∈W} a_{I,v}^{′Y}δ_{v} and Y_{I} = ∑_{v∈W} a_{I,v}^{Y}δ_{v} as in Lemma 3.5. Then

A_{I^{rev}}(x_{Π}f_{e}) = Y_{I^{rev}} • x_{Π}f_{e} = ∑_{v∈W} x_{Π}(a_{I,v}^{′Y}δ_{v} • f_{e}) =

∑_{v∈W} x_{Π}(a_{I,v}^{′Y} • f_{v^{−1}}) = ∑_{v∈W} x_{Π}v^{−1}(a_{I,v}^{′Y})f_{v^{−1}} = ∑_{v∈W} x_{Π}v(a_{I,v^{−1}}^{′Y})f_{v}.

The formula then follows by Lemma 3.5. □

5. Push-pull operators and elements

Let us now introduce and study a key notion of the present paper, the notion of push-pull operators (resp. elements) on Q (resp. in Q_{W}) with respect to given coset representatives in parabolic quotients of the Weyl group.

Let (Σ, Λ) be a root datum with a chosen set of simple roots Π. Let Ξ ⊆ Π and let W_{Ξ} denote the subgroup of the Weyl group W of the root datum generated by simple reflections s_{α}, α ∈ Ξ. We thus have W_{∅} = {e} and W_{Π} = W. Let Σ_{Ξ} := {α ∈ Σ | s_{α} ∈ W_{Ξ}} and let Σ_{Ξ}^{+} := Σ_{Ξ} ∩ Σ^{+}, Σ_{Ξ}^{−} := Σ_{Ξ} ∩ Σ^{−} be subsets of positive and negative roots respectively.

Given subsets Ξ′ ⊆ Ξ of Π, let Σ_{Ξ/Ξ′}^{+} := Σ_{Ξ}^{+} \ Σ_{Ξ′}^{+} and Σ_{Ξ/Ξ′}^{−} := Σ_{Ξ}^{−} \ Σ_{Ξ′}^{−}. We define

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

Lemma 5.1. Given subsets Ξ′ ⊆ Ξ of Π we have

v(Σ_{Ξ/Ξ′}^{−}) = Σ_{Ξ/Ξ′}^{−} and v(Σ_{Ξ/Ξ′}^{+}) = Σ_{Ξ/Ξ′}^{+} for any v ∈ W_{Ξ′}.

Proof. We prove the first statement only, the second one can be proven similarly.

Since v acts faithfully on Σ_{Ξ}, it suffices to show that for any α ∈ Σ_{Ξ/Ξ′}^{−}, the root β := v(α) ∉ Σ_{Ξ′} and is negative. Indeed, if β ∈ Σ_{Ξ′}, then so is α = v^{−1}(β) (as v^{−1} ∈ W_{Ξ′}), which is impossible. On the other hand, if β is positive, then

β = v(α) ∈ vΣ_{Ξ}^{−} ∩ Σ_{Ξ}^{+} = vΣ_{Ξ′}^{−} ∩ Σ_{Ξ′}^{+},

where the latter equality follows from (3.2) and the fact that v ∈ W_{Ξ′}. So α = v^{−1}(β) ∈ Σ_{Ξ′}, a contradiction. □

Corollary 5.2. For any v ∈ W_{Ξ′}, we have v(x_{Ξ/Ξ′}) = x_{Ξ/Ξ′}.

Definition 5.3. Given a set of left coset representatives W_{Ξ/Ξ′} of W_{Ξ}/W_{Ξ′} we define a push-pull operator on Q with respect to W_{Ξ/Ξ′} by

C_{Ξ/Ξ′}(q) := ∑_{w∈W_{Ξ/Ξ′}} w(\frac{q}{x_{Ξ/Ξ′}} ).

and a push-pull element with respect to W_{Ξ/Ξ′} by

Y_{Ξ/Ξ′} := ( ∑_{w∈W_{Ξ/Ξ′}} δ_{w})\frac{1}{x_{Ξ/Ξ′}} .

We set C_{Ξ} := C_{Ξ/∅} and Y_{Ξ} := Y_{Ξ/∅} (so they do not depend on the choice of W_{Ξ/∅} = W_{Ξ} in these two special cases).

By definition, we have C_{Ξ/Ξ′}(q) = Y_{Ξ/Ξ′} · q, where Y_{Ξ/Ξ′} acts on q ∈ Q by (4.3). Also in the trivial case where Ξ = Ξ′, then x_{Ξ/Ξ} = 1, while C_{Ξ/Ξ} = id_{Q} and Y_{Ξ/Ξ} = 1 if we choose e as representative of the only coset. Observe that for Ξ = {α_{i}} we have W_{Ξ} = {e, s_{i}} and C_{Ξ} = C_{i} (resp. Y_{Ξ} = Y_{i}) is the push-pull operator (resp. element) introduced before and preserves S.

Example 5.4. For the formal group law F_{e} and the root datum A_{2}, we have x_{Π} = x_{−α_{1}}x_{−α_{2}}x_{−α_{1}−α_{2}} and

C_{Π}(1) = ∑_{w∈W} w(\frac{1}{x_{Π}} ) = μ_{1}(\frac{1}{x_{−α_{2}} x_{−α_{1}−α_{2}}} + \frac{1}{x_{−α_{1}} x_{α_{2}}} + \frac{1}{x_{α_{1}} x_{α_{1}+α_{2}}} ) = μ_{1}^{3} + μ_{1}μ_{2}.

Lemma 5.5. The operator C_{Ξ/Ξ′} restricted to Q^{W_{Ξ′}} is independent of the choices of representatives W_{Ξ/Ξ′} and it maps Q^{W_{Ξ′}} to Q^{W_{Ξ}}.

Proof. Since 1/x_{Ξ/Ξ′} ∈ Q^{W_{Ξ′}} by corollary 5.2, the independence statement is clear. The second part follows, since for any v ∈ W_{Ξ}, and for any set of coset representatives W_{Ξ/Ξ′}, the set vW_{Ξ/Ξ′} is again a set of coset representatives. □

Actually, we will see in Corollary 10.4 that the operator C_{Ξ} sends S to S^{W_{Ξ}}.

Remark 5.6. The formula for the operator C_{Ξ} (with Ξ′ = ∅) had appeared before in related contexts, namely, in discussions around the Becker-Gottlieb transfer for topological complex-oriented theories (see [BE90, (2.1)] and [GR12, §4.1]). The definition of the element Y_{Ξ/Ξ′} can be viewed as a generalized algebraic analogue of this formula.

Lemma 5.7 (Composition rule). Given subsets Ξ′′ ⊆ Ξ′ ⊆ Ξ of Π and given sets of representatives W_{Ξ/Ξ′} and W_{Ξ′/Ξ′′}, take W_{Ξ/Ξ′′} := {wv | w ∈ W_{Ξ/Ξ′}, v ∈ W_{Ξ′/Ξ′′}} as the set of representatives of W_{Ξ}/W_{Ξ′′}. Then

C_{Ξ/Ξ′} ∘ C_{Ξ′/Ξ′′} = C_{Ξ/Ξ′′} and Y_{Ξ/Ξ′}Y_{Ξ′/Ξ′′} = Y_{Ξ/Ξ′′}.

Proof. We prove the formula for Y ’s, the one for C’s follows since C acts as Y, and the composition of actions corresponds to multiplication. We have Y_{Ξ/Ξ′}Y_{Ξ′/Ξ′′} =

( ∑_{w∈W_{Ξ/Ξ′}} δ_{w} \frac{1}{x_{Ξ/Ξ′}})( ∑_{v∈W_{Ξ′/Ξ′′}} δ_{v} \frac{1}{x_{Ξ′/Ξ′′}}) = ∑_{w∈W_{Ξ/Ξ′} , v∈W_{Ξ′/Ξ′′}} δ_{wv} \frac{1}{v^{−1}(x_{Ξ/Ξ′})x_{Ξ′/Ξ′′}} .

By Corollary 5.2, we have v^{−1}(x_{Ξ/Ξ′}) = x_{Ξ/Ξ′}. Therefore, v^{−1}(x_{Ξ/Ξ′})x_{Ξ′/Ξ′′} = x_{Ξ/Ξ′}x_{Ξ′/Ξ′′} = x_{Ξ/Ξ′′}. We conclude by definition of W_{Ξ/Ξ′′}. □

The following lemma is clear from the defining formula of C_{Ξ/Ξ′}.

