A COPRODUCT STRUCTURE ON THE FORMAL AFFINE DEMAZURE ALGEBRA

BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG

Abstract. In the present paper we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T -equivariant oriented cohomology of the variety of complete flags.

Contents

1. Formal group algebras  3
2. Root systems and regular elements  4
3. Localized formal group algebras  6
4. Formal Demazure operators  8
5. Twisted formal group algebra and Demazure elements  10
6. Invariant presentation of the formal affine Demazure algebra  12
7. Presentation in terms of generators and relations  13
8. Coproduct on the twisted formal group algebra  16
9. Coproduct on the formal affine Demazure algebra  19
10. Coproduct on the algebra of Demazure operators  21
11. The dual of a formal affine Demazure algebra  23
12. Appendix  26
References  27

In [KK86], [KK90] and [Ku02] Kostant and Kumar introduced the language of (nil-)Hecke rings and described in a purely algebraic way equivariant cohomology and K-theory of flag varieties. In particular, for a given simply connected semisimple split linear algebraic group G with a maximal split torus T and Borel subgroup B ⊃ T, they established two isomorphisms (see [KK90, Thm. 4.4], [Ku02, Cor. 11.3.17])

S(Λ) ⊗_{S(Λ)^{W}} S(Λ) ⟶^{≃} H_{T} (G/B; ℚ) and ℤ[Λ] ⊗_{ℤ[Λ]^{W}} ℤ[Λ] ⟶^{≃} K_{T} (G/B),

2010 Mathematics Subject Classification. 20C08, 14F43.

Key words and phrases. oriented cohomology, formal group law, Demazure operator, nil Hecke ring, algebraic cobordism, Schubert calculus.

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). The last author was supported by the Fields Institute and the NSERC grant of the second author.

where S(Λ) is the symmetric algebra of the group of characters Λ of T, ℤ[Λ] is the integral group ring, S(Λ)^{W} and ℤ[Λ]^{W} denote the respective subrings of invariants under the action of the Weyl group W, and H_{T} (G/B, ℚ), K_{T} (G/B) are the T - equivariant cohomology and the K-theory of the variety of Borel subgroups G/B, respectively.

More precisely, they constructed algebraic models Λ and Ψ of the mentioned equivariant theories, dual respectively to the nil-Hecke and Hecke rings, together with explicit maps inducing the ring isomorphisms

(*)  S(Λ) ⊗_{S(Λ)^{W}} S(Λ) ⟶^{≃} Λ and ℤ[Λ] ⊗_{ℤ[Λ]^{W}} ℤ[Λ] ⟶^{≃} Ψ,

where the product structure on Λ and Ψ is induced by the coproduct structure on the (nil-)Hecke ring. Indeed, they showed that all Hecke-type (Demazure, BGG) operators on S(Λ) and ℤ[Λ] actually sit in a bigger but simpler ring in the so called twisted group algebra Q_{W} and, hence, they defined the respective coproduct to be the restriction of a natural one given on Q_{W}.

One goal of the present paper is to generalize their construction of the duals Λ and Ψ and of the coproduct for arbitrary one-dimensional commutative formal group law F (e.g. additive for Λ, multiplicative for Ψ, or universal) over a coefficient ring R. Namely, we define the coproduct on the formal affine Demazure algebra D_{F}, already introduced in [HMSZ] as a generalization of the nil-Hecke ring, by restricting the natural coproduct given on a larger formal twisted group algebra.

As an application we provide a complete description of the algebra D_{F} in terms of generators and relations, hence, generalizing [HMSZ, Thm. 5.14]. Moreover, we show (see Theorem 11.4) that after inverting the torsion index 𝔱 of G or, more generally, under the assumption that the characteristic map is surjective, there is an isomorphism

R[[Λ]]_{F} ⊗_{R[[Λ]]_{F}^{W}} R[[Λ]]_{F} ⟶^{≃} D_{F}^{∗} ,

where R[[Λ]]_{F} is the formal group algebra introduced in [CPZ, §2] and D_{F}^{∗} is the dual of D_{F}. Note that specializing to the additive and multiplicative periodic formal group laws (see Example 1.1) we obtain the isomorphisms (*). Also note that the algebra D_{F}^{∗} is defined under very few restrictions on the base ring R, e.g. 2𝔱 is not a zero divisor in R, hence allowing us to consider various formal group laws over finite fields as well.

According to Levine-Morel [LM07], to any algebraic oriented cohomology theory h(−) one can associate a formal group law F over the coefficient ring R = h(pt); moreover, there is a universal algebraic oriented cohomology theory Ω(−), called algebraic cobordism, corresponding to the universal formal group law U over the Lazard ring 𝕃 = Ω(pt). The associated T -equivariant theory, denoted by Ω_{T} (G/B), has been intensively studied during the last years by Deshpande, Heller, Kiritchenko, Krishna and Malagon-Lopez. In particular, using comparison results and the Atiyah-Hirzebruch spectral sequence for complex cobordism it was shown [KiKr, Thm. 5.1] that after inverting the torsion index the ring Ω_{T} (G/B) can be identified with 𝕃[Λ]_{U} ⊗_{𝕃[Λ]_{U}^{W}} 𝕃[Λ]_{U}. We also refer to [HHH] for similar results concerning T -equivariant topological complex oriented theories and to [GR12] for further discussion about the Borel model for Ω_{T} (G/B). In this context, our (Hecketype) algebra D_{F}^{∗} can be viewed as an algebraic model for the algebraic oriented T -equivariant cohomology h_{T} (G/B).

This paper is based on the techniques developed in [CPZ] and especially in [HMSZ] devoted to the unification of various existing geometric approaches to Hecke-type algebras. It is organized as follows:

In section 1 we recall basic properties of the formal group algebra R[[Λ]]_{F} following [CPZ, §2]. In section 2 we discuss the regularity of elements of R[[Λ]]_{F} corresponding to roots. In section 3 we study properties of the localization of R[[Λ]]_{F}. In section 4 we recall the definition of formal Demazure operators. Section 5 is devoted to the twisted formal group algebra Q_{W} and the formal affine Demazure algebra D_{F}. There, we provide an important formula (Lemma 5.4) expressing a product of Demazure elements in terms of the canonical basis of Q_{W}. We finish our preparation by relating the formal affine Demazure algebra to the subalgebra D̃_{F} of Q_{W} stabilizing R[[Λ]]_{F} (section 6) and we show that they coincide in most cases.

In section 7 we describe the formal affine Demazure algebra D_{F} in terms of generators and relations, hence, generalizing [HMSZ, Thm. 5.14]. We also establish an isomorphism (Theorem 7.9) between D_{F} and the algebra of Demazure operators on R[[Λ]]_{F}. In key sections 8, 9 and 10 we introduce and study the coproduct structure on D_{F}. For instance, we show (section 10) that the coproduct on D_{F} induces the coproduct on the cohomology ring considered in [CPZ, §7]. In section 11 we apply the obtained results to construct the dual D_{F}^{∗} and to prove Theorem 11.4.

Acknowledgments: The first author would like to thank Lina and René Payet for their hospitality; most of his work on the present article has been done during a stay at their home. The first and the second authors are grateful to the Banff International Research Station for its hospitality in October 2012 which lead to a successful collaboration on the subject of the paper.

1. Formal group algebras

We recall definition and basic properties of a formal group algebra, following [CPZ, §2].

Let F be a one dimensional formal group law over a commutative (unital) ring R (see [Ha78], [LM07, p. 4]). Given an integer m ≥ 0 we use the notation

u +_{F} v = F (u, v), m ·_{F} u := \underbrace{u +_{F} ⋯ +_{F} u}_{m times}, and (−m) ·_{F} u := −_{F} (m ·_{F} u),

where −_{F} u denotes the formal inverse of u.

Let Λ be a lattice, i.e. a finitely generated free abelian group, and let R[x_{Λ}] denote the polynomial ring over R with variables indexed by Λ. Let ϵ: R[x_{Λ}] → R be the augmentation morphism which maps x_{λ} to 0 for each λ ∈ Λ. Let R[[x_{Λ}]] be the ker(ϵ)-adic completion of the polynomial ring R[x_{Λ}]. Let J_{F} be the closure of the ideal generated by x_{0} and elements of the form x_{λ_{1}+λ_{2}} − (x_{λ_{1}} +_{F} x_{λ_{2}}) for all λ_{1}, λ_{2} ∈ Λ.

Following [CPZ, Def. 2.4], we define the formal group algebra (or formal group ring) to be the quotient

R[[Λ]]_{F} := R[[x_{Λ}]]/J_{F} .

The class of x_{λ} in R[[Λ]]_{F} will be denoted identically. By definition, R[[Λ]]_{F} is a complete Hausdorff R-algebra with respect to the I_{F} -adic topology, where I_{F} = ker ϵ with ϵ: R[[Λ]]_{F} → R the induced augmentation map. We assume that I_{F}^{−i} = R[[Λ]]_{F} if i ≥ 0.

Recall from [CPZ, Cor. 2.13] that choosing a basis {λ_{1}, …, λ_{n}} of the lattice Λ gives a canonical isomorphism R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]] sending

x_{∑ m_{i}λ_{i}} ↦ m_{1} ·_{F} x_{1} +_{F} ⋯ +_{F} m_{n} ·_{F} x_{n} = m_{1}x_{1} + ⋯ + m_{n}x_{n} + I^{2},

where I = (x_{1}, …, x_{n}).

Example 1.1. (see [CPZ, Ex. 2.18 and Ex. 2.19])

For the additive formal group law over R given by F_{a}(u, v) = u + v we have an R-algebra isomorphism R[[Λ]]_{F} ≃ S_{R}(Λ)^{∧}, where S_{R}^{i}(Λ) is the i-th symmetric power of Λ over R, the completion is at the kernel of the augmentation map x_{λ} ↦ 0 and the isomorphism is induced by x_{λ} ↦ λ ∈ S_{R}^{1}(Λ).

Consider the group ring R[Λ] := { ∑_{j} a_{j}e^{λ_{j}} | a_{j} ∈ R, λ_{j} ∈ Λ}. Let tr: R[Λ] → R be the trace map, i.e. a R-linear map sending every e^{λ} to 1. Let R[Λ]^{∧} denote the completion of R[Λ] at ker(tr).

For the multiplicative periodic formal group law over R given by F_{m}(u, v) = u+v−βuv, where β is an invertible element in R, we have an R-algebra isomorphism R[[Λ]]_{F} ≃ R[Λ]^{∧} induced by x_{λ} ↦ β^{−1}(1 − e^{λ}) and e^{λ} ↦ (1 − βx_{λ}) = (1 − βx_{−λ})^{−1}.

Consider the universal formal group law U(u, v) = u + v + a_{11}uv + . . .. Its coefficient ring is the Lazard ring 𝕃. By universality there is a canonical ring homomorphism 𝕃[[Λ]]_{U} → S_{ℤ}(Λ)^{∧} (resp. 𝕃[[Λ]]_{U} → ℤ[β, β^{−1}][Λ]^{∧}) given by sending all the coefficients a_{ij} of U to 0 (resp. all coefficients except a_{11} to 0 and a_{11} ↦ −β).

We will need the following facts concerning formal group algebras. We refer to the appendix for the properties of regular elements.

Lemma 1.2. Let Λ ⊆ Λ′ be two lattices. If |Λ′/Λ| is regular in R, then the natural map R[[Λ]]_{F} → R[[Λ′]]_{F} is an injection.

Proof. Choose bases of Λ and Λ′, and let A be the matrix expressing the vectors of the first in terms of the second. Then by Lemma 12.6, det(A) = ±|Λ′/Λ|. The lemma follows from Corollary 12.5 through isomorphisms R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]] and R[[Λ′]]_{F} ≃ R[[x_{1}, …, x_{n}]] associated to the bases. □

Lemma 1.3. Let Λ = Λ_{1} ⊕ Λ_{2} be a direct sum of two lattices and let λ ∈ Λ_{1}. Then x_{λ} is regular in R[[Λ]]_{F} if and only if it is regular in R[[Λ_{1}]]_{F}.

Proof. By [CPZ, Thm. 2.1], there is a natural isomorphism of R[[Λ_{1}]]_{F} -algebras R[[Λ_{1} ⊕ Λ_{2}]]_{F} ≃ (R[[Λ_{1}]]_{F})[[Λ_{2}]]_{F}. The result then follows by Corollary 12.2 applied to f = x_{λ} ∈ R′ = R[[Λ_{1}]]_{F} and R′[[Λ_{2}]]_{F} ≃ R′[[x_{1}, …, x_{l}]] with l the rank of Λ_{2}. □

2. Root systems and regular elements

We recall several auxiliary facts concerning root datum following [SGA, Exp. XXI] and [Bo68]. We provide a criteria for regularity of a generator of the formal group algebra (see Lemma 2.2).

Following [SGA, Exp. XXI, §1.1] we define a root datum to be an embedding Σ ↪ Λ^{∨}, α ↦ α^{∨}, of a non-empty finite subset Σ of a lattice Λ into its dual Λ^{∨} such that

• Σ ∩ 2Σ = ∅, α^{∨}(α) = 2 for all α ∈ Σ, and • β −α^{∨}(β)α ∈ Σ and β^{∨} −β^{∨}(α)α^{∨} ∈ Σ^{∨} for all α, β ∈ Σ, where Σ^{∨} denotes

the image of Σ in Λ^{∨}. The elements of Σ (resp. Σ^{∨}) are called roots (resp. coroots).

