FORMAL HECKE ALGEBRAS

AND ALGEBRAIC ORIENTED COHOMOLOGY THEORIES

ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

Abstract. In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck’s K_{0}, connective Ktheory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.

Contents

Introduction 1 1. Formal group laws 4 2. Formal group algebras 6 3. Formal Demazure operators 8 4. Algebraic oriented cohomology theories and characteristic maps 9 5. Formal (affine) Demazure algebras: definitions 11 6. Formal (affine) Demazure algebras: examples and further properties 15 7. Formal (affine) Hecke algebras: definitions 17 8. Formal (affine) Hecke algebras: examples and further properties 21 References 25

Introduction

Geometric realizations of representations of algebras such as quantized enveloping algebras of Lie algebras and Hecke-type algebras have proved to be an exceptionally interesting and

2010 Mathematics Subject Classification. 20C08, 14F43. Key words and phrases. Hecke algebra, oriented cohomology, formal group law, Demazure operator. The work of the second two authors was supported by Discovery Grants from the Natural Sciences and

Engineering Research Council of Canada. The first two authors were supported by the Discovery Grants of the last two. The last author was also supported by an Early Researcher Award from the Government of Ontario.

2 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

useful tool in both representation theory and geometry. In particular, the field of geometric representation theory has produced such results as the proof of the Kazhdan-Lusztig conjecture and the construction of canonical bases in quantized enveloping algebras. Geometric realizations are also often a precursor to categorification, a current topic of great interest.

Two fundamental constructions in geometric representation theory are of particular relevance to the current paper. The first arises from so-called push-pull operators (coming from the projection from the flag variety G/B to the quotient G/P of G by a minimal parabolic) on the singular cohomology or K-theory (i.e. Grothendieck’s K_{0}) of the flag variety. If one works with singular cohomology, these operators generate the nil Hecke algebra. (When we use the term “nil Hecke algebra” here, we do not include the polynomial part.) If one works instead with K-theory, the push-pull operators generate the 0-Hecke algebra (the specialization of the Hecke algebra at q = 0). Adding the operators corresponding to multiplication by elements of the singular cohomology or K-theory, one obtains the affine analogues of the algebras above.

The above-mentioned algebras can also be realized in a more algebraic way. Let W be the Weyl group of a reduced root system, acting on the weight lattive Λ. In [KK86], Kostant and Kumar introduced a twisted group algebra Q_{W}, which is the smash product of the group ring ℤ[W] and the field of fractions Q of the polynomial ring S in Λ. Then they defined a subring R of Q_{W} generated by Demazure elements and elements of S and showed that R is similar to the 0-Hecke algebra: it satisfies the classical braid relation but a nilpotence relation instead of an idempotence one. For this reason, they called R the nil Hecke algebra. Following this approach, Evens and Bressler in [EB87] introduced the notion of a generalized Hecke ring (where the nilpotence/idempotence relation is replaced by a general quadratic one) which includes both 0-Hecke and nil Hecke algebras as examples. The Demazure elements play the role of a Hecke basis and have several geometric interpretations (as Demazure operators and push-pull operators) on the singular cohomology of the variety of Borel subgroups associated to the root system.

The second geometric construction relevant to the current paper is the realization of Hecke-type algebras via the geometry of the Steinberg variety. There is a natural structure of an algebra on the (co)homomology of the Steinberg variety via convolution. Again, the resulting algebra depends on the choice of (co)homology theory. Equivariant K-theory yields the affine Hecke algebra, equivariant singular cohomology yields the degenerate affine Hecke algebra, and top degree Borel-Moore homology yields the group algebra of the Weyl group. We refer the reader to [CG10, Gin] and the references therein for further details.

Another important ingredient of the current paper originates from algebraic topology and, especially, the cobordism theory of the 60s. The notion of an algebraic oriented cohomology theory was introduced by Levine-Morel [LM07] and by Panin-Smirnov [Pan03]. Roughly speaking, it is a cohomological-type functor h from the category of smooth algebraic varieties over a field to the category of commutative rings endowed with push-forward maps and characteristic classes (see §4). Basic examples of such functors are the Chow ring of algebraic cycles modulo rational equivalence, the Grothendieck group K_{0}, and the algebraic cobordism of Levine-Morel (see [LM07, §1.1] and [Pan03, §§2.1, 2.5, 3.8] for further examples). The theory of formal group laws has been used extensively in topology, especially in cobordism theory and, more generally, for studying topological oriented theories. The link between oriented cohomology and formal group laws is given by the Quillen formula expressing the first characteristic class c_{1}^{h} of a tensor product of two line bundles, c_{1}^{h}(L_{1}⊗L_{2}) = F (c_{1}^{h}(L_{1}), c_{1}^{h}(L_{2})) (see [LM07, Lem. 1.1.3]), where F is the one-dimensional commutative formal group law associated to h.

In [CPZ], the authors generalized the notions of Demazure operators and push-pull operators to the context of an arbitrary algebraic oriented cohomology theory h and the associated formal group law F, e.g. replacing singular cohomology by connective K-theory, elliptic cohomology or algebraic cobordism. It is thus natural to ask if one can extend the Kostant-Kumar construction of the nil Hecke algebra and the convolution construction of the affine Hecke algebra to the setting of an arbitrary algebraic oriented cohomology theory or formal group law. In the present paper we provide an affirmative answer to the first question. We also define algebras that we believe should be related to the more general convolution algebras of the second question.

Given a formal group law F (corresponding to some algebraic oriented cohomology theory), we introduce the notion of a twisted formal group algebra Q_{W}^{F}. To do this, we replace the polynomial ring S of [KK86, §4] by the formal group algebra associated to F. We then define the formal Demazure element to be the expression in Q_{W}^{F} corresponding to the formal Demazure operator. One of our key objects is the algebra generated by the formal Demazure elements and the elements of the formal group algebra. We call this the formal affine Demazure algebra and denote it D_{F}. The subalgebra generated by only the formal Demazure elements is called the formal Demazure algebra. Next, we “deform” these algebras by introducing an infinite cyclic group. Geometrically, this corresponds to introducing ℂ^{∗}-actions on the relevant varieties. We call the deformed algebra the formal (affine) Hecke algebra associated to the formal group law. Specializing to the additive and multiplicative periodic formal group laws, which correspond to (equivariant) singular cohomology and Ktheory respectively, we recover (completions of) all of the algebras mentioned above. This is summarized in the following table.

Additive FGL  Multiplicative FGL
Alg. Oriented Cohom. Theory (Equiv.) singular cohomology (Equiv.) K-theory
Formal Demazure alg. Nil Hecke alg. 0-Hecke alg.
Formal affine Demazure alg. Affine nil Hecke alg.  Affine 0-Hecke alg.
Formal Hecke alg.  Group alg. of the Weyl Group Hecke alg.
Formal affine Hecke alg.  Degenerate affine Hecke alg. Affine Hecke alg.

Therefore, the algebra D_{F} can be viewed as a deformation space between the generalized Hecke rings studied by Bressler and Evens. We see that D_{F} shares many properties with affine Hecke algebras. However, it does not always satisfy the braid relations. In general, the braid relations are satisfied only up to lower order terms (see Proposition 5.7). This reflects the fact that formal Demazure operators for a general algebraic oriented cohomology theory depend on a choice of reduced decomposition of an element of the Weyl goup.

Our construction provides two things. First, it gives a uniform presentation of the fundamental algebras appearing in both the push-pull and Steinberg variety constructions. Second, it generalizes to other formal groups laws and algebraic oriented cohomology theories, yielding new algebras in the process. These new algebras should be thought of as natural generalizations of the Hecke-type algebras appearing in the table above. Given the representation theoretic importance of these Hecke-type algebras, we expect the new algebras

4 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

defined here to be of interest to both geometers and representation theorists. For example, Hecke-type algebras have played crucial roles in the categorification of quantum groups and related algebras. Thus, it is natural to ask if the generalizations defined in the current paper can be used as building blocks in more general categorifications.

This paper is organized as follows. In the first four sections we recall basic definitions and facts used in the rest of the paper. We review the definition of a formal group law and the exponential map in Section 1. In Section 2, we recall the definition and basic properties of formal group rings/algebras following [CPZ, §2]. In Section 3, following [CPZ, §3], we recall the definition and basic properties of formal Demazure operators. Section 4 is devoted to algebraic oriented cohomology theories. We define the formal (affine) Demazure algebras and prove various facts about them in Sections 5 and 6. In particular, we describe them in terms of generators and relations in the case that the root system is simply-laced. We also show that they are all isomorphic over certain coefficient rings. In Section 7, we define the formal (affine) Hecke algebras and describe them in terms of generators and relations. We prove various properties about them in Section 8. In particular, we show that they are all isomorphic over certain coefficient rings (an analogue of a result of Lusztig ([Lus89, Thm. 9.3])).

Acknowledgements. The authors would like to thank Sam Evens, Iain Gordon, Anthony Licata and Erhard Neher for useful discussions.

1. Formal group laws

In the present section we recall the definition and properties of formal group laws (see [Frö68, Ch. 1, §3, Ch. III, §1] and [LM07, Ch. 1 and 2] for details).

Definition 1.1 (Formal group law). A one-dimensional commutative formal group law (FGL) is a pair (R, F), where R is a commutative ring, called the coefficient ring, and F = F (u, v) ∈ R⟦u, v⟧ is a power series satisfying the following axioms: (FG1) F (u, 0) = F (0, u) = u ∈ R⟦u⟧, (FG2) F (u, v) = F (v, u), and (FG3) F (u, F (v, w)) = F (F (u, v), w) ∈ R⟦u, v, w⟧.

Note that axioms (FG1) and (FG2) imply that

(1.1)  F (u, v) = u + v + ∑_{i,j≥1} a_{ij}u^{i}v^{j}, where a_{ij} = a_{ji} ∈ R.

Given an integer m ≥ 1 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, i.e. the unique power series in R⟦u⟧ such that u +_{F} (−_{F} u) = (−_{F} u) +_{F} u = 0 (see [Frö68, Ch. 1, §3, Prop. 1]). We define

(1.2)  μ_{F} (u) := \frac{−_{F} u}{−u} = 1 − a_{11}u + a_{11}^{2}u^{2} − (a_{11}^{3} + a_{12}a_{11} − a_{22} + 2a_{13})u^{3} + ⋯

(see [LM07, (2.7)]). Note that μ_{F} (u) has a multiplicative inverse since its constant term is invertible.

Throughout the current paper, whenever a particular FGL is denoted using a subscript (e.g. F_{A}, F_{M}, F_{L}, F_{U}), we will use the same subscript to denote various quantities associated to that FGL. Thus, we will write −_{A}u for −_{F_{A}}u, μ_{L} for μ_{F_{L}}, etc.

Example 1.2. (a) For the additive FGL (ℤ, F_{A}(u, v) = u + v) we have (see [LM07, Example 1.1.4])

−_{A}u = −u and μ_{A}(u) = 1.

(b) For the multiplicative FGL (R, F_{M}(u, v) = u + v − βuv), β ∈ R, β ≠ 0, we have (see [LM07, Example 1.1.5])

−_{M}u = −u ∑_{i≥0} β^{i}u^{i} and μ_{M}(u) = ∑_{i≥0} β^{i}u^{i}.

Observe that (1 − βu)μ_{M}(u) = 1, so μ_{M}(u)^{−1} = 1 − βu in R⟦u⟧. If β ∈ R^{×}, where R^{×} denotes the group of invertible elements of R, we say that the FGL is multiplicative periodic.

(c) The Lorentz FGL (R, F_{L}) is given by

F_{L}(u, v) = \frac{u+v}{1+βuv} = (u + v) ∑_{i≥0}(−βuv)^{i}, β ∈ R, β ≠ 0.

We have −_{L}u = −u and μ_{L}(u) = 1. Note that for β = 1/c^{2}, where c is the speed of light, the expression F_{L}(u, v) corresponds to the addition of relativistic parallel velocities.

(d) Let E be the elliptic curve defined by the Tate model ([Tat74, §3]):

(1.3)  E : v = u^{3} + a_{1}uv + a_{2}u^{2}v + a_{3}v^{2} + a_{4}uv^{2} + a_{6}v^{3}.

Here the coefficient ring is R = ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}]. The group law on E induces an elliptic FGL (R, F_{E}) with

F_{E}(u, v) = u + v − a_{1}uv − a_{2}(u^{2}v + uv^{2}) − 2a_{3}(u^{3}v + uv^{3}) + (a_{1}a_{2} − 3a_{3})u^{2}v^{2} + O(5)

(see [Lan87, Appendix 1, (3.6)]). We have

−_{E}u = \frac{−u}{1−a_{1}u−a_{3}v(u)}, μ_{E}(u) = \frac{1}{1−a_{1}u−a_{3}v(u)}

(see [Sil09, §IV.1,p. 120]), where v(u) is considered as an element in R⟦u⟧ after a recursive procedure in the Tate model.