Lemma 5.8 (Projection formula). We have

C_{Ξ/Ξ′}(qq′) = q C_{Ξ/Ξ′}(q′) for any q ∈ (Q_{W})^{W_{Ξ}} and q′ ∈ (Q_{W})^{W_{Ξ′}} .

Lemma 5.9. Given a subset Ξ of Π and α ∈ Ξ we have (a) Y_{Ξ} = Y′Y_{α} = Y_{α}Y′′ for some Y′ and Y′′ ∈ Q_{W},

(b) Y_{Ξ}X_{α} = X_{α}Y_{Ξ} = 0, Y_{α}Y_{Ξ} = κ_{α}Y_{Ξ} and Y_{Ξ}Y_{α} = Y_{Ξ}κ_{α}.

Proof. (a) The first identity follows from Lemma 5.7 applied to Ξ′ = {α} (in this case Y′ = Y_{Ξ/Ξ′}).

For the second identity, let ^{α}W_{Ξ} be set of right coset representatives of W_{α}\W_{Ξ}, thus each w ∈ W_{Ξ} can be written uniquely either as w = s_{α}u or as w = u with u ∈ ^{α}W_{Ξ}. Then

Y_{Ξ} = ∑_{u∈^{α}W_{Ξ}}(1 + δ_{α})δ_{u} \frac{1}{x_{Ξ}} = ∑_{u∈^{α}W_{Ξ}}(1 + δ_{α})\frac{1}{x_{−α}} x_{−α}δ_{u} \frac{1}{x_{Ξ}}
= ∑_{u∈^{α}W_{Ξ}} Y_{α}x_{−α}δ_{u} \frac{1}{x_{Ξ}} = Y_{α} ∑_{u∈^{α}W_{Ξ}} δ_{w} \frac{w^{−1}(x_{−α})}{x_{Ξ}}.

(b) then follows from (a) and (3.1). □

6. The push-pull operators on the dual

We now introduce and study the push-pull operators on the dual of the twisted formal group algebra Q_{W}^{∗}.

For w ∈ W, we define f_{w}^{Ξ} := ∑_{v∈wW_{Ξ}} f_{v}. Observe that f_{w}^{Ξ} = f_{w′}^{Ξ} if and only if wW_{Ξ} = w′W_{Ξ}. Consider the subring of invariants (Q_{W}^{∗})^{W_{Ξ}} by means of the ‘•’-action of W_{Ξ} on Q_{W}^{∗} and fix a set of representatives W_{Π/Ξ}. By Lemma 4.2, we then have the following lemma:

Lemma 6.1. The set {f_{w}^{Ξ}}_{w∈W_{Π/Ξ}} forms a basis of (Q_{W}^{∗})^{W_{Ξ}} as a left Q-module, and f_{w}^{Ξ}f_{v}^{Ξ} = δ_{w,v}f_{v}^{Ξ} for any w, v ∈ W_{Π/Ξ}.

In other words, {f_{w}}_{w∈W_{Π/Ξ}} is a set of pairwise orthogonal projectors, and the direct sum of their images is (Q_{W}^{∗})^{W_{Ξ}}.

Definition 6.2. Given subsets Ξ′ ⊆ Ξ of Π and a set of representatives W_{Ξ/Ξ′} we define a Q-linear operator on Q_{W}^{∗} by

A_{Ξ/Ξ′}(f) := Y_{Ξ/Ξ′} • f, f ∈ Q_{W}^{∗} ,

and call it the push-pull operator with respect to W_{Ξ/Ξ′}. It is Q-linear since so is the ′•′-action. We set A_{Ξ} = A_{Ξ/∅}.

Lemma 5.7 immediately implies:

Lemma 6.3 (Composition rule). Given subsets Ξ′′ ⊆ Ξ′ ⊆ Ξ of Π and sets of representatives W_{Ξ/Ξ′} and W_{Ξ′/Ξ′′}, let W_{Ξ/Ξ′′} = {wv | w ∈ W_{Ξ/Ξ′}, v ∈ W_{Ξ′/Ξ′′}}, then we have A_{Ξ/Ξ′} ∘ A_{Ξ′/Ξ′′} = A_{Ξ/Ξ′′}.

Here is an analogue of Lemma 5.5.

Lemma 6.4. The operator A_{Ξ/Ξ′} restricted to (Q_{W}^{∗})^{W_{Ξ′}} is independent of the choices of representatives W_{Ξ/Ξ′} and it maps (Q_{W}^{∗})^{W_{Ξ′}} to (Q_{W}^{∗})^{W_{Ξ}}.

Proof. Let f ∈ (Q_{W}^{∗})^{W_{Ξ′}}. For any w ∈ W and v ∈ W_{Ξ′}, by Corollary 5.2, we have

(δ_{wv} \frac{1}{x_{Ξ/Ξ′}} ) • f = (δ_{w} \frac{1}{x_{Ξ/Ξ′}} δ_{v}) • f = (δ_{w} \frac{1}{x_{Ξ/Ξ′}} ) • δ_{v} • f = (δ_{w} \frac{1}{x_{Ξ/Ξ′}} ) • f.

which proves that the action on f of any factor δ_{w}(\frac{1}{x_{Ξ/Ξ′}}) in Y_{Ξ/Ξ′} is independent of the choice of the coset representative w.

Now if v ∈ W_{Ξ}, we have

v(A_{Ξ/Ξ′}(f)) = δ_{v} • Y_{Ξ/Ξ′} • f = (δ_{v}Y_{Ξ/Ξ′}) • f = A_{Ξ/Ξ′}(f)

where the last equality holds since δ_{v}Y_{Ξ/Ξ′} is again an operator Y_{Ξ/Ξ′} corresponding to the set of coset representatives vW_{Ξ/Ξ′} instead of W_{Ξ/Ξ′}. This proves the second claim. □

Lemma 6.5. We have

A_{Ξ/Ξ′}(f_{v}) = \frac{1}{v(x_{Ξ/Ξ′})} ∑_{w∈W_{Ξ/Ξ′}} f_{vw^{−1}} = \frac{1}{v(x_{Ξ/Ξ′})} ∑_{w∈W_{Ξ/Ξ′}} w(f_{v}).

In particular, we have A_{Ξ/Ξ′}(f_{v}^{Ξ′}) = \frac{1}{v(x_{Ξ/Ξ′})}f_{v}^{Ξ} and A_{Π/Ξ}(f_{v}^{Ξ}) = \frac{1}{v(x_{Π/Ξ})}1.

Proof. By Lemma 4.2 we obtain A_{Ξ/Ξ′}(f_{v}) =

= ( ∑_{w∈W_{Ξ/Ξ′}} δ_{w} \frac{1}{x_{Ξ/Ξ′}} ) • f_{v} = ∑_{w∈W_{Ξ/Ξ′}} δ_{w} • (\frac{1}{v(x_{Ξ/Ξ′})}f_{v}) = \frac{1}{v(x_{Ξ/Ξ′})} ∑_{w∈W_{Ξ/Ξ′}} f_{vw^{−1}}.

In particular

A_{Ξ/Ξ′}(f_{v}^{Ξ′}) = ∑_{w∈W_{Ξ′}} \frac{1}{vw(x_{Ξ/Ξ′})} ∑_{u∈W_{Ξ/Ξ′}} f_{vwu^{−1}} = \frac{1}{v(x_{Ξ/Ξ′})} ∑_{w∈W_{Ξ′}} ∑_{u∈W_{Ξ/Ξ′}} f_{vwu^{−1}}

= \frac{1}{v(x_{Ξ/Ξ′})} ∑_{w∈vW_{Ξ}} f_{w} = \frac{1}{v(x_{Ξ/Ξ′})}f_{v}^{Ξ}

where the second equality follows from Corollary 5.2. □

Together with Lemma 6.1 we therefore obtain:

Corollary 6.6. We have A_{Ξ/Ξ′}((Q_{W}^{∗})^{W_{Ξ′}}) = (Q_{W}^{∗})^{W_{Ξ}}.

Lemma 6.7 (Projection formula). We have

A_{Ξ/Ξ′}(ff′) = fA_{Ξ/Ξ′}(f′) for any f ∈ (Q_{W}^{∗})^{W_{Ξ}} and f′ ∈ (Q_{W}^{∗})^{W_{Ξ′}} .