The sublattice of Λ generated by Σ is called the root lattice and is denoted by Λ_{r}. The rank of Λ_{ℚ} is called the rank of the root datum. A root datum is called irreducible if it is not a direct sum of root data of smaller ranks. The sublattice of Λ_{ℚ} = Λ ⊗_{ℤ} ℚ generated by all ω ∈ Λ_{ℚ} such that α^{∨}(ω) ∈ ℤ for all α ∈ Σ is called the weight lattice and is denoted by Λ_{w}. Observe that by definition Λ_{r} ⊆ Λ_{w} and Λ_{r} ⊗_{ℤ} ℚ = Λ_{w} ⊗_{Z} ℚ. A root datum is called semisimple if Λ_{ℚ} = Λ_{r} ⊗_{ℤ} ℚ. From now on by a root datum we will always mean a semisimple one. Observe that in this case Λ_{r} ⊆ Λ ⊆ Λ_{w}.

The root lattice Λ_{r} admits a basis Π = {α_{1}, …, α_{n}} such that each α ∈ Σ is a linear combination of α_{i}’s with either all positive or all negative coefficients. So the set Σ splits into two disjoint subsets Σ = Σ_{+} ⨿ Σ_{−}, where Σ_{+} (resp. Σ_{−}) is called the set of positive (resp. negative) roots. The roots α_{i} are called simple roots.

Given the set Π we define the set of fundamental weights {ω_{1}, …, ω_{n}} ⊂ Λ_{w} as α_{i}^{∨}(ω_{j}) = δ_{ij}, where δ_{ij} is the Kronecker symbol. Fundamental weights form a basis of the weight lattice Λ_{w}. The matrix expressing simple roots in terms of fundamental weights is called the Cartan matrix of the root datum.

If Λ = Λ_{w} (resp. Λ = Λ_{r}), then the root datum is called simply connected (resp. adjoint) and is denoted by D_{n}^{sc} (resp. D_{n}^{ad}), where D = A, B, C, D, E, F, G is one of the Dynkin types and n is its rank. Observe that an irreducible root datum is uniquely determined by its Dynkin type and the lattice Λ.

Determinant of the Cartan matrix of an irreducible root datum coincides with |Λ_{w}/Λ_{r}|, and can be found in the tables at the end of [Bo68] under the name “indice de connexion”. For future reference, we provide the list of determinants and the list of simply connected torsion primes, i.e. prime divisors of the torsion index of the associated simply connected root datum; these can be found in [D73, Prop. 8].

Type (s. connected) A_{l} B_{l}, l ≥ 3 C_{l} D_{l}, l ≥ 4 G_{2} F_{4} E_{6} E_{7} E_{8}
Determinant  l + 1  2  2  4  1 1  3  2  1
Torsion primes  ∅  2  ∅  2  2 2, 3 2, 3 2, 3 2, 3, 5

In several of the following statements, root datum C_{n}^{sc}, n ≥ 1 need to be treated separately. Note that A_{1}^{sc} = C_{1}^{sc} and B_{2}^{sc} = C_{2}^{sc} are parts of these special cases.

Lemma 2.1. Any root α ∈ Σ can be completed to a basis of Λ except if it is a long root in an irreducible component C_{n}^{sc}, n ≥ 1, of the root datum, in which case, it can still be completed to a basis of Λ over ℤ[\frac{1}{2}].

Proof. We may assume that the root datum is irreducible.

1. First, we can choose simple roots of Λ_{r} as Π = {α = α_{1}, α_{2}, …, α_{n}} by [Bo68, Ch. IV, §1, Prop. 15], hence, proving the statement in the adjoint case (Λ = Λ_{r}).

2. If Λ = Λ_{w}, we express the root α in terms of the fundamental weights corresponding to Π as α = ∑_{i=1}^{n} k_{i}ω_{i}. Looking at the Cartan matrix we see that k_{i} = −1 for some i except in type G_{2} or if α = α_{n} is the long root in type C_{n}, n ≥ 1. In other words, except in these two cases α can be completed to a basis of Λ_{w} by Lemma 12.7 as (k_{1}, ..., k_{n}) is unimodular. Since the G_{2} case is adjoint, it has already been considered. Finally, in the C_{n} case, since k_{i} = 2 for some i, the root α can be completed to a basis of Λ_{w} after inverting 2.

3. The remaining case Λ_{r} ⊊ Λ ⊊ Λ_{w} can only happen in type D_{n} or A_{n}, both with n ≥ 3, see the above table.

Pick a basis {λ_{1}, …, λ_{n}} of Λ and express α = ∑_{i=1}^{n} k_{i}λ_{i}. Then, complete α to a basis {α = e_{1}, e_{2}, …, e_{n}} of Λ_{w}, as in the previous step. Let A = (a_{ij}) be the matrix whose columns express the basis {λ_{i}} in terms of the basis {e_{i}}. Applying A to the vector (k_{i}) gives α = e_{1}. So ∑_{j=1}^{n} a_{j1}k_{j} = 1 which means that the row (k_{1}, …, k_{n}) is unimodular and, therefore, α can be completed to a basis of Λ by Lemma 12.7. □

Consider a formal group algebra R[[Λ]]_{F} associated to the lattice Λ of the root datum and a formal group law F over R.

Lemma 2.2. For any root α ∈ Σ the element x_{α} is regular in R[[Λ]]_{F}, except maybe if α is a long root in an irreducible component C_{n}^{sc}, n ≥ 1, of the root datum. In that case, x_{α} is regular if and only if the formal series 2 ·_{F} x is regular in R[[x[[, which always holds if 2 is regular.

Proof. If α is not a long root in C_{n}^{sc}, n ≥ 1, then by Lemma 2.1, it can be completed to a basis of Λ, and R[[Λ]]_{F} ≅ R[[x_{1}, ..., x_{n}]] with x_{1} = x_{α}, so x_{α} is regular by Lemma 12.3. On the other hand, if α is a long root in C_{n}^{sc}, then it is part of a basis of roots, with associated fundamental weights part of a basis of Λ. If n = 1, then α = 2ω where ω is the weight corresponding to α. If n ≥ 2, then α = 2(ω −ω′) where ω′ is another weight. In both cases, we can therefore find an isomorphism R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]] such that x_{α} is mapped to 2 ·_{F} x_{1}, which is regular if and only if it is already regular in R[[x_{1}]]. If 2 is regular, then 2 ·_{F} x_{1} = 2x_{1} + x_{1}^{2}y is regular by part (a) of Lemma 12.3. □

Remark 2.3. For some formal group laws x_{α} can indeed be a regular element even if 2 is a zero divisor and α is a long root in C_{n}^{sc}. Take the multiplicative formal group law F (u, v) = u + v − uv over R = ℤ/2 and the root datum C_{1}^{sc}. Then x_{α} = x_{2ω} = 2x_{ω} + x_{ω}^{2} = x_{ω}^{2} is regular.

Lemma 2.4. Let Λ ⊂ Λ′ be two lattices between Λ_{r} and Λ_{w}. If the determinant of the Cartan matrix is regular in R, then the natural map R[[Λ]]_{F} → R[[Λ′]]_{F} is injective.

Proof. Choose bases of Λ and Λ′, and let A be the matrix expressing the first basis in terms of the second. Then the determinant of A divides the determinant of the Cartan matrix by composition of inclusions Λ_{r} ⊆ Λ ⊆ Λ′ ⊆ Λ_{w}. The lemma then follows from Lemma 1.2. □

Remark 2.5. Here is a counter-example to the injectivity when the determinant of the Cartan matrix is not regular. For the type A_{2}, take R = ℤ/3 and the additive formal group law. The map from R[[Λ_{r}]]_{F} → R[[Λ_{w}]]_{F} is not injective: if {α_{1}, α_{2}} are simple roots, then x_{α_{1}+2α_{2}} = x_{α_{1}} + 2x_{α_{2}} ≠ 0 is sent to x_{3ω_{2}} = 3x_{ω_{2}} = 0.

3. Localized formal group algebras

Consider a (semisimple) root datum Σ ↪ Λ^{∨}. Various operators on the formal group algebra R[[Λ]]_{F}, such as formal Demazure operators, are defined using formulas involving formal division by elements x_{α}, α ∈ Σ, see for example [CPZ, Def. 3.5] in the case Λ = Λ_{w}. In the present section we study properties of the localization of R[[Λ]]_{F} at such elements. A key result here is Lemma 3.5 which will be used in the proof of Proposition 6.2.

Let Q_{R,Λ} denote the localization of R[[Λ]]_{F} at the multiplicative subset generated by elements x_{α} for all α ∈ Σ (if the multiplicative subset contains zero, then Q_{R,Λ} is trivial).

Remark 3.1. Observe that by [CPZ, Def. 3.12] x_{−α} = x_{α}(−1 + x_{−α}κ_{α}) for some κ_{α} ∈ R[[Λ]]_{F}. Therefore, we can replace Σ by Σ_{+} or Σ_{−}, since −1 + x_{−α}κ_{α} is invertible in R[[Λ]]_{F}.

Lemma 3.2. If 2 is regular in R or the root datum doesn’t have an irreducible component C_{n}^{sc}, n ≥ 1, then the localization map R[[Λ]]_{F} → Q_{R,Λ} is injective.

Proof. By Lemma 2.2, the element x_{α} is regular for any root α, so we are localizing at a set consisting of regular elements. □

Let r be a regular element of R. Then under the assumption of Lemma 3.2, there is a commutative diagram of inclusions

\begin{matrix}R[[Λ]]_{F} & ⊆ & ⊆ & R[[Λ]]_{F} [\frac{1}{r}] & ⊆ & ⊆ & R[\frac{1}{r}][[Λ]]_{F} & ⊆ & Q_{R,Λ} & ⊆ & Q_{R,Λ}[\frac{1}{r}] & ⊆ & Q_{R[ \frac{1}{r}],Λ} \\ (3.1) \end{matrix}

This can be seen by choosing a basis of Λ and, hence, identifying the respective formal group algebra with a ring of power series. Observe that if r is non-invertible, an element ∑_{i≥0} \frac{x^{i}}{r^{i}} while being in R[\frac{1}{r}][[x]] is not in R[[x]][\frac{1}{r}], so the inclusions R[[Λ]]_{F} ⊊ R[[Λ]]_{F} [\frac{1}{r}] ⊊ R[\frac{1}{r}][[Λ]]_{F} are proper.

Lemma 3.3. Let r be a regular element of R and let α be a root. Assume that 2 is invertible in R or α is not a long root of an irreducible component C_{n}^{sc}, n ≥ 1. If u ∈ R[\frac{1}{r}][[Λ]]_{F} and x_{α}u ∈ R[[Λ]]_{F} [\frac{1}{r}], then u ∈ R[[Λ]]_{F} [\frac{1}{r}].

Proof. By Lemma 2.1 we complete α to a basis of Λ. Observe that if α is a long root of C_{n}^{sc}, then x_{α} = 2x_{ω} + I_{F}^{2} . Therefore, the isomorphism R[[Λ]]_{F} ≃ R[[x_{1}, …, x_{n}]] determined by the choice of a basis can be modified so that it sends x_{α} to x_{1}.

The lemma then follows from the fact that if u ∈ R[\frac{1}{r}][[x_{1}, …, x_{n}]] and x_{1}u ∈ R[[x_{1}, …, x_{n}]][\frac{1}{r}], then u ∈ R[[x_{1}, …, x_{n}]][\frac{1}{r}]. □

Corollary 3.4. Let r be a regular element of R. Assume that 2 is invertible in R or the root datum doesn’t have an irreducible component C_{n}^{sc}, n ≥ 1. Then we have (R[\frac{1}{r}][[Λ]]_{F}) ∩ Q_{R,Λ}[\frac{1}{r}] = R[[Λ]]_{F} [\frac{1}{r}] in Q_{R[ \frac{1}{r}],Λ}, i.e. the right square of (3.1) is

cartesian.

Lemma 3.5. Let r be a regular element of R. Assume that 2 is invertible in R or the root datum doesn’t have an irreducible component C_{n}^{sc}, n ≥ 1. Then we have (R[[Λ]]_{F} [\frac{1}{r}]) ∩ Q_{R,Λ} = R[[Λ]]_{F} in Q_{R,Λ}[\frac{1}{r}], i.e. the left square of (3.1) is cartesian.

Proof. Since the localization map is injective by Lemma 3.2, it is enough to show that if u, v ∈ R[[Λ]]_{F} satisfy \frac{v}{∏_{α} x_{α}} = \frac{u}{r^{n}}, α ∈ Σ, or equivalently, r^{n}v = u ∏_{α} x_{α} in R[[Λ]]_{F}, then r^{n} divides u. In (R/r^{n})[[Λ]]_{F} = R[[Λ]]_{F} /r^{n}, we have r^{n}v = 0, and since by Lemma 2.2 the element x_{α} is regular in (R/r^{n})[[Λ]]_{F} for any α ∈ Σ, we also have u = 0. Thus u is divisible by r^{n}. □

4. Formal Demazure operators

We now introduce formal Demazure operators following the approach mentioned in [CPZ, Rem. 3.6]. The formula (4.1) will be extensively used in section 9 in various computations involving coproducts.