(e) We define the Lazard ring 𝕃 to be the commutative ring with generators a_{ij}, i, j ∈ ℕ_{+}, and subject to the relations that are forced by the axioms for formal group laws. The corresponding FGL (𝕃, F_{U}(u, v) = u + v + ∑_{i,j≥1} a_{ij}u^{i}v^{j}) is then called the universal FGL (see [LM07, §1.1]). The series μ_{U}(u) is given by (1.2).

Let (R, F) and (R, F′) be formal group laws. A morphism of formal group laws f: (R, F) → (R, F′) is a formal power series f ∈ R⟦u⟧ such that f(u +_{F} v) = f(u) +_{F′} f(v). Given a FGL F over R, there is an isomorphism of FGLs after tensoring with ℚ,

e_{F} : (R_{ℚ}, F_{A}) → (R_{ℚ}, F), R_{ℚ} = R ⊗_{ℤ} ℚ,

given by the exponential series e_{F} (u) ∈ R_{ℚ}⟦u⟧ which satisfies the property e_{F} (u + v) = e_{F} (u) +_{F} e_{F} (v) (see [Frö68, Ch. IV, §1]).

Example 1.3. (a) For a general FGL F (u, v) = u + v + a_{11}uv + a_{12}(u^{2}v + uv^{2}) + O(4)

we have

e_{F} (u) = u + \frac{a_{11}}{2!} u^{2} + \frac{a_{11}^{2}+2a_{12}}{3!} u^{3} + O(4).

(b) For the multiplicative FGL we have

e_{M}(u) = ∑_{i≥1}(−β)^{i−1} \frac{u^{i}}{i!} , so that βe_{M}(u) = 1 − exp(−βu).

(c) For the Lorentz FGL we have e_{L}(u) = \frac{e^{2u}−1}{e^{2u}+1}.

(d) For the elliptic FGL we have

e_{E}(u) = u − \frac{a_{1}}{2!} u^{2} + (\frac{3a_{1}^{2}−2(a_{1}^{2}+a_{2})}{3!}) u^{3} + O(4).

2. Formal group algebras

Following [CPZ, §2], we recall the definition and basic properties of formal group algebras. These will play a fundamental role in our definition of formal (affine) Demazure and Hecke algebras.

Definition 2.1 (Formal group algebra). Suppose (R, F) is a FGL and Λ is an abelian group. Let R[x_{Λ}] := R[{x_{λ} | λ ∈ Λ}] denote the polynomial ring over R with variables indexed by Λ. Let ε: R[x_{Λ}] → R be the augmentation homomorphism which maps all x_{λ}, λ ∈ Λ, to 0 and consider the (ker ε)-adic topology on R[x_{Λ}]. We define R⟦x_{Λ}⟧ to be the (ker ε)-adic completion of the polynomial ring R[x_{Λ}]. In particular, if Λ is finite of order n, then the ring R⟦x_{Λ}⟧ is the usual ring of power series in n variables.

Let J_{F} be the closure of the ideal generated by the elements x_{0} and x_{λ_{1}+λ_{2}} − (x_{λ_{1}} +_{F} x_{λ_{2}}) for all λ_{1}, λ_{2} ∈ Λ. We define the formal group algebra (or formal group ring) to be the quotient (see [CPZ, Def. 2.4])

R⟦Λ⟧_{F} := R⟦x_{Λ}⟧/J_{F} .

The class of x_{λ} in R⟦Λ⟧_{F} will be denoted by the same letter. By definition, R⟦Λ⟧_{F} is a complete Hausdorff R-algebra with respect to the (ker ε)-adic topology, where ε: R⟦Λ⟧_{F} → R is the induced augmentation map. We define the augmentation ideal I_{F} := ker ε to be the kernel of this induced map.

The assignment of the formal group algebra R⟦Λ⟧_{F} to the data (R, F, Λ) is functorial in the following ways (see [CPZ, Lem. 2.6]).

(a) Given a morphism f: (R, F) → (R, F′) of FGLs, there is an induced continuous ring homomorphism f^{⋆}: R⟦Λ⟧_{F′} → R⟦Λ⟧_{F}, x_{λ} ↦ f(x_{λ}). If f′: (R, F′) → (R, F′′) is another morphism of FGLs, then (f′f)^{⋆} = f^{⋆}(f′)^{⋆}.

(b) Given a group homomorphism f: Λ → Λ′, there is an induced continuous ring homomorphism f̂: R⟦Λ⟧_{F} → R⟦Λ′⟧_{F}, x_{λ} ↦ x_{f(λ)}. If f′: Λ′ → Λ′′ is another group homomorphism, then f̂′f = f̂′f̂.

Note that maps of the type f̂ commute with maps of the type f^{⋆}.

Example 2.2. The map x_{m} ↦ m ·_{F} x, m ∈ ℤ, defines R-algebra isomorphisms

R⟦ℤ⟧_{F} ≅ R⟦x⟧ and R⟦ℤ/nℤ⟧_{F} ≅ R⟦x⟧/(n ·_{F} x).

More generally, there is a (non-canonical) R-algebra isomorphism (see [CPZ, Cor. 2.12])

R⟦ℤ^{n}⟧_{F} ≅ R⟦x_{1}, …, x_{n}⟧,

where the right hand side is independent of F. This implies that if R is a domain, then so is R⟦ℤ^{n}⟧_{F}.

It follows from (1.1) that n ·_{F} x = nx + x^{2}p(x) for some p(x) ∈ R⟦x⟧. Thus, if n ∈ R^{×}, then n ·_{F} x is the product of x and a unit in R⟦x⟧, so (x) = (n ·_{F} x) and R⟦ℤ/nℤ⟧_{F} ≅ R.

Lemma 2.3. Given a FGL (R, F), we have μ_{F} (x_{λ})^{−1} = μ_{F} (x_{−λ}), for all λ ∈ Λ.

Proof. This follows immediately from the fact that −_{F} x_{λ} = x_{−λ} in R⟦Λ⟧_{F}. □

We now consider what happens at a finite (truncated) level in R⟦Λ⟧_{F}. Let R[Λ]_{F} denote the subalgebra of R⟦Λ⟧_{F} equal to the image of R[x_{Λ}] under the composition R[x_{Λ}] ↪ R⟦x_{Λ}⟧ ↠ R⟦Λ⟧_{F}. Then R⟦Λ⟧_{F} is the completion of R[Λ]_{F} at the ideal (ker ε)∩R[Λ]_{F}. As before, the assignment (R, F, Λ) ↦ R[Λ]_{F} is functorial with respect to group homomorphisms.

Example 2.4. (a) Suppose Λ is a free abelian group. Then for the additive FGL F_{A}(u, v) = u + v over R we have ring isomorphisms (cf. [CPZ, Example 2.19])

R⟦Λ⟧_{A} ≅ S_{R}^{∗}(Λ)^{∧} := ∏_{i=0}^{∞} S_{R}^{i}(Λ) and R[Λ]_{A} ≅ S_{R}^{∗}(Λ) := ⨁_{i=0}^{∞} S_{R}^{i}(Λ),

where S_{R}^{i}(Λ) is the i-th symmetric power of Λ over R, and the isomorphisms are induced by sending x_{λ} to λ ∈ S_{R}^{1}(Λ).

(b) Consider the group ring

R[Λ] := {∑_{j} r_{j}e^{λ_{j}} | r_{j} ∈ R, λ_{j} ∈ Λ} .

Let ε: R[Λ] → R be the augmentation map, i.e. the R-linear map sending all e^{λ}, λ ∈ Λ, to 1. Let R[Λ]^{∧} be the completion of R[Λ] at ker ε.

Assume that β ∈ R^{×}. Then for the multiplicative periodic FGL F_{M}(u, v) = u + v − βuv over R, we have R-algebra isomorphisms (cf. [CPZ, Example 2.20])

R⟦Λ⟧_{M} ≅ R[Λ]^{∧} and R[Λ]_{M} ≅ R[Λ]

induced by x_{λ} ↦ β^{−1}(1 − e^{−λ}) and e^{λ} ↦ (1 − βx_{−λ}) = (1 − βx_{λ})^{−1} respectively. Using this identification, along with Example 1.2(b) and Lemma 2.3, we obtain

μ_{M}(x_{λ})μ_{M}(x_{λ′}) = (1 − βx_{−λ})(1 − βx_{−λ′}) = e^{λ+λ′} = 1 − βx_{−λ−λ′} = μ_{M}(x_{λ+λ′}).

Example 2.5. Fix a generator γ of ℤ and let t = e^{γ} be the corresponding element in the group ring R[ℤ]. According to the previous examples we have R-algebra isomorphisms

R⟦ℤ⟧_{M} ≅ R[t, t^{−1}]^{∧} and R⟦ℤ⟧_{A} ≅ R⟦γ⟧,

where R[t, t^{−1}]^{∧} denotes the completion of R[t, t^{−1}] at the ideal generated by t − 1. At the truncated levels, we have

R[ℤ]_{M} ≅ R[t, t^{−1}] and R[ℤ]_{A} ≅ R[γ],

given by x_{nγ} ↦ β^{−1}(1 − t^{−n}) (with inverse map given by t ↦ 1 − βx_{−γ}) and x_{nγ} ↦ nγ respectively.

8 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

3. Formal Demazure operators

In the present section we introduce, following [CPZ, §3], the notion of formal Demazure operators. We also state some of their properties that will be needed in our constructions. For the remainder of the paper, we assume that R is a commutative domain.

Consider a reduced root system (Λ, Φ, ϱ) as in [Dem73, §1], i.e. a free ℤ-module Λ of finite rank (the weight lattice), a finite subset Φ of Λ whose elements are called roots, and a map ϱ: Λ → Λ^{∨} := Hom_{ℤ}(Λ, ℤ) associating a coroot α^{∨} ∈ Λ^{∨} to every root α, satisfying certain axioms. The reflection map λ ↦ λ − ⟨α^{∨}, λ⟩α is denoted by s_{α}. Here ⟨·, ·⟩ denotes the natural pairing between Λ^{∨} and Λ.

The Weyl group W associated to a reduced root system is the subgroup of linear automorphisms of Λ generated by the reflections s_{α}. We fix sets of simple roots {α_{i}}_{i∈I} and fundamental weights {ω_{i}}_{i∈I}. That is, ω_{i} ∈ Λ^{∨} satisfies ⟨ω_{i}, α_{j}⟩ = δ_{ij} for all i, j ∈ I. Let {s_{i} = s_{α_{i}}}_{i∈I} denote the corresponding set of simple reflections in W and let ℓ denote the usual length function on W. We say the root system is simply laced if ⟨α_{i}^{∨}, α_{j}⟩ ∈ {0, −1} for all i, j ∈ I, i ≠ j. For instance, the roots systems of type ADE are simply laced.

Fix a FGL (R, F). Since the Weyl group acts linearly on Λ, it acts by R-algebra automorphisms on R⟦Λ⟧_{F} via the functoriality in Λ of R⟦Λ⟧_{F} (see Section 2), i.e. we have

w(x_{λ}) = x_{w(λ)}, for all w ∈ W, λ ∈ Λ.

Definition 3.1 (Formal Demazure operator Δ_{α}^{F}). By [CPZ, Cor. 3.4], for any φ ∈ R⟦Λ⟧_{F} and root α ∈ Φ, the element φ − s_{α}(φ) is uniquely divisible by x_{α}. We define an R-linear operator Δ_{α}^{F} on R⟦Λ⟧_{F} (see [CPZ, Def. 3.5]), called the formal Demazure operator, by

Δ_{α}^{F} (φ) := \frac{φ−s_{α}(φ)}{x_{α}}, φ ∈ R⟦Λ⟧_{F} .

Observe that if F is the additive or multiplicative FGL, then Δ_{α}^{F} is the classical Demazure operator of [Dem73, §3 and §9]. We will often omit the superscript F when the FGL is understood.

Definition 3.2 (g^{F}, κ_{α}^{F} and C_{α}^{F}). Consider the power series g^{F} (u, v) defined by u +_{F} v = u + v − uvg^{F} (u, v) and, for α ∈ Φ, let

κ_{α}^{F} := g^{F} (x_{α}, x_{−α}) = \frac{1}{x_{α}} + \frac{1}{x_{−α}} ∈ R⟦Λ⟧_{F} .

We define an R-linear operator C_{α}^{F} on R⟦Λ⟧_{F} (see [CPZ, Def. 3.11]) by

C_{α}^{F} (φ) := κ_{α}^{F} φ − Δ_{α}^{F} (φ), φ ∈ R⟦Λ⟧_{F} .