Proof. Using (4.2) and Lemma 4.3, we compute

A_{Ξ/Ξ′}(ff′) = Y_{Ξ/Ξ′} • (ff′) = ( ∑_{w∈W_{Ξ/Ξ′}}δ_{w} \frac{1}{x_{Ξ/Ξ′}} ) • (ff′) = ∑_{w∈W_{Ξ/Ξ′}}δ_{w} • \frac{1}{x_{Ξ/Ξ′}} • (ff′)

= ∑_{w∈W_{Ξ/Ξ′}}δ_{w} • (f(\frac{1}{x_{Ξ/Ξ′}} • f′)) = ∑_{w∈W_{Ξ/Ξ′}}(δ_{w} • f)(δ_{w} • \frac{1}{x_{Ξ/Ξ′}} • f′)

= f ∑_{w∈W_{Ξ/Ξ′}}δ_{w} • \frac{1}{x_{Ξ/Ξ′}} • f′ = fA_{Ξ/Ξ′}(f′) □

Lemma 6.8. Given a sequence I in {1, …, n}, for any x, y ∈ S and f, f′ ∈ Q_{W}^{∗} we have

C_{Π}(Δ_{I}(x)y) = C_{Π}(xΔ_{I^{rev}}(y)) and A_{Π}(B_{I}(f)f′) = A_{Π}(fB_{I^{rev}}(f′)).

Proof. We prove the second formula only. The first one is obtained similarly. By Lemma 5.9.(b) we have Y_{Π}X_{α} = 0 for any α ∈ Π. By (4.5) we obtain

0 = A_{Π}(B_{α}(s_{α}(f)f′)) = A_{Π}(fB_{α}(f′) − B_{α}(f)f′).

Hence, A_{Π}(B_{α}(f)f′) = A_{Π}(fB_{α}(f′)). The formula then follows by iteration. □

Let {X_{I_{w}}^{∗}}_{w∈W} and {Y_{I_{w}}^{∗}}_{w∈W} be the Q-linear basis of Q_{W}^{∗} dual to {X_{I_{w}}}_{w∈W} and {Y_{I_{w}}}_{w∈W}, respectively, i.e. X_{I_{w}}^{∗}(X_{I_{v}}) = δ_{w,v} for w, v ∈ W. We have X_{I_{e}}^{∗} = 1. Indeed, from Lemma 3.1, we have δ_{v} = ∑_{w≤v} b_{v,w}^{X}X_{I_{w}} with b_{v,e}^{X} = 1. So for each v ∈ W we have X_{I_{e}}^{∗}(δ_{v}) = b_{v,e}^{X} = 1 = 1(δ_{v}).

Lemma 6.9. Let w_{0} be the longest element in W, of length N. We have

A_{Π}(X_{I_{w_{0}}}^{∗}) = (−1)^{N}1 and A_{Π}(Y_{I_{w_{0}}}^{∗}) = 1.

Proof. Consider the first formula. By Lemma 3.1 δ_{v} = ∑_{w≤v} b_{v,w}^{X}X_{I_{w}}, therefore X_{I_{w}}^{∗} = ∑_{v≥w} b_{v,w}^{X}f_{v}. Lemma 6.5 yields

A_{Π}(X_{I_{w}}^{∗}) = ∑_{v≥w} \frac{b_{v,w}^{X}}{v(x_{Π})}1.

If w = w_{0} is the longest element, then A_{Π}(X_{I_{w_{0}}}^{∗}) = \frac{b_{w_{0},w_{0}}^{X}}{w_{0}(x_{Π})}1. By Lemma 3.1 we have b_{w_{0},w_{0}}^{X} = (−1)^{N} ∏_{α∈Σ^{+}} x_{α} which by (3.2) equals to (−1)^{N}w_{0}(x_{Π}).

The second formula is obtained similarly using Lemma 3.2 instead. □

Definition 6.10. We define the characteristic map c: Q → Q_{W}^{∗} by q ↦ q • 1.

By the definition of the ‘•’ action, c is an R-algebra homomorphism given by c(q) = ∑_{w∈W} w(q)f_{w}. Note that c is Q_{W} -equivariant with respect to the action (4.3) and the ‘•’-action. Indeed, c(z · q) = (z · q) • 1 = z • (q • 1) = z • c(q). In particular, it is W -equivariant.

The following lemma provides an analogue of the push-pull formula of [CPZ, Theorem. 12.4].

Lemma 6.11. Given subsets Ξ′ ⊆ Ξ of Π, we have A_{Ξ/Ξ′} ∘ c = c ∘ C_{Ξ/Ξ′}.

Proof. By definition, we have

A_{Ξ/Ξ′}(c(q)) = Y_{Ξ/Ξ′} • c(q) = c(Y_{Ξ/Ξ′} · q) = c(C_{Ξ/Ξ′}(q)).  □

7. Another basis of the W_{Ξ}-invariant subring

Recall that {f_{w}^{Ξ}}_{w∈W_{Π/Ξ}} is a basis of the invariant subring (Q_{W}^{∗})^{W_{Ξ}}. In the present section we construct another basis {X_{I_{u}}^{∗}}_{u∈W^{Ξ}} of the subring (Q_{W}^{∗})^{W_{Ξ}}, which generalizes [KK90, Lemma 2.27].

Given a subset Ξ of Π we define

W^{Ξ} = {w ∈ W | ℓ(ws_{α}) > ℓ(w) for any α ∈ Ξ}.

Note that W^{Ξ} is a set of left coset representatives of W/W_{Ξ} such that each w ∈ W^{Ξ} is the unique representative of minimal length.

We will extensively use the following fact [Hu90, §1.10]:

(7.1)  For any w ∈ W there exist unique u ∈ W^{Ξ} and v ∈ W_{Ξ}
such that w = uv and ℓ(w) = ℓ(u) + ℓ(v).

Definition 7.1. Let Ξ be a subset of Π. We say that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible if for each w ∈ W and the unique factorization w = uv with u ∈ W^{Ξ} and v ∈ W_{Ξ}, ℓ(w) = ℓ(u) + ℓ(v) of (7.1) we have I_{w} = I_{u} ∪ I_{v}, i.e. I_{w} starts with I_{u} and ends by I_{v}.

Observe that there always exists a Ξ-compatible set of reduced sequences. Indeed, we can take any reduced sequence for w ∈ W^{Ξ} ∪ W_{Ξ} and then define I_{w} := I_{u} ∪ I_{v} for w = uv with u ∈ W^{Ξ} and v ∈ W_{Ξ}.

Lemma 7.2. For any z ∈ Q_{W}, we have X_{I_{e}}^{∗}(z) = z · 1, where e is the neutral element of W and z · 1 is defined in (4.3). In particular, for any sequence I with |I| ≥ 1, the coefficient of X_{I_{e}}^{∗}(X_{I}) = 0.

Proof. Since X_{α} · 1 = 0, for any sequence I with |I| ≥ 1, we have X_{I} · 1 = 0, and of course X_{I_{e}} · 1 = 1 · 1 = 1. Thus

z · 1 = ( ∑_{w∈W} X_{I_{w}}^{∗}(z)X_{I_{w}}) · 1 = X_{I_{e}}^{∗}(z).  □

It follows that if |I| ≥ 1 and we express X_{I} = ∑_{v∈W} q_{v}X_{I_{v}}, then q_{e} = 0.

Lemma 7.3. For any reduced sequence I of an element w and q ∈ Q we have

X_{I}q = ∑_{v≤w} ϕ_{I,v}(q)X_{I_{v}} for some ϕ_{I,I_{v}}(q) ∈ Q.

Proof. For any subsequence J of I (not necessarily reduced), we have w(J) ≤ w by [De77, Th. 1.1]. Thus, by developing all X_{i} = \frac{1}{x_{α_{i}}} (1 − δ_{α_{i}}), moving all coefficients to the left, and then using Lemma 3.1 and transitivity of the Bruhat order,

X_{I}q = ∑_{w≤v} ϕ̃_{I,w}(q)δ_{w} = ∑_{w≤v} ϕ_{I,w}(q)X_{I_{w}}

for some coefficients ϕ̃_{I,w}(q) and ϕ_{I,w}(q) ∈ Q. □

Theorem 7.4. Assume that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible. For any u ∈ W^{Ξ}, and for any sequence I of length at least 1 and in W_{Ξ} (i.e. α_{i} ∈ Ξ for any i appearing in the sequence I), we have

X_{I_{u}}^{∗}(zX_{I}) = 0 for all z ∈ Q_{W} .

Proof. Since {X_{I_{w}}}_{w∈W} is a basis of Q_{W}, we may assume that z = X_{I_{w}} for some w ∈ W. We proceed by induction on the length of w. First decompose X_{I} = ∑_{e<v∈W_{Ξ}} q_{v}X_{I_{v}} with q_{v} ∈ Q by Lemma 7.2.