Consider a root datum Σ ↪ Λ^{∨}. For each α ∈ Σ we define a ℤ-linear automorphism of Λ called a reflection by

s_{α}: λ ↦ λ − α^{∨}(λ)α, λ ∈ Λ.

The subgroup of linear automorphisms of Λ generated by reflections s_{α}, α ∈ Σ, is called the Weyl group of the root datum and is denoted by W. Observe that W depends only on the Dynkin type.

Fix a formal group law F over R. There is an induced action of the Weyl group W on R[[Λ]]_{F} via R-algebra automorphisms defined by w(x_{λ}) = x_{w(λ)} for all w ∈ W and λ ∈ Λ.

Lemma 4.1. If x_{α} is regular in R[[Λ]]_{F}, then the difference u − s_{α}(u) is uniquely divisible by x_{α} for each u ∈ R[[Λ]]_{F} and α ∈ Σ.

Proof. Divisibility follows from the proof of [CPZ, Cor. 3.4]. Uniqueness follows from the regularity of x_{α}. □

Definition 4.2. For each α ∈ Σ such that x_{α} is regular in R[[Λ]]_{F} we define an R-linear operator Δ_{α} on R[[Λ]]_{F}, called a formal Demazure operator,

Δ_{α}(u) := \frac{u−s_{α}(u)}{x_{α}}, u ∈ R[[Λ]]_{F} .

Remark 4.3. Observe that in [CPZ, Def. 3.5] we defined the Demazure operator on R[[Λ]]_{F} for Λ = Λ_{w} by specializing the Demazure operator defined over the Lazard ring. If x_{α} is regular, then this definition coincides with Definition 4.2.

Definition 4.4. We say that R[[Λ]]_{F} is Σ-regular if for each α ∈ Σ, the element x_{α} is regular in R[[Λ]]_{F}.

Unless otherwise stated, we shall always assume that the formal group algebra R[[Λ]]_{F} is Σ-regular. Observe that this immediately implies that R[[Λ]]_{F} injects into the localization Q_{R,Λ} (cf. Lemma 3.2).

Remark 4.5. Note that Lemma 2.2 implies that, R[[Λ]]_{F} is Σ-regular if 2 is regular in R or if the root datum doesn’t contain an irreducible component C_{n}^{sc}.

We fix a set of simple roots {α_{1}, …, α_{n}}. Let s_{i} = s_{α_{i}}, i ∈ [n] = {1, …, n} denote the corresponding (simple) reflections. One of the basic facts concerning the Weyl group W is that it is generated by simple reflections. Given a sequence I = (i_{1}, ..., i_{l}) of length |I| = l with i_{j} ∈ [n], we set w(I) = s_{i_{1}} … s_{i_{l}} to be the product of the respective simple reflections. A sequence I is called a reduced sequence of w ∈ W if I is of minimal length among all sequences J such that w = w(J). The length of I_{w} is called the length of w and is denoted by ℓ(w). Given a sequence I we define

Δ_{I} = Δ_{i_{1}} ∘ ... ∘ Δ_{i_{l}}, where Δ_{i} := Δ_{α_{i}}.

Remark 4.6. If F is the additive or multiplicative formal group law, then Δ_{I_{w}} doesn’t depend on a choice of the reduced sequence of w. In this case, it coincides with the classical Demazure operator Δ_{w} of [D73, §3 and §9]. For other formal group laws Δ_{I_{w}} depends on a choice of I_{w} (see [CPZ, Thm. 3.9]).

Definition 4.7. We define R-linear operators B_{i}^{(j)} : R[[Λ]]_{F} → R[[Λ]]_{F}, where j ∈ {−1, 0, 1} and i ∈ [n], as

B_{i}^{(−1)} := Δ_{i}, B_{i}^{(0)} := s_{i}, and B_{i}^{(1)} := multiplication by (−x_{i}) := −x_{α_{i}}.

Observe that we have B_{i}^{(j)}(I_{F}^{m}) ⊂ I_{F}^{m+j}, where I_{F} is the augmentation ideal.

Let I = (i_{1}, …, i_{l}) be a sequence of length l, and let E be a subset of [l]. We denote by I_{|E} the subsequence of I consisting of all i_{j}’s with j ∈ E.

Lemma 4.8. Let I = (i_{1}, …, i_{l}). Then for any u, v ∈ R[[Λ]]_{F} we have

(4.1)  Δ_{I}(uv) = ∑_{E_{1},E_{2}⊂[l]} p_{E_{1},E_{2}}^{I}Δ_{I_{|E_{1}}} (u)Δ_{I_{|E_{2}}} (v)

where p_{E_{1},E_{2}}^{I} = B_{1}∘...∘B_{l}(1) ∈ I_{F}^{|E_{1}|+|E_{2}|−l} with the operator B_{j}: R[[Λ]]_{F} → R[[Λ]]_{F} defined as

B_{j} = \begin{cases}B_{i_{j}}^{(1)} ∘ B_{i_{j}}^{(0)}, & if j ∈ E_{1} ∩ E_{2}, \\ B_{i_{j}}^{(−1)}, & if j ∉ E_{1} ∪ E_{2}, \\ B_{i_{j}}^{(0)}, & otherwise. \end{cases}

Proof. We proceed by induction on the length l of I. For l = 1 and I = (i) the formula holds since

Δ_{i}(uv) = Δ_{i}(u)v + uΔ_{i}(v) − x_{i}Δ_{i}(u)Δ_{i}(v)
= −Δ_{i}(1)uv + s_{i}(1)uΔ_{i}(v) + s_{i}(1)Δ_{i}(u)v − x_{i}Δ_{i}(u)Δ_{i}(v)

by [CPZ, Prop. 3.8.(4)] and because Δ_{i}(1) = 0 and s_{i}(1) = 1.

Setting I′ = (i_{2}, …, i_{l}), we obtain by induction

Δ_{I}(uv) = Δ_{i_{1}}(Δ_{I′}(uv)) = Δ_{i_{1}}( ∑_{E_{1},E_{2}⊂[l−1]} p_{E_{1},E_{2}}^{I′}Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)).

For fixed E_{1}, E_{2} ⊂ [l − 1], by [CPZ, Prop. 3.8.(4)] we have

Δ_{i_{1}}(p_{E_{1},E_{2}}^{I′}Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)) =Δ_{i_{1}}(p_{E_{1},E_{2}}^{I′})Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)
+ s_{i_{1}}(p_{E_{1},E_{2}}^{I′})Δ_{i_{1}}(Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)),

and

Δ_{i_{1}}(Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)) =(Δ_{i_{1}} ∘ Δ_{I′_{|E_{1}}} )(u)Δ_{I′_{|E_{2}}} (v) + Δ_{I′_{|E_{1}}} (u)(Δ_{i_{1}} ∘ Δ_{I′_{|E_{2}}} )(v)
− x_{i_{1}}(Δ_{i_{1}} ∘ Δ_{I′_{|E_{1}}} )(u)(Δ_{i_{1}} ∘ Δ_{I′_{E_{2}}} )(v).

Combining these two identities, we obtain

Δ_{i_{1}}(p_{E_{1},E_{2}}^{I′}Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)) = B_{i_{1}}^{(−1)}(p_{E_{1},E_{2}}^{I′}) · Δ_{I′_{|E_{1}}} (u)Δ_{I′_{|E_{2}}} (v)

+ B_{i_{1}}^{(0)}(p_{E_{1},E_{2}}^{I′}) · [(Δ_{i_{1}} ∘ Δ_{I′_{|E_{1}}} )(u)Δ_{I′_{|E_{2}}} (v) + Δ_{I′_{|E_{1}}} (u)(Δ_{i_{1}} ∘ Δ_{I′_{|E_{2}}} )(v)]

+ (B_{i_{1}}^{(1)} ∘ B_{i_{1}}^{(0)})(p_{E_{1},E_{2}}^{I′}) · (Δ_{i_{1}} ∘ Δ_{I′_{|E_{1}}} )(u)(Δ_{i_{1}} ∘ Δ_{I′_{|E_{2}}} )(v) and using the notation E + 1 = {e + 1 | e ∈ E} for a set of integers E we get

= B_{i_{1}}^{(−1)}(p_{E_{1},E_{2}}^{I′}) · Δ_{I_{|E_{1}+1}}(u)Δ_{I_{|E_{2}+1}}(v)

+ B_{i_{1}}^{(0)}(p_{E_{1},E_{2}}^{I′}) · [Δ_{I_{|{1}∪(E_{1}+1)}})(u)Δ_{I_{E_{2}+1}}(v) + Δ_{I_{|E_{1}+1}}(u)Δ_{I_{|{1}∪(E_{2}+1)}}(v)]

+ (B_{i_{1}}^{(1)} ∘ B_{i_{1}}^{(0)})(p_{E_{1},E_{2}}^{I′}) · (Δ_{I_{|{1}∪(E_{1}+1)}})(u)Δ_{I_{|{1}∪(E_{2}+1)}}(v).

Since a subset of [l] is of the form E + 1 or {1} ∪ (E + 1) with E ⊆ [l − 1], the result follows by induction. □

Remark 4.9. For any simple root α we have s_{α}(1) = 1 and Δ_{α}(1) = 0. Therefore, if there is an operator of type B^{(−1)} applied to 1 before applying all the operators of type B^{(1)} ∘ B^{(0)}, then p_{E_{1},E_{2}}^{I} = 0. In particular, if E_{1} ∩ E_{2} = ∅ and E_{1} ∪ E_{2} ≠ [l], then p_{E_{1},E_{2}}^{I} = 0.

5. Twisted formal group algebra and Demazure elements

We recall the notions of a twisted formal group algebra and a formal affine Demazure algebra introduced in [HMSZ, §5]. The purpose of this section is to generalize [KK86, Prop. 4.3] and [KK90, Prop. 2.6] to the context of arbitrary formal group laws (see Lemma 5.4), hence, providing a general formula expressing a product of Demazure elements in terms of the canonical basis of the twisted formal group algebra.

As in section 3, given a root datum Σ ↪ Λ^{∨} consider the localization Q = Q_{R,Λ} of the formal group algebra S = R[[Λ]]_{F}. Since the Weyl group W preserves the set Σ, its action on S extends to Q. Following to [KK86, §4.1] and [HMSZ, Def. 5.1] we define the twisted formal group algebra to be the R-module Q_{W} := Q ⊗_{R} R[W] with the multiplication given by

(qδ_{w})(q′δ_{w′}) = qw(q′)δ_{ww′} for all w, w′ ∈ W and q, q′ ∈ Q

(extended by linearity), where δ_{w} denotes the element in R[W] corresponding to w (so we have δ_{w}δ_{w′} = δ_{ww′} for w, w′ ∈ W) and δ_{1} denotes 1. Observe that Q_{W} is a free left Q-module (via the left multiplication) with basis {δ_{w}}_{w∈W}, but Q_{W} is not a Q-algebra as δ_{1}Q = Qδ_{1} is not central in Q_{W}.

Remark 5.1. The twisted formal group algebra can be defined for any W -action on Q, where W is a unital monoid and w(q_{1} · q_{2}) = w(q_{1}) · w(q_{2}) for any q_{1}, q_{2} ∈ Q and w ∈ W.

Remark 5.2. In [KK86, §4.1] (resp. [KK90, §2.1]) Q denoted the field of fractions of a symmetric algebra (resp. of an integral group ring) of the weight lattice Λ_{w} and Q_{W} was defined using the right Q-module notation, i.e. Q_{W} = ℤ[W] ⊗_{ℤ} Q. Using our terminology, the case of [KK86] (resp. of [KK90]) corresponds to the additive formal group law (resp. multiplicative periodic formal group law) and the simply connected root datum.

In [HMSZ], Q denoted the localization of R[[Λ_{w}]]_{F} at all elements x_{λ}, where λ ∈ Λ_{w}.

Definition 5.3. Following [KK86, I_{24}] and [HMSZ, Def. 5.2] for each root α ∈ Σ we define the Demazure element

X_{α} := x_{α}^{−1}(1 − δ_{s_{α}}) = x_{α}^{−1} − δ_{s_{α}}x_{−α}^{−1} ∈ Q_{W} .

Given a set of simple roots {α_{1}, …, α_{n}} for any sequence I = (i_{1}, ..., i_{l}) from [n] we denote

δ_{I} = δ_{s_{i_{1}} ...s_{i_{l}}} and X_{I} := X_{i_{1}} · . . . · X_{i_{l}}, where X_{i} = X_{α_{i}}.

There is an anti-involution (-)^{t} on the R-algebra Q_{W} defined by

qδ_{w} ↦ (qδ_{w})^{t} := w^{−1}(q)δ_{w^{−1}}, w ∈ W, q ∈ Q.

Observe that (qx)^{t} = x^{t}q for x ∈ Q_{W} and q ∈ Q, so it is neither right Q-linear nor left Q-linear.