We will often omit the superscript F when the FGL is understood.

Lemma 3.3. The following statements are equivalent.

(a) F (u, v) = (u + v)h(u, v) for some h(u, v) ∈ R⟦u, v⟧.

(b) κ_{α}^{F} = 0 for all α ∈ Φ.

(c) κ_{α}^{F} = 0 for some α ∈ Φ.

(d) μ_{F} (u) = 1. If these equivalent conditions are satisfied, we write κ^{F} = 0. If they are not satisfied, we write κ^{F} ≠ 0.

Proof. First suppose that F (u, v) = (u + v)h(u, v) for some h(u, v) ∈ R⟦u, v⟧. Then, for any α ∈ Φ, we have F (x_{α}, −x_{α}) = 0. By the uniqueness of the formal inverse, this implies that x_{−α} = −_{F} x_{α} = −x_{α}. Thus κ_{α}^{F} = 0 and so (a) implies (b). Clearly (b) implies (c).

Now suppose that κ_{α}^{F} = 0 for some α ∈ Φ. Then x_{α} ∈ I_{F} \ {0} and F (x_{α}, −x_{α}) = 0 in R⟦Λ⟧_{F}. By the definition of a FGL, we have

F (x_{α}, −x_{α}) = x_{α}^{2} ∑_{i,j≥1}(−1)^{j}a_{ij}x_{α}^{i+j−2} = x_{α}^{2} ∑_{n≥0} b_{n}x_{α}^{n} ∈ I_{F}^{2} ,
where b_{n} = ∑_{i+j=n+2}(−1)^{j}a_{ij}.

We claim that b_{n} = 0 for all n ≥ 0. Indeed, let n_{0} be the smallest n such that b_{n_{0}} ≠ 0. Then

0 = F (x_{α}, −x_{α}) = x_{α}^{2} ∑_{n≥n_{0}} b_{n}x_{α}^{n} = x_{α}^{2+n_{0}} ∑_{n≥n_{0}} b_{n}x_{α}^{n−n_{0}}.

Since x_{α} ≠ 0 (this follows from [CPZ, Lem. 4.2]) and R⟦Λ⟧_{F} is a domain, we have

∑_{n≥0} b_{n}x_{α}^{n−n_{0}} = 0.

Applying the augmentation map, we obtain b_{n_{0}} = 0, contradicting our choice of n_{0}.

Now let F_{i}(u, v) be the i-th homogeneous component of F, i ≥ 2. Since F_{i}(u, −u) = b_{i−2}u^{i} = 0, F_{i}(u, v) is divisible by (u+v) for every i ≥ 2. Since the homogeneous components of degree zero and one for any FGL are 0 and u+v, this implies that F (u, v) = (u+v)h(u, v) for some h(u, v) ∈ R⟦u, v⟧. Thus (c) implies (a).

Now, if μ_{F} (u) = 1, then −x_{α} = −_{F} x_{α} = x_{−α} and so κ_{α}^{F} = 0 for all α ∈ Φ. Thus (d) implies (b).

Finally, if F (u, v) = (u + v)h(u, v) for some h(u, v) ∈ R⟦u, v⟧, then −_{F} u = −u by the uniqueness of the formal inverse (as above). Thus μ_{F} (u) = 1. Hence (a) implies (d). □

As in the case of the usual Demazure operators, the operators Δ_{α}^{F} and C_{α}^{F} satisfy Leibniztype properties (see [CPZ, Props. 3.8 and 3.12]).

4. Algebraic oriented cohomology theories and characteristic maps

We now recall several facts concerning algebraic oriented cohomology theories. We refer the reader to [LM07] and [Pan03] for further details and examples.

An algebraic oriented cohomology theory (AOCT) is a contravariant functor h from the category of smooth projective varieties over a field k to the category of commutative unital rings which satisfies certain properties (see [LM07, §1.1]). Given a morphism f: X → Y of varieties, the map h(f) will be denoted f^{∗} and called the pullback of f. One of the characterizing properties of h is that, for any proper map f: X → Y, there is an induced map f_{∗}: h(X) → h(Y) of h(Y)-modules called the push-forward (here h(X) is an h(Y)-module via f^{∗}). A morphism of AOCTs is a natural transformation of functors that also commutes with push-forwards. Basic examples of AOCTs are Chow groups CH and Grothendieck’s K_{0} (see [Pan03, §§2.1, 2.5, 3.8] for further examples).

The connection between algebraic oriented cohomology theories and FGLs is as follows. Given two line bundles L_{1} and L_{2} over X, we have (see [LM07, Lem. 1.1.3])

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

where c_{1}^{h} is the first characteristic class with values in h and F is a one-dimensional commutative FGL over the coefficient ring R = h(Spec k) associated to h.

10 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

There is an AOCT Ω defined over a field of characteristic zero, called algebraic cobordism (see [LM07, §1.2]), that is universal in the following sense: Given any AOCT h there is a unique morphism Ω → h of AOCTs. The FGL associated to Ω is the universal FGL F_{U}.

Moreover, given a FGL F over a ring R together with a morphism 𝕃 → R, we define a functor X ↦ h(X) := Ω(X) ⊗_{𝕃} R. If F satisfies certain conditions, the functor h gives an AOCT.

Example 4.1. In the above terms, the additive FGL corresponds to the theory of Chow groups. The multiplicative periodic FGL with β ∈ R^{×} corresponds to Grothendieck’s K_{0}. The multiplicative FGL with β ∉ R^{×} corresponds to connective K-theory.

Let G be a split simple simply connected linear algebraic group over a field k corresponding to the root system (Λ, Φ, ϱ). Fix a split maximal torus T and a Borel subgroup B so that T ⊆ B ⊆ G. Let G/B be the variety of Borel subgroups of G and let F be the FGL over R associated to an AOCT h satisfying the assumptions of [CPZ, Thm. 13.12]. Consider the formal group algebra R⟦Λ⟧_{F}. Then there is a ring homomorphism, called the characteristic map (see [CPZ, §6]),

𝔠_{F} : R⟦Λ⟧_{F} → h(G/B), x_{λ} ↦ c_{1}^{h}(L(λ)),

where L(λ) is the line bundle associated to λ ∈ Λ. Note that this map is neither injective nor surjective in general. Its kernel contains the ideal generated by W -invariant elements, and h(G/B) modulo the ideal generated by the image of 𝔠_{F} is isomorphic to h(G) (see [GZ12, Prop. 5.1]).

Example 4.2. (a) The characteristic map for the theory of Chow groups, i.e. corresponding to the additive FGL, is given by

𝔠_{A}: ℤ⟦Λ⟧_{A} → CH(G/B), x_{λ} ↦ c_{1} (L(λ)) ,

which recovers the usual characteristic map for Chow groups (see [Dem74, §1.5]).

(b) The characteristic map for Grothendieck’s K_{0}, i.e. corresponding to the multiplicative periodic FGL, is given by

𝔠_{M}: ℤ⟦Λ⟧_{M} → K_{0}(G/B), x_{λ} ↦ 1 − [L(λ)^{∨}].

Restricting to the integral group ring ℤ[Λ] and using the identification of Example 2.4(b), we recover the usual characteristic map for K_{0} ([Dem74, §1.6]) which maps e^{λ} to [L(λ)].

(c) Algebraic cobordism Ω defined over a field of characteristic 0 satisfies the assumptions of [CPZ, Thm.13.12]. Therefore, we have the characteristic map

𝔠_{U}: 𝕃⟦Λ⟧_{U} → Ω(G/B), x_{λ} ↦ c_{1}^{Ω}(L(λ)).

Let G/P_{i} be the projective homogeneous variety, where P_{i} is the minimal parabolic subgroup of G corresponding to the simple root α_{i}, i ∈ I. Then

p: G/B = ℙ_{G/P_{i}}(1 ⊕ L(ω_{i})) → G/P_{i}

is the projective bundle associated to the vector bundle 1 ⊕ L(ω_{i}), there 1 denotes the trivial bundle of rank one (see, for example, [CPZ, §10.3]). Then the operators C_{α}^{F} introduced in Definition 3.2 have the following geometric interpretation in terms of push-pull operators (generalizing [PR99, Prop.]).

Proposition 4.3 ([CPZ, Prop. 10.10(4)]). We have

p^{∗}p_{∗}(𝔠_{F} (χ)) = 𝔠_{F} (C_{α}^{F} (χ)), for all χ ∈ R⟦Λ⟧_{F} .

5. Formal (affine) Demazure algebras: definitions

In the present section we introduce the notion of a twisted formal group algebra and a particular subalgebra, called the formal (affine) Demazure algebra, which is one of our main objects of interest. Our method is inspired by the approach of [KK86, §4.1].

Definition 5.1 (Twisted formal group algebra). Let Q^{F} = Q^{(R,F)} denote the field of fractions of R⟦Λ⟧_{F}. The action of the Weyl group W on R⟦Λ⟧_{F} induces an action by automorphisms on Q^{F}. We define the twisted formal group algebra to be the smash product Q_{W}^{F} := R[W]⋉_{R}Q^{F}. (This is sometimes denoted by R[W]#Q^{F}.) In other words, Q_{W}^{F} is equal to R[W] ⊗_{R} Q^{F} as an R-module, with multiplication given by

(δ_{w′}ψ′)(δ_{w}ψ) = δ_{w′w}w^{−1}(ψ′)ψ for all w, w′ ∈ W, ψ, ψ′ ∈ Q^{F}

(extended by linearity), where δ_{w} denotes the element in R[W] corresponding to w (so we have δ_{w′}δ_{w} = δ_{w′w} for w, w′ ∈ W).

Observe that Q_{W}^{F} is a free right Q^{F} -module (via right multiplication) with basis {δ_{w}}_{w∈W}. Note that Q_{W}^{F} is not a Q^{F} -algebra (but only an R-algebra) since δ_{e}Q^{F} = Q^{F} δ_{e} is not central in Q_{W}^{F}. We denote δ_{e} (the unit element of Q_{W}^{F}) by 1.

Definition 5.2 (Formal Demazure element). For each root α ∈ Φ, we define the corresponding formal Demazure element

Δ_{α}^{F} := \frac{1}{x_{α}} (1 − δ_{s_{α}}) = \frac{1}{x_{α}} − δ_{s_{α}} \frac{1}{x_{−α}} ∈ Q_{W}^{F}

(cf. [KK86, (I_{24})]). We will omit the superscript F when the FGL is clear from the context.

We can now define our first main objects of study.

Definition 5.3 (Formal (affine) Demazure algebra). The formal Demazure algebra D_{F} is the R-subalgebra of Q_{W}^{F} generated by the formal Demazure elements Δ_{i}^{F}. The formal affine Demazure algebra D_{F} is the R-subalgebra of Q_{W}^{F} generated by D_{F} and R⟦Λ⟧_{F}. When we wish to specify the coefficient ring, we write D_{R,F} (resp. D_{R,F}) for D_{F} (resp. D_{F}).

Remark 5.4. Suppose h is an algebraic oriented cohomology theory satisfying the assumptions of [CPZ, Thm. 13.12] and with FGL F (see Section 4). By Proposition 4.3, we see that, under the characteristic map 𝔠_{F}, the affine Demazure algebra D_{F} corresponds to the algebra of operators on h(G/B) generated by left multiplication (by elements of h(G/B)) and the push-pull operators p^{∗}p_{∗}.

Lemma 5.5 (cf. [KK86, Prop. 4.2]). For all ψ ∈ Q and α ∈ Φ, we have

ψΔ_{α} = Δ_{α}s_{α}(ψ) + Δ_{α}(ψ),

where Δ_{α}(ψ) = \frac{ψ−s_{α}(ψ)}{x_{α}} ∈ Q is the formal Demazure operator applied to ψ (see Definition 3.1).

Proof. We have

ψΔ_{α} = ψ (\frac{1}{x_{α}} − δ_{s_{α}} \frac{1}{x_{−α}} ) = \frac{ψ−s_{α}(ψ)}{x_{α}} + (\frac{1}{x_{α}} − δ_{s_{α}} \frac{1}{x_{−α}} ) s_{α}(ψ) = Δ_{α}(ψ) + Δ_{α}s_{α}(ψ). □

12 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

Proceeding from Lemma 5.5 by induction, we obtain the following general formula (cf. [KK86, (I_{26})])

ψΔ_{β_{1}}Δ_{β_{2}} ⋯ Δ_{β_{s}} = ∑_{(1≤i_{1}<···<i_{r}≤s)} Δ_{β_{i_{1}}}Δ_{β_{i_{2}}} ⋯ Δ_{β_{ir}} ϕ(i_{1}, …, i_{r}),

where β_{1}, β_{2}, …, β_{s} ∈ Φ are roots and ϕ(i_{1}, …, i_{r}) ∈ Q^{F} is defined to be the composition Δ_{β_{s}} ∘ Δ_{β_{s−1}} ∘ ⋯ ∘ Δ_{β_{1}} applied to ψ, where the Demazure operators at the places i_{1}, …, i_{r} are replaced by the respective reflections.

