The formal affine Demazure algebra and real finite reflection groups

TL;DR

This paper extends the formal affine Demazure algebra framework to all real finite reflection groups, providing new algebraic structures and explicit computations of structure coefficients for specific groups.

Contribution

It generalizes the formal affine Demazure algebra and formal group ring to all real finite reflection groups, including explicit calculations for complex cases.

Findings

Generalized formal group ring for all real finite reflection groups

Defined and studied formal Demazure operators on these rings

Computed structure coefficients for specific reflection groups

Abstract

In this paper, we generalize the formal affine Demazure algebra of Hoffnung-Malag\'on-L\'opez-Savage-Zainoulline to all real finite reflection groups. We begin by generalizing the formal group ring of Calm\`es-Petrov-Zainoulline to all real finite reflection groups. We then define and study the formal Demazure operators that act on the formal group ring. Using these results and constructions, we define and study the formal affine Demazure algebra for all real finite reflection groups. Finally, we compute several structure coefficients that appear in a braid relation among the formal Demazure elements, and we conclude this paper by computing all structure coefficients for the reflection groups $I_2(5)$, $I_2(7)$, $H_3$, and $H_4$.

Equations

$$c_1({\mathcal{L}}_1\otimes {\mathcal{L}}_2)=F(c_1({\mathcal{L}}_1),c_1({\mathcal{L}}_2)),$$

$${\mathfrak{c}}_{G/B}:{\widetilde{R[[\Lambda ]]}}_F\to {\mathrm{h}}^{\ast }(G/B).$$

$$s_{\alpha }(v)=v-2\frac{(\alpha ,v)}{(\alpha ,\alpha )}\alpha ,v\in V.$$

$$(s_i{s}_j{)}^{m_{i,j}}=1,{\alpha }_i,{\alpha }_j\in \Delta ,$$

$$v({\Sigma }^{-})\cap {\Sigma }^{+}=\{{\theta }_{i_j}:1\le j\le t\},$$

$${\Sigma }^{+}=\{{\theta }_{i_j}:1\le j\le t\}.$$

$${\Sigma }_{i,j}^{+}:$$

$$s_j\left({\Sigma }_{i,j}^{+}\right)$$

$${\alpha }_1=\left(a,-\frac{1}{2},b,0\right);{\alpha }_2=\left(-a,\frac{1}{2},b,0\right);{\alpha }_3=\left(\frac{1}{2},b,-a,0\right);{\alpha }_4=\left(-\frac{1}{2},-a,0,b\right).$$

$$\alpha_{i}^{\vee}(\alpha_{j})=\begin{cases}2,\quad{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}o}\text{ if $i=j$};\\ -\tau,\quad\text{if $(i,j)\in\{(1,2),(2,1)\}$};\\ -1,\quad\text{if $(i,j)\in\{(2,3),(3,2),(3,4),(4,3)\}$};\\ 0,\quad{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}o}\text{ if $(i,j)\in\{(1,3),(3,1),(1,4),(4,1),(2,4),(4,2)\}$},\end{cases}$$

$${\alpha }_1=\left(a,-\frac{1}{2},b\right);{\alpha }_2=\left(-a,\frac{1}{2},b\right);{\alpha }_3=\left(\frac{1}{2},b,-a\right).$$

$${\alpha }_1=(-1,0);{\alpha }_2=\left(\text{cos}\left(\frac{\pi }{m}\right),\text{sin}\left(\frac{\pi }{m}\right)\right).$$

$$\alpha_{i}^{\vee}(\alpha_{j})=\begin{cases}2,\quad\quad\quad\quad{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}....}\text{if $i=j$};\\ -2\textrm{cos}\left(\tfrac{\pi}{m}\right),\quad\text{if $(i,j)\in\{(1,2),(2,1)\}$}.\end{cases}$$

$$\left\{1,2\cos \left(\frac{\pi }{m}\right),{\left(2\cos \left(\frac{\pi }{m}\right)\right)}^2,\dots ,{\left(2\cos \left(\frac{\pi }{m}\right)\right)}^{l-1}\right\}$$

$$x^{-\frac{\varphi (2m)}{2}}{Q}_{2m}(x)={\psi }_m(x+x^{-1}),$$

$${\alpha }_1;s_2({\alpha }_1)={\alpha }_1+{\alpha }_2;s_1{s}_2({\alpha }_1)={\alpha }_2;$$

$$s_2{s}_1{s}_2({\alpha }_1)=-{\alpha }_2;s_1{s}_2{s}_1{s}_2({\alpha }_1)=-{\alpha }_1-{\alpha }_2;s_2{s}_1{s}_2{s}_1{s}_2({\alpha }_1)=-{\alpha }_1.$$

$${\alpha }_1;s_2({\alpha }_1)={\alpha }_1+\sqrt{2}{\alpha }_2;s_1{s}_2{s}_1{s}_2({\alpha }_1)=-{\alpha }_1;s_2{s}_1{s}_2{s}_1{s}_2({\alpha }_1)=-{\alpha }_1-\sqrt{2}{\alpha }_2;$$

$${\alpha }_2;s_1({\alpha }_2)=\sqrt{2}{\alpha }_1+{\alpha }_2;s_2{s}_1{s}_2{s}_1({\alpha }_2)=-{\alpha }_2;s_1{s}_2{s}_1{s}_2{s}_1({\alpha }_2)=-\sqrt{2}{\alpha }_1-{\alpha }_2.$$

$${\alpha }_1;s_2({\alpha }_1)={\alpha }_1+\tau {\alpha }_2;s_1{s}_2({\alpha }_1)=\tau {\alpha }_1+\tau {\alpha }_2;s_2{s}_1{s}_2({\alpha }_1)=\tau {\alpha }_1+{\alpha }_2;$$

$$s_{1,2,\dots }^{(4)}({\alpha }_1)={\alpha }_2;s_{2,1,\dots }^{(5)}({\alpha }_1)=-{\alpha }_2;s_{1,2,\dots }^{(6)}({\alpha }_1)=-\tau {\alpha }_1-{\alpha }_2;$$

$$s_{2,1,\dots }^{(7)}({\alpha }_1)=-\tau {\alpha }_1-\tau {\alpha }_2;s_{1,2,\dots }^{(8)}({\alpha }_1)=-{\alpha }_1-\tau {\alpha }_2;s_{2,1,\dots }^{(9)}({\alpha }_1)=-{\alpha }_1.$$

$${\alpha }_1;s_2({\alpha }_1)={\alpha }_1+\zeta {\alpha }_2;s_1{s}_2({\alpha }_1)=({\zeta }^2-1){\alpha }_1+\zeta {\alpha }_2;s_2{s}_1{s}_2({\alpha }_1)=({\zeta }^2-1){\alpha }_1+({\zeta }^2-1){\alpha }_2;$$

$$s_{1,2,\dots }^{(4)}({\alpha }_1)=\zeta {\alpha }_1+({\zeta }^2-1){\alpha }_2;s_{2,1,\dots }^{(5)}({\alpha }_1)=\zeta {\alpha }_1+{\alpha }_2;s_{1,2,\dots }^{(6)}({\alpha }_1)={\alpha }_2;s_{2,1,\dots }^{(7)}({\alpha }_1)=-{\alpha }_2;$$

$$s_{1,2,\dots }^{(8)}({\alpha }_1)=-\zeta {\alpha }_1-{\alpha }_2;s_{2,1,\dots }^{(9)}({\alpha }_1)=-\zeta {\alpha }_2+(1-{\zeta }^2){\alpha }_2;s_{1,2,\dots }^{(10)}({\alpha }_1)=(1-{\zeta }^2){\alpha }_1+(1-{\zeta }^2){\alpha }_2;$$

$$s_{2,1,\dots }^{(11)}({\alpha }_1)=(1-{\zeta }^2){\alpha }_1-\zeta {\alpha }_2;s_{1,2,\dots }^{(12)}({\alpha }_1)=-{\alpha }_1-\zeta {\alpha }_2;s_{2,1,\dots }^{(13)}({\alpha }_1)=-{\alpha }_1.$$

$${\exp }_F(u)=u+\sum _{i\ge 2}{a}_i{u}^i;{\log }_F(u)=u+\sum _{i\ge 2}{b}_i{u}^i.$$

$${\log }_{F_m}(x)=\log (1-x)=\sum _{i\ge 1}(-1{)}^{i-1}\frac{x^i}{i};{\exp }_{F_m}(x)=\exp (x)=\sum _{i\ge 1}\frac{x^i}{i!}.$$

$${\log }_{F_l}(x)=\tan {\mathrm{h}}^{-1}(x)=\sum _{i\ge 1}\frac{x^{2i-1}}{2i-1};{\exp }_{F_l}(x)=\tanh (x)=\sum _{i\ge 1}\frac{2^{2i}(2^{2i}-1)B_{2i}}{(2i)!}{x}^{2i-1},$$

$$F(x,y)=(x+y)g(x,y),g(x,y)\in R[[x,y]].$$

Full text

1. 1 Introduction
2. 2 Real finite reflection groups and the real root lattice
3. 3 Formal group laws and the classical formal group ring
4. 4 Construction of the formal group ring
5. 5 Formal Demazure operators
6. 6 Endomorphisms of the formal group ring
7. 7 The formal affine Demazure algebra
8. 8 Presentation in terms of generators and relations
9. 9 Computations of structure coefficients
10. 10 Applications
11. 11 Appendix

1. Introduction

Let $\mathrm{h}^{*}$ be an algebraic oriented cohomology theory in the sense of Levine-Morel [22]. If $X$ is a smooth projective variety over a field $k$, then, given a vector bundle $E\to X$ over $X$ of rank $n$, there is a set of Chern classes $c_{i}(E)\in\mathrm{h}^{i}(X)$, $i\in\{0,\dotsc,n\}$. Any two line bundles $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ over $X$ satisfy Quillen’s formula,

[TABLE]

where $F$ is the formal group law associated to $\mathrm{h}^{*}$. Suppose $G$ is a split semisimple simply connected linear algebraic group over a field $k$, and let $T$ be a maximal split torus sitting inside a Borel subgroup $B$ of $G$, so that $G/B$ is a complete flag variety. In [7], Calmès-Petrov-Zainoulline constructed the formal group ring, which we will denote $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$, where $F$ is a one-dimensional commutative formal group law over a commutative unital ring $R$. If $\mathrm{h}^{*}$ is the Chow theory, then $F$ is the additive formal group law over $R=\mathbb{Z}$, and $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ is the completion of the symmetric algebra $S_{\mathbb{Z}}^{*}(\Lambda)$, where $\Lambda$ is the character lattice of $T$. When $R=\mathrm{h}^{*}(\mathrm{Spec}(k))$, the authors showed that there is a ring homomorphism, called the chararacteristic map,

[TABLE]

Using this map, the authors constructed an algebraic model $\mathcal{H}(\Lambda)_{F}$ for the algebraic oriented cohomology ring $\mathrm{h}^{*}(G/B)$ in terms of augmented Demazure operators. Specializing $\mathrm{h}^{*}$ to the Chow theory, $\mathcal{H}(\Lambda)_{F}$ is the algebraic model for the Chow theory of the flag variety $G/B$ constructed by Demazure in [11].