We obtain the following generalization of [KK86, Prop. 4.3]:

Lemma 5.4. Given a reduced sequence I_{v} of v ∈ W of length l let

X_{I_{v}} = ∑_{w∈W} a_{v,w}δ_{w} = ∑_{w∈W} δ_{w}a′_{v,w}

for some a_{v,w}, a′_{v,w} ∈ Q. Then (a) a_{v,w} = 0 unless w ≤ v with respect to the Bruhat order on W,

(b) a_{v,v} = (−1)^{l} ∏_{α∈v(Σ_{−})∩Σ_{+}} x_{α}^{−1} = a′_{v,v^{−1}},

(c) a′_{v,w} = 0 unless w ≥ v^{−1}.

Proof. We proceed by induction on the length l of v.

The lemma obviously holds for I_{v} = ∅ the empty sequence, since X_{∅} = 1.

Let I_{v} = (i_{1}, … i_{l}) be a reduced sequence of v and let β = α_{i_{1}}. Then (i_{2}, …, i_{n}) is a reduced sequence of v′ = s_{β}v and we have

(1) w ≤ v′ implies w ≤ v and s_{β}w ≤ v;

(2) {β} ∪ s_{β}(v′(Σ_{−}) ∩ Σ_{+}) = v(Σ_{−}) ∩ Σ_{+};

(3) w^{−1} ≤ v if and only w ≥ v^{−1}. Indeed, the properties (1) and (3) are consequences of the fact that elements smaller than v are the elements w obtained by taking a subsequence of a reduced sequence of v, by [De77, Th. 1.1, III, (ii)]. Property (2) follows from [Bo68, Ch. VI, §1, No 6, Cor. 2].

We then compute

\begin{aligned}X_{I_{v}} & = x_{β}^{−1}(1 − δ_{β}) ∑_{w≤v′} a_{v′,w}δ_{w} = ∑_{w≤v′} x_{β}^{−1}a_{v′,w}δ_{w} − ∑_{w≤v′} x_{β}^{−1}s_{β}(a_{v′,w})δ_{s_{β}w} \\ & =^{(1)} −x_{β}^{−1}s_{β}(a_{v′,v′})δ_{v} + ∑_{w<v} a_{w,v}δ_{w}. \end{aligned}

So part (a) holds and a_{v,v} = −x_{β}^{−1}s_{β}(a_{v′,v′}). Hence, the expression of a_{v,v} in part (b) follows from property (2). Property (3) implies (c) by applying the antiinvolution sending δ_{w} to δ_{w^{−1}} and, thus, identifying a′_{v,w} = a_{v,w^{−1}}. □

Remark 5.5. Observe that the coefficient a_{v,v} doesn’t depend on the choice of a reduced sequence I_{v} of v.

Corollary 5.6. (cf. [KK86, Cor. 4.5]) For each w ∈ W, let I_{w} be a reduced sequence of w. The elements (X_{I_{w}})_{w∈W} form a basis of Q_{W} as a left (resp. right) Q-module. The element δ_{v} decomposes as ∑_{w≤v} b_{w,v}X_{I_{w}} with b_{w,v} in Q. Furthermore,

b_{w,w} = ∏_{α∈v(Σ_{−})∩Σ_{+}}(−1)^{l}x_{α}.

Proof. The matrix (a_{v,w})_{(v,w)∈W^{2}}, resp. (a′_{v,w})_{(v,w)∈W^{2}}, is triangular with invertible coefficients on the diagonal (resp. the anti-diagonal) by Lemma 5.4. It expresses elements X_{I_{w}} in terms of the basis (δ_{w})_{w∈W} of the left (resp. right) Q-module Q_{W}.

The decomposition of δ_{w} follows from the fact that the inverse of a triangular matrix is also triangular with inverse coefficients on the diagonal. □

We are now ready to define a key object of the present paper.

Definition 5.7. We define the formal affine Demazure algebra D_{F} to be the Rsubalgebra of Q_{W} generated by elements of the formal group algebra S and by Demazure elements X_{α} for all α ∈ Σ.

The following lemma shows that it is the same object as the one considered in [HMSZ, Definition 5.3].

Lemma 5.8. The Demazure algebra D_{F} coincides with the R-subalgebra of Q_{W} generated by elements of S and Demazure elements X_{i}, for i ∈ [l].

Proof. Since the Weyl group is generated by reflections and δ_{s_{i}} = 1 − x_{α_{i}}X_{i}, we have R[W] ⊆ D_{F}. Since any α ∈ Σ can be written as α = w(α_{i}) for some simple root α_{i}, the lemma then follows from the formula δ_{w}X_{α_{i}}δ_{w^{−1}} = X_{w(α_{i})}. □

6. Invariant presentation of the formal affine Demazure algebra

In the present section using the action of Demazure elements on R[[Λ]]_{F} via Demazure operators we consider the algebra D̃_{F} of elements of Q_{W} fixing S. It is closely related to the formal affine Demazure algebra D_{F}. We show that the algebra D̃_{F} is a free R[[Λ]]_{F} -module with a basis given by products of Demazure elements (see Corollary 6.3), hence, generalizing [KK86, Thm.4.6.(a)].

The localization Q = Q_{R,Λ} of S = R[[Λ]]_{F} has a structure of left Q_{W} -module defined by (cf. [KK86, I_{33}])

(δ_{w}q)q′ = w(qq′) for w ∈ W and q, q′ ∈ Q.

Let D̃_{F} denote the R-subalgebra of Q_{W} preserving S when acting on the left, i.e.

D̃_{F} := {x ∈ Q_{W} | x · S ⊆ S}.

By definition we have S ⊂ D̃_{F} and X_{α} ∈ D̃_{F}, since X_{α} acts on S by the Demazure operator Δ_{α}. Therefore, in this case D_{F} ⊆ D̃_{F}.

Let 𝔱 be the torsion index of the root datum as defined in [D73, §5]. Its prime factors are the torsion primes listed in the table in §2 together with the prime divisors of |Λ_{w}/Λ|, by [D73, §5, Prop. 6].

Let I_{0} be a reduced sequence of the longest element w_{0} and let N = ℓ(w_{0}). Recall that by [CPZ, 5.2] there exists an element u_{0} ∈ I_{F}^{N} ⊂ S such that ϵΔ_{I_{0}}(u_{0}) = 𝔱. Moreover, u_{0} satisfies the following property: if |I| ≤ N, then

ϵΔ_{I}(u_{0}) = \{ \begin{matrix}𝔱, & if I is a reduced sequence of w_{0}, \\ 0, & otherwise. \end{matrix}

Let us recall a key result for future reference.

Lemma 6.1. For each w ∈ W, let I_{w} be a reduced sequence of w. Then if the torsion index 𝔱 is invertible in R, the matrix (Δ_{I_{v}}Δ_{I_{w}}(u_{0}))_{(v,w)∈W ×W} with coefficients in S is invertible. Thus, if 𝔱 is just regular, the kernel of this matrix is trivial.

Proof. It readily follows from the above property of u_{0}, see [CPZ, Prop. 6.6] and its proof. □

Proposition 6.2. Assume that 𝔱 is regular in R. Also assume that 2 ∈ R^{×} or that either all or none of the irreducible components of the root datum are C_{n}^{sc}, n ≥ 1.

Then the R-algebra D̃_{F} is the left S-submodule of Q_{W} generated by {X_{I_{w}}}_{w∈W}.

Proof. Observe that the hypotheses of the proposition imply that S is Σ-regular, so D̃_{F} is well-defined.

By Corollary 5.6 we can write y ∈ D̃_{F} as y = ∑_{v} c_{v}X_{I_{v}}, where c_{v} ∈ Q. It is enough to prove that c_{v} ∈ S for each v. Apply y successively to the elements Δ_{I_{w}}(u_{0}), for all w ∈ W. By definition of D_{F} all resulting elements ∑_{v} c_{v}Δ_{I_{v}}Δ_{I_{w}}(u_{0}) are in S. By Lemma 6.1, the matrix (Δ_{I_{v}}Δ_{I_{w}}(u_{0}))_{(v,w)∈W ×W} with coefficients in S becomes invertible after inverting 𝔱. This implies that c_{v} ∈ S[\frac{1}{𝔱}] for each v ∈ W.

If all irreducible components of the root system are C_{n}^{sc}, n ≥ 1, we are finished since 𝔱 = 1. In the remaining two cases (2 is invertible or there are no components C_{n}^{sc}) we have S[\frac{1}{𝔱}] ∩ Q = S by Lemma 3.5. □

Corollary 6.3. Under the hypotheses of Proposition 6.2 the elements {X_{I_{w}}}_{w∈W} form a basis of the left S-module D̃_{F}.

Proof. The elements X_{I_{w}} are independent over S, since they are independent over Q and the map S → Q is injective by Lemma 3.2. □

Example 6.4. Here is an example where the torsion index 𝔱 = 2 is regular in R = ℤ while the remaining assumptions of Proposition 6.2 are not satisfied and its conclusion does not hold.

Consider the additive formal group law over ℤ and a direct sum of root data C_{1}^{ad} and C_{1}^{sc} with the respective lattices Λ_{r} and Λ′_{w}. Let α be the simple root of Λ_{r}, and let ω′ be the fundamental weight of Λ′_{w}. Consider the isomorphism ℤ[[Λ ⊕ Λ′]]_{F} ≃ ℤ[[x, y]] defined by x_{α} ↦ x and x_{ω′} ↦ y. Observe that Q ≃ ℤ[[x, y]][\frac{1}{x}, \frac{1}{2y}], so it contains \frac{1}{2}.

Consider the simple reflection s_{α}. By definition s_{α}(y) = y and s_{α}(x) = −x. Therefore, Δ_{α} acts on Q linearly over ℤ[[y]] and Δ_{α}(x^{i}) = (x^{i} + (−x)^{i})/x for each i ≥ 1. So if z = ∑_{i≥0} a_{i}x^{i} with a_{i} ∈ ℤ[[y]], then Δ_{α}(z) = ∑_{i≥1} a_{i}(1 + (−1)^{i})x^{i−1} is divisible by 2. Hence, \frac{1}{2}X_{α} is in D̃_{F}, while it is not in the left S-submodule, since \frac{1}{2} ∉ ℤ[[x, y]]. Observe that the proposition holds if ℤ is replaced by ℤ[\frac{1}{2}].

7. Presentation in terms of generators and relations

In the present section we describe the formal affine Demazure algebra D_{F} in terms of generators and relations. The main result (Theorem 7.5) generalizes [HMSZ, Thm. 5.14]. We also establish an isomorphism (Theorem 7.9) between the formal affine Demazure algebra and the algebra of Demazure operators on R[[Λ]]_{F} of [CPZ, §4].

Lemma 7.1. Assume that 𝔱 is regular in R. Let I and I′ be reduced sequences of the same element w ∈ W. Then in the formal affine Demazure algebra D_{F} we have

X_{I} − X_{I′} = ∑_{v<w} c_{v}X_{I_{v}} for some c_{v} ∈ S.

Proof. Since the definition of the twisted formal group algebra is functorial in the lattice, we may assume that each irreducible component of our root datum is adjoint. In particular, it is not C_{n}^{sc} and, hence, satisfies all the hypotheses of 6.2. Then, by Proposition 6.2, the difference X_{I} − X_{I′} ∈ D̃_{F} can be written as a linear combination ∑_{w∈W} c_{v}X_{I_{v}}, c_{v} ∈ S. Since S injects into Q by Lemma 3.2, it suffices to check that c_{v} = 0 unless v ≤ w (in the Bruhat order) when this linear combination is the one obtained in Q_{W} by Corollary 5.6. We therefore compute:

X_{I} − X_{I′} =^{Lem. 5.4 (a)} ∑_{u≤w} a_{u}δ_{u} − ∑_{u≤w} a′_{u}δ_{u} =^{Lem. 5.4 (b)} ∑_{u<w}(a_{u} − a′_{u})δ_{u}

=^{Cor. 5.6} ∑_{u<w}(a_{u} − a′_{u}) ∑_{v≤u} b_{v}X_{I_{v}} = ∑_{v<w} b_{v}( ∑_{v≤u<w}(a_{u} − a′_{u}))X_{I_{v}}. □

Let us examine commutation relations between Demazure elements of simple roots of root datum of rank 1 or 2.

Example 7.2. In the rank 1 case we have [HMSZ, (5.1)]

(7.1)  X_{α}^{2} = κ_{α}X_{α} = X_{α}κ_{α}

where κ_{α} = \frac{1}{x_{α}} + \frac{1}{x_{−α}} actually lives in S [CPZ, Def. 3.12].

Example 7.3. For the rank 2 case let α_{1} and α_{2} be simple roots and let W be the Weyl group generated by s_{1} = s_{α_{1}} and s_{2} = s_{α_{2}}. Let m = 2, 3, 4 or 6 be the order of s_{1}s_{2}, i.e. the length of the longest element

w_{0}^{α_{1},α_{2}} = \underbrace{s_{1}s_{2}s_{1} ⋯}_{m times} = \underbrace{s_{2}s_{1}s_{2} ⋯}_{m times} .