Recall the elements κ_{α}, α ∈ Φ, from Definition 3.2. It is easy to verify that κ_{α}δ_{s_{α}} = δ_{s_{α}}κ_{α}, κ_{α}Δ_{α} = Δ_{α}κ_{α}, and

(5.1)  Δ_{α}^{2} = Δ_{α}κ_{α}.

Example 5.6. (a) For the additive and Lorentz FGLs we obtain the nilpotence relation

Δ_{α}^{2} = 0 since κ_{α}^{A} = κ_{α}^{L} = 0.

(b) For the multiplicative FGL we obtain the relation Δ_{α}^{2} = βΔ_{α}, since κ_{α}^{M} = β. In particular, if β = 1 we obtain the idempotence relation Δ_{α}^{2} = Δ_{α}.

(c) For the elliptic FGL we have, in the notation of Example 1.2(d),

Δ_{α}^{2} = \frac{a_{1}x_{α}+a_{3}v(x_{α})}{x_{α}}Δ_{α}.

For example, if a_{3} = 0, then Δ_{α}^{2} = a_{1}Δ_{α}.

To simplify notation in what follows, for i, j, i_{1}, …, i_{k} ∈ I, we set

(5.2)  x_{±i} = x_{±α_{i}}, x_{±i±j} = x_{±α_{i}±α_{j}}, δ_{i_{1}i_{2}...i_{k}} = δ_{s_{i_{1}}s_{i_{2}}···s_{i_{k}}} , Δ_{i} = Δ_{α_{i}}, κ_{i} = κ_{α_{i}}.

Furthermore, when we write an expression such as \frac{δ_{w}}{φ} for w ∈ W, φ ∈ R⟦Λ⟧_{F}, we interpret this as being equal to δ_{w} \frac{1}{φ}. That is, we consider the numerators of rational expressions to be to the left of their denominators.

Proposition 5.7. Suppose i, j ∈ I and let m_{ij} be the order of s_{i}s_{j} in W. Then

(5.3)  \underbrace{Δ_{j}Δ_{i}Δ_{j} ⋯}_{m_{ij} terms} − \underbrace{Δ_{i}Δ_{j}Δ_{i} ⋯}_{m_{ij} terms} = ∑_{w∈W, ℓ(w)<m_{ij}} Δ_{w}η_{ij}^{w}

for some η_{ij}^{w} ∈ Q^{F}. In particular, we have the following:

(a) If ⟨α_{i}, α_{j}⟩ = 0, so that m_{ij} = 2, then Δ_{i}Δ_{j} = Δ_{j}Δ_{i}.

(b) If ⟨α_{i}, α_{j}⟩ = −1, so that m_{ij} = 3, then

Δ_{j}Δ_{i}Δ_{j} − Δ_{i}Δ_{j}Δ_{i} = Δ_{i}κ_{ij} − Δ_{j}κ_{ji},

where

(5.4)  κ_{ij} = \frac{1}{x_{i+j}} (\frac{1}{x_{j}} − \frac{1}{x_{−i}} ) − \frac{1}{x_{i}x_{j}} ∈ R⟦Λ⟧_{F} .

Proof. We have

\underbrace{Δ_{j}Δ_{i}Δ_{j} ⋯}_{m_{ij} terms} = \frac{1}{x_{j}} (1 − δ_{j})\frac{1}{x_{i}} (1 − δ_{i})\frac{1}{x_{j}} (1 − δ_{j}) · · · .

Since the δ_{w}, w ∈ W, form a basis of Q_{W} as a right Q-module, this can be written as a sum of (right) Q-multiplies of δ_{w}. The leading term (with respect to the length of w) is

(−1)^{m_{ij}}δ\underbrace{s_{j}s_{i}s_{j} ⋯ s_{n}s_{k}}_{m_{ij} terms}(x_{− \underbrace{s_{k}s_{n} ⋯ s_{j}s_{i}}_{m_{ij} −1 terms}(α_{j})} ⋯ x_{−s_{k}(α_{n})}x_{−α_{k}})^{−1},

where n = i and k = j (resp. n = j and k = i) if m_{ij} is odd (resp. even). Now, by [Bou81, Cor. 2 de la Prop. 17],

α_{k}, s_{k}(α_{n}), …, \underbrace{s_{k}s_{n} ⋯ s_{j}s_{i}}_{m_{ij−1} terms}(α_{j})

are precisely the positive roots mapped to negative roots by δ_{s_{j}s_{i}s_{j}···} (m_{ij} reflections in the subscript). Since δ_{s_{j}s_{i}s_{j}···} = δ_{s_{i}s_{j}s_{i}} (m_{ij} reflections in each subscript), we see that the highest order terms in Δ_{j}Δ_{i}Δ_{j} ⋯ − Δ_{i}Δ_{j}Δ_{i} ⋯ cancel, proving the first part of the proposition.

Under the assumptions of (a), we have s_{i}(α_{j}) = α_{j} and

Δ_{i}Δ_{j} = (\frac{1}{x_{i}} − \frac{δ_{i}}{x_{−i}} ) (\frac{1}{x_{j}} − \frac{δ_{j}}{x_{−j}} ) = \frac{1}{x_{i}x_{j}} − \frac{δ_{i}}{x_{−i}x_{j}} − \frac{1}{x_{i}} \frac{δ_{j}}{x_{−j}} + \frac{δ_{i}}{x_{−i}} \frac{δ_{j}}{x_{−j}} = \frac{1}{x_{i}x_{j}} − \frac{δ_{i}}{x_{−i}x_{j}} − \frac{δ_{j}}{x_{i}x_{−j}} + \frac{δ_{ij}}{x_{−i}x_{−j}} .

Since the final expression is symmetric in i and j, we have Δ_{i}Δ_{j} = Δ_{j}Δ_{i}.

It remains to prove (b). We have s_{i}(α_{j}) = s_{j}(α_{i}) = α_{i} + α_{j} and s_{i}s_{j}(α_{i}) = α_{j}. Thus

Δ_{j}Δ_{i}Δ_{j} = (\frac{1}{x_{j}} − \frac{δ_{j}}{x_{−j}} ) (\frac{1}{x_{i}} − \frac{δ_{i}}{x_{−i}} ) (\frac{1}{x_{j}} − \frac{δ_{j}}{x_{−j}} )

= \frac{1}{x_{i}x_{j}^{2}} − \frac{1}{x_{j}x_{i}} \frac{δ_{j}}{x_{−j}} − \frac{1}{x_{j}} \frac{δ_{i}}{x_{−i}x_{j}} − \frac{δ_{j}}{x_{−j}x_{i}x_{j}} + \frac{1}{x_{j}} \frac{δ_{i}}{x_{−i}} \frac{δ_{j}}{x_{−j}} + \frac{δ_{j}}{x_{−j}x_{i}} \frac{δ_{j}}{x_{−j}} + \frac{δ_{j}}{x_{−j}} \frac{δ_{i}}{x_{−i}x_{j}} − \frac{δ_{j}}{x_{−j}} \frac{δ_{i}}{x_{−i}} \frac{δ_{j}}{x_{−j}}

= \frac{1}{x_{i}x_{j}^{2}} − \frac{δ_{j}}{x_{i+j}x_{−j}^{2}} − \frac{δ_{i}}{x_{i+j}x_{−i}x_{j}} − \frac{δ_{j}}{x_{−j}x_{i}x_{j}} + \frac{δ_{ij}}{x_{i}x_{−i−j}x_{−j}} + \frac{1}{x_{i+j}x_{−j}x_{j}} + \frac{δ_{ji}}{x_{−i−j}x_{−i}x_{j}} − \frac{δ_{jij}}{x_{−i−j}x_{−i}x_{−j}}

= Δ_{j} (\frac{1}{x_{i}x_{j}} + \frac{1}{x_{i+j}x_{−j}} ) + Δ_{i} \frac{1}{x_{i+j}x_{j}} + \frac{δ_{ij}}{x_{i}x_{−i−j}x_{−j}} + \frac{δ_{ji}}{x_{−i−j}x_{−i}x_{j}} − \frac{δ_{jij}}{x_{−i−j}x_{−i}x_{−j}} − \frac{1}{x_{i}x_{i+j}x_{j}} .

Using the fact that s_{i}s_{j}s_{i} = s_{j}s_{i}s_{j}, we obtain

(5.5) Δ_{j}Δ_{i}Δ_{j} − Δ_{i}Δ_{j}Δ_{i} = Δ_{i} (\frac{1}{x_{i+j}x_{j}} − \frac{1}{x_{i+j}x_{−i}} − \frac{1}{x_{i}x_{j}} ) − Δ_{j} (\frac{1}{x_{i+j}x_{i}} − \frac{1}{x_{i+j}x_{−j}} − \frac{1}{x_{i}x_{j}} ) .

It remains to prove that κ_{ij} ∈ R⟦Λ⟧_{F}. We have

g(x_{i+j}, x_{−i}) = \frac{x_{i+j}+x_{−i}−x_{j}}{x_{i+j}x_{−i}} = \frac{1}{x_{−i}} + \frac{1}{x_{i+j}} − \frac{x_{j}}{x_{i+j}x_{−i}} ∈ R⟦Λ⟧_{F} .

and, hence,

κ_{ij} = \frac{g(x_{i+j},x_{−i})−g(x_{i},x_{−i})}{x_{j}},

where g(x_{i}, x_{−i}) = \frac{1}{x_{i}} + \frac{1}{x_{−i}} ∈ R⟦Λ⟧_{F}. Therefore, it suffices to show that g(x_{i+j}, x_{−i}) − g(x_{i}, x_{−i}) is divisible by x_{j}. The latter follows (taking u_{1} = x_{i}, u_{2} = x_{j} and v = x_{−i}) from the congruence u_{1} +_{F} u_{2} ≡ u_{1} (mod u_{2}), which implies that g(u_{1} +_{F} u_{2}, v) ≡ g(u_{1}, v) (mod u_{2}). □

Remark 5.8. Note that

κ_{ij} = \frac{x_{i}(x_{−i}−x_{j})−x_{i+j}x_{−i}}{x_{i}x_{−i}x_{j}x_{i+j}} .

By [BE90, p. 809], the numerator of the above expression equals zero if and only if F (u, v) = u + v + a_{11}uv for some a_{11} ∈ R (i.e. if and only if F is the additive or multiplicative FGL). Therefore, contrary to the situation for the additive and multiplicative FGLs, the formal Demazure elements do not satisfy the braid relations in general (cf. [BE90, Thm. 3.7]).

14 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

Remark 5.9. Observe that the key difference between our setting and the setting of [BE90] is that we deal with algebraic theories for which the groups h(BT) and their properties, which are used extensively in [BE90], are not well-defined or remain unknown. Instead, we rely on the formal group algebra R⟦Λ⟧_{F} (as a replacement for h(BT)) and techniques introduced in [CPZ].

For each w ∈ W, fix a reduced decomposition w = s_{i_{1}} ⋯ s_{i_{k}} and set

(5.6)  Δ_{w} = Δ_{i_{1}} ⋯ Δ_{i_{k}}.

Note that, in general, Δ_{w} depends on the choice of reduced decomposition.

Definition 5.10 (R̃ and R⟦Λ⟧_{F}^{∼}). Let R̃ be the subalgebra of Q^{F} defined by

(5.7)  R̃ := R[W] · R[η_{ij}^{w} | i, j ∈ I, w ∈ W, ℓ(w) < m_{ij}],

where R[W]· denotes the natural action of the group algebra R[W] of W on Q^{F}. Similarly, define

(5.8)  R⟦Λ⟧_{F}^{∼} := R[W] · R⟦Λ⟧_{F} [η_{ij}^{w} | i, j ∈ I, w ∈ W, ℓ(w) < m_{ij}].

Note that R⟦Λ⟧_{F}^{∼} = R⟦Λ⟧_{F} if the root system is simply laced (since κ_{ij} ∈ R⟦Λ⟧_{F}).

The following lemma is an easy generalization of [KK86, Thm. 4.6] (which considers the case of the additive FGL).

Lemma 5.11. The set {Δ_{w} | w ∈ W } forms a basis of D_{F} ⊗_{R} R̃ as a right (or left) R̃-module and a basis of D_{F} ⊗_{R⟦Λ⟧_{F}} R⟦Λ⟧_{F}^{∼} as a right (or left) R⟦Λ⟧_{F}^{∼}-module.