In [19, 20], Kostant-Kumar described the equivariant cohomology and $K$-theory of flag varieties using the techniques of nil-Hecke and 0-Hecke algebras. These algebras are generated by Demazure operators, which satisfy a braid relation. Calmès, Hoffnung, Malagón-López, Savage, Zainoulline, and Zhong used the formal group ring $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ to generalize the nil-Hecke and [math]-Hecke algebras of Kostant-Kumar to an arbitrary algebraic oriented cohomology theory [8, 9, 10, 17]. In particular, they constructed and studied the formal affine Demazure algebra, $\mathbf{D}_{F}$, which is generated by formal Demazure elements that satisfy a twisted braid relation (see [17, Prop. 6.8]). After specializing $F$ to the additive and multiplicative formal group laws, $\mathbf{D}_{F}$ equals completion of the nil-Hecke and 0-Hecke algebra, respectively.

Let $W$ be a real finite reflection group, let $\Sigma$ be a root system of $W$, and let $\Delta$ be a simple system of $\Sigma$. The first goal of this paper is to generalize the formal group ring of [7] to all real finite reflection groups. We begin by defining the real root lattice $\Lambda$ with respect to the pair $(\Sigma,\Delta)$. When $\Sigma$ is crystallographic, $\Lambda$ is the crystallographic root lattice, as described in [9, §2]. Using the data of a formal group law $F$ over a subring $R$ of $\mathbb{C}$ satisfying the conditions of 4.6, and the real root lattice $\Lambda$, we define the formal group ring $R\llbracket\Lambda\rrbracket_{F}$. When $\Sigma$ is crystallographic, $R\llbracket\Lambda\rrbracket_{F}$ is the formal group ring $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ of [7]. Once $R\llbracket\Lambda\rrbracket_{F}$ is defined, we study the subalgebra $\mathcal{D}_{(R,F)}(\Lambda)$ of the endomorphism algebra of $R\llbracket\Lambda\rrbracket_{F}$ generated by formal Demazure operators and by multiplication by elements in $R\llbracket\Lambda\rrbracket_{F}$. In particular, we show that $\mathcal{D}_{(R,F)}(\Lambda)$ is a free finitely-generated $R\llbracket\Lambda\rrbracket_{F}$-module. When $\Sigma$ is crystallographic, the $R\llbracket\Lambda\rrbracket_{F}$-algebra $\mathcal{D}_{(R,F)}(\Lambda)$ is the endomorphism algebra studied in [7]. When $F$ is the additive formal group law and $R=\mathbb{C}$, the $R\llbracket\Lambda\rrbracket_{F}$-algebra $\mathcal{D}_{(R,F)}(\Lambda)$ is the completion of the endomorphism algebra studied by Hiller in [16, Ch. IV]. When $F$ is the additive formal group law, $R=\mathbb{Z}$, and $\Sigma$ is crystallographic, the $R\llbracket\Lambda\rrbracket_{F}$-algebra $\mathcal{D}_{(R,F)}(\Lambda)$ is the completion of the endomorphism algebra studied by Demazure in [11].

The second goal of this paper is to extend the formal affine Demazure algebra of [17] to all real finite reflection groups. We first define the formal affine Demazure algebra $\mathbf{D}_{F}$ for all real finite reflection groups using the formal group ring $R\llbracket\Lambda\rrbracket_{F}$. We then show that several results in [9, 17] regarding the crystallographic formal affine Demazure algebra extend to all real finite reflection groups. In particular, we describe $\mathbf{D}_{F}$ in terms of generators and relations in 8.9, as was done in [9, Thm. 7.9]. We then compute several structure coefficients that appear in a braid relation among the formal Demazure elements (see 9.10), and we introduce a seemingly new formula involving the formal Demazure elements in the process (see 9.4). In addition, we compute all structure coefficients for the dihedral groups $I_{2}(5)$ and $I_{2}(7)$ in 10.4 and 10.5, and the reflection groups $H_{3}$ and $H_{4}$ in 10.11 and 10.12. The formulas for the structure coefficients that we obtain in this paper generalize several formulas that appear in the literature, such as the formulas in [17, Prop. 6.8] and [15, Section 8], to all real finite reflection groups.

This paper is organized as follows. In Section 2, we recall the definition and basic properties of real finite reflection groups and their associated root systems. We then define the real root lattice $\Lambda$, generalizing the crystallographic root lattice to noncrystallographic root systems. In Section 3, we recall the definition and basic properties of formal groups laws, and we discuss the formal group ring constructed in [7] with respect to the real root lattice. In Section 4, we define the formal group ring for all real finite reflection groups. In Section 5, we define and discuss the formal Demazure operators acting on the formal group ring. In Section 6, we define the associated graded ring with respect to the formal group ring and use it to prove a basis theorem for a subalgebra of the endomorphism algebra of the formal group ring containing the formal Demazure operators. This section closely follows [7, §4]. In Section 7, we define and discuss the formal affine Demazure algebra for all real finite reflection groups. In Section 8, we prove a presentation for the formal affine Demazure algebra in terms of the formal Demazure elements. This section closely follows [9, §6 and §7]. In Section 9, we compute several structure coefficients that appear in a braid relation among the formal Demazure elements. In Section 10, we analyze the structure coefficients derived in Section 9 after specializing to certain formal group laws, and we compute all structure coefficients for the groups $I_{2}(5)$, $I_{2}(7)$, $H_{3}$, and $H_{4}$. In Section 11, we provide computations for products of up to seven formal Demazure elements for all real finite reflection groups.

Acknowledgements: The author is grateful to K. Zainoulline for proposing the project of generalizing the formal affine Demazure algebra to all real finite reflection groups, for providing many comments and suggestions in the construction of the generalized formal group ring, and for his encouragement to carry out this project to its completion. The author would also like to thank A. Savage for his careful reading of the paper and for his comments on earlier drafts of this work. Finally, the author would like to thank K. Besrour, J. Champaign, C. Kioulos, and S. Pilon for some useful discussions. This project was partially supported by an Undergraduate Student Research Award and a Canada Graduate Scholarship - M.Sc. from the Natural Sciences and Engineering Research Council of Canada. It was also supported by an Ontario Graduate Scholarship and by the University of Ottawa.

2. Real finite reflection groups and the real root lattice

In this section, we recall the definition and basic properties of real finite reflection groups and their associated root systems. We then define the real root lattice $\Lambda$, generalizing the crystallographic root lattice to noncrystallographic root systems.

Let $V=\mathbb{R}^{n}$, and let $(\cdot{,}\cdot)$ be the standard inner product on $V$. For any $\alpha\in V$, the reflection across $\alpha$ is the linear operator $s_{\alpha}$ defined by the formula

[TABLE]

One can easily compute that $s_{\alpha}^{2}=1$.

Definition 2.1**.**

(see [18, pp. 6] or [16, Ch. I, §3, Def. 3.1]) A root system $\Sigma$ in $V$ is a finite set of nonzero vectors in $V$ satisfying the conditions:

1. (1)

$\Sigma\cap\mathbb{R}\alpha=\{\alpha,-\alpha\}$ for all $\alpha\in\Sigma$; 2. (2)

$s_{\alpha}(\Sigma)=\Sigma$ for all $\alpha\in\Sigma$; 3. (3)

The roots $\alpha\in\Sigma$ generate $V$.

The rank of a root system in $V$ is the dimension of $V$ as an $\mathbb{R}$-vector space. Fix any root system $\Sigma$ in $V$, and let $\alpha\in\Sigma$ be any root. Let $\alpha^{\vee}:V\to\mathbb{R}$ be the linear function defined by $\alpha^{\vee}(\beta)=2\tfrac{(\alpha,\beta)}{(\alpha,\alpha)}$, $\beta\in V$. We call $\alpha^{\vee}$ the coroot of $\alpha$. We adopt the following naming convention:

• •

If $\alpha^{\vee}(\beta)\in\mathbb{Z}$ for all $\alpha,\beta\in\Sigma$, we call $\Sigma$ a crystallographic root system. Otherwise, we call $\Sigma$ a noncrystallographic root system.

Let $W$ be the group generated by the reflections $s_{\alpha}$ over all roots $\alpha\in\Sigma$. We call $W$ the real finite reflection group of $\Sigma$. If $\Sigma$ is crystallographic, we call $W$ a Weyl group. Let $\Delta=\{\alpha_{1},\dotsc,\alpha_{n}\}$ be a subset of $\Sigma$. We call $\Delta$ a simple system of $\Sigma$ if it is an $\mathbb{R}$-basis of $V$, and if every root $\alpha\in\Sigma$ can be written as an $\mathbb{R}$-linear combination of elements in $\Delta$ with all coefficients nonnpositive or all coefficients nonnegative. The elements in $\Delta$ are called simple roots and the reflection across a simple root is called a simple reflection. It is shown in [18, Ch. I, Thm. 1.3] that every root system contains a simple system. If a root $\alpha$ can be written as a linear combination of the simple roots with all coefficients nonpositive (resp. nonnegative), then $\alpha$ is called a negative root (resp. positive root). The positive system $\Sigma^{+}$ is the set of positive roots of $\Sigma$ with respect to $\Delta$, and the negative system $\Sigma^{-}$ is the set of negative roots of $\Sigma$ with respect to $\Delta$. If $\alpha_{i}$ is a simple root, we denote a reflection along $s_{\alpha_{i}}$ by $s_{i}$, and we set $S=\{s_{i}\mid\alpha_{i}\in\Delta\}$.

Lemma 2.2**.**

(see* [18, Ch. I, Cor. 1.5]) For any $\alpha\in\Sigma$, there is a reflection $w\in W$ and a simple root $\alpha_{i}\in\Delta$ such that $\alpha=w(\alpha_{i}).$*

Theorem 2.3**.**

(see* [18, Ch. I, Thm. 1.9]) $W$ is generated by the set of simple reflections $S$, subject only to the relations:*

[TABLE]

where $m_{i,j}$ is the order of $s_{i}s_{j}$ in $W$.

Remark 2.4**.**

2.2 and 2.3 tell us that, if $\Delta$ is a simple system of a root system $\Sigma$, then $\Sigma$ consists of the elements $w(\alpha_{j})$, where $w=s_{i_{1}}\dotsb s_{i_{k}}$ is a product of simple reflections and $\alpha_{j}\in\Delta$.

Theorem 2.5**.**

(see* [18, Ch. I, §2]) Let $\Sigma$ be a noncrystallographic root system, and let $W$ be the real finite reflection group of $\Sigma$. Then $W$ is either the dihedral group $I_{2}(m)$, $m\geq 3$, of order $2m$, the full icosahedral group $H_{3}$, or the symmetry group $H_{4}$ of the hyperdodecahedron.*

Definition 2.6**.**

(see [18, Ch. I, §1.6 and Ch. II, §5.9]) By 2.3, we can write $w\in W$ as a product $w=s_{i_{1}}\dotsb s_{i_{r}}$ of simple reflections. The length $l(w)$ of $w$ is the minimal $r$ for which such an expression exists, and we call such an expression a reduced decomposition of $w$. The longest word in $W$ is the unique word in $W$ of maximal length. Write $w^{\prime}\to w$ if $l(w)>l(w^{\prime})$ and $w=w^{\prime}u$ for some $u\in W$. We define the Bruhat order on $W$ to be the partial ordering $<$ on $W$ defined by $w^{\prime}<w$ if and only if there is a sequence $w^{\prime}=w_{0}\to w_{1}\to\dotsb\to w_{m}=w$.

We will refer to the next few results frequently throughout this paper. We use the notation $s_{i,j,\dotsc}^{(k)}$ to denote the product of $k$ simple reflections $s_{i}s_{j}s_{i}s_{j}s_{i}\dotsb$.

Proposition 2.7**.**

(see* [16, Ch. I, Prop. 3.6]) Let $v=s_{i_{1}}\dotsb s_{i_{t}}$ be a reduced decomposition of $v\in W$. Then*

[TABLE]

where $\theta_{i_{1}}=\alpha_{i_{1}}$, and $\theta_{i_{j}}=s_{i_{1}}\dotsb s_{i_{j-1}}(\alpha_{i_{j}})$ if $j>1$.