When ℓ(w) = 0, we have X_{I_{w}} = X_{I_{e}} = 1. Since W_{Ξ}∩W^{Ξ} = {e}, for any v ∈ W_{Ξ}, v ≠ e, we have X_{I_{u}}^{∗}(X_{I_{v}}) = 0 so this case is clear.

Then, the induction step goes as follows: since the sequences are Ξ-compatible, we have X_{I_{w}}X_{I} = X_{I_{w′}} X_{I_{v′}} X_{I} = X_{I_{w′}} X_{I′} with w′ ∈ W^{Ξ}, v′ ∈ W_{ξ} and I′ ∈ W_{Ξ} with ℓ(I′) ≥ ℓ(I) ≥ 1. We can thus assume that w ∈ W^{Ξ}. Then, by Lemma 7.3,

X_{I_{w}}X_{I} = ∑_{v≠e} X_{I_{w}}q_{v}X_{I_{v}} = ∑_{w′≤w,v≠e} ϕ_{I_{w,w′}} (q_{v})X_{I_{w′}} X_{I_{v}}.

Now X_{I_{u}}^{∗}(X_{I_{w}}X_{I_{v}}) = X_{I_{u}}^{∗}(X_{I_{wv}}) = 0 since wv is not minimal (v ≠ e) so wv ≠ u. Applying X_{I_{u}}^{∗} to other terms in the above summation gives zero by induction. □

Remark 7.5. The proof will not work if we replace X’s by Y ’s, because constant terms appear.

Corollary 7.6. Assume that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible. The elements {X_{I_{u}}^{∗}}_{u∈W^{Ξ}} form a Q-module basis of (Q_{W}^{∗})^{W_{Ξ}}.

Proof. For every α_{i} ∈ Ξ we have

(δ_{i} • X_{I_{u}}^{∗})(z) = X_{I_{u}}^{∗}(zδ_{i}) = X_{I_{u}}^{∗}(z(1 − x_{i}X_{i})) = X_{I_{u}}^{∗}(z), z ∈ Q_{W} ,

where the last equality follows by Theorem 7.4. Therefore, X_{I_{u}}^{∗} is W_{Ξ}-invariant. Let σ ∈ (Q_{W}^{∗})^{W_{Ξ}}, i.e. for each α_{i} ∈ Ξ we have σ = s_{i}(σ) = δ_{i} • σ. Then

σ(zX_{i}) = σ(z \frac{1}{x_{α_{i}}} (1 − δ_{α_{i}})) = σ(z \frac{1}{x_{α_{i}}}) − (δ_{i} • σ)(z \frac{1}{x_{i}}) = (σ − δ_{i} • σ)(z \frac{1}{x_{i}}) = 0

for any z ∈ Q_{W}. Write σ = ∑_{w∈W} c_{w}X_{I_{w}}^{∗} for some c_{w} ∈ S. If w ∉ W^{Ξ}, then I_{w} ends by some i such that α_{i} ∈ Ξ which implies that

c_{w} = σ(X_{I_{w}}) = σ(X_{I_{w}\i}X_{i}) = 0,

where I_{w} \ i is the sequence obtained by deleting the last i in I_{w}. So σ is a linear combination of {X_{I_{u}}^{∗}}_{u∈W^{Ξ}}. □

Corollary 7.7. Assume that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible. Then we have b_{wv,u}^{X} = b_{w,u}^{X} for any v ∈ W_{Ξ}, u ∈ W^{Ξ} and w ∈ W, where b_{wv,u}^{X} are the coefficients of Lemma 3.1.

Proof. From Lemma 3.1 we have X_{I_{u}}^{∗} = ∑_{w≥u} b_{w,u}^{X}f_{w}. By Lemma 4.2 we obtain that v(X_{I_{u}}^{∗}) = ∑_{w≥u} b_{w,u}^{X}f_{wv^{−1}} for any v ∈ W_{Ξ}. Since X_{I_{u}}^{∗} is W_{Ξ}-invariant by Corollary 7.7 and {f_{w}} is a basis of Q_{W}^{∗}, this implies that b_{wv^{−1},u}^{X} = b_{w,u}^{X}. □

8. The formal affine Demazure algebra

In the present section we recall the definition and basic properties of the formal (affine) Demazure algebra D_{F} following [HMSZ], [CZZ] and [Zh13]. We show that for the special elliptic formal group law F_{e}, the formal Demazure algebra is related to the classical Iwahori-Hecke algebra.

Following [HMSZ], we define the formal affine Demazure algebra D_{F} to be the R-subalgebra of the twisted formal group algebra Q_{W} generated by elements of S and the Demazure elements X_{i} for all i ∈ {1, …, n}. By [CZZ, Lemma 5.8], D_{F} is also generated by S and all X_{α} for all α ∈ Σ. Since κ_{α} ∈ S, the algebra D_{F} is also generated by Y_{α}’s and elements of S. Finally, since δ_{α} = 1 − x_{α}X_{α}, all elements δ_{w} are in D_{F}, and D_{F} is a sub-S_{W} -module of Q_{W}, both on the left and on the right.

Remark 8.1. Since {X_{I_{w}}}_{w∈W} is a Q-linear basis of Q_{W}, restricting the action (4.3) of Q_{W} onto D_{F} we obtain an isomorphism between the algebra D_{F} and the R-subalgebra D(Λ)_{F} of End_{R}(S) generated by operators Δ_{α} (resp. C_{α}) for all α ∈ Σ, and multiplications by elements from S. This isomorphism maps X_{α} ↦ Δ_{α} and Y_{α} ↦ C_{α}. Therefore, for any identity or statement involving elements X_{α} or Y_{α} there is an equivalent identity or statement involving operators Δ_{α} or C_{α}.

According to [HMSZ, Theorem 6.14] (or [CZZ, 7.9] when the ring R is not necessarily a domain), in type A_{n}, the algebra D_{F} is generated by the Demazure elements X_{i}, i ∈ {1, …, n}, and multiplications by elements from S subject to the folowing relations:

(a) X_{i}^{2} = X_{i}

(b) X_{i}X_{j} = X_{j}X_{i} for |i − j| > 1,

(c) X_{i}X_{j}X_{i} − X_{j}X_{i}X_{j} = κ_{ij}(X_{j} − X_{i}) for |i − j| = 1 and

(d) X_{i}q = s_{i}(q)X_{i} + Δ_{i}(q),

Recall that the Iwahori-Hecke algebra H of the symmetric group S_{n+1} is an ℤ[t, t^{−1}]-algebra with generators T_{i}, i ∈ {1, …, n}, subject to the following relations:

(A) (T_{i} + t)(T_{i} − t^{−1}) = 0 or, equivalently, T_{i}^{2} = (t^{−1} − t)T_{i} + 1, (B) T_{i}T_{j} = T_{j}T_{i} for |i − j| > 1 and (C) T_{i}T_{j}T_{i} = T_{j}T_{i}T_{j} for |i − j| = 1.

(The T_{i}’s appearing in the definition of the Iwahori-Hecke algebra in [CG10, Def. 7.1.1] correspond to tT_{i} in our notation, where t = q^{−1/2}.)

Following [HMSZ, Def. 6.3] let D_{F} denote the R-subalgebra of D_{F} generated by the elements X_{i}, i ∈ {1, …, n}, only. By [HMSZ, Prop. 7.1], over R = ℂ, if F = F_{a} (resp. F = F_{m}), then D_{F} is isomorphic to the completion of the nil-Hecke algebra (resp. the 0-Hecke algebra) of Kostant-Kumar. The following observation provides another motivation for the study of formal (affine) Demazure algebras.

Let us consider the special elliptic formal group law of example 2.2 with coefficient μ_{1} = 1. Then its formal inverse is x/(x − 1), and since (1 + μ_{2}x_{i}x_{j})x_{i+j} = x_{i} + x_{j} − x_{i}x_{j}, the coefficient κ_{ij} of relation (c) is simply μ_{2}:

(8.1) κ_{ij} = \frac{1}{x_{i+j}x_{j}} − \frac{1}{x_{i+j}x_{−i}} − \frac{1}{x_{i}x_{j}} = \frac{x_{i}+x_{j}−x_{i}x_{j}−x_{i+j}}{x_{i}x_{j}x_{i+j}} = \frac{(1+μ_{2}x_{i}x_{j})x_{i+j}−x_{i+j}}{x_{i}x_{j}x_{i+j}} = μ_{2}

Proposition 8.2. Let F_{e} be a normalized (i.e. μ_{1} = 1) special elliptic formal group law over an integral domain R containing ℤ[t, t^{−1}], and let a, b ∈ R. Then the following are equivalent

(1) The assignment T_{i} ↦ aX_{i} + b, i ∈ {1, …, n}, defines an isomorphism of R-algebras H ⊗_{ℤ[t,t^{−1}]} R → D_{F}.