Any element w ≠ w_{0}^{α_{1},α_{2}} in W has a unique reduced sequence, of the form either (1, 2, 1 ⋯) or (2, 1, 2 ⋯). Let I_{w} be that sequence. In this case, Lemma 7.1 says that

(7.2)  \underbrace{X_{1}X_{2}X_{1} ⋯}_{m times} − \underbrace{X_{2}X_{1}X_{2} ⋯}_{m times} = ∑_{w<w_{0}^{α_{1},α_{2}}} η_{w}^{α_{1},α_{2}}X_{I_{w}}, where η_{w}^{α_{1},α_{2}} ∈ S.

Remark 7.4. In [HMSZ, Proposition 5.8] several explicit formulas were given for the coefficients η_{w}^{α_{i},α_{j}} (i ≠ j) appearing in similar decompositions with coefficients on the right. Moreover, case by case it was shown that η_{w}^{α_{1},α_{2}} ∈ S for root data of types A_{2}, B_{2} and D_{2} (the G_{2} case was left open). The formula (7.2) provides a uniform proof of this fact for all types including G_{2}.

The following theorem describes the twisted formal group algebra Q_{W} and the formal affine Demazure algebra D_{F} in terms of generators and relations. It generalizes [HMSZ, Theorem 5.14].

Theorem 7.5. Let Σ ↪ Λ^{∨} be a root datum and let F be a formal group law over R. Assume that the formal group algebra S = R[[Λ]]_{F} is Σ-regular and the torsion index 𝔱 is regular in R. Let Q = Q_{R,Λ} denote the localization of S. Given a set of simple roots {α_{1}, …, α_{n}} with associated simple reflections {s_{1}, …, s_{n}}, let m_{i,j} denote the order of the product s_{i}s_{j} in the Weyl group.

Then elements q ∈ Q (resp. q ∈ S) and the Demazure elements X_{i} = X_{α_{i}} satisfy the following relations (∀i, j ∈ [n])

X_{i}q = Δ_{i}(q) + s_{i}(q)X_{i}, (7.3)

X_{i}^{2} = κ_{α_{i}}X_{i}, (7.4)

(7.5)  \underbrace{X_{i}X_{j}X_{i} ⋯}_{m_{i,j} times} − \underbrace{X_{j}X_{i}X_{j} ⋯}_{m_{i,j} times} = ∑_{w<w_{0}^{α_{i},α_{j}}} η_{w}^{α_{i},α_{j}}X_{I_{w}}, η_{w}^{α_{i},α_{j}} ∈ S.

Here w_{0}^{α_{1},α_{2}} is defined in Example 7.3. These relations together with the ring law in S and the fact that the X_{i} are R-linear form a complete set of relations in the twisted formal group algebra Q_{W} (resp. the formal affine Demazure algebra D_{F}).

Proof. The third relation follows from (7.2). The first two relations follow from [HMSZ, Theorem 5.14]. The presentation of the twisted formal group algebra Q_{W} follows from Theorem 5.14, loc. cit. The presentation of the formal affine Demazure algebra D_{F} follows similarly, because of the fact that η_{w}^{α_{i},α_{j}} ∈ S. □

Corollary 7.6. If 𝔱 is regular in R and S is Σ-regular, then for any sequence I, we can write X_{I} = ∑_{v∈W} a_{I,I_{v}}X_{I_{v}} for some a_{I,I_{v}} ∈ S such that:

(1) If I is a reduced decomposition of w ∈ W, then a_{I,I_{v}} = 0 unless v ≤ w, and a_{I,I_{w}} = 1.

(2) If I is not reduced, then a_{I,I_{v}} = 0 for all v such that ℓ(v) ≥ |I|. and this decomposition is unique.

Proof. Uniqueness holds because the (X_{I_{w}})_{w∈W} are linearly independent since they are linearly independent over Q, and the map S → Q is injective. (1) holds by Theorem 7.1 with I′ = I_{w}. If I is not reduced, then we proceed by induction on its length, which is at least 2. We write I as (i) ∪ I′. If I′ is not reduced we are done by induction and applying identity (7.3) to move coefficients to the left of X_{i}. If I′ is a reduced sequence of w′ ∈ W, then

X_{i}X_{I′} =^{(1)} X_{i} ∑_{v≤w′} a_{v}X_{I_{v}} =^{(7.3)} ∑_{v<w′} X_{i}(a_{v})X_{I_{v}} + X_{i}X_{I_{w′}} + ∑_{v<w′} a_{v}X_{i}X_{I_{v}}.

By induction, the third term is irrelevant. Since (i) ∪ I_{w′} is not reduced, there is a reduced sequence I′′ of w′ with first term i by the exchange property [Bo68, Ch. IV, §1, no 5, Prop. 4]. Using part (1), and identity (7.4), X_{i}X_{I_{w′}} also decomposes as a linear combination of X_{I_{v}} with v of length at most ℓ(I_{w′}) = |I′| and we are done. □

As an immediate consequence we obtain the following generalization of [HMSZ, Lem. 5.13] and [KK86, Thm. 4.6.(a)].

Proposition 7.7. If 𝔱 is regular in R and S is Σ-regular, then the R-algebra D_{F} is free as a left S-submodule of Q_{W}, with basis (X_{I_{w}})_{w∈W}.

Proof. We’ve already seen that the (X_{I_{w}}) are linearly independent. Let P denote the left S-submodule of Q_{W} generated by {X_{I_{w}}}_{w∈W}. By definition we have P ⊆ D_{F}. It suffices to prove that P is a R-subalgebra, which reduces to showing that X_{i}qX_{I_{w}} ∈ P for any i ∈ [n], w ∈ W and q ∈ S. Using (7.3), it reduces further to showing that X_{i}X_{I_{w}} ∈ P, which holds by Corollary 7.6. □

Remark 7.8. Under the hypotheses of 6.2, Proposition 6.2 and Proposition 7.7 imply that D_{F} = D̃_{F}.

We now explain the relationship between the algebras D_{F}, D̃_{F}, Q_{W} and R-linear endomorphisms of Q and S. Following [CPZ, Definition 4.5] let D(Λ)_{F} denote the subalgebra of End_{R}(S) generated by Demazure operators Δ_{α} for all α ∈ Σ and left multiplications by elements in S.

Let ϕ: Q_{W} → End_{R}(Q) be the R-algebra homomorphism induced by the left action of Q_{W} on Q. By definition the formal group algebra S acts on the left on both R-algebras, ϕ is S-linear, and ϕ(X_{α}) = Δ_{α} for each α ∈ Σ.

We then have the following commutative diagram, where Stab(S) is the Rsubalgebra of End_{R}(Q) mapping S into itself, res_{S} is the restriction of an endomorphism of Q to S and ϕ_{D_{F}} is the restriction of ϕ to D_{F}

\xymatrix{D_{F} \ar@{->>}[dd]_{ϕ_{D_{F}}} & ⊆ & D̃_{F} \ar[d] & ⊆ & Q_{W} \ar[d]^{ϕ} \\  &  & Stab(S) \ar[d] & ⊆ & End_{R}(Q) \ar[d]^{_{res_{S}}} \\ D(Λ)_{F} & ⊆ & End_{R}(S) & ⊆ & Hom_{R}(S, Q)}

(7.6)

Observe that the image of ϕ_{D_{F}} is D(Λ)_{F} since R-algebra generators of D_{F} map to R-algebra generators of D(Λ)_{F}.

Theorem 7.9. In Diagram (7.6), if 𝔱 is regular in R and S is Σ-regular, then the map res_{S} ∘ ϕ is injective and, therefore, ϕ: Q_{W} → End_{R}(Q) is injective and the map ϕ_{D_{F}} : D_{F} → D(Λ)_{F} is an isomorphism.

Proof. Since Q_{W} is a free Q-module, it injects into the respective twisted formal group algebra over R[1/𝔱], so we can assume that the torsion index is invertible in R. Let y be an element acting by zero on S. We can express it as y = ∑_{v∈W} a_{v}X_{I_{v}} by Corollary 5.6. Applying y successively to all Δ_{I_{w}}(u_{0}), w ∈ W, we conclude that a_{v} = 0 for each v ∈ W by Lemma 6.1. This shows that res_{S} ∘ ϕ is injective, and the other two facts follow by diagram chase. □

Remark 7.10. Note that by [CPZ, Theorem 4.11], once reduced sequences I_{w} have been chosen for all w ∈ W, the family {Δ_{I_{w}}}_{w∈W} forms a basis of D(Λ)_{F} as a S-module, so the S-module P ⊆ D_{F} from the proof of Proposition 7.7 already surjects to D(Λ)_{F} via ϕ_{D_{F}}, even if 𝔱 is not regular in R.

8. Coproduct on the twisted formal group algebra

In the present section we introduce and study various (co)products on the twisted formal group algebra Q_{W}. We follow the ideas of [KK86, (4.14)] and [KK90, (2.14)]. In this section we don’t assume that S is Σ-regular.

Let W be a group and let R be a commutative ring. The group ring R[W] is a Hopf R-algebra endowed with a cocommutative coproduct △_{R[W]} : R[W] → R[W] ⊗ R[W] defined by δ_{w} ↦ δ_{w} ⊗ δ_{w}. Its counit ε_{R[W]} sends any δ_{w} to 1 (unless otherwise specified, all modules and tensor products are over R).

Assume that W acts on Q by means of R-linear automorphisms. Let Q_{W} = Q ⊗ R[W] be the twisted formal group algebra considered in §5. Let υ_{Q}: Q → Q_{W} and υ_{R[W]} : R[W] → Q_{W} denote the natural inclusions of R-algebras defined by q ↦ q ⊗ 1 and δ_{w} ↦ 1 ⊗ δ_{w} respectively.

Definition 8.1. We define a coproduct △_{Q_{W}} : Q_{W} → Q_{W} ⊗ Q_{W} to be the composition of R-linear morphisms

Q ⊗ R[W] ⟶^{id_{Q}⊗△_{R[W]}} Q ⊗ R[W] ⊗ R[W] ⟶^{id_{Q⊗R[W]}⊗υ_{R[W]}} (Q ⊗ R[W]) ⊗ (Q ⊗ R[W]).

We endow the R-module Q_{W} ⊗Q_{W} with the usual tensor product algebra structure, i.e. the product is given by (z_{1} ⊗ z′_{1}) · (z_{2} ⊗ z′_{2}) ↦ (z_{1}z_{2} ⊗ z′_{1}z′_{2}).

Lemma 8.2. The coproduct △_{Q_{W}} is associative and is a morphism of R-algebras, i.e. △_{Q_{W}} (z · z′) = △_{Q_{W}} (z) · △_{Q_{W}} (z′).

Proof. By linearity, it suffices to check these properties on elements of the form qδ_{w} with q ∈ Q and w ∈ W. By definition we have △_{Q_{W}} (qδ_{w}) = qδ_{w} ⊗ δ_{w} Therefore, we obtain

△_{Q_{W}} (qδ_{w} · q′δ_{w′}) = △_{Q_{W}} (qw(q′)δ_{ww′}) = qw(q′)δ_{ww′} ⊗ δ_{ww′} =
qδ_{w} · q′δ_{w′} ⊗ δ_{w} · δ_{w′} = △_{Q_{W}} (qδ_{w}) · △_{Q_{W}} (q′δ_{w′}).  □

Remark 8.3. The coproduct △_{Q_{W}} has no counit in general, and it is not cocommutative, because coefficients in Q are arbitrarily put on the left.

Lemma 8.4. The map υ_{R[W]} is a map of R-coalgebras (without counits), i.e. △_{Q_{W}} ∘ υ_{R[W]} = (υ_{R[W]} ⊗ υ_{R[W]}) ∘ △_{R[W]}.

Proof. It is straightforward on R-basis elements δ_{w} of R[W]. □

Lemma 8.5. The map id_{Q} ⊗ △_{R[W]} : Q_{W} → Q_{W} ⊗ R[W] is a morphism of Ralgebras (with units), and so is the coproduct △_{Q_{W}}.

Proof. The first claim can be checked directly on elements qδ_{w}, and the second one follows since the second map in the composition defining △_{Q_{W}} is a morphism of R-algebras. □

Remark 8.6. Observe that the product in Q_{W} is, therefore, a morphism of Rcoalgebras (without units), since it is the same condition. Similarly, the unit of Q_{W} preserves coproducts.

We now define a coproduct on Q_{W} viewed as a left Q-module. Beware that Q is not central in Q_{W}, so Q_{W} is not a Q-algebra in the usual sense. The convention is still that unlabeled tensor products are over R.

Note that the tensor product Q_{W} ⊗_{Q} Q_{W} of left Q-modules has a natural structure of left Q-module via action q · (z ⊗ z′) = qz ⊗ z′ = z ⊗ qz′.

Let π : Q_{W} ⊗ Q_{W} → Q_{W} ⊗_{Q} Q_{W} be the projection map. It is a morphism of left Q-modules for both actions of Q on the source, on the left or right factor. The Q-submodule ker(π) is generated by elements of the form qz ⊗ z′ − z ⊗ qz′ = (q ⊗ 1 − 1 ⊗ q) · (z ⊗ z′). Let ι = id_{Q_{W}} ⊗ υ_{R[W]}. The morphism of left Q-modules

ψ = π ∘ ι: Q_{W} ⊗ R[W] → Q_{W} ⊗_{Q} Q_{W}

is an isomorphism.