Proof. Since R is a domain, so are R̃ and R⟦Λ⟧_{F}^{∼}. By (5.1) and Proposition 5.7, we can write any product of formal Demazure elements as a R̃-linear combination of the elements Δ_{w}, w ∈ W. Combined with Lemma 5.5, we can write any product of formal Demazure elements and elements of R⟦Λ⟧_{F}^{∼} as an R⟦Λ⟧_{F}^{∼}-linear combination of the elements Δ_{w}, w ∈ W. Thus {Δ_{w} | w ∈ W } is a spanning set of the modules in the statement of the lemma. Now, it is easy to see from the definition of the formal Demazure elements (Definition 5.2) that, for all w ∈ W,

Δ_{w} = ∑_{v: ℓ(v)≤ℓ(w)} δ_{v}a_{w},

where the sum is over elements v ∈ W with length less than or equal to the length of w, a_{v} ∈ Q^{F} for all v, and a_{w} ≠ 0. Thus, since {δ_{w} | w ∈ W } is a basis for Q_{W}^{F} as a right (or left) Q^{F} -module, we see that {Δ_{w} | w ∈ W } is also a basis for this module. In particular, the set {Δ_{w} | w ∈ W } is linearly independent over Q^{F} and hence over R̃ or R⟦Λ⟧_{F}^{∼}. □

Theorem 5.12. Given a formal group law (R, F), the formal affine Demazure algebra D_{F} is generated as an R-algebra by R⟦Λ⟧_{F} and the formal Demazure elements Δ_{i}, i ∈ I, and satisfies the following relations:

(a) φΔ_{i} = Δ_{i}s_{i}(φ) + Δ_{α_{i}}(φ) for all i ∈ I and φ ∈ R⟦Λ⟧_{F};

(b) Δ_{i}^{2} = Δ_{i}κ_{i} for all i ∈ I, where κ_{i} = \frac{1}{x_{i}} + \frac{1}{x_{−i}} ∈ R⟦Λ⟧_{F};

(c) Δ_{i}Δ_{j} = Δ_{j}Δ_{i} for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ = 0; (d) Δ_{j}Δ_{i}Δ_{j} − Δ_{i}Δ_{j}Δ_{i} = Δ_{i}κ_{ij} − Δ_{j}κ_{ji} for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ = −1, where

κ_{ij} = \frac{1}{x_{i+j}} (\frac{1}{x_{j}} − \frac{1}{x_{−i}} ) − \frac{1}{x_{i}x_{j}} ∈ R⟦Λ⟧_{F}

(note that, in general, κ_{ij} ≠ κ_{ji});

(e) relation (5.3) for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ ≤ −2. Furthermore, if the root system is simply laced, then (a)–(d) form a complete set of relations. In arbitrary type, (a)–(e) form a complete set of relations (over R⟦Λ⟧_{F}^{∼}) for D_{F} ⊗_{R⟦Λ⟧_{F}} R⟦Λ⟧_{F}^{∼}.

Proof. The first part of the theorem follows immediately from (5.1), Lemma 5.5 and Proposition 5.7.

Since R is a domain, so is R⟦Λ⟧_{F}^{∼}. Let D̃_{F} be the R-algebra generated by R⟦Λ⟦_{F}^{∼} and elements Δ′_{i}, i ∈ I, subject to the relations given in the theorem. Then we have a surjective ring homomorphism ρ : D̃_{F} → D_{F} ⊗_{R⟦Λ⟧_{F}} R⟦Λ⟦_{F}^{∼} which is the identity on R⟦Λ⟦_{F}^{∼} and maps Δ′_{i} to Δ_{i}. We wish to show that this map is an isomorphism.

For w ∈ W, define Δ′_{w} as in (5.6). The relations among the Δ′_{w} allow us to write any element of D̃_{F} in the form

∑_{w∈W} Δ′_{w}a_{w}, a_{w} ∈ R⟦Λ⟦_{F}^{∼}.

Suppose such an element is in the kernel of ρ. Then

0 = ρ (∑_{w∈W} Δ′_{w}a_{w}) = ∑_{w∈W} Δ_{w}a_{w}.

By Lemma 5.11, this implies that a_{w} = 0 for all w. Thus ρ is injective and hence an isomorphism. This completes the proof of the proposition once we recall that R⟦Λ⟦_{F}^{∼} ≅ R⟦Λ⟧_{F} in simply laced type. □

6. Formal (affine) Demazure algebras: examples and further properties

In this section we specialize the definition of the formal (affine) Demazure algebra to various FGLs. We then prove several important facts about these algebras in general. The first proposition demonstrates that our definition recovers classical objects in the additive and multiplicative cases.

Proposition 6.1. (a) For the additive FGL over R = ℤ, the formal affine Demazure algebra D_{A} is isomorphic to the completion of the nil Hecke ring of [KK86, Def. 4.12]. In this case, all the relations among the Δ_{i} are given by the braid relations and the nilpotence relations Δ_{i}^{2} = 0 (even in non-simply laced type).

(b) For the additive FGL over R = ℂ, the formal Demazure algebra D_{A} is isomorphic to the nil-Hecke ring of [EB87, Def. 3].

(c) For the multiplicative FGL over R = ℂ with β = 1, the formal Demazure algebra D_{M} is the completion of the 0-Hecke algebra, which is the classical Hecke algebra specialized at q = 0. In this case all the relations among the Δ_{i} are given by the braid relations and the idempotence relations (even in non-simply laced type).

Proof. For the additive FGL over R = ℤ, Q is the field of fractions of the ring S^{∗}(Λ)^{∧} ≅ R⟦Λ⟧_{A} and Δ_{i} is the −x_{i} of [KK86, (I_{24})]. This proves part (a). Similarly, if R = ℂ, then our Δ_{i} corresponds to the X_{i} of [EB87], proving part (b).

16 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

For the multiplicative FGL over R = ℂ with β = 1, we have that Q is the field of fractions of ℂ[Λ]^{∧} and −Δ_{α_{i}} is the B_{i} of [EB87, §2] if we identify the simple root α_{i} of [EB87] with our −α_{i}. This proves part (c). □

We now consider some other FGLs, where our definition gives new algebras.

Example 6.2 (Lorentz affine Demazure algebra). Consider the Lorentz FGL (R, F_{L}). Then x_{i+j} = \frac{x_{i}+x_{j}}{1+βx_{i}x_{j}}, β ∈ R. Since x_{−λ} = −x_{λ} for all λ ∈ Λ, we have

(6.1)  κ_{ij} = \frac{1+βx_{i}x_{j}}{x_{i}+x_{j}} · \frac{x_{i}+x_{j}}{x_{i}x_{j}} − \frac{1}{x_{i}x_{j}} = β

for i, j ∈ I with ⟨α_{i}, α_{j}⟩ = −1. Therefore, relation (d) of Theorem 5.12 becomes

Δ_{j}Δ_{i}Δ_{j} − Δ_{i}Δ_{j}Δ_{i} = β(Δ_{i} − Δ_{j}) for all i, j such that ⟨α_{i}, α_{j}⟩ = −1.

Example 6.3 (Elliptic affine Demazure algebra). Consider the elliptic FGL (R, F_{E}). Set μ_{i} = μ_{E}(x_{i}) and g_{ij}^{E} = g^{E}(x_{i}, x_{j}). Then, for i, j ∈ I with ⟨α_{i}, α_{j}⟩ = −1, we have

κ_{ij} = \frac{x_{i}(x_{−i}−x_{j})−x_{i+j}x_{−i}}{x_{i}x_{−i}x_{j}x_{i+j}} = \frac{x_{−i}−x_{j}+x_{i+j}μ_{i}}{x_{−i}x_{j}x_{i+j}} = \frac{x_{−i}−x_{j}+μ_{i}(x_{i}+x_{j}−x_{i}x_{j}g_{ij}^{E})}{x_{−i}x_{j}x_{i+j}} = \frac{−x_{j}+μ_{i}x_{j}−μ_{i}x_{i}x_{j}g_{ij}^{E}}{x_{−i}x_{j}x_{i+j}}

= \frac{μ_{i}−1−μ_{i}x_{i}g_{ij}^{E}}{x_{−i}x_{i+j}} = \frac{μ_{−i}^{−1}−1−μ_{i}x_{i}g_{ij}^{E}}{x_{−i}x_{i+j}} = \frac{−a_{1}x_{−i}−a_{3}v(x_{−i})+x_{−i}g_{ij}^{E}}{x_{i+j}x_{−i}} = \frac{g_{ij}^{E}−a_{1}}{x_{i+j}} − \frac{a_{3}v(x_{−i})}{x_{i+j}x_{−i}}.

Theorem 6.4. For any two formal group laws (R, F) and (R, F′) over the same ring R, we have D_{R_{ℚ},F} ≅ D_{R_{ℚ},F′} as algebras, where R_{ℚ} = R ⊗_{ℤ} ℚ.

Proof. It suffices to prove the result for the special case where F′ = F_{A}. There is an isomorphism of FGLs

e_{F} : (R_{ℚ}, F_{A}) → (R_{ℚ}, F)

given by the exponential series e_{F} (u) ∈ R_{ℚ}⟦u⟧ (see Section 1). This induces an isomorphism of formal group algebras

e_{F}^{⋆} : R_{ℚ}⟦Λ⟧_{F} → R_{ℚ}⟦Λ⟧_{A}.

This map commutes with the action of W and thus we have an induced isomorphism of twisted formal group algebras

e_{F}^{⋆} : Q_{W}^{(R_{ℚ},F)} → Q_{W}^{(R_{ℚ},A)}.

Thus D_{R_{ℚ},F} is isomorphic to its image D′ := e_{F}^{⋆} (D_{R_{ℚ},F}) under this map. Now, D′ is generated over R_{ℚ}⟦Λ⟧_{A} by the elements

e_{F}^{⋆} (Δ_{i}^{F}) = \frac{1}{e_{F} (x_{i})}(1 − δ_{i}) = \frac{x_{i}}{e_{F} (x_{i})}Δ_{i}^{A}, i ∈ I.

Since e_{F} (x_{i})/x_{i} ∈ R_{ℚ}⟦Λ⟧_{A} is invertible in R_{ℚ}⟦Λ⟧_{A} (because its constant term is invertible in R_{ℚ} – see Example 1.3(a)), D′ is also generated over R_{ℚ}⟦Λ⟧_{A} by Δ_{i}^{A}, i ∈ I, and thus isomorphic to D_{R_{ℚ},A}. □

Remark 6.5. Note that while Theorem 6.4 shows that all affine Demazure algebras are isomorphic when the coefficient ring is R_{ℚ}, the isomorphism is not the naive one sending Δ_{i}^{F} to Δ_{i}^{F′}. Furthermore, the completion (with respect to the augmentation map) is crucial. No assertion is made regarding an isomorphism (even over R_{ℚ}) of truncated versions.

7. Formal (affine) Hecke algebras: definitions

In the present section we define deformed versions of the formal (affine) Demazure algebras. Our goal is to construct generalizations of the usual (affine) Hecke algebra and its degenerate analogue. These two examples correspond to the multiplicative periodic and additive FGL cases of our more general construction. We begin by reminding the reader of the definition of these classical objects.

Definition 7.1 (Hecke algebra). The (classical) Hecke algebra H associated to the Weyl group W is the ℤ[t, t^{−1}]-algebra with 1 generated (as a ℤ[t, t^{−1}]-algebra) by elements T_{i} := T_{s_{i}}, i ∈ I, modulo

(a) the quadratic relations (T_{i} + t^{−1})(T_{i} − t) = 0 for all i ∈ I, and

(b) the braid relations T_{i}T_{j}T_{i} ⋯ = T_{j}T_{i}T_{j} ⋯ (m_{ij} factors in both products) for any i ≠ j in I with s_{i}s_{j} of order m_{ij} in W.

Definition 7.2 (Affine Hecke algebra). The (classical) affine Hecke algebra H is H ⊗_{ℤ[t,t^{−1}]} ℤ[t, t^{−1}][Λ], where the factors H and ℤ[t, t^{−1}][Λ] are subalgebras and the relations between the two factors are given by

e^{λ}T_{i} − T_{i}e^{s_{i}(λ)} = (t − t^{−1})\frac{e^{λ}−e^{s_{i}(λ)}}{1−e^{−α_{i}}} , λ ∈ Λ, i ∈ I.

Remark 7.3. In Defintions 7.1 and 7.2, we have followed the conventions found, for instance, in [CMHL02, pp. 71–72] (except that we use t in place of v and T_{i} in place of T̃_{i}). These conventions differ somewhat from those found in other places in the literature. For instance, H as defined above is isomorphic to H′ ⊗_{ℤ[q,q^{−1}]} ℤ[q^{1/2}, q^{−1/2}], where H′ is the Hecke algebra as defined in [Hum90, §7.4] or [CG10, Def. 7.1.1]. The T_{s_{i}} appearing in [Hum90, CG10] correspond to tT_{s_{i}} in our notation, where t corresponds to q^{1/2}.