If $v$ is the longest word in $W$, then

[TABLE]

Lemma 2.8**.**

We have $\alpha_{j}=\begin{cases}s_{i,j,\dotsc}^{(m-1)}(\alpha_{j}),\quad\textit{m {even}};\\ s_{i,j,\dotsc}^{(m-1)}(\alpha_{i}),\quad\textit{m {odd}}.\end{cases}$

Proof.

Let $m_{i,j}$ be the order of $s_{i}s_{j}$ in $W$, and suppose $m_{i,j}$ is odd. Since $s_{i,j,\dotsc}^{(m_{i,j})}=s_{j,i,\dotsc}^{(m_{i,j})}$, it follows from 2.7 that

[TABLE]

Thus,

[TABLE]

Since $s_{j}(\Sigma^{+})\cap\Sigma^{-}=\{s_{j}(\alpha_{j})\}$, we must have that $s_{i,j,\dotsc}^{(m-1)}(\alpha_{i})=\alpha_{j}$. The proof for even $m_{i,j}$ is similar. ∎

Proposition 2.9**.**

(see [18, Ch. I, §1.7, Ex. 2])* Let $w=s_{i_{1}}\dotsb s_{i_{r}}$ be a (not necessarily reduced) decomposition of $w\in W$ in terms of simple reflections. If $l(sw)<l(w)$ for some simple reflection $s$, then there is an index $i_{j}$ for which $sw=ss_{i_{1}}\dotsb\hat{s}_{i_{j}}\dotsb s_{i_{r}}$.*

Theorem 2.10**.**

(see [2, Ch. 2, §2.2, Thm. 2.2.2])* Let $w=s_{i_{1}}\dotsb s_{i_{q}}$ be a reduced decomposition of $w\in W$. Then $u\leq w$ if and only if there is a reduced decomposition $u=s_{i_{r_{1}}}s_{i_{r_{2}}}\dotsb s_{i_{r_{t}}}$ of $u$ such that $1\leq r_{1}<r_{2}<\dotsb<r_{t}\leq q$.*

Proposition 2.11**.**

(see [18, Ch. I, Prop. 1.2])* For any root $\beta\in\Sigma$ and reflection $w\in W$, we have $ws_{\alpha}w^{-1}=s_{w(\alpha)}$.*

Definition 2.12**.**

(cf. [16, pp. 22]) If $\Sigma$ is a root system in $V$ with finite reflection group $W$, and $\Delta$ is a simple system of $\Sigma$, then we call the pair $(\Sigma,\Delta)$ a geometric realization of $W$ in $V$.

Remark 2.13**.**

As stated in [18, Ch. I, §1.1, Ex. 1], the reflections form a single conjugacy class in $I_{2}(m)$ when $m$ is odd, but form two conjugacy classes when $m$ is even. This tells us that, given a geometric realization of $I_{2}(m)$, all roots have the same length when $m$ is odd. However, only the lengths of the roots in each conjugacy class are guaranteed to be the equal when $m$ is even.

Remark 2.14**.**

The rank $2$ root systems are $A_{1}\times A_{1}$, $A_{2}$, $B_{2}$, $G_{2}$, and the root systems of $I_{2}(m)$, $m\geq 3$.

Let $W$ be a real finite reflection group, and let $(\Sigma,\Delta)$ be a geometric realization of $W$ in $V$, with simple system $\Delta=\{\alpha_{1},\dotsc,\alpha_{n}\}$. Let $\alpha\in\Sigma$ be any root. By definition, there exist unique elements $c_{i}^{\alpha}\in\mathbb{R}$ such that $\alpha=c_{1}^{\alpha}\alpha_{1}+\dotsb+c_{n}^{\alpha}\alpha_{n}$. We would like to determine whether the subring $\mathcal{R}$ of $\mathbb{R}$ generated by the elements $c_{i}^{\alpha}$ over all $i=1,\dotsc,n$ and $\alpha\in\Sigma$ is a free finitely-generated $\mathbb{Z}$-module with a power basis, i.e., a basis of the form $\{1,\beta,\beta^{2},\dotsc,\beta^{l-1}\}$ for some $l\geq 1$, where $\beta\in\mathcal{R}$. As $c_{i}^{\alpha_{i}}=1$, the ring $\mathcal{R}$ must contain $1$. If $\alpha$ is not a simple root, then 2.2 and 2.3 imply that $\alpha=s_{i_{1}}\dotsb s_{i_{k}}(\alpha_{j})$, where $\alpha_{j}\in\Delta$ and the $s_{i_{r}}$ are simple reflections. Thus, $\mathcal{R}$ is the unital subring of $\mathbb{R}$ generated by the elements $\alpha_{i}^{\vee}(\alpha_{j})$ over all $\alpha_{i},\alpha_{j}\in\Delta$.

For the remainder of this paper, we will work under the following Assumption:

Assumption 2.15**.**

$W$* is a real finite reflection group, and $(\Sigma,\Delta)$ is a geometric realization of $W$ in $V$, such that $\mathcal{R}$ is a free finitely-generetated $\mathbb{Z}$-module with a power basis.*

We will show the existence of a geometric realization for $W$ satisfying the conditions of 2.15 for every real finite reflection group $W$. If $W$ is a Weyl group, then $\mathcal{R}=\mathbb{Z}$, since the elements $\alpha_{i}^{\vee}(\alpha_{j})\in\mathbb{Z}$ for all $\alpha_{i},\alpha_{j}\in\Delta$.

The geometric realizations of $W=H_{3}$ and $W=H_{4}$ that follow are taken from [18, §2.13], and the geometric realizations of $W=I_{2}(m)$, $m\geq 3$, that follow are taken from [18, Ch. §1.1]. First, we will consider the case $W=H_{4}$. In this case, $V=\mathbb{R}^{4}$. Set $a:=\tfrac{1+\sqrt{5}}{4}$ and $b:=\tfrac{-1+\sqrt{5}}{4}$. Let $(\Sigma,\Delta)$ be the geometric realization of $W=H_{4}$ in $V$ with simple roots:

[TABLE]

A computation gives the following result:

[TABLE]

where $\tau=\tfrac{1+\sqrt{5}}{2}$ is the well-known golden section. The constant $\tau$ is a root of the characteristic equation $x^{2}-x-1=0$. Thus, we may take $\mathcal{R}=\mathbb{Z}[\tau]$, which has a power basis $\{1,\tau\}$ over $\mathbb{Z}$.

Now we will consider the case $W=H_{3}$. In this case, $V=\mathbb{R}^{3}$. Let $(\Sigma,\Delta)$ be the geometric realization of $W=H_{3}$ in $V$ with simple roots:

[TABLE]

By the earlier computation, we may take $\mathcal{R}=\mathbb{Z}[\tau]$ in this case as well.

Next we will consider the dihedral groups $W=I_{2}(m)$, where $m\geq 3$. In this case, $V=\mathbb{R}^{2}$. Let $(\Sigma,\Delta)$ be the geometric realization of $W=I_{2}(m)$ in $V$ with simple roots:

[TABLE]

A computation gives the following result:

[TABLE]

Thus, $\mathcal{R}=\mathbb{Z}[2\textrm{cos}(\tfrac{\pi}{m})]$, and $\mathcal{R}$ is indeed a free finitely-generated $\mathbb{Z}$-module with a power basis. To see this, we first note that $\left(\textrm{cos}(\tfrac{\pi}{m}),\textrm{sin}(\tfrac{\pi}{m})\right)=\left(\textrm{cos}(\tfrac{2\pi}{2m}),\textrm{sin}(\tfrac{2\pi}{2m})\right)$ is a primitive $2m$-th root of unity. Thus, by [25], $\mathcal{R}$ is the ring of integers of the number field $\mathbb{Q}(2\textrm{cos}(\tfrac{\pi}{m}))$. It is well known that the ring of integers $\mathcal{O}_{K}$ of a number field $K$ of degree $l$ has a power basis $\{1,\beta,\beta^{2},\dotsc,\beta^{l-1}\}$, whenever $\mathcal{O}_{K}=\mathbb{Z}[\beta]$ for some $\beta\in\mathcal{O}_{K}$. Therefore,

[TABLE]

is a $\mathbb{Z}$-basis of $\mathcal{R}$, where $l$ is the degree of the field extension $\mathbb{Q}\subseteq\mathbb{Q}(2\mathrm{cos}(\tfrac{\pi}{m}))$. In [21, Thm. 1], it is shown that $l=\tfrac{\phi(2m)}{2}$, where $\phi$ is the Euler totient function, and that the minimal polynomial $\psi_{m}(x)$ of $2\mathrm{cos}(\tfrac{\pi}{m})$ over $\mathbb{Q}$ satisfies

[TABLE]

where $Q_{2m}$ is the $2m$-th cyclotomic polynomial.

We now go over four concrete examples of the geometric realization of dihedral groups constructed above. Some of the formulas we compute in these examples are used in Section 10.

Example 2.16**.**

Suppose $m=3$. Since $2\textrm{cos}\left(\tfrac{\pi}{3}\right)=1$, we have $\mathcal{R}=\mathbb{Z}$. Clearly, $\{1\}$ is a $\mathbb{Z}$-basis of $\mathcal{R}$. The roots of $\Sigma$ are:

[TABLE]

[TABLE]

Example 2.17**.**

Suppose $m=4$. Since $2\textrm{cos}\left(\tfrac{\pi}{4}\right)=\sqrt{2}$, we have $\mathcal{R}=\mathbb{Z}[\sqrt{2}]$. Moreover, $\{1,\sqrt{2}\}$ is a $\mathbb{Z}$-basis of $\mathcal{R}$, since $\sqrt{2}$ is a root of the irreducible quadratic polynomial $x^{2}-2$ over $\mathbb{Q}$. The roots of $\Sigma$ are:

[TABLE]

[TABLE]

Example 2.18**.**

Suppose $m=5$. Since $2\textrm{cos}\left(\tfrac{\pi}{5}\right)=\tfrac{1+\sqrt{5}}{2}=\tau$ (this is the golden section, which was mentioned earlier), we have $\mathcal{R}=\mathbb{Z}[\tau]$. Moreover, $\{1,\tau\}$ is a $\mathbb{Z}$-basis of $\mathcal{R}$, since $\tau$ is a root of the irreducible quadratic polynomial $x^{2}-x-1$ over $\mathbb{Q}$. The roots of $\Sigma$ are:

[TABLE]

[TABLE]

[TABLE]

Example 2.19**.**

Suppose $m=7$. Set $\zeta=2\textrm{cos}\left(\tfrac{\pi}{7}\right)$. We have $\mathcal{R}=\mathbb{Z}[\zeta]$. Moreover, $\{1,\zeta,\zeta^{2}\}$ is a $\mathbb{Z}$-basis of $\mathcal{R}$, since $\zeta$ is a root of the irreducible cubic polynomial $x^{3}-x^{2}-2x+1$ over $\mathbb{Q}$. The roots of $\Sigma$ are:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For the remainder of this paper, we will work under the following Assumption:

Assumption 2.20**.**

$B=\{e_{1},\dotsc,e_{l}\}$* is a power basis of $\mathcal{R}$, and $e_{1}=1$.*

Definition 2.21**.**

Let $\Lambda$ be the $\mathcal{R}$-module generated by the roots $\alpha\in\Sigma$. We call $\Lambda$ a real root lattice. If $\Sigma$ is crystallographic, then we call $\Lambda$ a crystallographic root lattice.