(2) We have a = t+t^{−1} or −t−t^{−1} and b = −t or t^{−1} respectively. Furthermore μ_{2}(t + t^{−1})^{2} = −1; in particular, the element t + t^{−1} is invertible in R.

Proof. Assume there is an isomorphism of R-algebras given by T_{i} ↦ aX_{i} + b. Then relations (b) and (B) are equivalent and relation (A) implies that

0 = (aX_{i} +b)^{2} +(t−t^{−1})(aX_{i} +b)−1 = [a^{2} +2ab+a(t−t^{−1})]X_{i} +b^{2} +b(t−t^{−1})−1.

Therefore b = −t or t^{−1} and a = t^{−1} − t − 2b = t + t^{−1} or −t − t^{−1} respectively, since 1 and X_{i} are S-linearly independent in D_{F} ⊆ D_{F}.

Relations (C) and (a) then imply

\begin{aligned}0 & = (aX_{i} + b)(aX_{j} + b)(aX_{i} + b) − (aX_{j} + b)(aX_{i} + b)(aX_{j} + b) \\ & = a^{3}(X_{i}X_{j}X_{i} − X_{j}X_{i}X_{j}) + (a^{2}b + ab^{2})(X_{i} − X_{j}). \end{aligned}

Therefore, by relation (c) and (8.1), we have a^{3}μ_{2} − a^{2}b − ab^{2} = 0 which implies that 0 = a^{2}μ_{2} − ab − b^{2} = (t + t^{−1})^{2}μ_{2} + 1.

Conversely, by substituting the values of a and b, it is easy to check that the assignment is well defined, essentially by the same computations. It is an isomorphism since a = ±(t + t^{−1}) is invertible in R. □

Remark 8.3. The isomorphism of Theorem 8.2 provides a presentation of the Iwahori-Hecke algebra with t + t^{−1} inverted in terms of the Demazure operators on the formal group algebra R[[Λ]]_{F_{e}}.

9. The algebraic restriction to the fixed locus

In the present section we define the algebraic counterpart of the restriction to T -fixed points of G/B.

Let S_{W} be the twisted group algebra of S and the group ring R[W], i.e. S_{W} = S ⊗_{R} R[W] as a R-module and the multiplication is defined by

(xδ_{w})(x′δ_{w′}) = xw(x′)δ_{ww′}, x, x′ ∈ S, w, w′ ∈ W.

The algebra S_{W} is a free S-module with basis {δ_{w}}_{w∈W}. Since S is Σ-regular, it injects into its localization Q. Therefore, S_{W} injects into Q_{W} via δ_{w} ↦ δ_{w}.

Since δ_{α} = 1 − x_{α}X_{α} for each α ∈ Σ, there is a natural inclusion of S-modules η: S_{W} ↪ D_{F}. By [CZZ, Prop. 7.7] the elements {X_{I_{w}}}_{w∈W} and, hence, {Y_{I_{w}}}_{w∈W} form a basis of D_{F} as a left S-module. Tensoring η by Q we obtain an isomorphism η_{Q}: Q_{W} →^{≃} Q ⊗_{S} D_{F}, because both are free Q-modules and their bases {X_{I_{w}}}_{w∈W} are mapped to each other. Observe that by definition D_{F} injects into Q ⊗_{S} D_{F} ≃ Q_{W}.

Consider the S-linear dual S_{W}^{⋆} = Hom_{S}(S_{W} , S). Since {δ_{w}}_{w∈W} is a basis for both S_{W} and Q_{W}, S_{W}^{⋆} can be identified with the free S-submodule of Q_{W}^{∗} with basis {f_{w}}_{w∈W} or, equivalently, with the subset {f ∈ Q_{W}^{∗} | f(S_{W}) ⊆ S}. Consider the S-linear dual D_{F}^{⋆} = Hom_{S}(D_{F} , S).

Definition 9.1. The induced map η^{⋆}: D_{F}^{⋆} → S_{W}^{⋆} (composition with η) will be called the algebraic restriction to the fixed locus, because of its geometric interpretation, given in the introduction.

Lemma 9.2. The map η^{⋆} is injective and its image in S_{W}^{⋆} ⊆ Q_{W}^{∗} = Q ⊗_{S} S_{W}^{⋆} coincides with the subset

{f ∈ Q_{W}^{∗} | f(D_{F}) ⊆ S}.

Proof. There is a commutative diagram

\xymatrix{D_{F}^{⋆} \ar[d] \ar[r]^{η^{⋆}} & S_{W}^{⋆} \ar[d] \\ Q ⊗_{S} D_{F}^{⋆} \ar[r]^{η_{Q}^{⋆}}_{≃} & Q ⊗_{S} S_{W}^{⋆}}

where the vertical maps are injective by freeness of the modules and because S injects into Q. The description for the image then follows from the fact that {X_{I_{w}}}_{w∈W} is a basis for both D_{F} and Q_{W}. □

Remark 9.3. Observe that η^{⋆} is not surjective, unless the root datum is trivial. Indeed, since X_{α} = \frac{1}{x_{α}} − \frac{1}{x_{α}} δ_{α}, we have f_{s_{α}}(X_{α}) = −\frac{1}{x_{α}} ∉ S, so f_{s_{α}} is not in the image of η^{∗}.

Since D_{F} is a subring of Q_{W}, the ‘•’-action of Q_{W} on Q_{W}^{∗} restricts to an S-linear action of D_{F} on D_{F}^{⋆} given by the same formula: (z′ • σ)(z) = σ(zz′) for σ ∈ D_{F}^{⋆} and z, z′ in D_{F}. Thus, the action of W on Q_{W}^{∗} restricts to an action on D_{F}.

By [CZZ, Theorem 9.2] the coproduct △ on Q_{W} restricts to a coproduct on D_{F}. Hence, the dual D_{F}^{⋆} becomes a subring of Q_{W}^{∗}.

10. The push-pull operators on D_{F}^{⋆}

In this section we restrict the push-pull operators onto the dual of the formal affine Demazure algebra D_{F}^{⋆}, and define a non-degenerate pairing on it.

Lemma 10.1. For any subset Ξ of Π we have Y_{Ξ} ∈ D_{F}.

Proof. The ring Q_{W} is functorial in the root datum (i.e. along morphisms of lattices that send roots to roots) and in the formal group law. This functoriality sends elements X_{α} (or Y_{α}) to themselves, so it restricts to a functoriality the subring D_{F}. We can therefore assume that the root datum is adjoint, and that the formal group law is the universal one over the Lazard ring, in which all integers are regular, since it is a polynomial ring over ℤ.

Consider the involution ι on Q_{W} given by qδ_{w} ↦ (−1)^{ℓ(w)}w^{−1}(q)δ_{w^{−1}}. It satisfies ι(zz′) = ι(z′)ι(z). Since ι(X_{α}) = Y_{−α}, it restricts to an anti-automorphism on D_{F}. Hence, it suffices to show that ι(Y_{Ξ}) ∈ D_{F}. By definition we have

ι(Y_{Ξ}) = ∑_{w∈W_{Ξ}}(−1)^{ℓ(w)} \frac{1}{x_{Ξ}} δ_{w^{−1}} = \frac{1}{x_{Ξ}} ∑_{w∈W_{Ξ}}(−1)^{ℓ(w)}δ_{w}.

Since the root datum is adjoint, D_{F} = {z ∈ Q_{W} | z · S ⊆ S} by [CZZ, Remark 7.8]. It therefore suffices to show that ι(Y_{Ξ}) · x ∈ S for any x ∈ S. We have

ι(Y_{Ξ}) · x = \frac{1}{x_{Ξ}} ∑_{w∈W_{Ξ}}(−1)^{ℓ(w)}w(x).