Definition 7.4 (Degenerate affine Hecke algebra). Let ϵ be an indeterminate. The degenerate affine Hecke algebra H_{deg} is the unital associative ℤ[ϵ]-algebra that is ℤ[W] ⊗_{ℤ} S_{ℤ[ϵ]}^{∗}(Λ) as a ℤ[ϵ]-module and such that the subspaces ℤ[W] and S_{ℤ[ϵ]}^{∗}(Λ) are subalgebras and the following relations hold:

δ_{i}λ − s_{i}(λ)δ_{i} = −ϵ⟨α_{i}^{∨}, λ⟩, i ∈ I, λ ∈ Λ.

Fix a free abelian group Γ of rank 1 with generator γ. Denote by R_{F} the formal group algebra R⟦Γ⟧_{F}. For instance, ℤ_{M} ≅ ℤ[t, t^{−1}]^{∧} and ℤ_{A} ≅ ℤ⟦γ⟧ (see Example 2.5). Let Q′ := Q^{(R_{F} ,F)} denote the fraction field of R_{F} ⟦Λ⟧_{F} and let Q′_{W} ≅ R_{F} [W] ⋉_{R} Q′ be the respective twisted formal group algebra over R_{F} (see Definition 5.1). We will continue to use the shorthand (5.2). We are now ready to define our second main objects of study.

Definition 7.5 (Formal (affine) Hecke algebra). The formal Hecke algebra H_{F} is the R_{F} - subalgebra of Q′_{W} generated by the elements

(7.1)  T_{i}^{F} := \begin{cases}Δ_{i}^{F} \frac{Θ_{F}}{κ_{i}^{F}} + δ_{i}μ_{F} (x_{γ}) & if κ^{F} ≠ 0, \\ 2Δ_{i}^{F} x_{γ} + δ_{i} & if κ^{F} = 0, \end{cases}

18 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

for all i ∈ I, where Θ_{F} := μ_{F} (x_{γ}) − μ_{F} (x_{−γ}) ∈ R_{F}. The formal affine Hecke algebra H_{F} is the R_{F} -subalgebra of Q′_{W} generated by H_{F} and

R_{F} ⟦Λ⟧_{F}^{κ} := \begin{cases}R_{F} ⟦Λ⟧_{F} [(κ_{α}^{F})^{−1} | α ∈ Φ] & if κ^{F} ≠ 0, \\ R_{F} ⟦Λ⟧_{F} & if κ^{F} = 0. \end{cases}

We sometimes write T_{i} when the FGL is understood from the context. When we wish to specify the coefficient ring, we write H_{R,F} (resp. H_{R,F}) for H_{F} (resp. H_{F}).

Remark 7.6. (a) If a_{11} (in the notation of (1.1)) is invertible in R, then κ_{α}^{F} is invertible

in R_{F} ⟦Λ⟧_{F} for all α ∈ Φ. Thus, R_{F} ⟦Λ⟧_{F}^{κ} = R_{F} ⟦Λ⟧_{F}.

(b) The coefficients \frac{Θ_{F}}{κ_{i}}, μ_{F} (x_{γ}), and x_{γ} −x_{−γ} appearing in Definition 7.5 are all invariant under the action of s_{i}.

(c) In the multiplicative case we have

\frac{Θ_{F}}{κ_{i}} = \frac{β(x_{γ}−x_{−γ})}{β} = x_{γ} − x_{−γ}.

Since the additive FGL is the β → 0 limit of the multiplicative, this motivates the choice of coefficient of Δ_{i}^{F} in the case κ^{F} = 0. More generally, one can show that when \frac{Θ_{F}}{κ_{i}^{F}} is expanded as a power series in x_{i} and x_{γ}, the leading term is equal to 2x_{γ}. Similarly, when κ^{F} = 0, we have x_{−γ} = −x_{γ} and so μ_{F} (x_{γ}) = 1, the coefficient of δ_{i}.

Lemma 7.7. For all ψ ∈ Q′ and i ∈ I, we have

(7.2)  ψT_{i} − T_{i}s_{i}(ψ) = \begin{cases}\frac{Θ_{F}}{κ_{i}} Δ_{α_{i}}^{F}(ψ) & if κ^{F} ≠ 0, \\ 2x_{γ}Δ_{α_{i}}^{F}(ψ) & if κ^{F} = 0. \end{cases}

In particular, φT_{i} − T_{i}s_{i}(φ) ∈ R_{F} ⟦Λ⟧_{F}^{κ} for all φ ∈ R_{F} ⟦Λ⟧_{F}^{κ}.

Proof. Let a and b be the coefficients of Δ_{i} and δ_{i} in (7.1), so that T_{i} = Δ_{i}a + δ_{i}b. By Lemma 5.5, we have

ψT_{i} = ψ(Δ_{i}a + δ_{s_{i}}b) = (Δ_{i}s_{i}(ψ) + Δ_{α_{i}}(ψ))a + δ_{s_{i}}s_{i}(ψ)b = T_{i}s_{i}(ψ) + aΔ_{α_{i}}(ψ).

The last statement is an easy verification left to the reader. □

Lemma 7.8. The elements T_{i}, i ∈ I, satisfy the quadratic relation

(7.3)  T_{i}^{2} = T_{i}Θ_{F} + 1.

Thus T_{i} is invertible and T_{i}^{−1} = T_{i} − Θ_{F}. Furthermore

(7.4)  (T_{i} + μ_{F} (x_{−γ}))(T_{i} − μ_{F} (x_{γ})) = 0.

Proof. Let a and b be the coefficients of Δ_{i} and δ_{i} in (7.1), so that T_{i} = Δ_{i}a+δ_{i}b. Using (5.1) and the fact that Δ_{i}δ_{i} + δ_{i}Δ_{i} = (δ_{i} − 1)κ_{i}^{F}, we have

T_{i}^{2} = Δ_{i}^{2}a^{2} + (Δ_{i}δ_{i} + δ_{i}Δ_{i})ab + b^{2} = Δ_{i}κ_{i}^{F} a^{2} + (δ_{i} − 1)κ_{i}^{F} ab + b^{2} = T_{i}(κ_{i}^{F} a) + b(b − κ_{i}^{F} a).

One readily verifies that in both cases in (7.1), we have κ_{i}^{F} a = Θ_{F} and b(b − κ_{i}^{F} a) = 1, completing the proof of the first statement in the lemma. The second two statements follow easily. □

Remark 7.9. Because of (7.4) and the fact that μ_{F} (x_{−γ}) = μ_{F} (x_{γ})^{−1}, one may think of the power series μ_{F} (x_{γ}) as a generalization of the deformation parameter t of the classical Hecke algebra (see Definition 7.1(a)).

Proposition 7.10. Suppose i, j ∈ I and let m_{ij} be the order of s_{i}s_{j} in W. Then

(7.5)  \underbrace{T_{j}T_{i}T_{j} ⋯}_{m_{ij} terms} − \underbrace{T_{i}T_{j}T_{i} ⋯}_{m_{ij} terms} = ∑_{w∈W, ℓ(w)<m_{ij}} T_{w}τ_{ij}^{w}

for some τ_{ij}^{w} ∈ Q′. In particular, we have the following.

(a) If ⟨α_{i}, α_{j}⟩ = 0, so that m_{ij} = 2, then T_{i}T_{j} = T_{j}T_{i}.

(b) If ⟨α_{i}, α_{j}⟩ = −1, so that m_{ij} = 3, then

(7.6)  T_{j}T_{i}T_{j} − T_{i}T_{j}T_{i} = (T_{i} − T_{j})σ_{ij}, σ_{ij} = χ_{i+j}(χ_{j} − χ_{−i}) − χ_{i}χ_{j},

where

χ_{α} = \begin{cases}\frac{Θ_{F}}{x_{α}κ_{α}} & if κ^{F} ≠ 0, \\ \frac{2x_{γ}}{x_{α}} & if κ^{F} = 0, \end{cases} α ∈ Φ.

(We use the usual convention that χ_{±i} = χ_{±α_{i}} and χ_{±i±j} = χ_{±α_{i}±α_{j}}.) Moreover, σ_{ij} = σ_{ji} commutes with δ_{i} and δ_{j} (and hence with T_{i} and T_{j}). If κ^{F} = 0, then

σ_{ij} = 4x_{γ}^{2}κ_{ij} ∈ R_{F} ⟦Λ⟧_{F}.

Proof. Set μ = μ_{F} (x_{γ}). (Thus μ = 1 iff κ^{F} = 0 by Lemma 3.3.) In both cases (i.e. κ^{F} ≠ 0 or κ^{F} = 0), we have T_{j} = χ_{j} + (μ − χ_{j})δ_{j}. Now

\underbrace{T_{j}T_{i}T_{j} ⋯}_{m_{ij} terms} = (χ_{j} + (μ − χ_{j})δ_{j})(χ_{i} + (μ − χ_{i})δ_{i})(χ_{j} + (μ − χ_{j})δ_{j}) · · · .

Since the δ_{w}, w ∈ W, form a basis of Q′_{W} as a right Q′-module, this can be written as a sum of (right) Q′-multiplies of δ_{w}. The leading term (with respect to the length of w) is

δ\underbrace{s_{j}s_{i}s_{j} ⋯ s_{n}s_{k}}_{m_{ij} terms}(μ − χ_{− \underbrace{s_{k}s_{n} ⋯ s_{j}s_{i}}_{m_{ij} −1 terms}(α_{j})}) ⋯ (μ − χ_{−s_{k}(α_{n})})(μ − χ_{−α_{k}}),

where n = i and k = j (resp. n = j and k = i) if m_{ij} is odd (resp. even). As in the proof of Proposition 5.7, we see that the highest order terms in T_{j}T_{i}T_{j} ⋯−T_{i}T_{j}T_{i} ⋯ cancel, proving the first part of the proposition.

Part (a) follows immediately from Proposition 5.7(a) and the facts that, under the assumptions, δ_{i}δ_{j} = δ_{j}δ_{i}, Δ_{i}δ_{j} = δ_{j}Δ_{i}, and Δ_{j}δ_{i} = δ_{i}Δ_{j}.

It remains to prove (b). We have

T_{j}T_{i}T_{j} = (χ_{j} + (μ − χ_{j})δ_{j})(χ_{i} + (μ − χ_{i})δ_{i})(χ_{j} + (μ − χ_{j})δ_{j})
= χ_{i}χ_{j}^{2} + (μ − χ_{j})(μ − χ_{−j})χ_{i+j} + ((μ − χ_{j})χ_{i+j}χ_{−j} + (μ − χ_{j})χ_{i}χ_{j})δ_{j}
+ (μ − χ_{i})χ_{j}χ_{i+j}δ_{i} + (μ − χ_{j})(μ − χ_{i+j})χ_{i}δ_{j}δ_{i} + (μ − χ_{i})(μ − χ_{i+j})χ_{j}δ_{i}δ_{j}
+ (μ − χ_{j})(μ − χ_{i})(μ − χ_{i+j})δ_{j}δ_{i}δ_{j}.

20 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

Using the fact that δ_{j}δ_{i}δ_{j} = δ_{j}δ_{i}δ_{j}, we see that

T_{j}T_{i}T_{j} − T_{i}T_{j}T_{i} = χ_{i}χ_{j}^{2} − χ_{i}^{2}χ_{j} + (μ − χ_{j})(μ − χ_{−j})χ_{i+j} − (μ − χ_{i})(μ − χ_{−i})χ_{i+j}
− σ_{ji}(μ − χ_{j})δ_{j} + σ_{ij}(μ − χ_{i})δ_{i}

= σ_{ij}χ_{i} − σ_{ji}χ_{j} − σ_{ji}(μ − χ_{j})δ_{j} + σ_{ij}(μ − χ_{i})δ_{i} + μχ_{i+j}(χ_{i} + χ_{−i} − χ_{j} − χ_{−j})

= σ_{ij}T_{i} − σ_{ji}T_{j} + μχ_{i+j}(χ_{i} + χ_{−i} − χ_{j} − χ_{−j})
= σ_{ij}T_{i} − σ_{ji}T_{j} + μ(σ_{ji} − σ_{ij})
= σ_{ij}(T_{i} − μ) − σ_{ji}(T_{j} − μ).