Definition 8.7. We define the product ⊙ on Q_{W} ⊗_{Q} Q_{W} by transporting the product of the tensor product algebra Q_{W} ⊗ R[W] through the isomorphism ψ, i.e.

z ⊙ z′ := ψ(ψ^{−1}(z) · ψ^{−1}(z′)) for z, z′ in Q_{W} ⊗_{Q} Q_{W}.

Remark 8.8. Note that the morphism ψ is a morphism of R-algebras, and so is ι, but π does not respect products in general. We therefore have an inclusion ι of R-algebras, that is split by ψ^{−1} ∘ π, a map of Q-modules, but not a map of R-algebras. Still, π restricted to im(ι) is a morphism of R-algebras, because it is the inverse of the morphism of R-algebras ι.

Definition 8.9. We define a coproduct △: Q_{W} → Q_{W} ⊗_{Q} Q_{W} on Q_{W} viewed as a Q-module to be the morphism of left Q-modules given by the composition π ∘ △_{Q_{W}} = ψ ∘ (id_{Q} ⊗ △_{R[W]}).

We therefore have △(qδ_{w}) = qδ_{w} ⊗ δ_{w} = δ_{w} ⊗ qδ_{w}.

Proposition 8.10. The coproduct △ is Q-linear, associative and cocommutative, with counit ε : Q_{W} → Q defined by x ↦ x(1) (action of Q_{W} on Q defined in section 6). Therefore, Q_{W} is a cocommutative coalgebra in the category of left Q-modules

Proof. Since all maps involved are morphisms of left Q-modules, it suffices to check it on Q-basis elements δ_{w} of Q_{W}. All three properties are then straightforward. □

Remark 8.11. The counit ε satisfies that for any q ∈ Q and any x ∈ Q_{W}, we have ε(xq) = x(q).

Lemma 8.12. The coproduct △ is a morphism of R-algebras.

Proof. We have △ = ψ ∘(id_{Q} ⊗△_{R[W]}) and both maps are maps of R-algebras. □

Lemma 8.13. The kernel ker(π) is a right ideal, and is also stable by multiplication on the left by elements in the image of △_{Q_{W}}. In particular ker(π) ∩ im(△_{Q_{W}}) is a double-sided ideal of the R-subalgebra im(△_{Q_{W}}).

Proof. It is a right ideal, since it is additively generated by elements of the form (q ⊗ 1 − 1 ⊗ q) · (z ⊗ z′), with q ∈ Q and z, z′ ∈ Q_{W}. To prove the second part of the claim, it suffices to prove that any element of the form △_{Q_{W}} (x) · (q ⊗ 1 − 1 ⊗ q) with q ∈ Q and x ∈ Q_{W} is a sum of elements of the form (q′ ⊗ 1 − 1 ⊗ q) · △_{Q_{W}} (x) for some q′ ∈ Q and x′ ∈ Q_{W}. It is enough to check it when x = q_{w}δ_{w}, in which case one has q′ = w(q). □

Remark 8.14. Note that this left stability uses the cocommutativity of △_{R[W]} in an essential way.

Corollary 8.15. The R-submodule im(△_{Q_{W}}) + ker(π) ⊆ Q_{W} ⊗ Q_{W} is an Rsubalgebra and the morphism π restricted to im(△_{Q_{W}}) + ker(π) is a morphism of R-algebras.

Proposition 8.16. For elements y ∈ Q_{W} ⊗ R[W] and z ∈ Q_{W} ⊗ Q_{W}, we have

(8.1)  π(z) ⊙ ψ(y) = π(z · ι(y))

and if ι(y) · ker(π) ⊆ ker(π), we also have

(8.2)  ψ(y) ⊙ π(z) = π(ι(y) · z).

In particular, for any x ∈ Q_{W}, we have

(8.3) π(z) ⊙ △(x) = π(z · △_{Q_{W}} (x)) and △(x) ⊙ π(z) = π(△_{Q_{W}} (x) · z).

Proof. We have

π(z) ⊙ ψ(y) = ψ(ψ^{−1}π(z) · y) = πι(ψ^{−1}π(z) · y)

= π(ιψ^{−1}π(z) · ι(y)) = π((z + ker(π)) · ι(y)) = π(z · ι(y) + ker(π)) = π(z · ι(y)).

Here we have used that ker(π) is a right ideal. Similarly, reversing the product we obtain (8.2). For identity (8.3), just set y = (id_{Q} ⊗ △_{R[W]})(x) and use Lemma 8.13 to satisfy the extra assumption. □

9. Coproduct on the formal affine Demazure algebra

In the present section we define the coproduct on the formal affine Demazure algebra D_{F} by restricting the coproduct △ on Q_{W} (see Theorem 9.2) and compute it for Demazure elements (see Proposition 9.5).

All results of the present section assume that 𝔱 is regular in R and the formal group algebra S is Σ-regular (Definition 4.4).

Lemma 9.1. The induced map D_{F} ⊗_{S} D_{F} → Q_{W} ⊗_{Q} Q_{W} of left S-modules is injective.

Proof. The S-module D_{F} ⊗_{S} D_{F} is free with basis (X_{I_{v}} ⊗X_{I_{w}})_{(v,w)∈W^{2}}, by Proposition 7.7. On the other hand this basis is also a Q-basis of Q_{W} ⊗_{Q} Q_{W} by Corollary 5.6. Since S is Σ-regular, by Lemma 3.2, S injects in Q and we are done. □

We can therefore identify D_{F} ⊗_{S} D_{F} with the S-submodule image π(D_{F} ⊗ D_{F}) in Q_{W} ⊗_{Q} Q_{W}. Beware, however, that through this identification, the product ⊙ does not correspond to the usual product of D_{F} ⊗D_{F}, and D_{F} ⊗_{S} D_{F} is not stable by the product ⊙.

By direct computation we obtain in Q_{W} ⊗ Q_{W} that △_{Q_{W}} (X_{α}) =

\begin{matrix}X_{α} ⊗ 1 + 1 ⊗ X_{α} − x_{α}X_{α} ⊗ X_{α} + (\frac{1}{x_{α}} ⊗ 1 − 1 ⊗ \frac{1}{x_{α}})(δ_{s_{α}} ⊗ 1 − δ_{s_{α}} ⊗ δ_{s_{α}}) \\ = X_{α} ⊗ 1 + 1 ⊗ X_{α} − X_{α} ⊗ x_{α}X_{α} + (\frac{1}{x_{α}} ⊗ 1 − 1 ⊗ \frac{1}{x_{α}})(1 ⊗ 1 − 1 ⊗ δ_{s_{α}}). \end{matrix} ; (9.1)

It implies that in Q_{W} ⊗_{Q} Q_{W} we have

(9.2)  △(X_{α}) = X_{α} ⊗ 1 + 1 ⊗ X_{α} − x_{α}X_{α} ⊗ X_{α}.

Theorem 9.2. The coproduct △: Q_{W} → Q_{W} ⊗_{Q} Q_{W} restricts to an S-linear coproduct △: D_{F} → D_{F} ⊗_{S} D_{F} with counit ε: D_{F} → S obtained by restricting the counit ε: Q_{W} → Q. One has ε(X_{I_{w}}) = δ_{w,1} (the Kronecker symbol).

Proof. Since △ is Q-linear and is a morphism of R-algebras, it suffices to show that △(X_{α}) ⊙ π(z) ∈ π(D_{F} ⊗ D_{F}) for any root α and any z ∈ D_{F} ⊗ D_{F} (in particular for z = 1 ⊗ 1). We thus compute

△(X_{α}) ⊙ π(z) =^{(8.3)} π(△_{Q_{W}} (X_{α}) · z)

= π((1 ⊗ X_{α} + X_{α} ⊗ 1 − x_{α}X_{α} ⊗ X_{α} + ker(π)) · z)

=^{Lem. 8.13} π((1 ⊗ X_{α} + Δ_{α} ⊗ 1 − x_{α}X_{α} ⊗ X_{α}) · z + ker(π))

= π((1 ⊗ X_{α} + Δ_{α} ⊗ 1 − x_{α}X_{α} ⊗ X_{α}) · z)

and the last term is in π(D_{F} ⊗D_{F}), since the product is term by term in Q_{W} ⊗Q_{W}.

The counit ε on Q_{W} obviously maps D_{F} to S by remark 8.11 since D_{F} ⊆ D̃_{F}. One has ε(X_{I_{w}}) = Δ_{I_{w}}(1) = δ_{w,1}. □

Given a sequence I in [n] we want to provide a formula for the coproduct of the element X_{I}. By [HMSZ, Lemma 5.5] we have the following relation

(9.3)  X_{α} · q = Δ_{α}(q) + s_{α}(q) · X_{α}, for all q ∈ Q and α ∈ Σ.

and, more generally, we obtain

Lemma 9.3. For any sequence I and for any q ∈ Q, we have

(9.4)  X_{I} · q = ∑_{E⊆[l]} ϕ_{I,E}(q) · X_{I_{|E}}

where ϕ_{I,E} = B_{1} ∘ ⋯ ∘ B_{l} with

B_{j} = \begin{cases}B_{i_{j}}^{(0)}, & if j ∈ E, \\ B_{i_{j}}^{(−1)} & otherwise. \end{cases}

Proof. Induction using (9.3). □

Lemma 9.4. For any sequence I = (i_{1}, …, i_{l}) in [n] we have in Q_{W} ⊗ Q_{W}

(9.5) ∏_{k=1}^{l}(X_{i_{k}} ⊗ 1 + 1 ⊗ X_{i_{k}} − x_{i_{k}}X_{i_{k}} ⊗ X_{i_{k}}) = ∑_{E_{1},E_{2}⊂[l]}(p_{E_{1},E_{2}}^{I}X_{I_{|E_{1}}} ) ⊗ X_{I_{|E_{2}}}

where the element p_{E_{1},E_{2}}^{I} ∈ S is defined in Lemma 4.8.

Proof. We prove the formula by induction on the length l of I. For l = 1, it holds by identity (9.2). Let I′ = (i_{2}, …, i_{l}). For fixed E_{1}, E_{2} ⊆ [l − 1], and for p ∈ S we have (1 ⊗ X_{i_{1}} + X_{i_{1}} ⊗ 1 − x_{i_{1}}X_{i_{1}} ⊗ X_{i_{1}}) · ((pX_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}} )

= (pX_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} ) + (X_{i_{1}}pX_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}} − (x_{i_{1}}X_{i_{1}}pX_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} )

=^{(9.3)} (pX_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} ) + (Δ_{i_{1}}(p)X_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}} + (s_{i_{1}}(p)X_{i_{1}}X_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}}

− (x_{i_{1}}Δ_{i_{1}}(p)X_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} ) − (x_{i_{1}}s_{i_{1}}(p)X_{i_{1}}X_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} )

= (Δ_{i_{1}}(p)X_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}} + (s_{i_{1}}(p)X_{i_{1}}X_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}}

+ (s_{i_{1}}(p)X_{i_{1}}X_{I′_{|E_{1}}} ) ⊗ X_{I′_{|E_{2}}} − (x_{i_{1}}s_{i_{1}}(p)X_{i_{1}}X_{I′_{|E_{1}}} ) ⊗ (X_{i_{1}}X_{I′_{|E_{2}}} )

and the result follows by induction as in the end of the proof of Lemma 4.8. □

Proposition 9.5 (cf. Proposition 4.15 [KK86]). For any I = (i_{1}, …, i_{l}) we have

(9.6)  △(X_{I}) = ∑_{E_{1},E_{2}⊂[l]} p_{E_{1},E_{2}}^{I}X_{I_{|E_{1}}} ⊗ X_{I_{|E_{2}}} .

Proof. We have △(X_{I}) = △(X_{i_{1}}) ⊙ ⋯ ⊙ △(X_{i_{l}}) = π△_{Q_{W}} (X_{i_{1}}) ⊙ ⋯ ⊙ π△_{Q_{W}} (X_{i_{l}})

=^{(9.2)} π(1 ⊗ X_{i_{1}} + X_{i_{1}} ⊗ 1 − x_{i_{1}}X_{i_{1}} ⊗ X_{i_{1}}) ⊙ ⋯ ⊙ π(1 ⊗ X_{i_{l}} + X_{i_{l}} ⊗ 1 − x_{i_{l}}X_{i_{l}} ⊗ X_{i_{l}})

=^{Cor. 8.15} π((1 ⊗ X_{i_{1}} + X_{i_{1}} ⊗ 1 − x_{i_{1}}X_{i_{1}} ⊗ X_{i_{1}}) ⋯ (1 ⊗ X_{i_{l}} + X_{i_{l}} ⊗ 1 − x_{i_{l}}X_{i_{l}} ⊗ X_{i_{l}}))

=^{(9.5)} ∑_{E_{1},E_{2}⊂[l]} p_{E_{1},E_{2}}^{I}X_{I_{|E_{1}}} ⊗ X_{I_{|E_{2}}} . □

Remark 9.6. Note that the only subtlety in the proof of the previous formula is to check that products of elements △(X_{i}) can indeed be computed term-wise in the tensor product, which is the content of Corollary 8.15.