For any α ∈ Σ_{Ξ}^{−} let ^{α}W_{Ξ} = {w ∈ W_{Ξ} | ℓ(s_{α}w) > ℓ(w)}. Then W_{Ξ} = ^{α}W_{Ξ} ⨿ s_{α}^{α}W_{Ξ} and the sum

∑_{w∈W_{Ξ}}(−1)^{ℓ(w)}w(x) = ∑_{w∈^{α}W_{Ξ}}(−1)^{ℓ(w)}(w(x)−s_{α}(w(x))) = x_{α} ∑_{w∈^{α}W_{Ξ}}(−1)^{ℓ(w)}Δ_{α}(w(x))

is divisible by x_{α}. By using the next lemma recursively, we then conclude that x_{Ξ} divides the sum and thus that (ι(Y_{Ξ})) · x ∈ D_{F}. □

Lemma 10.2. Assume that all integers are regular in R, and that the root datum is adjoint. Let α and β be roots such that α ≠ ±β and let x′ ∈ S. Then if x_{α} | x_{β}x′, we have x_{α} | x′.

Proof. By [CZZ, Lemma 2.1], the root α can be completed as a basis (e_{1} = α, e_{2}, …, e_{n}) of the lattice, giving an R-algebra isomorphism ϕ : S → R[[x_{1}, …, x_{n}[[, sending x_{α} to x_{1}, by [CPZ, Cor 2.13]. Since β ≠ ±α, we have β = ∑_{i} n_{i}e_{i} with n_{i} ≠ 0 for some i ≠ 1 and ϕ(x_{β}) = ∑_{i} n_{i}x_{i} + z, where z ∈ I^{2}, the square of the augmentation ideal (generated by the variables). Since R[[x_{1}, …, x_{n}]]/(x_{1}) ≃ R[[x_{2}, …, x_{n}]], the result follows from the regularity of the class of x_{β} in that quotient, which in turn follows from [CZZ, Lemma 12.3]. □

Corollary 10.3. The operator Y_{Ξ} (resp. A_{Ξ}) restricted to S (resp. to D_{F}^{⋆}) defines an operator on S (resp. on D_{F}^{⋆}). Moreover, we have

Y_{Ξ}(S) ⊆ S^{W_{Ξ}} and A_{Ξ}(D_{F}^{⋆}) ⊆ (D_{F}^{⋆})^{W_{Ξ}}.

Proof. Here Y_{Ξ} acts on S ⊆ Q via (4.3). Since Y_{Ξ} ∈ D_{F} ⊆ {z ∈ Q_{W} | z · S ⊆ S} by [CZZ, Remark 7.8] and Y_{Ξ} · Q ⊆ (Q)^{W_{Ξ}}, the result follows.

As for A_{Ξ}, by Lemma 9.2 any f ∈ D_{F}^{⋆} has the property that f(D_{F}) ⊆ S. Therefore, (A_{Ξ}(f))(D_{F}) = (Y_{Ξ} • f)(D_{F}) = f(D_{F} Y_{Ξ}) ⊆ S, so A_{Ξ}(f) ∈ D_{F}^{⋆}. The result then follows by Lemma 6.7. □

Corollary 10.4. Suppose that the root datum has no irreducible component of type C_{n}^{sc} or that 2 is invertible in R. Then if |W_{Ξ′}| is regular in R, for any Ξ′ ⊆ Ξ ⊆ Π, we have

Y_{Ξ/Ξ′}(S^{W_{Ξ′}}) ⊆ S^{W_{Ξ}}.

Proof. Let x ∈ (S)^{W_{Ξ′}}, then |W_{Ξ′}| · x = ∑_{w∈W_{Ξ′}} w(x). So we have

|W_{Ξ′}| · Y_{Ξ/Ξ′}(x) = Y_{Ξ/Ξ′}(|W_{Ξ′}| · x) = ∑_{u∈W_{Ξ/Ξ′}} u(\frac{|W_{Ξ′}| · x}{x_{Ξ/Ξ′}})

= ∑_{u∈W_{Ξ/Ξ′}} ∑_{v∈W_{Ξ′}} uv(\frac{x}{x_{Ξ/Ξ′}}) = ∑_{w∈W_{Ξ}} w(\frac{xx_{Ξ′}}{x_{Ξ}} ) ∈ S^{W_{Ξ}}.

Thus |W_{Ξ′}| · Y_{Ξ/Ξ′}(x) ∈ S, which implies that Y_{Ξ/Ξ′}(x) ∈ S by [CZZ, Lemma 3.5]. Besides, it is fixed by W_{Ξ} by 5.5. □

For each v ∈ W define

f̃_{v} := x_{Π}f_{v} ∈ S_{W}^{⋆} , i.e. f̃_{v}( ∑_{w∈W} q_{w}δ_{w}) = x_{Π}q_{v}.

Lemma 10.5. For any v ∈ W we have f̃_{v} ∈ D_{F}^{⋆}.

Proof. We first show that f̃_{w_{0}} ∈ D_{F}^{⋆}. By Lemma 3.1, we have

X_{I_{v}} = ∑_{w≤v} a_{v,w}^{X}δ_{w}, where a_{v,v}^{X} = ∏_{α∈(vΣ^{−})∩Σ^{+}}(−\frac{1}{x_{α}}).

Since w_{0}Σ^{−} = Σ^{+}, we have

f̃_{w_{0}}(X_{I_{v}}) = f̃_{w_{0}}(∑_{w≤v} a_{v,w}^{X}δ_{w}) = (x_{Π}a_{w_{0},w_{0}}^{X})δ_{v,w_{0}} = ∏_{α∈Σ^{+}}(−\frac{x_{−α}}{x_{α}})δ_{v,w_{0}} ∈ S.

By Lemma 9.2, we have f̃_{w_{0}} ∈ D_{F}^{⋆}. For an arbitrary v ∈ W, by Lemma 4.2, we obtain

f̃_{v} = x_{Π}f_{w_{0}w_{0}^{−1}v} = v^{−1}w_{0}(x_{Π}f_{w_{0}}) = v^{−1}w_{0}(f̃_{w_{0}}) ∈ D_{F}^{⋆} .  □

Corollary 10.6. For any X ∈ D_{F} we have x_{Π}X ∈ S_{W}.

Proof. It suffices to show that for any sequence I_{v}, x_{Π}X_{I_{v}} ∈ S_{W}. And indeed,

x_{Π}X_{I_{v}} = x_{Π}(∑_{w≤v} a_{v,w}^{X}δ_{w}) = ∑_{w≤v}(x_{Π}a_{v,w}^{X})δ_{w} = ∑_{w≤v} f̃_{w}(X_{I_{v}})δ_{w} ∈ S_{W} .  □

Theorem 10.7. For any v, w ∈ W, we have

A_{Π}(Y_{I_{v}}^{∗}A_{I_{w}^{rev}}(x_{Π}f_{e})) = δ_{w,v}1 = A_{Π}(X_{I_{v}}^{∗}B_{I_{w}^{rev}}(x_{Π}f_{e})).

Consequently, the pairing

A_{Π} : D_{F}^{⋆} × D_{F}^{⋆} → (D_{F}^{⋆})^{W} ≅ S, (σ, σ′) ↦ A_{Π}(σσ′)

is non-degenerate and satisfies that {A_{I_{w}^{rev}}(x_{Π}f_{e})}_{w∈W} is dual to the basis {Y_{I_{v}}^{∗}}_{v∈W}, and {B_{I_{w}^{rev}}(x_{Π}f_{e})}_{w∈W} is dual to the basis {X_{I_{v}}^{∗}}_{v∈W}.

Proof. We prove the first identity. The second identity is obtained similarly.

Let Y_{I_{w}^{rev}} = ∑_{v∈W} a′_{w,v}δ_{v} and Y_{I_{w}} = ∑_{v∈W} a_{w,v}δ_{v}. Let δ_{w} = ∑_{v∈W} b_{w,v}Y_{I_{v}} so that ∑_{v∈W} a_{w,v}b_{v,u} = δ_{w,u} and Y_{I_{u}}^{∗} = ∑_{v∈W} b_{v,u}f_{v}.

Combining the formula of Lemma 4.6 with the formula A_{Π}(f_{v}) = \frac{1}{v(x_{Π})}1 of Lemma 6.5, we obtain

A_{Π}(Y_{I_{u}}^{∗}A_{I_{w}^{rev}}(x_{Π}f_{e})) = ∑_{v∈W} b_{v,u}v(x_{Π})a_{w,v}A_{Π}(f_{v}) = ∑_{v∈W} b_{v,u}a_{w,v}1 = δ_{w,u}1. □

The characteristic map c introduced in 6.10 restricts to a homomorphism of Ralgebras c_{S}: S → D_{F}^{⋆} which maps x ∈ S to the evaluation at x using the action of D_{F} on S, that is c_{S}(x)(z) = z · x for z ∈ D_{F}. In particular, we have

c_{S}(x)(X_{I}) = Δ_{I}(x) and c_{S}(x)(δ_{w}) = w(x), w ∈ W.