If κ^{F} = 0, then clearly σ_{ij} = 4x_{γ}^{2}κ_{ij}. Since, in this case, κ_{ij} = κ_{ji} (use the fact that x_{−i} = −x_{i} in (5.4)), we have σ_{ij} = σ_{ji}. If κ^{F} ≠ 0, we have

σ_{ij} = Θ_{F}^{2} \frac{x_{i}x_{−i}κ_{i}−x_{i}x_{j}κ_{j}−x_{−i}x_{i+j}κ_{i+j}}{x_{i}x_{−i}x_{j}x_{i+j}κ_{i}κ_{j}κ_{i+j}}
= Θ_{F}^{2} \frac{(x_{i}+x_{−i})x_{−j}x_{−i−j}−(x_{j}+x_{−j})x_{i}x_{−i−j}−(x_{i+j}+x_{−i−j})x_{−i}x_{−j}}{(x_{i}+x_{−i})(x_{j}+x_{−j})(x_{i+j}+x_{−i−j})}
= −Θ_{F}^{2} \frac{x_{i}x_{j}x_{−i−j}+x_{−i}x_{−j}x_{i+j}}{(x_{i}+x_{−i})(x_{j}+x_{−j})(x_{i+j}+x_{−i−j})} = −Θ_{F}^{2} (\frac{1}{x_{i}x_{j}x_{−i−j}} + \frac{1}{x_{−i}x_{−j}x_{i+j}} ) \frac{1}{κ_{i}κ_{j}κ_{i+j}} ,

which implies that σ_{ij} = σ_{ji}. Thus we have

T_{j}T_{i}T_{j} − T_{i}T_{j}T_{i} = σ_{ij}(T_{i} − T_{j})

The fact that σ_{ij} commutes with δ_{i} and δ_{j} is an easy verification left to the reader. □

For each w ∈ W, fix a reduced decomposition w = s_{i_{1}} ⋯ s_{i_{k}} and set

(7.7)  T_{w} = T_{i_{1}} ⋯ T_{i_{k}}.

Note that, in general, T_{w} depends on the choice of reduced decomposition.

Definition 7.11 (R̃_{F} and R_{F} ⟦Λ⟧_{F}^{∼}). Let R̃_{F} be the subalgebra of Q′ defined by

(7.8)  R̃_{F} := R_{F} [W] · R_{F} [τ_{ij}^{w} | i, j ∈ I, w ∈ W, ℓ(w) < m_{ij}].

where R_{F} [W]· denotes the natural action of the group algebra R_{F} [W] of W on Q′. Similarly, define

(7.9)  R_{F} ⟦Λ⟧_{F}^{∼} := R_{F} [W] · R_{F} ⟦Λ⟧_{F}^{κ} [τ_{ij}^{w} | i, j ∈ I, w ∈ W, ℓ(w) < m_{ij}].

Note that R_{F} ⟦Λ⟧_{F}^{∼} = R_{F} ⟦Λ⟧_{F} if the root system is simply laced and κ^{F} = 0 (since σ_{ij} = 4x_{γ}^{2}κ_{ij} ∈ R_{F} ⟦Λ⟧_{F} in that case).

Lemma 7.12. The set {T_{w} | w ∈ W } forms a basis of H_{F} ⊗_{R_{F}} R̃_{F} as a right (or left) R̃_{F} -module and a basis of H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R_{F} ⟦Λ⟧_{F}^{∼} as a right (or left) R_{F} ⟦Λ⟧_{F}^{∼}-module.

Proof. The proof is analogous to that of Lemma 5.11 and will be omitted. □

Theorem 7.13. Given a formal group law (R, F), the formal affine Hecke algebra H_{F} is generated as an R_{F} -algebra by R_{F} ⟦Λ⟧_{F}^{κ} and the elements T_{i}, i ∈ I, and satisfies

(a) relation (7.2) for all i ∈ I and φ ∈ R_{F} ⟦Λ⟧_{F}^{κ},

(b) (T_{i} + μ_{F} (x_{−γ}))(T_{i} − μ_{F} (x_{γ})) = 0 for all i ∈ I,

(c) T_{i}T_{j} = T_{j}T_{i} for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ = 0,

(d) relation (7.6) for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ = −1, and

(e) relation (7.5) for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ ≤ −2. Furthermore, (a)–(e) form a complete set of relations (over R_{F} ⟦Λ⟧_{F}^{∼}) for H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R_{F} ⟦Λ⟧_{F}^{∼}.

Proof. The first part of the theorem follows immediately from Lemmas 7.7 and 7.8 and Proposition 7.10. The second part is analogous to the proof of Theorem 5.12 and will be omitted. □

8. Formal (affine) Hecke algebras: examples and further properties

In this final section we specialize the definition of the formal (affine) Hecke algebra to various FGLs, yielding classical algebras as well as new ones. We then prove several important facts about these algebras in general.

As in Section 5, we have a map Q′_{W} → End_{R_{F}} Q′. Since the operators T_{i}^{F} preserve R_{F} ⟦Λ⟧_{F}^{κ}, we have an induced map H_{F} → End_{R_{F}} R_{F} ⟦Λ⟧_{F}^{κ} of R_{F} -algebras. Recall that if a_{11} (in the notation of (1.1)) is invertible in R, then κ_{α}^{F} is invertible for all α ∈ Φ, and so R_{F} ⟦Λ⟧_{F}^{κ} = R_{F} ⟦Λ⟧_{F}.

Proposition 8.1. If a_{11} is invertible in R, then the map H_{F} → End_{R_{F}} R_{F} ⟦Λ⟧_{F} described above is injective. In other words, the natural action of H_{F} on R_{F} ⟦Λ⟧_{F} is faithful.

Proof. Suppose, contrary to the statement of the proposition, that the given map is not injective. Let a ∈ H_{F} be in the kernel of this map, with a ≠ 0. In other words, a acts by zero on R_{F} ⟦Λ⟧_{F} under the associated action. By Lemma 7.12, we may write

a = ∑_{w∈W} T_{w}a_{w}, a_{w} ∈ R_{F} ⟦Λ⟧_{F}^{∼}.

Now, clearly aφ also acts by zero on R_{F} ⟦Λ⟧_{F} for all φ ∈ R_{F} ⟦Λ⟧_{F}. Choosing φ to be a common denominator of all the a_{w}, we see that we may assume that a_{w} ∈ R_{F} ⟦Λ⟧_{F} for all w ∈ W.

For φ ∈ R_{F} ⟦Λ⟧_{F}, define the degree of φ to be

deg φ := max{m ∈ ℤ_{≥0} | φ ∈ I_{F}^{m}},

where I_{F} is the kernel of the augmentation map ε: R_{F} ⟦Λ⟧_{F} → R_{F} (i.e. the element x_{γ} is not mapped to zero). We adopt the convention that deg 0 = −∞. Then the formal Demazure operators lower degree and the coefficients μ_{F} (x_{γ}), \frac{Θ_{F}}{κ_{i}^{F}} and x_{γ} appearing in Definition (7.1) of T_{i} preserve degree. Thus, if deg φ = m, we have

T_{i}(φ) = μ_{F} (x_{γ})s_{i}(φ) + (terms of degree < m).

Furthermore, deg(φφ′) = deg φ + deg φ′ for φ, φ′ ∈ R_{F} ⟦Λ⟧_{F}. Indeed, it follows by definition that deg(φφ′) ≥ deg φ + deg φ′. If deg(φφ′) > deg φ + deg φ′, then in the associate graded algebra we have φφ′ = 0, where φ ≠ 0 and φ′ ≠ 0. Identifying the associated graded algebra with the polynomial algebra (by [CPZ, Lem. 4.2]) we obtain a contradiction as the polynomial algebra is a domain.

Let m be the maximum degree of the a_{w}, w ∈ W, and set W′ = {w ∈ W | deg a_{w} = m}. Then, for all φ ∈ R_{F} ⟦Λ⟧_{F}, we have

0 = a(φ) = ∑_{w∈W′} T_{w}(a_{w}φ) + ∑_{w∈W \W′} T_{w}(a_{w}φ)
= ∑_{w∈W′} μ_{F} (x_{γ})^{ℓ(w)}s_{w}(a_{w}φ) + b,

22 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

where the last summation lies in I_{F}^{m+deg φ} and b ∉ I_{F}^{m+deg φ}. It follows that

∑_{w∈W′} μ_{F} (x_{γ})^{ℓ(w)}s_{w}(a_{w}φ) = ∑_{w∈W′} s_{w} (μ_{F} (x_{γ})^{ℓ(w)}a_{w}φ) = 0 for all φ ∈ R_{F} ⟦Λ⟧_{F} .

The above sum is therefore also equal to zero in I_{F}^{m+deg φ}/I_{F}^{m+deg φ+1}. But ⨁_{n} I_{F}^{n}/I_{F}^{n+1} ≅ S_{R_{F}}^{∗} (Λ), by [CPZ, Lem. 4.2]. Since the action of R_{F} ⟦W ⟧ ⋉ S_{R_{F}}^{∗} (Λ) on S_{R_{F}}^{∗} (Λ) is faithful (see, for example, the argument in [Kle05, Second Proof of Thm. 3.2.2]), we have that μ_{F} (x_{γ})^{ℓ(w)}a_{w} = 0 (hence a_{w} = 0) for all w ∈ W′. But this contradicts the choice of m. □

Remark 8.2. In the additive and multiplicative cases, Proposition 8.1 reduces to known embeddings of the (degenerate) affine Hecke algebra into endomorphism rings. See the proof of Proposition 8.3.

The following proposition demonstrates that our definition of the formal (affine) Hecke algebra recovers classical objects in the additive and multiplicative cases.

Proposition 8.3. Suppose R = ℤ.

(a) For the additive FGL, we have the following isomorphisms of algebras:

H_{A} ≅ H_{deg}^{∧} := H_{deg} ⊗_{ℤ[ϵ]} ℤ⟦γ⟧, H_{A} ≅ ℤ_{A}[W] ≅ ℤ[W] ⊗_{ℤ} ℤ⟦γ⟧,

where ϵ = −2γ.

(b) For the multiplicative periodic FGL, we have the following isomorphisms of algebras:

H_{M} ≅ H ⊗_{ℤ[t,t^{−1}]} ℤ[t, t^{−1}]^{∧}, H_{M} ≅ H ⊗_{ℤ[t,t^{−1}]} ℤ[t, t^{−1}]^{∧}.

Proof. It is easy to see that for the additive and multiplicative FGLs in simply laced type, the relations of Theorem 7.13 become the relations of the respective algebras in the statement of the proposition. However, we provide a proof that remains valid in all types (i.e. not necessarily simply laced). Note that in the both the additive case (where κ^{F} = 0) and multiplicative periodic case (where a_{11} = β is invertible and hence all κ_{α}, α ∈ Φ, are invertible), we have R_{F} ⟦Λ⟧_{F}^{κ} = R_{F} ⟦Λ⟧_{F}.

Consider first the additive FGL. Recall the identification ℤ_{A} ≅ ℤ⟦γ⟧ of Example 2.5. The injective map H_{A} ↪ End_{ℤ_{A}} ℤ_{A}⟦Λ⟧_{A} is given on the T_{i} by

T_{i} = 2γΔ_{i}^{A} + δ_{i} ↦ s_{i} + 2γ \frac{1}{α}(s_{i} − 1).

Thus H_{A} is isomorphic to the subalgebra H′_{A} of End_{ℤ_{A}} ℤ_{A}⟦Λ⟧_{A} generated by multiplication by elements of ℤ_{A}⟦Λ⟧_{A} and the operators s_{i} + 2γ \frac{1}{α}(s_{i} − 1).

Observe that, in the notation of [Gin, §12], the algebra S_{ℤ_{A}}^{∗}(Λ)^{∧} ⊗_{ℤ} ℂ = S_{ℤ}^{∗}(Λ)^{∧} ⊗_{ℤ} ℂ⟦γ⟧ can be identified with the completion of the algebra ℂ[𝔥, γ] of polynomial functions on 𝔥 with coefficients in ℂ[γ]. If we let ϵ = −2γ, then we see that H′_{A} is precisely the completion of the image of H_{deg} under the faithful action on ℂ[𝔥, ϵ] given by Demazure-Lusztig type operators (see [Gin, Prop. 12.2] or [Kle05, Second Proof of Thm. 3.2.2]). This proves the first isomorphism of part (a). The second follows by considering the subalgebra generated by the T_{i}.

Now consider the multiplicative periodic FGL F_{M}(u, v) = u + v − βuv, β ∈ ℤ^{×}. We have (see Example 1.2(b))

μ_{M}(x_{γ}) = 1 − βx_{−γ} = t and Θ_{M} = β(x_{γ} − x_{−γ}) = t − t^{−1} ∈ ℤ[t, t^{−1}]^{∧},

under the identifications of Example 2.5. Using the above and the identifications of Example 2.4(b), the injective map H_{M} ↪ End_{ℤ_{M}} ℤ_{M}⟦Λ⟧_{M} is given on the T_{i} by

(8.1)  T_{i} = Δ_{i}^{M} \frac{Θ_{M}}{κ_{i}^{M}} + δ_{i}μ_{M}(x_{γ}) = \frac{t−t^{−1}}{1−e^{−α_{i}}} (1 − δ_{i}) + tδ_{i} ↦ t^{−1} \frac{1−s_{i}}{e^{−α_{i}}−1} − t\frac{1−e^{−α_{i}}s_{i}}{e^{−α_{i}}−1} .