The group $\Lambda$ is a free finitely-generated $\mathbb{Z}$-module of rank $nl$. Since $\Delta$ is an $\mathcal{R}$-basis of $\Lambda$, and $B$ is a $\mathbb{Z}$-basis of $\mathcal{R}$, we take $\{e_{i}\alpha_{j}\}$ as a $\mathbb{Z}$-basis of $\Lambda$. Moreover, since $e_{1}=1$, this basis contains $\Delta$.

As $W$ acts on $\Sigma$ by permuting the roots, there is a natural action of $W$ on $\Lambda$ given by $w\cdot(r\alpha)=r(w(\alpha))$ for all $w\in W$, $\alpha\in\Sigma$, and $r\in\mathcal{R}$. By linearity, the action of $w\in W$ on $\Lambda$ is a group homomorphism.

3. Formal group laws and the classical formal group ring

In this section, we recall the definition of a one-dimensional commutative formal group law, and we discuss several of its properties. We then analyze the formal group ring $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ of [7] associated to the real root lattice $\Lambda$ and the formal group law $(R,F)$.

Definition 3.1**.**

(see [22, pp. 4]) A one-dimensional commutative formal group law $(R,F)$ over a commutative unital ring $R$ is a power series $F(u,v)\in R\llbracket u,v\rrbracket$ satisfying the following axioms:

1. (1)

$F(u,0)=F(0,u)=u\in R\llbracket u\rrbracket$; 2. (2)

$F(u,v)=F(v,u)$; 3. (3)

$F(u,F(v,w))=F(F(u,v),w)\in R\llbracket u,v,w\rrbracket$.

By [14, Ch. I, §3, Prop. 1], given a formal group law $(R,F)$, there is a unique power series $i(u)\in R\llbracket u\rrbracket$ such that $F(u,i(u))=F(i(u),u)=0$. The series $i(u)$ is called the formal inverse of $u$ with respect to the formal group law $(R,F)$.

Following [6, pp. 3], a morphism $f\colon(R,F)\to(R,F^{\prime})$ of formal group laws over $R$ is a power series $f(u)\in R\llbracket u\rrbracket$ such that $f(u+_{F}v)=f(u)+_{F^{\prime}}f(v)$ and $f(0)=0$. The morphism $f$ is called a strong isomorphism if its formal derivative satisfies $f^{\prime}(0)=1$. The condition $f^{\prime}(0)=1$ implies that there is a morphism $f^{-1}:(R,F^{\prime})\to(R,F)$ such that $f(f^{-1}(u))=f^{-1}(f(u))=u$.

Let $(\mathbb{C},F)$ be a one-dimensional commutative formal group law over $\mathbb{C}$. Write $F(u,v)=u+v+\sum_{i,j\geq 1}t_{i,j}u^{i}v^{j}$, where $t_{i,j}\in\mathbb{C}$, and suppose $(\mathbb{C},F_{a})$ is the additive formal group law over $\mathbb{C}$, i.e., $F_{a}(u,v)=u+v$. In [23, Ch. IV.5], the author constructs a strong isomorphism of formal group laws $\mathrm{log}_{F}:(\mathbb{C},F)\to(\mathbb{C},F_{a})$ called the logarithm of $(\mathbb{C},F)$. The inverse of the logarithm is called the exponential of $(\mathbb{C},F)$ and is denoted $\mathrm{exp}_{F}$. In [23, Ch. IV.4], it is shown that $\mathrm{log}_{F}(u)=\int\omega(u)$, where $\omega(v)=F_{u}(0,v)dv$ is the normalized invariant differential of $(\mathbb{C},F)$ and $F_{u}(x,y)$ is the partial derivative of $F(u,v)$ with respect to $u$, evaluated at $(x,y)$. There exist $a_{i},b_{i}\in\mathbb{C}$, $i\geq 2$, such that

[TABLE]

Note that $\mathrm{exp}_{F_{a}}(u)=u$ and $\mathrm{log}_{F_{a}}(u)=u$. Set $C_{F}:=\{a_{i}\}_{i\geq 2}\cup\{b_{i}\}_{i\geq 2}\cup\{t_{i,j}\}_{i,j\geq 1}$.

Definition 3.2**.**

If $R$ is any subring of $\mathbb{C}$ containing $C_{F}$, then we call $R$ an ample ring with respect to the formal group law $(\mathbb{C},F)$.

If $R$ is an ample ring with respect to $(\mathbb{C},F)$, then we may view $(\mathbb{C},F)$ as a formal group law $(R,F)$ over $R$. In particular, there is a strong isomorphism of formal group laws $\mathrm{log}_{F}:(R,F)\to(R,F_{a})$, with inverse $\mathrm{exp}_{F}:(R,F_{a})\to(R,F)$, induced by the logarithm and exponential of $(\mathbb{C},F)$. We call $\mathrm{log}_{F}:(R,F)\to(R,F_{a})$ and $\mathrm{exp}_{F}:(R,F_{a})\to(R,F)$ the logarithm and exponential of $(R,F)$, respectively.

Example 3.3**.**

(see [22, Ex. 1.1.4]) The additive formal group law $(R,F_{a})$ over $R$ is given by $F_{a}(x,y)=x+y$. The formal inverse of $x$ under $(R,F_{a})$ is $i(x)=-x$.

If $R$ is an ample ring with respect to $(\mathbb{C},F_{a})$, then the logarithm of $(R,F_{a})$ is $\mathrm{log}_{F_{a}}(x)=x$, and the exponential of $(R,F_{a})$ is $\mathrm{exp}_{F_{a}}(x)=x$.

Example 3.4**.**

(see [22, Ex. 1.1.5] and [23, Ch. IV, §9, Ex. 5.1]) The multiplicative formal group law $(R,F_{m})$ over $R$ is given by $F_{m}(x,y)=x+y+xy$. The formal inverse of $x$ under $(R,F_{m})$ is $i(x)=\tfrac{-x}{1-x}:=-x\sum_{i\geq 0}x^{i}$.

If $R$ is an ample ring with respect to $(\mathbb{C},F_{m})$, then the logarithm and exponential of $(R,F_{m})$ are given by the formulas

[TABLE]

Example 3.5**.**

(see [24, Ch. IV, §9, Ex. 3.4(3)]) The Lorentz formal group law $(R,F_{l})$ over $R$ is given by $F_{l}(x,y)=\tfrac{x+y}{1+xy}:=(x+y)\sum_{i\geq 2}(xy)^{i}$. The formal inverse of $x$ under $(R,F_{l})$ is $i(x)=-x$.

If $R$ is an ample ring with respect to $(\mathbb{C},F_{l})$, then the logarithm and exponential of $(R,F_{l})$ are given by the formulas

[TABLE]

where $B_{2i}$ is the $2i$-th Bernoulli number.

Example 3.6**.**

(see [22, §1.1]) Let $\mathbb{L}$ be Lazard ring, i.e., the commutative unital ring generated by some elements $r_{i,j}$, $i,j\geq 1$, subject only to the relations imposed by the definition of the formal group law. The universal formal group law $(\mathbb{L},F_{u})$ over the Lazard ring $\mathbb{L}$ is given by $F_{u}(x,y)=x+y+\sum_{i,j\geq 1}r_{i,j}x^{i}y^{j}$. The formal inverse of $x$ under $(\mathbb{L},F_{u})$ is a power series $i(x)=-x-c_{2}x^{2}-c_{3}x^{3}-\dotsb\in\mathbb{L}\llbracket x\rrbracket$, where the $c_{i}$ can be computed explicitly in terms of the $r_{i,j}$ from the relation $F_{u}(x,i(x))=0$.

Definition 3.7**.**

Let $(R,F)$ be a formal group law of the form

[TABLE]

We say that $(R,F)$ is of additive type. In particular, the formal inverse of $x$ under $(R,F)$ is $i(x)=-x$.

Example 3.8**.**

The additive formal group law and the Lorentz formal group law are examples of formal group laws of additive type.

We now state and prove some standard facts about topological rings.

Lemma 3.9**.**

Let $S$ be a topological ring, and let $I$ be an ideal in $S$. The topological closure $\hat{I}$ of $I$ in $S$ is an ideal in $S$.

Proof.

Let $x,y\in\hat{I}$ and $s\in S$. The difference $x-y\in\hat{I}$ by the proof of [5, Ch. III, §2.1, Prop. 1]. To see that $sx\in\hat{I}$, let $U$ be a neighbourhood of $sx$. Since $x\in\hat{I}$, every neighbourhood $X$ of $x$ intersects $I$ nontrivially. The inverse image of $U$ under the multiplication map $(a,b)\mapsto ab$ is open in $S\times S$ by continuity of multiplication, and contains $(s,x)$. Thus, there is a neighbourhood $X$ of $x$ such that $(s\cdot X)\cap I$ is nonempty and $(s\cdot X)\subseteq U$. So $U\cap I$ is nonempty, as required. ∎

Theorem 3.10**.**

Suppose $S$ is a metrizable complete Hausdorff topological ring, and $I$ is a closed ideal in $S$. Then the quotient $S/I$ is a complete Hausdorff ring.

Proof.

Since $S$ is a commutative metrizable complete Hausdorff topological group, and $I$ is a closed normal subgroup of $S$, the quotient group $S/I$ is Hausdorff by [5, Ch. III, §2, Prop. 18] and complete by [3, Ch. IX, §3, Prop. 4]. Thus, the quotient $S/I$ is a complete Hausdorff topological ring. ∎

Remark 3.11**.**

(see [1, §10]) Let $S$ be a ring, and let $I$ be an ideal in $S$. The $I$-adic topology on $S$ is the topology generated by elements of the form $x+I^{n}$, where $x\in S$ and $n\geq 1$. In particular, $S$ is first-countable with respect to this topology. If $S$ is Hausdorff, then Urysohn’s metrization theorem implies that $S$ is metrizable. If, in addition to being Hausdorff, $S$ is complete, then 3.10 implies that $S/J$ is Hausdorff and complete, where $J$ is any closed ideal in $S$. The ring $S$ is complete in the $I$-adic topology if and only if the canonical ring homomorphism $S\to\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{i}}}(S/I^{i})$ is a ring isomorphism, and $S$ is Hausdorff in the $I$-adic topology if and only if $\cap_{i\geq 0}I^{i}=(0)$.

The following construction of the formal group ring was introduced in [7, §2]. Let $R$ be a commutative unital ring and $(R,F)$ a one-dimensional commutative formal group law over $R$. Let $R[x_{\Lambda}]$ be the polynomial ring over $R$ with variables indexed by elements of the real root lattice $\Lambda$. Here, we view $\Lambda$ as a free finitely-generated $\mathbb{Z}$-module with basis $\{e_{i}\alpha_{j}\}$. The augmentation map $\epsilon:R[x_{\Lambda}]\to R$ sends $x_{\lambda}\mapsto 0$ for each $\lambda\in\Lambda$ and fixes $R$. Let $R\llbracket x_{\Lambda}\rrbracket$ be the $\mathrm{ker}(\epsilon)$-adic completion of the polynomial ring $R[x_{\Lambda}]$. Given $u,v\in R\llbracket x_{\Lambda}\rrbracket$, $m\in\mathbb{Z}_{\geq 0}$, we define the following notation:

[TABLE]

where $-_{F}u$ is the formal inverse of $u$ under $(R,F)$.

Let $\widetilde{\mathcal{J}}_{F}$ be the closure of the ideal in $R\llbracket x_{\Lambda}\rrbracket$ generated by the elements

[TABLE]

over all $\lambda_{1},\lambda_{2}\in\Lambda$.

Definition 3.12**.**

The quotient

[TABLE]

is the formal group ring of [7, Def. 2.4]. For the remainder of this paper, we call $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ the classical formal group ring.