Lemma 10.8. For any sequence I and x ∈ S, we have

A_{Π}(c_{S}(x)A_{I^{rev}}(x_{Π}f_{e})) = C_{I}(x)1 and A_{Π}(c_{S}(x)B_{I^{rev}}(x_{Π}f_{e})) = Δ_{I}(x)1.

Proof. We prove the first formula only. The second formula is obtained similarly. Let Y_{I} = ∑_{v∈W} a_{I,v}^{Y}δ_{v}. Since c_{S}(x) = ∑_{v∈W} v(x)f_{v}, by Lemma 4.6, we get

A_{Π}(c_{S}(x)A_{I^{rev}}(x_{Π}f_{e})) = A_{Π}( ∑_{v∈W}v(x)v(x_{Π})a_{I,v}^{Y}f_{v}) = ∑_{v∈W} a_{I,v}^{Y}v(x)1 = C_{I}(x)1. □

11. The non-degenerate pairing on the W_{Ξ}-invariant subring

In this last section, we construct a non-degenerate pairing on the subring of invariants (D_{F}^{⋆})^{W_{Ξ}}. Using this pairing we provide several S-module bases of (D_{F}^{⋆})^{W_{Ξ}}.

Lemma 11.1. Suppose that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible, then the set {X_{I_{u}}^{∗}}_{u∈W^{Ξ}} is a basis of (D_{F}^{⋆})^{W_{Ξ}}.

Proof. This follows immediately from Lemma 6.1 and the identity (Q_{W}^{⋆})^{W_{Ξ}} ∩D_{F}^{⋆} = (D_{F}^{⋆})^{W_{Ξ}}. □

Given representatives u, u′ ∈ W^{Ξ} we set

d_{u,u′}^{Y} := u′(x_{Π/Ξ}) ∑_{v∈W_{Ξ}} a_{u,u′v}^{Y}, d_{u,u′}^{X} := u′(x_{Π/Ξ}) ∑_{v∈W_{Ξ}} a_{u,u′v}^{X}, η_{Ξ} = ∏_{w∈W^{Ξ}} w(x_{Π/Ξ})

where a_{w,v}^{X} and a_{w,v}^{Y} are the coefficients introduced in Lemma 3.1 and 3.2.

Lemma 11.2. For any u ∈ W^{Ξ} we have

A_{Ξ}(A_{I_{u}^{rev}}(x_{Π}f_{e})) = ∑_{u′∈W^{Ξ}} d_{u,u′}^{Y}f_{u′}^{Ξ}, A_{Ξ}(B_{I_{u}^{rev}}(x_{Π}f_{e})) = ∑_{u′∈W^{Ξ}} d_{u,u′}^{X}f_{u′}^{Ξ}.

Proof. We prove the first formula only. The second formula follows similarly. By Lemma 4.6 and 6.5 we obtain

A_{Ξ}(A_{I_{u}^{rev}}(x_{Π}f_{e})) = A_{Ξ}( ∑_{w∈W} w(x_{Π})a_{u,w}^{Y}f_{w}) = ∑_{w∈W} w(x_{Π/Ξ})a_{u,w}^{Y}f_{w}^{Ξ} =

by (7.1), representing w = u′v, and Lemma 5.1 we get

= ∑_{u′∈W^{Ξ}, v∈W_{Ξ}} u′v(x_{Π/Ξ})a_{u,u′v}^{Y}f_{u′v}^{Ξ} = ∑_{u′∈W^{Ξ}, v∈W_{Ξ}} u′(x_{Π/Ξ})a_{u,u′v}^{Y}f_{u′}^{Ξ}.  □

Theorem 11.3. Assume that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible, then the sets {A_{Ξ}(A_{I_{u}^{rev}}(x_{Π}f_{e}))}_{u∈W^{Ξ}} and {A_{Ξ}(B_{I_{u}^{rev}}(x_{Π}f_{e}))}_{u∈W^{Ξ}} are S-module bases of (D_{F}^{⋆})^{W_{Ξ}}.

Proof. Observe that by Corollary 10.3 our sets are in the S-module (D_{F}^{⋆})^{W_{Ξ}}. To show that they are bases, it suffices to show that the respective matrices M_{Ξ}^{Y} and M_{Ξ}^{X} expressing them on the basis {X_{I_{u}}^{∗}}_{u∈W^{Ξ}} of Lemma 11.1 have invertible determinants (in S).

If u′ ∈ W^{Ξ} and v ∈ W_{Ξ}, we have u′ ≤ u′v where the equality holds if and only if v = e. By Lemma 3.2, we get a_{u,u′v}^{Y} = 0 unless u′ ≤ u and a_{u,uv}^{Y} = 0 if v ≠ e. This implies that d_{u,u′}^{Y} = 0 unless u′ ≤ u, and that

d_{u,u}^{Y} = u(x_{Π/Ξ}) ∑_{v∈W_{Ξ}} a_{u,uv}^{Y} = u(x_{Π/Ξ})a_{u,u}^{Y} = u(x_{Π/Ξ})\frac{1}{b_{u,u}^{Y}} .

Hence, the matrix D_{Ξ}^{Y} := (d_{u,u′}^{Y})_{u,u′∈W^{Ξ}} is lower triangular with determinant η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{b_{u,u}^{Y}}. Similarly, the matrix D_{Ξ}^{X} := (d_{u,u′}^{X})_{u,u′∈W^{Ξ}} is lower triangular

with determinant η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{b_{u,u}^{X}}.

On the other hand, for u ∈ W^{Ξ}, we have

X_{I_{u}}^{∗} = ∑_{w∈W} b_{w,u}^{X}f_{w} = ∑_{u′∈W^{Ξ}} ∑_{v∈W_{Ξ}} b_{u′v,u}^{X}f_{u′v}.

By Corollary 7.7, and because X_{I_{u}}^{∗} is fixed by W_{Ξ}, we have b_{u′v,u}^{X} = b_{u′,u}^{X}. Therefore,

X_{I_{u}}^{∗} = ∑_{u′∈W^{Ξ}} b_{u′,u}^{X} ∑_{v} f_{u′v} = ∑_{u′∈W^{Ξ}} b_{u′,u}^{X}f_{u′}^{Ξ}.

By Lemma 3.1, b_{u′,u}^{X} = 0 unless u′ ≥ u, so the matrix E_{Ξ}^{X} := {b_{u′,u}^{X}}_{u′,u∈W^{Ξ}} is lower triangular with determinant ∏_{u∈W^{Ξ}} b_{u,u}^{X} = ∏_{u∈W^{Ξ}}(−1)^{ℓ(u)}b_{u,u}^{Y}.

The matrix M_{Ξ}^{X} = (E_{Ξ}^{X})^{−1}D_{Ξ}^{X} has determinant

η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{(b_{u,u}^{X})^{2}} = η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{(b_{u,u}^{Y})^{2}}

which is invertible in S by Lemma 11.5 below. Since the determinant of M_{Ξ}^{Y} = (E_{Ξ}^{X})^{−1}D_{Ξ}^{Y} differs by sign only, it is invertible as well. □

Recall the definition of Σ_{Ξ} from the beginning of section 5, and let w_{0}^{Ξ} be the longest element of W_{Ξ}.

Lemma 11.4. For any w ∈ W_{Ξ}, we have b_{w,w}^{Y}b_{ww_{0}^{Ξ},ww_{0}^{Ξ}}^{Y} = w_{0}^{Ξ}(x_{Ξ}). In particular, if Ξ = Π we have b_{w,w}^{Y}b_{ww_{0},ww_{0}}^{Y} = w_{0}(x_{Π}).

Proof. Recall from Lemma 3.2 that b_{w,w}^{Y} = ∏_{wΣ^{−}∩Σ^{+}} x_{α}. By (3.2), it also equals ∏_{wΣ_{Ξ}^{−}∩Σ_{Ξ}^{+}} x_{α}. Since w_{0}^{Ξ}Σ_{Ξ}^{−} = Σ_{Ξ}^{+}, we have ww_{0}^{Ξ}Σ_{Ξ}^{−} ∩ Σ_{Ξ}^{+} = wΣ_{Ξ}^{+} ∩ Σ_{Ξ}^{+}. Moreover,

(wΣ_{Ξ}^{−} ∩ Σ_{Ξ}^{+}) ∩ (wΣ_{Ξ}^{+} ∩ Σ_{Ξ}^{+}) ⊂ wΣ_{Ξ}^{−} ∩ wΣ_{Ξ}^{+} = w(Σ_{Ξ}^{−} ∩ Σ_{Ξ}^{+}) = ∅

and their union is Σ_{Ξ}^{+}. □

Lemma 11.5. For any Ξ ⊂ Π the product η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{(b_{u,u}^{Y})^{2}} is an invertible element in S.