We identify ℤ_{M}⟦Λ⟧_{M} with ℤ[q, q^{−1}][P] ⊗_{ℤ[q,q^{−1}]} ℤ[t, t^{−1}]^{∧} in the notation of [Lus85] (where the P and q of [Lus85] are our Λ and t^{2}, respectively) via the map e^{λ} ↦ −λ (see Example 2.4(b)). (The negative sign in front of the α_{i} arises from the twisting of the action of ℤ[q, q^{−1}][P] on itself by a sign in [Lus85, (8.2)].) Under this identification, the right hand side of (8.1) corresponds to the Demazure-Lusztig operator [Lus85, (8.1)], where the T_{s} of [Lus85] corresponds to our tT_{i}, where s = s_{i} (see Remark 7.3). Therefore, the actions of H_{M} and H on ℤ_{M}⟦Λ⟧_{M} ≅ ℤ[q, q^{−1}][P] ⊗_{ℤ[q,q^{−1}]} ℤ[t, t^{−1}]^{∧} coincide. The action of H_{M} is faithful by Proposition 8.1 and the action of H is also known to be faithful (see, for example, [Gin, Prop. 12.2(i)] or note that the action of H specializes to the standard action of ℤ[W] ⋉ ℤ[Λ] on ℤ[Λ] when q = 1). Thus we have the first isomorphism of part (b). The second follows by considering the subalgebra generated by the T_{i}. □

For other FGLs our definition gives new algebras as the following examples indicate.

Example 8.4 (Lorentz case). For the Lorentz FGL F_{L}, we have μ_{L}(u) = 1, Θ_{L} = 0, and κ = 0. Since κ_{ij} = β (see (6.1)), we have σ_{ij} = 4βx_{γ}^{2}. Thus the relations (a)–(d) of Theorem 7.13 become

(a) φT_{i} − T_{i}s_{i}(φ) = 2x_{γ}Δ_{α_{i}}^{L}(φ) for all φ ∈ R_{F} ⟦Λ⟧_{F} , i ∈ I.

(b) T_{i}^{2} = 1 for all i ∈ I,

(c) T_{i}T_{j} = T_{j}T_{i} for all i, j ∈ I such that ⟨α_{i}, α_{j}^{∨}⟩ = 0,

(d) T_{i}T_{j}T_{i} − T_{j}T_{i}T_{j} = 4βx_{γ}^{2}(T_{i} − T_{j}) for all i, j ∈ I such that ⟨α_{i}, α_{j}⟩ = −1. These form a complete set of relations in the simply laced case.

Example 8.5 (Elliptic case). For the elliptic FGL F_{E}, we have

μ_{E}(u) = \frac{1}{1−a_{1}u−a_{3}v(u)}, Θ_{E} = \frac{2ψ−ψ^{2}}{1−ψ} = \frac{ψ}{1−ψ} + ψ,

where ψ = a_{1}x_{γ} + a_{3}v(x_{γ}) (see Example 1.2(d)). If, for example, a_{3} = 0, then

μ_{E}(u) = \frac{1}{1−a_{1}u}, Θ_{E} = \frac{2a_{1}x_{γ}−a_{1}^{2}x_{γ}^{2}}{1−a_{1}x_{γ}} , κ_{i} = a_{1} for all i ∈ I,

and so

T_{i} = Δ_{i}^{E} \frac{2x_{γ}−a_{1}x_{γ}^{2}}{1−a_{1}x_{γ}} + δ_{i} \frac{1}{1−a_{1}u} for all i ∈ I.

Furthermore, when a_{3} = 0, we have

χ_{i} = \frac{Θ_{E}}{x_{i}a_{1}}

and so

σ_{ij} = \frac{Θ_{E}}{x_{i}a_{1}} \frac{Θ_{E}}{x_{j}a_{1}} + \frac{Θ_{E}}{x_{i+j}a_{1}} (\frac{Θ_{E}}{x_{−i}a_{1}} − \frac{Θ_{E}}{x_{j}a_{1}} ) = −\frac{Θ_{E}^{2}}{a_{1}^{2}} κ_{ij}.

Example 8.6 (Universal formal Hecke algebra). We call the formal Hecke algebra H_{U} corresponding to the universal FGL F_{U} the universal formal Hecke algebra. Observe that H_{U} is an algebra over 𝕃_{U}, where 𝕃 is the Lazard ring. Note that in this case we have

Θ_{U} = −a_{11}(x_{γ} − x_{−γ}) + a_{11}^{2}(x_{γ}^{2} − x_{−γ}^{2}) − (a_{11}^{3} + a_{12}a_{11} − a_{22} + 2a_{13})(x_{γ}^{3} − x_{−γ}^{3}) + ⋯
= −2a_{11}x_{γ} − 2(a_{11}^{3} + a_{11}a_{12} − a_{22} + 2a_{13})x_{γ}^{3} + · · · .

24 ALEX HOFFNUNG, JOSÉ MALAGÓN LÓPEZ, ALISTAIR SAVAGE, AND KIRILL ZAINOULLINE

Theorem 8.7. Suppose (R, F) and (R, F′) are FGLs over the same ring R, with either κ^{F} = 0 or a_{11} invertible in R (in the notation of (1.1)). Then

H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R′_{F} ⟦Λ⟧_{F} ≅ H_{F′} ⊗_{R_{F′}⟦Λ⟧_{F′}} R′_{F′}⟦Λ⟧_{F′}

as algebras, where R′_{F} = (R ⊗_{ℤ} ℚ)_{F} ⊗_{ℚ} ℚ[x_{γ}^{−1}] (and similarly, with F replaced by F′).

Proof. It suffices to prove the result when F′ = F_{A}. Let R_{ℚ} = R ⊗_{ℤ} ℚ. As in the proof of Theorem 6.4, we have an isomorphism of twisted formal group algebras

e_{F}^{⋆} : Q_{W}^{(R′_{F} ,F)} → Q_{W}^{(R′_{A},A)}.

Since

e_{F}^{⋆} (x_{γ}^{−1}) = \frac{1}{e_{F} (x_{γ})} = \frac{x_{γ}}{e_{F} (x_{γ})}x_{γ}^{−1}

and \frac{x_{γ}}{e_{F} (x_{γ})} ∈ (R_{ℚ})_{A} is invertible in (R_{ℚ})_{A}, we see that e_{F}^{⋆} (R′_{F}) = R′_{A} and so e_{F}^{⋆} (R′_{F} ⟦Λ⟧_{F}) = R′_{A}⟦Λ⟧_{A}. The algebra H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R′_{F} ⟦Λ⟧_{F} is isomorphic to its image H′ := e_{F}^{⋆} (H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R′_{F} ⟦Λ⟧_{F}) under e_{F}^{⋆}.

We first consider the case where κ^{F} = 0. Then H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R′_{F} ⟦Λ⟧_{F} is generated over R′_{F} ⟦Λ⟧_{F} by (the element 1 and) the elements

T_{i}^{F} − 1 = 2Δ_{i}^{F} x_{γ} + δ_{i} − 1 = Ξ_{i}^{F} (δ_{i} − 1), i ∈ I,

where

Ξ_{i}^{F} = 1 − \frac{2x_{γ}}{x_{i}} ∈ Q_{W}^{(R_{F} ,F)}.

We see that

e_{F}^{⋆} (T_{i}^{F} − 1) = e_{F}^{⋆} (Ξ_{i}^{F})(δ_{i} − 1) = \frac{e_{F}^{⋆} (Ξ_{i}^{F})}{Ξ_{i}^{A}}(T_{i}^{A} − 1).

Thus it suffices to show that e_{F}^{⋆} (Ξ_{i}^{F})/Ξ_{i}^{A} lies in R′_{A}⟦Λ⟧_{A} and is invertible in R′_{A}⟦Λ⟧_{A} (i.e. has invertible constant term). Now,

\frac{e_{F}^{⋆} (Ξ_{i}^{F})}{Ξ_{i}^{A}} = (1 − \frac{2e_{F} (x_{γ})}{e_{F} (x_{i})}) (1 − \frac{2x_{γ}}{x_{i}} )^{−1}
= (\frac{x_{i}}{e_{F} (x_{i})} \frac{e_{F} (x_{γ})}{x_{γ}} − \frac{x_{i}}{2x_{γ}} ) (1 − \frac{x_{i}}{2x_{γ}} )^{−1} .

Note that e_{F} (x_{i})/x_{i}, e_{F} (x_{γ})/x_{γ} ∈ R′_{A}⟦Λ⟧_{A} are invertible in R′_{A}⟦Λ⟧_{A} (with constant term one). Since 1 − x_{i}/(2x_{γ}) ∈ R′_{A}⟦Λ⟧_{A} is also clearly invertible in R′_{A}⟦Λ⟧_{A}, we are done.

Now consider the case where κ^{F} ≠ 0 and a_{11} is invertible in R (hence in R′_{A}). The elements

T_{i}^{F} − μ_{F} (x_{γ}) = ρ_{i}(δ_{i} − 1), i ∈ I,

where

ρ_{i} = μ_{F} (x_{γ}) − \frac{Θ_{F}}{x_{i}κ_{i}^{F}} ∈ Q_{W}^{(R_{F} ,F)}

generate H_{F} ⊗_{R_{F} ⟦Λ⟧_{F}} R′_{F} ⟦Λ⟧_{F} over R′_{F} ⟦Λ⟧_{F} (along with the element 1). Since

e_{F}^{⋆} (T_{i}^{F} − μ_{F} (x_{γ})) = \frac{e_{F}^{⋆} (ρ_{i})}{Ξ_{i}^{A}}(T_{i}^{A} − 1),

it follows as in the κ^{F} = 0 case that it suffices to show that e_{F}^{⋆} (ρ_{i})/Ξ_{i}^{A} lies in R′_{A}⟦Λ⟧_{A} and is invertible in R′_{A}⟦Λ⟧_{A} (i.e. has invertible constant term).

For any x ∈ R′_{A}⟦Λ⟧_{A} we set

ψ(x) = \frac{1−μ_{F} (e_{F} (x))}{x} = a_{11} + O(1) ∈ R′_{A}⟦Λ⟧_{A}

so that μ_{F} (e_{F} (x)) = 1 − xψ(x). Then (as x_{−λ} = −x_{λ} in R_{A}⟦Λ⟧_{A})

\frac{e_{F}^{⋆} (ρ_{i})}{Ξ_{i}^{A}} = (μ_{F} (e_{F} (x_{γ})) − \frac{μ_{F} (e_{F} (x_{γ}))−μ_{F} (e_{F} (−x_{γ}))}{1−μ_{F} (e_{F} (x_{−i}))}) (1 − \frac{2x_{γ}}{x_{i}} )^{−1}
= (1 − x_{γ}ψ(x_{γ}) − \frac{x_{γ}}{x_{i}} · \frac{ψ(−x_{γ})+ψ(x_{γ})}{ψ(−x_{i})}) (1 − \frac{2x_{γ}}{x_{i}} )^{−1}
= (\frac{ψ(−x_{γ})+ψ(x_{γ})}{ψ(−x_{i})} + x_{i}ψ(x_{γ}) − \frac{x_{i}}{x_{γ}} ) (2 − \frac{x_{i}}{x_{γ}} )^{−1} .

Since a_{11} is invertible, we have

\frac{ψ(−x_{γ})+ψ(x_{γ})}{ψ(−x_{i})} = \frac{2a_{11}+O(1)}{a_{11}+O(1)} = 2 + O(1).

Combining all of the above computations, we see that

\frac{e_{F}^{⋆} (ρ_{i})}{Ξ_{i}^{A}} = 1 + O(1) ∈ R′_{A}⟦Λ⟧_{A}

is invertible in R′_{A}⟦Λ⟧_{A} as desired. □

Remark 8.8. (a) It is known that certain localizations or completions of the affine Hecke algebra and degenerate affine Hecke algebra are isomorphic (see [Lus89, Thm. 9.3] and [Rou, §3.1.7]). Theorem 8.7 can be seen as an analogue of these results.

(b) Note that while Theorem 8.7 shows that all affine Hecke algebras (satisfying the hypotheses of the proposition) become isomorphic over appropriate rings, the isomorphism is not the naive one sending T_{i}^{F} to T_{i}^{F′}. Furthermore, the completion (with respect to the augmentation map) is crucial. No assertion is made regarding an isomorphism (even over ℚ) of truncated versions. See Remark 6.5.

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Department of Mathematics and Statistics, University of Ottawa