Remark 3.13**.**

If $\Sigma$ is crystallographic, then $\Lambda$ is a crystallographic root lattice, and $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ is a special case of the formal group ring studied in [9].

We denote the image of $x_{\lambda}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ by the same symbol. We will now review several facts about the classical formal group ring.

The ring $R\llbracket x_{\Lambda}\rrbracket$ is a complete Hausdorff ring with respect to the $\mathrm{ker}(\epsilon)$-adic topology, and $\widetilde{\mathcal{J}}_{F}$ is contained in $\mathrm{ker}(\epsilon)$. Thus, 3.9 and 3.10 imply that the classical formal group ring $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ is a complete Hausdorff ring with respect to the $\widetilde{\mathcal{I}}_{F}$-adic topology, where $\widetilde{\mathcal{I}}_{F}$ is the kernel of the induced augmentation map $\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to R$. By [7, Cor. 2.13], there is a continuous ring isomorphism $\varrho\colon\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to R\llbracket x_{1},\dotsc,x_{nl}\rrbracket$ such that

[TABLE]

In particular, the isomorphism sends $x_{e_{i}\alpha_{j}}\mapsto x_{(i-1)n+j}$. If $R$ is an integral domain, then so is $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$.

Since $W$ acts on $\Lambda$, it follows from [7, Lem. 2.6] that $W$ also acts on the classical formal group ring by

[TABLE]

In fact, since each $w\in W$ sends $\widetilde{\mathcal{I}}_{F}$ to itself, the action of $w\in W$ on $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ is a continuous ring homomorphism.

Let $(R,F^{\prime})$ be another formal group law over $R$. By [7, Lem. 2.6], any morphism $f\colon(R,F)\to(R,F^{\prime})$ of formal group laws over $R$ induces a continuous ring homomorphism $f^{*}\colon\widetilde{R\llbracket\Lambda\rrbracket}_{F^{\prime}}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ sending $x_{\lambda}$ to $f(x_{\lambda})$. To see that $f^{*}$ is well-defined, note that

[TABLE]

for all $\lambda$, $\mu\in\Lambda$. In addition, $f^{*}$ is $W$-equivariant: write $f(u)=\sum_{i\geq 1}a_{i}u^{i}$ for some $a_{i}\in R$. Given $w\in W$ and $\lambda\in\Lambda$, we have

[TABLE]

If $(\mathbb{C},F)$ is a formal group law over $\mathbb{C}$, and $R$ is an ample ring with respect to $(\mathbb{C},F)$, then the exponential of the formal group law $(R,F)$ induces a $W$-equivariant continuous ring isomorphism

[TABLE]

This is summarized by the following commutative diagram, which commutes for any $w\in W$:

[TABLE]

Remark 3.14**.**

When $\Sigma$ is crystallographic, it is shown in [7, Cor. 3.4] that $x_{\alpha}$ divides the element $u-s_{\alpha}(u)$ for all $u\in\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ and $\alpha\in\Sigma$. Therefore, one can define a formal Demazure operator $\Delta_{\alpha}$ on $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ by the formula

[TABLE]

However, when $\Sigma$ is noncrystallographic, it is not necessarily true that $x_{\alpha}$ divides the element $u-s_{\alpha}(u)$ for all $u\in\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ and $\alpha\in\Sigma$. We provide an example to demonstrate this. Suppose $(R,F_{a})$ is the additive formal group law over a commutative unital ring $R$, the reflection group $W=I_{2}(5)$, and $(\Sigma,\Delta)$ is the geometric realization of $I_{2}(5)$ given in 2.18. Then, in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$, we have

[TABLE]

If $x_{\alpha_{2}}$ divides $x_{\alpha_{1}}-s_{\alpha_{2}}(x_{\alpha_{1}})=-x_{\tau\alpha_{2}}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$, then the isomorphism ${\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}}\simeq R\llbracket x_{1},\dotsc,x_{4}\rrbracket$ of Eq. 1 implies that $x_{2}$ divides $-x_{4}$ in $R\llbracket x_{1},\dotsc,x_{4}\rrbracket$, which is a contradiction.

The focus of the next two sections is to resolve this issue. Let $W$ be any real finite reflection group, and let $R$ be subring of $\mathbb{C}$ satisfying the hypotheses of 4.6. In Section 4, we show that there is a closed ideal $\mathcal{J}_{F}$ of $R\llbracket x_{\Lambda}\rrbracket$, such that the quotient $R\llbracket\Lambda\rrbracket_{F}:=R\llbracket x_{\Lambda}\rrbracket/\mathcal{J}_{F}$ is an integral domain, and such that there is a well-defined $W$-action on $R\llbracket\Lambda\rrbracket_{F}$, and such that $x_{\alpha_{j}}$ divides $x_{e_{i}\alpha_{j}}$ in $R\llbracket\Lambda\rrbracket_{F}$ for all $e_{i}\in B$ and $\alpha_{j}\in\Delta$ (see 4.10). Here, we are denoting the image of $x_{\lambda}$ in $R\llbracket\Lambda\rrbracket_{F}$ by the same symbol. In Section 5, we show that $x_{\alpha}$ divides $u-s_{\alpha}(u)$ in $R\llbracket\Lambda\rrbracket_{F}$ for all $u\in R\llbracket\Lambda\rrbracket_{F}$ and $\alpha\in\Sigma$, using the fact that $x_{\alpha_{j}}$ divides $x_{e_{i}\alpha_{j}}$ for all $e_{i}\in B$ and $\alpha_{j}\in\Delta$. (see 5.1). This allows us to define the formal Demazure operator on $R\llbracket\Lambda\rrbracket_{F}$ (see 5.2). We call $R\llbracket\Lambda\rrbracket_{F}$ the formal group ring with respect to $(R,F)$ and $\Lambda$. When $\Sigma$ is crystallographic, $R\llbracket\Lambda\rrbracket_{F}$ collapses to classical formal group ring $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ with respect to the crystallographic root lattice $\Lambda$.

4. Construction of the formal group ring

The purpose of this section is to construct the formal group ring $R\llbracket\Lambda\rrbracket_{F}$ for all real finite reflection groups. This construction will allow us to define formal Demazure operators on $R\llbracket\Lambda\rrbracket_{F}$ in Section 5. See 3.14 for a detailed motivation for this construction.

From here up to 4.6, we assume that $\Sigma$ is a noncrystallographic root system. Let $(\mathbb{C},F)$ be a formal group law over $\mathbb{C}$, and let $R$ be an ample ring with respect to $(\mathbb{C},F)$. Furthermore, assume that $R$ contains $\mathcal{R}$. The exponential $\mathrm{exp}_{F}$ (resp. logarithm $\mathrm{log}_{F}$) of $(R,F)$ induces a $W$-equivariant continuous ring isomorphism $\mathrm{exp}_{F}^{*}:\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$ (resp. $\mathrm{log}_{F}^{*}:\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F}$). The isomorphisms $\mathrm{exp}_{F}^{*}$ and $\mathrm{log}_{F}^{*}$ are inverse to each other. Since $R$ contains $\mathcal{R}$, we let ${\mathcal{J}}^{\star}_{F}$ be the closure of the ideal in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ generated by the elements

[TABLE]

over all $e_{i}\in B$, and $\alpha_{j}\in\Delta$.

The ideal ${\mathcal{J}}^{\star}_{F}$ is contained in the kernel of the augmentation map $\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to R$, and, hence, the map $\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to R$ factors through the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$. Therefore, 3.9 and 3.10 imply that $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$ is a complete Hausdorff ring with respect to the ${\mathcal{I}}^{\star}_{F}$-adic topology, where ${\mathcal{I}}^{\star}_{F}$ is the kernel of the induced augmentation map $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}\to R$. We will denote the image of $x_{\lambda}$ in the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$ by the same symbol.

Lemma 4.1**.**

The ideal ${\mathcal{J}}^{\star}_{F}$ in ${R\llbracket\Lambda\rrbracket}_{F}$ is $W$-invariant.

Proof.

Under the continuous $W$-equivariant ring isomorphism $\mathrm{exp}_{F}^{*}\colon\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$, we have $\mathrm{exp}_{F}^{*}(\widetilde{\mathcal{J}}_{F})=\widetilde{\mathcal{J}}_{F_{a}}$. To see this, we first note that, by continuity, $\mathrm{exp}_{F}^{*}(\widetilde{\mathcal{J}}_{F})$ is contained in the closure $\widetilde{\mathcal{J}}_{F_{a}}$. Since $\mathrm{exp}_{F}^{*}$ is an isomorphism of topological rings, $\mathrm{exp}_{F}^{*}(\widetilde{\mathcal{J}}_{F})$ is closed in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$, so it must equal $\widetilde{\mathcal{J}}_{F_{a}}$.

Since $\mathrm{exp}_{F}^{*}$ is $W$-equivariant, we see that $\widetilde{\mathcal{J}}_{F}$ is $W$-invariant in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}$ if and only if $\widetilde{\mathcal{J}}_{F_{a}}$ is $W$-invariant in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$. Thus, we will show that $\widetilde{\mathcal{J}}_{F_{a}}$ is $W$-invariant in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$. Let $\mathcal{J}$ be the ideal in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$ generated by the elements $e_{i}x_{\alpha_{j}}-x_{e_{i}\alpha_{j}},$ over all $e_{i}\in B$ and $\alpha_{j}\in\Delta$. The closure of $\mathcal{J}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$ is ${\mathcal{J}}^{\star}_{F_{a}}$. If $\mathcal{J}$ is $W$-invariant, then so is the closure ${\mathcal{J}}^{\star}_{F_{a}}$ by continuity of the action of $w\in W$ on $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$. In other words, suppose $\mathcal{J}$ is $W$-invariant. If $x\in{\mathcal{J}}^{\star}_{F_{a}}$ and $(x_{i})_{i\geq 0}$ is a sequence of elements in $\mathcal{J}$ converging to $x$, then the sequence $(w(x_{i}))_{i\geq 0}$ in $\mathcal{J}$ converges to $w(x)$. So $w(x)$ lies in the closure ${\mathcal{J}}^{\star}_{F_{a}}$ as well. Thus, it is enough to show that $\mathcal{J}$ is $W$-invariant in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$. To do this, we will show that the image of $w(e_{i}x_{\alpha_{j}}-x_{e_{i}\alpha_{j}})$ in the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}/\mathcal{J}$ is zero for all $e_{i}\in B$, $\alpha_{j}\in\Delta$, and $w\in W$. Here, we are denoting the image of $x_{\lambda}$ in the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}/\mathcal{J}$ by the same symbol.