10. Coproduct on the algebra of Demazure operators

In the present section we lift the coproduct on ϵD(Λ)_{F} defined in [CPZ] to a natural coproduct on D(Λ)_{F}, and show it is isomorphic to the coproduct △ on D_{F} through the isomorphism ϕ_{D_{F}} of Theorem 7.9.

Unless otherwise specified, all tensor products are still over R, all Hom and End groups are homomorphisms of R-modules. When M and N are topological Rmodules, let Hom^{0}(N, M) (resp. End^{0}(M)) denote the R-submodule of Hom(N, M) (resp. of End(M)) of continuous homomorphisms. When M is a topological module over S, and R acts through S, Hom(N, M) has an obvious structure of left S-module, by s · f = (x ↦ sf(x)). Then Hom^{0}(N, M) is an S-submodule of Hom(N, M), and in particular End^{0}(S) is an S-submodule of End(S). By construction, D(Λ)_{F} is an S-submodule of End^{0}(S). Let Υ_{1} : D(Λ)_{F} → End^{0}(S) denote the inclusion.

Let S ⊗S (resp. S ⊗S ⊗S) be endowed with the topology defined by the I_{F}^{i} ⊗I_{F}^{j}, i, j ∈ ℕ (by the I_{F}^{i} ⊗ I_{F}^{j} ⊗ I_{F}^{k}, i, j, k ∈ ℕ) as a basis of neighborhoods of zero. Let

\begin{aligned}Υ_{2} : & D(Λ)_{F} ⊗_{S} D(Λ)_{F} & → & Hom^{0}(S ⊗ S, S) \\ & D_{1} ⊗ D_{2} & ↦ & ((r ⊗ s) ↦ D_{1}(r)D_{2}(s)) \end{aligned}

and similarly

\begin{aligned}Υ_{3} : & D(Λ)_{F} ⊗_{S} D(Λ)_{F} ⊗_{S} D(Λ)_{F} & → & Hom^{0}(S ⊗ S ⊗ S, S) \\ & D_{1} ⊗ D_{2} ⊗ D_{3} & ↦ & ((r ⊗ s ⊗ t) ↦ D_{1}(r)D_{2}(s)D_{3}(t)). \end{aligned}

Lemma 10.1. If the torsion index 𝔱 is regular in R, the maps Υ_{1}, Υ_{2} and Υ_{3} are injective.

Proof. By definition, Υ_{1} is an inclusion. For Υ_{2}, by [CPZ, Theorem 4.11], once reduced sequences I_{w} have been chosen for all w ∈ W, the family {Δ_{I_{w}}}_{w∈W} forms a basis of D(Λ)_{F} as an S-module. Let us assume that ∑_{v,w} a_{v,w}Δ_{I_{v}}⊗Δ_{I_{w}} is mapped to the zero morphism. Thus, for any r, s ∈ S, we have ∑_{w} ∑_{v} a_{v,w}Δ_{I_{v}}(r)Δ_{I_{w}}(s) = 0. Replacing successively s by Δ_{I_{w′}} (u_{0}) for all w′ ∈ W, we obtain that for any w′, ∑_{w} ζ_{w}(r)Δ_{I_{w}}(Δ_{I_{w′}} (u_{0})) = 0 with ζ_{w}(r) = ∑_{v} a_{v,w}Δ_{I_{v}}(r). So, given w, for any r ∈ S, we have ζ_{w}(r) = 0 by Lemma 6.1. Replacing r successively by Δ_{I_{v′}} (u_{0}) for all v′ ∈ W, we similarly obtain a_{v,w} = 0.

The proof for Υ_{3} is the same, but with one extra step for the last factor. □

Another way of phrasing the previous lemma is that an element of D(Λ)_{F} ⊗_{S} D(Λ)_{F} is characterized by how it acts on S ⊗ S (with values in S), and similarly with three factors.

Lemma 10.2. If 𝔱 is regular in R, then the preimage by Υ_{1} of Hom^{0}(S, I_{F}) is I_{F} D(Λ)_{F}.

Proof. Let d ∈ D(Λ)_{F} such that d(s) ∈ I_{F} for any s ∈ S. Decompose d = ∑_{w} a_{w}Δ_{I_{w}} by [CPZ, Theorem 4.11]. After inverting the torsion index in R, by Lemma 6.1, we obtain that for any w ∈ W, the element a_{w} is in I_{F,R[1/𝔱]}, the augmentation ideal of R[\frac{1}{𝔱}][[Λ]]_{F}. Since I_{F,R[1/𝔱]} ∩ S = I_{F} inside R[\frac{1}{𝔱}][[Λ]]_{F}, we are done. □

Let χ : End^{0}(S) → Hom^{0}(S ⊗ S, S) be the morphism of left S-modules sending f ∈ End^{0}(S) to ((u ⊗ v) ↦ f(uv)).

Theorem 10.3. Assume that 𝔱 is regular. Along Υ_{2}, the morphism χ factors (uniquely) through a morphism of left S-modules △_{D(Λ)_{F}} : D(Λ)_{F} → D(Λ)_{F} ⊗_{S} D(Λ)_{F}, which is thus the unique such morphism satisfying the formula

(10.1)  Υ_{2}(△_{D(Λ)_{F}} (f))(u ⊗ v) = f(uv).

It is an associative and commutative S-linear coproduct, with counit ε_{D(Λ)_{F}} mapping f to f(1) (evaluation at 1).

Proof. Since D(Λ)_{F} ⊗_{S} D(Λ)_{F} injects by Υ_{2} in Hom^{0}(S ⊗S, S), uniqueness is clear. Now χ restricts by [CPZ, Lemma 7.1 (1)]. Associativity follows from the fact that both compositions involved D(Λ)_{F} → D(Λ)_{F} ⊗_{S} D(Λ)_{F} ⊗_{S} D(Λ)_{F} composed with Υ_{3} send D ∈ D(Λ)_{F} to ((u, v, w) ↦ D(uvw)), so they are equal by injectivity of Υ_{3}. Commutativity follows similarly with Υ_{2} instead of Υ_{3}, and the fact that ε_{D(Λ)_{F}} is a (left) counit holds because (ε_{D(Λ)_{F}} ⊗ id_{S}) ∘ △_{D(Λ)_{F}} (f) sends f to (v ↦ f(1 · v) = f(v)). It is also a right counit by the same argument on the right side. □

Theorem 10.4. When the torsion index 𝔱 is regular in R and S is Σ-regular, the map ϕ_{D_{F}} : D_{F} → D(Λ)_{F} of Theorem 7.9 is an isomorphism of S-coalgebras with counits.

Proof. Since 𝔱 is regular, the (X_{I_{w}})_{w∈W} are an S-basis of D_{F}. The (Δ_{I_{w}})_{w∈W} are an S-basis of D(Λ)_{F} in any case. The identification of △ and △_{D(Λ)_{F}} through ϕ_{D_{F}} then follows from formulas (9.6) and (4.1), via the uniqueness of Theorem 10.3. □

The coproduct that appears in [CPZ] is then constructed as follows. The augmentation map ϵ : R[[Λ]]_{F} → R induces a map of R-modules ϵ_{∗}: D(Λ)_{F} → Hom_{R}(S, R) given by f ↦ ϵ ∘ f. Let ϵD(Λ)_{F} be the image of D(Λ)_{F} by this map. As in the proof of [CPZ, Thm. 7.3] we define an R-linear coproduct

△^{ϵ}: ϵD(Λ)_{F} → ϵD(Λ)_{F} ⊗_{R} ϵD(Λ)_{F}

by

△^{ϵ}(f)(u ⊗ v) := f(uv), f ∈ ϵD(Λ)_{F} , u, v ∈ R[[Λ]]_{F} .

Here the property (which follows from [CPZ, Prop. 3.8])

ϵΔ_{i}(u · v) = ϵΔ_{i}(u) · ϵ(v) + ϵ(u) · ϵΔ_{i}(v), u, v ∈ R[[Λ]]_{F}

guarantees that the coproduct △^{ϵ} is well defined (cf. [CPZ, Lem. 7.1]) and

△^{ϵ}(ϵΔ_{i}) = ϵΔ_{i} ⊗ ϵ + ϵ ⊗ ϵΔ_{i}.

Through ϵ : S → R, we can restrict ϵD(Λ)_{F} to an S-module. Then ϵD(Λ)_{F} ⊗_{R} ϵD(Λ)_{F} = ϵD(Λ)_{F} ⊗_{S} ϵD(Λ)_{F}, and △^{ϵ} is an S-linear coproduct. The morphism ϵ_{∗} : D(Λ)_{F} → ϵD(Λ)_{F} is clearly a morphism of S-coalgebras. In other words, the diagram

\xymatrix{D_{F} \ar[r]^{ϕ_{D_{F}}} \ar[d]^{△} & D(Λ)_{F} \ar[r]^{ϵ_{∗}} \ar[d]^{△_{D(Λ)_{F}}} & ϵD(Λ)_{F} \ar[d]^{△^{ϵ}} \\ D_{F} ⊗_{S} D_{F} \ar[r]^{ϕ_{D_{F}}^{⊗2}} & D(Λ)_{F} ⊗_{S} D(Λ)_{F} \ar[r]^{ϵ_{∗}^{⊗2}} & ϵD(Λ)_{F} ⊗_{R} ϵD(Λ)_{F}}

(10.2)

commutes.

Remark 10.5. The significance of the coproduct △^{ϵ} follows from the following fact:

Following [CPZ, §6] consider the dual ϵD(Λ)_{F}^{∗} = Hom_{R}(ϵD(Λ)_{F} , R) (in loc. cit. ϵD(Λ)_{F}^{∗} = H(M)_{F}). Then by [CPZ, Thm. 7.3 and Thm. 13.12] the coproduct △^{ϵ} on ϵD(Λ)_{F} induces (by duality) an R-algebra structure on ϵD(Λ)_{F}^{∗}. Moreover, under the hypotheses of [CPZ, Thm. 13.3] there is a natural ring isomorphism ϵD(Λ)_{F}^{∗} ≃ h(G/B), where h(G/B) is the algebraic oriented cohomology of the variety of complete flags G/B.

11. The dual of a formal affine Demazure algebra

In the present section we consider the dual (D(Λ)_{F})^{⋆} of the algebra of Demazure operators, or equivalently the dual of the formal affine Demazure algebra D_{F}^{⋆} and prove Theorem 11.4.

We will need the following classical facts, that the reader can easily verify anyway. Let S be a commutative unital ring and let D be a free left S-module of finite

type. Assume that D is an S-coalgebra, with linear coproduct △: D → D ⊗_{S} D and counit ε : D → S. Then the dual D^{⋆} = Hom_{S}(D, S) has a natural structure of S-algebra with unit, given by dualizing the coproduct and using the natural isomorphism D^{⋆} ⊗_{S} D^{⋆} ≃ (D ⊗_{S} D)^{⋆}, and dualizing the counit, i.e. ε, seen as an element of D^{⋆}, is the unit. It is commutative if and only if D is cocommutative. Let p_{θ_{1},θ_{2}}^{θ} ∈ S be the coefficients of the coproduct on a basis {x_{θ}}_{θ∈Θ} of D (Θ is therefore finite), i.e. for every θ ∈ Θ,

△(x_{θ}) = ∑_{θ_{1},θ_{2}∈Θ} p_{θ_{1},θ_{2}}^{θ}x_{θ_{1}} ⊗ x_{θ_{2}}.

On the dual basis (x_{θ}^{⋆})_{θ∈Θ} of (x_{θ})_{θ∈Θ}, the product of D^{⋆} is given by

x_{θ_{1}}^{⋆}x_{θ_{2}}^{⋆} = ∑_{θ∈Θ} p_{θ_{1},θ_{2}}^{θ}x_{θ}^{⋆}.

For the rest of this section, we assume that 2𝔱 is regular in R. It implies that S is Σ-regular by Lemma 2.2.

We apply the above construction to D = D(Λ)_{F}, with its coalgebra structure given by Theorem 10.3. This yields a commutative algebra with unit D(Λ)_{F}^{⋆}. Given choices of a reduced decompositions I_{w} for each element w ∈ W, we have a basis (Δ_{I_{w}})_{w∈W} of D(Λ)_{F}.

Remark 11.1. Instead of D(Λ)_{F}, we can use D_{F}, with its coalgebra structure given by Theorem 9.2 and its basis (X_{I_{w}})_{w∈W}. Since we have D_{F} ≃ D(Λ)_{F} as S-coalgebras by Theorem 10.4, we dually have (D(Λ)_{F})^{⋆} ≃ D_{F}^{⋆} as algebras, and it won’t make any difference. In particular, Theorem 11.4 also holds for D_{F}.

The R-dual (ϵD(Λ)_{F})^{∗} = Hom_{R}(ϵD(Λ)_{F} , R) is an R-algebra with unit by dualizing the coproduct △^{ϵ} of section 10. By [CPZ, Prop. 5.4], the ϵΔ_{I_{w}}, w ∈ W, form a basis of ϵD(Λ)_{F}. Using ϵ : S → R, we can restrict (ϵD(Λ)_{F})^{∗} to an S-algebra with unit.

Lemma 11.2. The map ξ : (D(Λ)_{F})^{⋆} → (ϵD(Λ)_{F})^{∗} sending f to the map ϵd ↦ ϵf(d) is well defined, and it is a surjective map of S-algebras with units.