Proof. We already know that this product is in S, since it is the determinant of the matrix M_{Ξ}^{X} whose coefficients are in S. Consider the R-linear involution u ↦ ū on S = R[[Λ]]_{F} induced by λ ↦ −λ, λ ∈ Λ. Observe that it is W -equivariant.

Set b_{v} := b_{v,v}^{Y}. For any α ∈ Ξ, we have

x_{Ξ} = s_{α}(x_{Ξ})x_{−α}x_{α}^{−1} = s_{α}(x_{Ξ})b̄_{s_{α}}b_{s_{α}}^{−1}

and, therefore, by induction x_{Ξ} = w(x_{Ξ})b̄_{v}b_{v}^{−1} for any v ∈ W_{Ξ}. In particular, x_{Π} = w(x_{Π})b̄_{w}b_{w}^{−1} for any w ∈ W. Then

x_{Ξ}^{|W_{Ξ}|} = ∏_{v∈W_{Ξ}} v(x_{Ξ})b̄_{v}b_{v}^{−1} and x_{Π}^{|W |} = ∏_{w∈W} w(x_{Π})b̄_{w}b_{w}^{−1}.

If w = uv with ℓ(w) = ℓ(u) + ℓ(v), by (3.2), we see that

wΣ^{−} ∩ Σ^{+} = (uΣ^{−} ∩ Σ^{+}) ⊔ u(vΣ^{−} ∩ Σ^{+}),

so b_{uv} = b_{u}u(b_{v}) and b̄_{uv} = b̄_{u}u(b̄_{v}). Hence

x_{Π}^{|W |} = ∏_{w∈W} w(x_{Π})b̄_{w}b_{w}^{−1} = ∏_{u∈W^{Ξ}} ∏_{v∈W_{Ξ}} uv(x_{Π/Ξ}x_{Ξ})b̄_{uv}b_{uv}^{−1}

=^{5.2} ∏_{u∈W^{Ξ}} u(x_{Π/Ξ}^{|W_{Ξ}|}) ∏_{v∈W_{Ξ}} uv(x_{Ξ})b̄_{u}u(b̄_{v})b_{u}^{−1}u(b_{v}^{−1})

= η_{Ξ}^{|W_{Ξ}|} ∏_{u∈W^{Ξ}} (b̄_{u}b_{u}^{−1})^{|W_{Ξ}|}u( ∏_{v∈W_{Ξ}} v(x_{Ξ})b̄_{v}b_{v}^{−1})

= η_{Ξ}^{|W_{Ξ}|} ∏_{u∈W^{Ξ}} (b̄_{u}b_{u}^{−1})^{|W_{Ξ}|}u(x_{Ξ})^{|W_{Ξ}|}

(11.1)

On the other hand, by Lemma 11.4,

x̄_{Ξ}^{|W_{Ξ}|} = w_{0}^{Ξ}(x_{Ξ})^{|W_{Ξ}|} = ∏_{v∈W_{Ξ}} b_{v}b_{vw_{0}^{Ξ}} = ∏_{v∈W_{Ξ}} b_{v}^{2}

and, in particular, x̄_{Ξ}^{|W |} = ∏_{w∈W} b_{w}^{2}. So, we obtain

x̄_{Π}^{|W |} = ∏_{w∈W} b_{w}^{2} = ∏_{u∈W^{Ξ}} ∏_{v∈W_{Ξ}} b_{uv}^{2} = ∏_{u∈W^{Ξ}} ∏_{v∈W_{Ξ}} b_{u}^{2}u(b_{v}^{2})

= ( ∏_{u∈W^{Ξ}} b_{u}^{2|W_{Ξ}|})( ∏_{u∈W^{Ξ}} u( ∏_{v∈W_{Ξ}} b_{v}^{2})) = ( ∏_{u∈W^{Ξ}} b_{u}^{2})^{|W_{Ξ}|}( ∏_{u∈W^{Ξ}} u(x̄_{Ξ}))^{|W_{Ξ}|}.

Combining this with equation (11.1), we obtain

(η_{Ξ}^{−1} ∏_{u∈W^{Ξ}} b_{u}^{2})^{|W_{Ξ}|} = x̄_{Π}^{|W |}x_{Π}^{−|W |}( ∏_{u∈W^{Ξ}} u(x̄_{Ξ}^{−1}x_{Ξ})b̄_{u}b_{u}^{−1})^{|W_{Ξ}|}

which is an element of S, since it is a product of elements of the form x_{α}x_{−α}^{−1} ∈ S. Therefore η_{Ξ} ∏_{u∈W^{Ξ}} \frac{1}{b_{u}^{2}} is invertible, since so is its |W_{Ξ}|-th power. □

Corollary 11.6. Given Ξ′ ⊆ Ξ ⊆ Π we have A_{Ξ}(D_{F}^{⋆}) = (D_{F}^{⋆})^{W_{Ξ}}. For any set of coset representatives W_{Ξ/Ξ′} the operator A_{Ξ/Ξ′} induces a surjection (D_{F}^{⋆})^{W′_{Ξ}} → (D_{F}^{⋆})^{W_{Ξ}} (independent of the choices of W_{Ξ/Ξ′} by Lemma 6.4).

Proof. By Corollary 10.3 and Theorem 11.3, we obtain the first part. To prove the second part, let σ ∈ (D_{F}^{⋆})^{W_{Ξ′}}. By the first part, there exists σ′ ∈ D_{F}^{⋆} such that σ = A_{Ξ′}(σ′), so by Lemma 6.3 we have

A_{Ξ/Ξ′}(σ) = A_{Ξ/Ξ′}(A_{Ξ′}(σ′)) = A_{Ξ}(σ′) ∈ (D_{F}^{⋆})^{W_{Ξ}}.

Hence, A_{Ξ/Ξ′} restricts to A_{Ξ/Ξ′}: (D_{F}^{⋆})^{W_{Ξ′}} → (D_{F}^{⋆})^{W_{Ξ}}. Since A_{Ξ}(D_{F}^{⋆}) = (D_{F}^{⋆})^{W_{Ξ}}, we also have A_{Ξ/Ξ′}((D_{F}^{⋆})^{W_{Ξ′}}) = (D_{F}^{⋆})^{W_{Ξ}}. □

Theorem 11.7. Assume that the set of reduced sequences {I_{w}}_{w∈W} is Ξ-compatible. If u ∈ W^{Ξ}, then A_{Π/Ξ}(X_{I_{u}}^{∗}A_{Ξ}(B_{I_{w}}^{rev}(x_{Π}f_{e}))) = δ_{w,u}1. Consequently, the pairing

(D_{F}^{⋆})^{W_{Ξ}} × (D_{F}^{⋆})^{W_{Ξ}} → (D_{F}^{⋆})^{W} ≅ S, (σ, σ′) ↦ A_{Π/Ξ}(σσ′)

is non-degenerate, and {A_{Ξ}(B_{I_{u}^{rev}}(x_{Π}f_{e}))}_{u∈W^{Ξ}}, {X_{I_{u}}^{∗}}_{u∈W^{Ξ}} are dual S-bases of (D_{F}^{⋆})^{W_{Ξ}}.

Proof. By Corollary 11.6, the pairing is well-defined (i.e. it does map into S). By Lemma 6.7, Lemma 6.3 and Theorem 10.7, we obtain

A_{Π/Ξ}(X_{I_{u}}^{∗}A_{Ξ}(B_{I_{w}^{rev}}(x_{Π}f_{e}))) = A_{Π/Ξ}(A_{Ξ}(X_{I_{u}}^{∗}B_{I_{w}^{rev}}(x_{Π}f_{e})))
= A_{Π}(X_{I_{u}}^{∗}B_{I_{w}^{rev}}(x_{Π}f_{e})) = δ_{w,u}1.  □

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Baptiste Calmès, Université d’Artois, Laboratoire de Mathématiques de Lens, France

   E-mail address: baptiste.calmes@univ-artois.fr

Kirill Zainoulline, Department of Mathematics and Statistics, University of Ottawa, Canada

   E-mail address: kirill@uottawa.ca

Changlong Zhong, Department of Mathematical and Statistical Sciences, University of Alberta, Canada

   E-mail address: zhongusc@gmail.com