Fix $w\in W$. Since $w(\alpha_{j})\in\Lambda$, we write $w(\alpha_{j})=\sum_{k=1}^{l}\sum_{r=1}^{n}c_{k,r}e_{k}\alpha_{r}$ for some $c_{k,r}\in\mathbb{Z}$. Furthermore, we can write $e_{i}e_{k}=\sum_{t=1}^{l}d_{t}^{(i,k)}e_{t}$ for some $d_{t}^{(i,k)}\in\mathbb{Z}$, since $\{e_{1},\dotsc,e_{l}\}$ is a $\mathbb{Z}$-basis of $\mathcal{R}$. Thus, in the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}/{\mathcal{J}}$, we have

[TABLE]

Since $(R,F_{a})$ is the additive formal group law, we have

[TABLE]

Finally, since $e_{i}x_{\alpha_{j}}=x_{e_{i}\alpha_{j}}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}/{\mathcal{J}}$, we have

[TABLE]

Therefore, ${\mathcal{J}}$ is a $W$-invariant ideal in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$. ∎

Remark 4.2**.**

4.1 implies that there is a well-defined $W$-action on the quotient $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$, given by

[TABLE]

In the proof of 4.1, we showed that the continuous $W$-equivariant ring isomorphism $\mathrm{exp}_{F}^{*}\colon\widetilde{R\llbracket\Lambda\rrbracket}_{F}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$ exchanges the $W$-invariant ideals ${\mathcal{J}}^{\star}_{F}$ and ${\mathcal{J}}^{\star}_{F_{a}}$. Thus, $\mathrm{exp}_{F}^{*}$ induces a continuous $W$-equivariant ring isomorphism on the quotients,

[TABLE]

This is summarized by the following diagram, which commutes for all $w\in W$:

[TABLE]

Lemma 4.3**.**

There is a continuous ring isomorphism

[TABLE]

sending $x_{e_{i}\alpha_{j}}\mapsto\mathrm{exp}_{F}(e_{i}x_{j})$. In particular, $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$ is an integral domain.

Proof.

There is a continuous ring isomorphism

[TABLE]

Suppose there is a continuous ring isomorphism

[TABLE]

Then the composition

[TABLE]

is a continuous ring isomorphism such that $(\gamma\circ\widetilde{\mathrm{exp}}_{F}^{*})(x_{e_{i}\alpha_{j}})=\mathrm{exp}_{F}(e_{i}x_{j})$, which proves the Lemma. Therefore, we will show the existence of the isomorphism $\gamma$.

Recall that ${\mathcal{J}}^{\star}_{F_{a}}$ is the closure of the ideal in $\widetilde{R\llbracket\Lambda\rrbracket}_{F_{a}}$ generated by the elements $x_{e_{i}\alpha_{j}}-e_{i}x_{\alpha_{j}}$, over all $e_{i}\in B$ and $\alpha_{j}\in\Delta$. In addition, recall that there is a continuous ring isomorphism

[TABLE]

for all $e_{i}\in B$, $\alpha_{j}\in\Delta$, and $m\in\mathbb{Z}$ (see Eq. 1). The image $\varrho({\mathcal{J}}^{\star}_{F_{a}})$ is the closure of the ideal in $R\llbracket x_{1},\dotsc x_{nl}\rrbracket$ generated by the elements $x_{(i-1)n+j}-e_{i}x_{j}$, over all $i=1,\dotsc,l$, and $j=1,\dotsc,n$. By 3.9 and 3.10, the quotient $R\llbracket x_{1},\dotsc,x_{nl}\rrbracket/\varrho({\mathcal{J}}^{\star}_{F_{a}})$ is a complete Hausdorff ring with respect to the topology induced by the kernel of the augmentation map $R\llbracket x_{1},\dotsc,x_{nl}\rrbracket/\varrho({\mathcal{J}}^{\star}_{F_{a}})\to R$. Thus, there is a continuous ring isomorphism on the quotients,

[TABLE]

Here we denote the image of $x_{r}$ in the quotient $R\llbracket x_{1},\dotsc,x_{nl}\rrbracket/\varrho({\mathcal{J}}^{\star}_{F_{a}})$ by the same symbol.

Suppose there is a continuous ring isomorphism

[TABLE]

Then the composition

[TABLE]

is a continuous ring isomorphism such that $({\kappa}\circ\tilde{\varrho})(x_{e_{i}\alpha_{j}})=e_{i}x_{j}$. In particular, we can take $\gamma:=\kappa\circ\tilde{\varrho}$. Therefore, all we need to do is show the existence of the isomorphism ${\kappa}$.

The ring homomorphism

[TABLE]

extends to a continuous ring homomorphism

[TABLE]

Since $e_{1}=1$, we have

[TABLE]

Therefore, $\tilde{\kappa}$ induces a continuous ring homomorphism

[TABLE]

In the other direction, the ring homomorphism

[TABLE]

extends to a continuous ring homomorphism

[TABLE]

We see that $({\kappa}\circ\iota)(x_{j})={\kappa}(x_{j})=x_{j}$ and that $(\iota\circ{\kappa})(x_{(i-1)n+j})=\iota(e_{i}x_{j})=e_{i}x_{j}=x_{(i-1)n+j}$. Since the $x_{j}$, $j=1,\dotsc,n$, generate $R\llbracket x_{1},\dotsc,x_{n}\rrbracket$ as a ring, and the $x_{k}$, $k=1,\dotsc,nl$, generate $R\llbracket x_{1},\dotsc,x_{nl}\rrbracket/\varrho({\mathcal{J}}^{\star}_{F_{a}})$ as a ring, it follows from the continuity of $\iota$ and ${\kappa}$ that $\iota$ and ${\kappa}$ are continuous ring isomorphisms that are inverse to each other. This proves the Lemma. ∎

Lemma 4.4**.**

The element $x_{\alpha_{j}}$ divides $x_{e_{i}\alpha_{j}}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$ for any $\alpha_{j}\in\Delta$ and $e_{i}\in B$.

Proof.

Recall that $\mathrm{log}_{F}$ satisfies

[TABLE]

for some $b_{k}\in R$. The series $g(u):=(1+\sum_{k\geq 2}b_{k}u^{k-1})$ is invertible in $R\llbracket u\rrbracket$ since its constant term is 1. By definition, the relation

[TABLE]

holds in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$. Therefore,

[TABLE]

Thus, $x_{\alpha_{j}}$ divides $x_{e_{i}\alpha_{j}}$ in $\widetilde{R\llbracket\Lambda\rrbracket}_{F}/{\mathcal{J}}^{\star}_{F}$. ∎

Let $\mathcal{J}_{F}^{\circ}$ be the closure of the ideal in $R\llbracket x_{\Lambda}\rrbracket$ generated by the elements

[TABLE]

over all $e_{i}\in B$, $\alpha_{j}\in\Delta$, and $\lambda_{1},\lambda_{2}\in\Lambda$. By similar reasoning as earlier, the quotient $R\llbracket x_{\Lambda}\rrbracket/\mathcal{J}_{F}^{\circ}$ is a complete Hausdorff ring with respect to the $\mathcal{I}_{F}^{\circ}$-adic topology, where $\mathcal{I}_{F}^{\circ}$ is the kernel of the induced augmentation map $R\llbracket x_{\Lambda}\rrbracket/\mathcal{J}_{F}^{\circ}\to R$.

Lemma 4.5**.**

There is a canonical $W$-equivariant continuous ring isomorphism

[TABLE]

sending $x_{\lambda}+\mathcal{J}_{F}^{\circ}\mapsto\left(x_{\lambda}+\widetilde{{\mathcal{J}}}_{F}\right)+{\mathcal{J}}_{F}^{\star}$ for all $\lambda\in\Lambda$.

Proof.

The ring homomorphism

[TABLE]

extends to a continuous ring homomorphism

[TABLE]

By continuity of $\hat{\phi}$ and Hausdorff completeness of $R\llbracket x_{\Lambda}\rrbracket$ and $R\llbracket x_{\Lambda}\rrbracket/{{\mathcal{J}}}_{F}^{\circ}$, we see that

[TABLE]

Thus, $\hat{\phi}$ induces a continuous ring homomorphism

[TABLE]

By continuity of $\bar{\phi}$ and Hausdorff completeness of $R\llbracket x_{\Lambda}\rrbracket/\widetilde{{\mathcal{J}}}_{F}$ and $R\llbracket x_{\Lambda}\rrbracket/{{\mathcal{J}}}_{F}^{\circ}$, we see that

[TABLE]

Thus, $\bar{\phi}$ induces a continuous ring homomorphism

[TABLE]

In the other direction, the ring homomorphism

[TABLE]

extends to a continuous ring homomorphism

[TABLE]

By continuity of $\hat{\psi}$ and Hausdorff completeness of $R\llbracket x_{\Lambda}\rrbracket/\widetilde{{\mathcal{J}}}_{F}$ and $\left(R\llbracket x_{\Lambda}\rrbracket/\widetilde{{\mathcal{J}}}_{F}\right)/{\mathcal{J}}_{F}^{\star}$ and $R\llbracket x_{\Lambda}\rrbracket$, we see that

[TABLE]

In addition, we have

[TABLE]

Thus, $\hat{\psi}$ induces a continuous ring homomorphism

[TABLE]

The continuous ring homomorphisms ${\phi}$ and ${\psi}$ are $W$-equivariant and inverse to each other. Therefore, we have proven the Lemma. ∎

Suppose $R$ is a subring of $\mathbb{C}$, and $(R,F)$ is a one-dimensional commutative formal group law over $R$. Then we may view $(R,F)$ as a formal group law $(\mathbb{C},F)$ over $\mathbb{C}$. We now allow $\Sigma$ to be crystallographic or noncrystallographic. We work under 4.6 for the remainder of this paper.

Assumption 4.6**.**

$R$* is a subring of $\mathbb{C}$, and $(R,F)$ is a one-dimensional commutative formal group law over $R$. In addition, if $\Sigma$ is noncrystallographic, then $R$ is an ample ring with respect to $(\mathbb{C},F)$, such that $R$ contains $\mathcal{R}$.*

We define elements $f_{i,j}\in R\llbracket x_{\Lambda}\rrbracket$, $i=1,\dotsc,l$, and $j=1,\dotsc,n$, as follows:

[TABLE]

Let $\mathcal{J}_{F}$ be the closure of the ideal in $R\llbracket x_{\Lambda}\rrbracket$ generated by the elements

[TABLE]

over all $i=1,\dotsc,l$, and $j=1,\dotsc,n$, and $\lambda_{1},\lambda_{2}\in\Lambda$.

Definition 4.7**.**

The quotient

[TABLE]

is the formal group ring with respect to $(R,F)$ and $\Lambda$.

We denote the image of $x_{\lambda}$ in ${R\llbracket\Lambda\rrbracket}_{F}$ by the same symbol. By similar reasoning as earlier, ${R\llbracket\Lambda\rrbracket}_{F}$ is a complete Hausdorff ring with respect to the $\mathcal{I}_{F}$-adic topology, where $\mathcal{I}_{F}$ is the kernel of the augmentation map ${R\llbracket\Lambda\rrbracket}_{F}\to R$ sending $x_{\lambda}\mapsto 0$ for all $\lambda\in\Lambda$.

Remark 4.8**.**

If $\Sigma$ is crystallographic, then $\mathcal{J}_{F}$ is the closure of the ideal in $R\llbracket x_{\Lambda}\rrbracket$ generated by the elements

[TABLE]

over all $\lambda,\mu\in\Lambda$. In this case, ${R\llbracket\Lambda\rrbracket}_{F}$ is the classical formal group ring with respect to the crystallographic root lattice $\Lambda$.

Remark 4.9**.**

In [7], the formal group ring is defined over any commutative unital ring $S$. However, if $\Sigma$ is noncrystallographic, then the definition of $\mathcal{J}_{F}$ requires that $S$ contains $\mathcal{R}$. In order to provide a simple unified exposition of the formal group ring for all finite root systems, we restrict $S$ to a subring of $\mathbb{C}$ in this paper (subject to 4.6).

Lemma 4.10**.**

The ring ${R\llbracket\Lambda\rrbracket}_{F}$ satisfies the following properties:

1. (1)

There is a well-defined $W$-action on ${R\llbracket\Lambda\rrbracket}_{F}$ given by

[TABLE] 2. (2)

There is a continuous ring isomorphism

[TABLE]

In particular, ${R\llbracket\Lambda\rrbracket}_{F}$ is an integral domain. 3. (3)

$x_{\alpha_{j}}$* divides $x_{e_{i}\alpha_{j}}$ in ${R\llbracket\Lambda\rrbracket}_{F}$ for all $e_{i}\in B$ and $\alpha_{j}\in\Delta$.*

Proof.