Proof. If ϵd_{1} = ϵd_{2}, as elements of Hom^{0}(S, R), or in other words (d_{1} − d_{2}) ∈ Hom^{0}(S, I_{F}), then Lemma 10.2 shows that d_{1} − d_{2} ∈ I_{F} D(Λ)_{F}, and thus ϵf(d_{1}) = ϵf(d_{2}) in R for any f ∈ Hom_{S}(D(Λ)_{F} , S). This proves that ξ is well-defined.

It is clearly a morphism of S-modules. Checking that it respects the products is a simple exercise from their definitions as duals to the coproducts, using the commutativity of Diagram (10.2) and the fact that ϵ is a morphism of rings.

Finally, ξ is surjective, because it sends the (dual) basis Δ_{I_{w}}^{⋆} of (D(Λ)_{F})^{⋆} to the (dual) basis ϵΔ_{I_{w}}^{∗} of (ϵD(Λ)_{F})^{∗}. □

For any s ∈ S, let ev_{s} ∈ D_{F}^{⋆} be the map d ↦ d(s).

Lemma 11.3. The product on D(Λ)_{F} satisfies ev_{s_{1}} · ev_{s_{2}} = ev_{s_{1}s_{2}}.

Proof. By Theorem 10.3, the coproduct △_{D(Λ)_{F}} on D(Λ)_{F} satisfies formula (10.1). Thus

(ev_{s_{1}} · ev_{s_{2}})(d) = (ev_{s_{1}} ⊗ ev_{s_{2}})(△_{D(Λ)_{F}} (d)) = Υ_{2}(△_{D(Λ)_{F}} (d))(s_{1} ⊗ s_{2})
= d(s_{1}s_{2}) = ev_{s_{1}s_{2}}(d).

On D_{F}^{⋆}, the property then follows using Theorem 10.4. □

Let c_{S} : S → D(Λ)_{F}^{⋆} be the R-linear map sending s ∈ S to ev_{s}. By the previous lemma, it is actually an R-algebra map. Since the subring S^{W} ⊆ S ⊆ D(Λ)_{F} of elements fixed by the Weyl group W is central^{1} in D(Λ)_{F}, the map c_{S} is in fact S^{W} -linear.

The map c_{S} would correspond in cohomology to an equivariant characteristic map. In [CPZ, Def. 6.1], there is a (non-equivariant) characteristic map c_{R} : S → (ϵD(Λ)_{F})^{∗}, sending s ∈ S to ev_{s} : ϵd ↦ ϵd(s). They are related by c_{R} = ξ ∘ c_{S}. Note that c_{R} is a map of R-algebras, either by [CPZ, Theorem 7.3] or by Lemma 11.3 and the fact that ξ is a morphism of S-algebras and S acts on ϵD(Λ)_{F}^{∗} via R through ϵ.

The tensor product S ⊗_{S^{W}} S is an S-algebra via the first component. We now consider the S-algebra map

ρ_{S} : S ⊗_{S^{W}} S → D(Λ)_{F}^{⋆} , s_{1} ⊗ s_{2} ↦ s_{1}c_{S}(s_{2})

and the R-algebra map

ρ_{R} : R ⊗_{S^{W}} S → ϵD(Λ)_{F}^{∗} , r ⊗ s ↦ rc_{R}(s_{2}).

where R is considered an S^{W} -module via the augmentation map ϵ. These two maps are related by the commutative diagram

\xymatrix{S ⊗_{S^{W}} S \ar[r]^{ρ_{S}} \ar@{->>}[d]_{ϵ⊗id_{S}} & D(Λ)_{F}^{⋆} \ar@{->>}[d]^{ξ} \\ R ⊗_{S^{W}} S \ar[r]^{ρ_{R}} & (ϵD(Λ)_{F})^{∗}}

The following theorem generalizes [KK90, Thm.4.4] and [Ku02, Cor. 11.3.17] to the context of an arbitrary formal group law. It is also related to [KiKr, Theorem 4.7], see Remark 11.6.

Theorem 11.4. If 2𝔱 is regular in R, the following conditions are equivalent.

1D(Λ)_{F} is by definition generated by elements that all commute with S^{W}, see [CPZ, Prop.

3.14]. This is where we first use that 2 is regular.

(a) The map ρ_{S} : S ⊗_{S^{W}} S → D(Λ)_{F}^{⋆} is an isomorphism (of S-algebras).

(b) The map ρ_{R} : R ⊗_{S^{W}} S → (ϵD(Λ)_{F})^{∗} is an isomorphism (of R-algebras).

(c) The characteristic map c_{R} (of R-algebras) is surjective. In particular all are true if the torsion index 𝔱 is invertible, by [CPZ, Theorem 6.4].

Proof. By the above diagram, if ρ_{S} is surjective, then so is ρ_{R}. Furthermore, by definition of ρ_{R}, we have c_{R} = ρ_{R} ∘ (1 ⊗ id_{S}). Since the map 1 ⊗ id_{S} is surjective (because S^{W} surjects to R via ϵ), we have that c_{R} is surjective if and only if ρ_{R} is surjective. We now need the following lemma.

Lemma 11.5. Let N be the length of w_{0}, the longest element in W. If c_{R} is surjective, then there is an element u′_{0} ∈ S such that for any sequence I, we have

ϵΔ_{I}(u′_{0}) = \begin{cases}0, & if I is of length < N or is of length N but non reduced, \\ 1, & if I is of length N and is reduced. \end{cases}

Proof. Choose a reduced decomposition I_{w} for every w ∈ W. By definition of c_{R}, if it is surjective, there is an element u′_{0} such that ϵX_{I_{w}}(u′_{0}) = 1 if w = w_{0} and zero if not. Then, decomposing X_{I} as in Corollary 7.6, we have X_{I}(u′_{0}) = ∑_{v} ϵ(a_{I,I_{v}})ϵX_{I_{v}}(u′_{0}) = ϵ(a_{I,I_{w_{0}}} ) and the conditions of Corollary 7.6 imply the result. □

Returning to the proof of Theorem 11.4, let us conversely show that if c_{R} is surjective, then both ρ_{R} and ρ_{S} are isomorphisms. The lemma implies that the matrix (ϵX_{I_{v}}X_{I_{w}}(u′_{0}))_{(v,w)∈W ×W} (with coefficients in R) is invertible, since it is triangular with 1 on the diagonal. Thus, the matrix (X_{I_{v}}X_{I_{w}}(u′_{0}))_{(v,w)∈W ×W} (with coefficients in S) is invertible, since the kernel of ϵ : S → R is in the radical of S. As in [CPZ, Theorem 6.7], this implies that the Δ_{I_{w}}(u′_{0}) form a basis of S as an S^{W} -module. Thus, the map ρ_{S} (resp. ρ_{R}) is between two free S-modules (resp. Rmodules) of the same rank |W |, so it is an isomorphism if it is surjective. It suffices to show the surjectivity of ρ_{S}, since it implies that of ρ_{R}. The image ρ_{S}(1⊗X_{I_{v}}(u′_{0})) of a basis element of S ⊗_{S^{W}} S decomposes as ∑_{w} X_{I_{w}}X_{I_{v}}(u′_{0})X_{I_{w}}^{⋆} on the (dual) basis of (D(Λ)_{F})^{⋆}, so by invertibility of the above matrix, we are done. □

Remark 11.6. When F = U is the universal formal group law over the Lazard ring, by [KiKr, Thm. 5.1] the algebra S⊗_{S^{W}} S can be identified (up to a completion) with the equivariant algebraic cobordism Ω_{T} (G/B) after inverting 𝔱. Therefore, one may think of the dual algebra (D(Λ)_{F})^{⋆} or D_{U}^{⋆} as an algebraic model of Ω_{T} (G/B). In [KiKr, Theorem 4.6], a theorem similar to Theorem 11.4 is shown, using topological inputs and inverting the torsion index. Here, apart from the fact that the treatment is completely algebraic, the main improvement is that for other formal group laws, it might not be necessary to invert the torsion index for the equivalent conditions of Theorem 11.4 to hold. For example, in the case of the periodic multiplicative formal group law x+y−xy of K-theory, the characteristic map is already surjective over ℤ for simply connected root data [Pi72].

Remark 11.7. Note that in Lemma 11.5, the element u′_{0} has no reason to be in I_{F}^{N}. For example, for the periodic multiplicative formal group law x + y − xy over ℤ, for simply connected root data with non-trivial torsion index (ex: G_{2}^{sc}), such an element u′_{0} cannot be found in I_{F}^{N}.

Remark 11.8. In this section the algebra (D(Λ)_{F})^{⋆} can be replaced everywhere by D_{F}^{⋆}.

12. Appendix

Recall that an element of a commutative, associative unital ring R is called regular if it is not a zero divisor or, in other words, that multiplication by this element is an injective ring endomorphism.

Lemma 12.1. Let f be a zero divisor in the polynomial ring R[x_{1}, …, x_{n}]. Then there exists an element r ≠ 0 in R such that rf = 0.

Proof. See [MC42], Theorem 3 and the comment following it. □

Let I = (x_{1}, …, x_{n}) denote the ideal in the ring R[[x_{1}, …, x_{n}]] of formal power series generated by x_{1}, …, x_{n}.

Corollary 12.2. If f ∈ R[[x_{1}, …, x_{n}]] is a zero divisor, then there exists an element r ≠ 0 in R such that all coefficients of the lowest degree part of f are annihilated by r.

Proof. Take a nonzero g ∈ R[[x_{1}, …, x_{n}]] such that fg = 0. Let f = f_{i} + I^{i+1} and g = g_{j} + I^{j+1} where f_{i} and g_{j} are the lowest degree parts (homogeneous and nonzero). Then fg = f_{i}g_{j} + I^{i+j+1} and, hence, f_{i}g_{j} = 0. The result then follows by Lemma 12.1. □

Lemma 12.3. Let f = a_{1}x_{1}+⋯+a_{n}x_{n}+I^{2}, a_{i} ∈ R, be an element of R[[x_{1}, …, x_{n}]]. (a) If a_{i} is regular in R for some i, then f is regular in R[[x_{1}, …, x_{n}]].

(b) If the vector of coefficients (a_{1}, …, a_{n}) can be completed to a basis of R^{n}, then f is regular in R[[x_{1}, …, x_{n}]].

Proof. Assume that f is a zero divisor in R[[x_{1}, …, x_{n}]]. Then by Corollary 12.2 there is r ≠ 0 in R that annihilates each a_{i}. This contradicts the assumptions of (a) and (b). □

Observe that there exist examples of vectors consisting of zero divisors that can be completed to a basis, so that (a) does not imply (b).

Let A = (a_{ij}) be an n × n-matrix with coefficients in R. Consider a map Ψ_{A}: R[[x_{1}, …, x_{n}]] → R[[x_{1}, …, x_{n}]] defined by sending each x_{i} to an element of the form ∑ a_{ij}x_{j} + I^{2}.

Lemma 12.4. The matrix A is invertible if and only if the map Ψ_{A} is invertible.

Proof. See [Bo58, Ch. IV, §4, No 7, Prop. 10]. □

Corollary 12.5. If the determinant of the matrix A is regular, then the map Ψ is injective.

Proof. Let T be the total ring of fractions of R. There is a canonical injection R ↪ T. The matrix A viewed as a matrix with coefficients in T has an invertible determinant. Therefore, by Lemma 12.4 the respective map Ψ_{A}: T [[x_{1}, …, x_{n}]] → T [[x_{1}, …, x_{n}]] is invertible. The injectivity of Ψ_{A} over R then follows by the commutative diagram

\xymatrix{R[[x_{1}, …, x_{n}]] \ar[r]^{Ψ_{A}} \ar@{^{(}->}[d] & R[[x_{1}, …, x_{n}]] \ar@{^{(}->}[d] \\ T [[x_{1}, …, x_{n}]] \ar[r]^{Ψ_{A}}_{≃} & T [[x_{1}, …, x_{n}]]} .  □

By a lattice we call a finitely generated free Abelian group, i.e. a free ℤ-module of finite rank.

Lemma 12.6. Given an inclusion of lattices Λ ⊆ Λ′ of the same rank n, an n × n matrix A expressing a basis of Λ in terms of a basis of Λ′ satisfies det(A) = ±|Λ′/Λ|, the order of the group Λ′/Λ up to a sign.

Proof. It follows from [Bo58, Ch. VII, §4, No 6, Cor. 1]. □

An n-tuple of integers (k_{1}, …, k_{n}) is called a unimodular row if there exist integers (a_{1}, …, a_{n}) such that ∑_{i=1}^{n} k_{i}a_{i} = 1.

Lemma 12.7. Any unimodular row (k_{1}, …, k_{n}) can be completed to a basis of ℤ^{n}.

Proof. The kernel of the map ℤ^{n} → ℤ sending (a_{1}, …, a_{n}) to ∑_{i} k_{i}a_{i} is a direct factor of ℤ^{n}, and it is torsion free so it is free. □

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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, Ottawa Ontario, K1N6N5, Canada

   E-mail address: kirill@uottawa.ca

Changlong Zhong, Department of Mathematics and Statistics, University of Ottawa

   E-mail address: zhongusc@gmail.com