First assume $\Sigma$ is noncrystallographic. Property (1) follows from 4.2 and 4.5. Property (2) follows from 4.3 and 4.5. Property (3) follows from 4.4 and 4.5.

Now assume $\Sigma$ is crystallographic. Then $B=\{e_{1}=1\}$. Property (1) follows from Eq. 2. Property (2) follows from Eq. 1. Property (3) is trivial. ∎

Remark 4.11**.**

Since $e_{1}=1$, the isomorphism of 4.10 (2) sends $x_{\alpha_{j}}\mapsto\mathrm{exp}_{F}(x_{j})$ for each $\alpha_{j}\in\Delta$.

For a finite sequence of simple roots $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$, we set $x_{I}:=x_{\alpha_{i_{1}}}\dotsb x_{\alpha_{i_{r}}}$. We define the length of $I$ to be $l(I)=r$. We say a sequence $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$ of simple roots is ordered if $i_{1}\leq i_{2}\leq\dotsb\leq i_{r}$. Let $\Upsilon$ be the set of all sequences $I$ of simple roots such that $l(I)\geq 1$, and let $\Upsilon_{r}$ be the subset of $\Upsilon$ consisting of sequences of simple roots of length $r$.

Remark 4.12**.**

Let $\lambda\in\Lambda$. Since $\{e_{i}\alpha_{j}\}$ is a $\mathbb{Z}$-basis for $\Lambda$, we can write $\lambda=\sum_{j=1}^{n}\sum_{i=1}^{l}c_{i,j}e_{i}\alpha_{j}$ for some $c_{i,j}\in\mathbb{Z}$. The relations in $R\llbracket\Lambda\rrbracket_{F_{a}}$ allow us to write $x_{\lambda}\in R\llbracket\Lambda\rrbracket_{F_{a}}$ in the form $x_{\lambda}=\sum_{j=1}^{n}\left(\sum_{i=1}^{l}c_{i,j}e_{i}\right)x_{\alpha_{j}}$, where $\sum_{i=1}^{l}c_{i,j}e_{i}\in\mathcal{R}$ for each fixed $j$. In particular, $x_{\lambda}$ can be written as an $\mathcal{R}$-linear combination of the elements $x_{\alpha_{j}}$, where $\alpha_{j}\in\Delta$. Since $e_{1}=1$, the isomorphism of 4.10 (2) implies that $x_{\lambda}$ can be written uniquely as an $\mathcal{R}$-linear combination of the elements $x_{\alpha_{j}}$, where $\alpha_{j}\in\Delta$. Thus, any product of $r$ $x_{\lambda}$’s, where $r\geq 1$, can be written uniquely as an $\mathcal{R}$-linear combination of elements of the form $x_{I}$, where $I\in\Upsilon_{r}$.

Remark 4.13**.**

Since $e_{1}=1$, the isomorphism of 4.10 (2) implies that any element in $R\llbracket\Lambda\rrbracket_{F_{a}}$ can be written uniquely as an ${R}$-linear combination of $1$ and the elements $x_{I}$, where $I\in\Upsilon$.

Example 4.14**.**

(cf. [7, Ex. 2.19]) Let $S_{R}^{i}(\Lambda)$ be the $i$-th symmetric power of the $R$-module $R\otimes_{\mathcal{R}}\Lambda$, where we view $\Lambda$ as a free finitely-generated $\mathcal{R}$-module with basis $\Delta$. For $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})\in\Upsilon$, set $\alpha_{I}:=\alpha_{i_{1}}\dotsb\alpha_{i_{r}}\in S_{R}^{r}(\Lambda)$. By definition, every element in $S_{R}^{r}(\Lambda)$ can be written uniquely as an $R$-linear combination of $1$ and the elements $\alpha_{I}$, where $I\in\Upsilon_{r}$. The $R$-algebra $(S_{R}^{*}(\Lambda))^{\wedge}:=\prod\limits_{i=0}^{\infty}S_{R}^{i}(\Lambda)$ is the completion of the symmetric algebra $S_{R}^{*}(\Lambda):=\bigoplus\limits_{i=0}^{\infty}S_{R}^{i}(\Lambda)$ at the ideal generated by the kernel of the augmentation map $\alpha_{I}\mapsto 0$, $I\in\Upsilon$. By 4.13, every element in $R\llbracket\Lambda\rrbracket_{F_{a}}$ can be written uniquely as an ${R}$-linear combination of $1$ and the elements $x_{I}$, where $I\in\Upsilon$. Thus, there is a continuous $R$-linear ring isomorphism

[TABLE]

sending $x_{I}\mapsto\alpha_{I}$ for all $I\in\Upsilon$, and extended by $R$-linearity.

Example 4.15**.**

(cf. [7, Ex. 2.20]) Consider the group ring

[TABLE]

Let $\mathrm{tr}:R[\Lambda]\to R$ be the $R$-linear trace map that sends $e^{\lambda}\mapsto 1$ for all $\lambda\in\Lambda$. Let $R[\Lambda]^{\wedge}$ be the $\mathrm{ker}(\mathrm{tr})$-adic completion of $R[\Lambda]$. There is a continuous ring isomorphism

[TABLE]

where $\hat{h}(x_{\lambda})=e^{\lambda}-1$ and $\hat{h}^{-1}(e^{\lambda})=1+x_{\lambda}$.

Suppose that $\Sigma$ is noncrystallographic. Let $\mathcal{E}$ be the closure of the ideal in $R[\Lambda]^{\wedge}$ generated by the elements

[TABLE]

over all $e_{i}\in B$ and $\alpha_{j}\in\Delta$. We have $\hat{h}(\widetilde{\mathcal{J}}_{F_{m}})=\mathcal{E}$. To see this, note that, by continuity, $\hat{h}(\widetilde{\mathcal{J}}_{F_{m}})\subseteq\mathcal{E}$. Since $\hat{h}$ is an isomorphism of topological rings, $\hat{h}(\widetilde{\mathcal{J}}_{F_{m}})$ is closed, so we get equality. Through the continuous ring isomorphism $R\llbracket\Lambda\rrbracket_{F_{m}}\to\widetilde{R\llbracket\Lambda\rrbracket}_{F_{m}}/\mathcal{J}_{F_{m}}$ of 4.5, we see that $\hat{h}$ induces a continuous ring isomorphism

[TABLE]

such that ${h}(x_{\lambda})=e^{\lambda}-1$ and ${h}^{-1}(e^{\lambda})=1+x_{\lambda}$.

5. Formal Demazure operators

In this section, we show that the element $u-s_{\alpha}(u)$ is divisible by $x_{\alpha}$ in $R\llbracket\Lambda\rrbracket_{F}$ for all $u\in R\llbracket\Lambda\rrbracket_{F}$ and $\alpha\in\Sigma$. Using this fact, we define the formal Demazure operator $\Delta_{\alpha}^{(R,F)}$ in this section. When $(R,F)=(R,F_{a})$ is the additive formal group law over $R$, the isomorphism $\phi$ of 4.14 exchanges the formal Demazure operator $\Delta_{\alpha}^{(R,F_{a})}$ on $R\llbracket\Lambda\rrbracket_{F_{a}}$ with an operator $(\Delta_{\alpha}^{R})^{\wedge}$ on $(S_{R}^{*}(\Lambda))^{\wedge}$. We conclude this section by studying the restriction of $(\Delta_{\alpha}^{R})^{\wedge}$ to the symmetric algebra $S_{R}^{*}(\Lambda)$.

The following result will allow us to define the formal Demazure operator on $R\llbracket\Lambda\rrbracket_{F}$.

Lemma 5.1**.**

(cf.* [7, Cor. 3.4]) For any $u\in R\llbracket\Lambda\rrbracket_{F}$ and root $\alpha\in\Sigma$, the element $u-s_{\alpha}(u)$ is divisible by $x_{\alpha}$ in $R\llbracket\Lambda\rrbracket_{F}$.*

Proof.

First we assume that $\alpha=\alpha_{j}$ is a simple root. Since $\alpha_{j}^{\vee}(\lambda)\in\mathcal{R}$ for any $\lambda\in\Lambda$, it follows that $\alpha_{j}^{\vee}(\lambda)=\sum_{i=1}^{l}c_{i}e_{i}$ for some $c_{i}\in\mathbb{Z}$. We have

[TABLE]

Set $r=((-c_{1})\cdot_{F}x_{e_{1}\alpha_{j}})+_{F}\dotsb+_{F}((-c_{l})\cdot_{F}x_{e_{l}\alpha_{j}})$. By (3) of 4.10, $x_{\alpha_{j}}$ divides each $x_{e_{i}\alpha_{j}}$. Hence, $x_{\alpha_{j}}$ divides each $(-c_{i})\cdot_{F}x_{e_{i}\alpha_{j}}$. Thus, $x_{\alpha_{j}}$ divides $r$. Write $u+_{F}v=u+vg(u,v)$, where $g(u,v)\in R\llbracket u,v\rrbracket$. Then

[TABLE]

So $x_{\alpha_{j}}$ divides $x_{\lambda}-s_{j}(x_{\lambda})$. Then, by the formula

[TABLE]

the element $x_{\alpha_{j}}$ divides $u-s_{j}(u)$ for any monominal $u\in R[x_{\Lambda}].$ Finally, $x_{\alpha_{j}}$ divides $u-s_{j}(u)$ for any $u\in R\llbracket\Lambda\rrbracket_{F}$ by density of $R\llbracket\Lambda\rrbracket_{F}$.

Now let $\alpha\in\Sigma$ be any root. By 2.2, the root $\alpha$ can be written $\alpha=w(\alpha_{j})$ for some $\alpha_{j}\in\Delta$ and $w\in W$. Furthermore, 2.11 says that $ws_{\beta}w^{-1}=s_{w(\beta)}$ for any $w\in W$ and $\beta\in\Sigma$. Thus, given $u\in R\llbracket\Lambda\rrbracket_{F}$, we see that

[TABLE]

∎

Definition 5.2**.**

Following [17, Def. 4.1], for each root $\alpha\in\Sigma$, we define an $R$-linear operator $\Delta_{\alpha}^{(R,F)}$ on $R\llbracket\Lambda\rrbracket_{F}$ by the formula

[TABLE]

We call the operator $\Delta_{\alpha}^{(R,F)}$ a formal Demazure operator. We will usually write $\Delta_{\alpha}$ instead of $\Delta_{\alpha}^{(R,F)}$, omitting the superscript.

Proposition 5.3**.**

(see* [7, Prop. 3.8] and [11, §3]) The following formulas hold for any $u,v\in R\llbracket\Lambda\rrbracket_{F}$, $\alpha\in\Sigma$, and $w\in W.$*

1. (1)

$\Delta_{\alpha}(1)=0$,  $\Delta_{\alpha}(u)x_{\alpha}=u-s_{\alpha}(u)$; 2. (2)

$\Delta_{\alpha}^{2}(u)x_{\alpha}=\Delta_{\alpha}(u)+\Delta_{-\alpha}(u)$,  $\Delta_{\alpha}(u)x_{\alpha}=\Delta_{-\alpha}(u)x_{-\alpha}$; 3. (3)

$s_{\alpha}\Delta_{\alpha}(u)=-\Delta_{-\alpha}(u),$*  $\Delta_{\alpha}s_{\alpha}(u)=-\Delta_{\alpha}(u)$;* 4. (4)

$\Delta_{\alpha}(uv)=\Delta_{\alpha}(u)v+u\Delta_{\alpha}(v)-\Delta_{\alpha}(u)\Delta_{\alpha}(v)x_{\alpha}=\Delta_{\alpha}(u)v+s_{\alpha}(u)\Delta_{\alpha}(v)$; 5. (5)

$w\Delta_{\alpha}w^{-1}(u)=\Delta_{w(\alpha)}(u)$.

Proof.

The formulas (1)-(4) are straightforward computations using the definition of the formal Demazure operator. Relation (5) follows from 2.11, which says that $ws_{\alpha}w^{-1}=s_{w(\alpha)}$. ∎

Let $(\Delta_{\alpha}^{R})^{\wedge}\colon(S_{R}^{*}(\Lambda))^{\wedge}\to(S_{R}^{*}(\Lambda))^{\wedge}$ be the $R$-linear operator corresponding to the operator $\Delta_{\alpha}:R\llbracket\Lambda\rrbracket_{F_{a}}\to R\llbracket\Lambda\rrbracket_{F_{a}}$ with respect to the isomorphism $\phi:R\llbracket\Lambda\rrbracket_{F_{a}}\to(S_{R}^{*}(\Lambda))^{\wedge}$ of 4.14. The operator $(\Delta_{\alpha}^{R})^{\wedge}$ restricts to an $R$-linear operator $\Delta_{\alpha}^{R}\colon S_{R}^{*}(\Lambda)\to S_{R}^{*}(\Lambda)$. If $\Sigma$ is crystallographic and $R=\mathbb{Z}$, then $\Delta_{i}^{\mathbb{Z}}$ is the classical Demazure operator of [11], which acts on the symmetric algebra $S_{\mathbb{Z}}(\Lambda):=\bigoplus\limits_{i=1}^{\infty}S_{\mathbb{Z}}^{i}(\Lambda)$. If $R=\mathbb{C}$, then $\Delta_{i}^{\mathbb{C}}$ is the Demazure operator of [16, Ch. IV, pp. 135], which acts on the symmetric algebra $S(V)=S_{\mathbb{C}}(\Lambda):=\bigoplus\limits_{i=1}^{\infty}S_{\mathbb{C}}^{i}(\Lambda)$, where $V$ is the complex vector space generated by the roots $\beta\in\Sigma$.

For $\alpha_{i}\in\Delta$, we will use the notation $\Delta_{i}:=\Delta_{\alpha_{i}}$ and $\Delta_{i}^{R}:=\Delta_{\alpha_{i}}^{R}$. For a finite sequence of simple roots $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$, we set:

[TABLE]

We also set $\alpha_{I}:=\alpha_{i_{1}}\dotsb\alpha_{i_{r}}\in S_{R}^{r}(\Lambda)$. We say that $I$ is reduced if $s_{I}$ is reduced in $W$. Fix a reduced decomposition of a reflection $w=s_{i_{1}}\dotsb s_{i_{r}}$, where $i_{j}\in\{1,\dotsc,n\}$. We call $I_{w}:=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$ a reduced sequence of $w$.

Lemma 5.4**.**

Let $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{k}})\in\Upsilon_{k}$. We can write

[TABLE]

where the $c_{I^{\prime}}^{R}\in\mathcal{R}$ are independent of $R$.

Proof.

First note that, for any $\lambda\in\Lambda$, we have $\Delta_{i}^{R}(\lambda)=\alpha_{i}^{\vee}(\lambda)\in\mathcal{R}$. Thus, since $\mathcal{R}$ is a ring, the result follows by induction on the degree of monomials using (4) of 5.3. ∎

Proposition 5.5**.**

(see* [11, §4, Thm. 1] and [16, Ch. IV, Prop. 1.7]) Let $I_{w}$ and $I_{w}^{\prime}$ be reduced sequences for the same element $w\in W$. Then*

[TABLE]

Proof.

By [16, Ch. IV, Prop. 1.7], this result holds when $R=\mathbb{C}$. Now the Proposition follows from 5.4, and from the fact that $\{1\}\cup\{\alpha_{I}\}_{I\in\Upsilon}$ is an $R$-basis of $S_{R}^{*}(\Lambda)$. ∎

Remark 5.6**.**

5.5 implies that $\Delta_{I_{w}}^{R}$ is independent of the reduced sequence $I_{w}$ of $w\in W$. By the density of $(S_{R}^{*}(\Lambda))^{\wedge}$, the operator $(\Delta_{I_{w}}^{R})^{\wedge}$ is independent of the reduced sequence $I_{w}$ of $w\in W$ as well. Thus, the operator $\Delta_{I_{w}}^{(R,F_{a})}$ is independent of the reduced sequence $I_{w}$ of $w\in W$. In general, $\Delta_{I_{w}}^{(R,F)}$ depends on the reduced sequence $I_{w}$ of $w\in W$.

From now on, we will simply write $\Delta_{w}^{R}$ to denote the operator $\Delta_{I_{w}}^{R}$, where $I_{w}$ is any reduced sequence of $w\in W$.

Proposition 5.7**.**

(see* [11, §4, Prop. 3(a)] and [16, Ch. IV, Lem. 2.2]) Let $w,w^{\prime}\in W$. We have*

[TABLE]

Proof.

By [16, Ch. IV,  Lem. 2.2], this results holds when $R=\mathbb{C}$. Now the Proposition follows from 5.4, and from the fact that $\{1\}\cup\{\alpha_{I}\}_{I\in\Upsilon}$ is an $R$-basis of $S_{R}^{*}(\Lambda)$. ∎

Since $S_{R}^{*}(\Lambda)$ injects into the integral domain $(S_{R}^{*}(\Lambda))^{\wedge}\simeq R\llbracket\Lambda\rrbracket_{F}$, the symmetric algebra $S_{R}^{*}(\Lambda)$ is an integral domain. We let $\overline{S_{R}^{*}(\Lambda)}$ be its field of fractions.

Lemma 5.8**.**

(cf.* [11, Lem. 3]) Set $\Sigma_{w}:=\Sigma^{+}\cap w(\Sigma^{-})$ and $q_{w}=\left(\prod\limits_{\alpha\in\Sigma_{w}}\alpha\right)$. We have*

[TABLE]

where the $a(I_{w^{\prime}})\in\overline{S_{R}^{*}(\Lambda)}$, and the ordering $w^{\prime}<w$ is with respect to the Bruhat order on $W$.

Proof.

This proof is the same as the proof of [11, Lem. 3]; however, the proof of [11, Lem. 3] uses the result [4, Ch. VI, §1, No 6, Cor. 2], which is a property of the Weyl group. Since we are working with real finite reflection groups, we provide an updated reference. The result [4, Ch. VI, §1, No 6, Cor. 2] can be replaced by 2.7. ∎

Definition 5.9**.**

Let $\mathcal{D}_{R}(\Lambda)$ be the $R$-algebra of endomorphisms of $S_{R}^{*}(\Lambda)$ generated by the Demazure operators $\Delta_{i}^{R}$ and by multiplication by elements in $S_{R}^{*}(\Lambda)$.

Corollary 5.10**.**

(cf.* [11, Cor. 1]) The algebra $\mathcal{D}_{R}(\Lambda)$ is free as a left $S_{R}^{*}(\Lambda)$-module with basis $\{\Delta_{w}^{R}\}_{w\in W}$.*

Proof.

Since 5.3 and 5.8 hold, this proof is the same as the proof of [11, Cor. 1]. ∎

6. Endomorphisms of the formal group ring

In this section, we define the associated graded ring $\mathcal{G}r_{(R,F)}^{*}(\Lambda)$, and we discuss the subalgebra $\mathcal{D}_{(R,F)}(\Lambda)$ of the endomorphism algebra of $R\llbracket\Lambda\rrbracket_{F}$ generated by the formal Demazure operators and by multiplication by elements in $R\llbracket\Lambda\rrbracket_{F}$. This section closely follows [7, §4].

If $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$ is a sequence of simple roots, we set $x_{I}:=x_{\alpha_{i_{1}}}\dotsb x_{\alpha_{i_{r}}}$. Denote by $\mathcal{I}$ the kernel of the augmentation map $R\llbracket x_{\Lambda}\rrbracket\to R$. Recall the kernel $\mathcal{I}_{F}$ of the augmentation map $R\llbracket\Lambda\rrbracket_{F}\to R$. By convention, we set $\mathcal{I}_{F}^{i}=R\llbracket\Lambda\rrbracket_{F}$ for $i\leq 0$. Recall that $R\llbracket\Lambda\rrbracket_{F}=R\llbracket x_{\Lambda}\rrbracket/\mathcal{J}_{F}$, where $\mathcal{J}_{F}$ is the closure of the ideal in $R\llbracket x_{\Lambda}\rrbracket$ generated by the elements

[TABLE]

over all $e_{i}\in B$, $\alpha_{j}\in\Delta$, and $\lambda_{1},\lambda_{2}\in\Lambda$, where

[TABLE]

When $\Sigma$ is noncrystallographic, observe that

[TABLE]

where $g_{i,j}=\sum_{k\geq 2}b_{k}(e_{i}x_{\alpha_{j}}^{k}-x_{e_{i}\alpha_{j}}^{k})\in{\mathcal{I}}_{F}^{2}$. Here the $b_{k}\in R$ are the coefficients that appear in the series $\mathrm{log}_{F}(u)=u+\sum_{k\geq 2}b_{k}u^{k}$.

We define the associated graded ring,

[TABLE]

We have

[TABLE]

Set $\mathcal{T}_{k}:=\mathcal{J}_{F}\cap{\mathcal{I}}^{k}+{\mathcal{I}}^{k+1}$. When $\Sigma$ is crystallographic, $\mathcal{T}_{k}/{\mathcal{I}}^{k+1}$ is generated by elements of the form

[TABLE]

where $\alpha_{j}\in\Delta$, $\lambda,\mu\in\Lambda$, and $\rho$ is a monomial of degree $k-1$. When $\Sigma$ is noncrystallographic, $\mathcal{T}_{k}/{\mathcal{I}}^{k+1}$ is generated by elements of the form

[TABLE]

where $e_{i}\in B$, $\alpha_{j}\in\Delta$, $\lambda,\mu\in\Lambda$, and $\rho$ is a monomial of degree $k-1$. In either case, the generators of $\mathcal{T}_{k}/{\mathcal{I}}^{k+1}$ allow us to write any element in

[TABLE]

as an $R$-linear combination of elements of the form $x_{I}+\mathcal{I}_{F}^{k+1}$, where $I\in\Upsilon_{k}$.

Lemma 6.1**.**

(cf.* [7, Lem. 4.2]) The morphism of graded $R$-algebras*

[TABLE]

defined by sending $\alpha_{I}$ to $x_{I}+\mathcal{I}_{F}^{k+1}$, $I\in\Upsilon_{k}$, and extended by $R$-linearity, is an isomorphism.

Proof.

Every element in $S_{R}^{k}(\Lambda)$, $k>0$, can be written uniquely as an $R$-linear combination of monomials of the form $\alpha_{I}$, where $I\in\Upsilon_{k}$. The map $\psi$ sends the monomial $\alpha_{I}$ to the element $x_{I}+\mathcal{I}_{F}^{k+1}\in\mathcal{G}r_{(R,F)}^{*}(\Lambda)$, and, therefore, $\psi$ is well-defined. Since $\mathcal{G}r_{(R,F)}^{*}(\Lambda)$ is generated as a graded unital $R$-algebra by the elements $x_{\alpha_{i}}+\mathcal{I}_{F}^{2}$, where $\alpha_{i}\in\Delta$, the map $\psi$ is surjective. We define a map in the other direction.

Define the map of $R$-modules

[TABLE]

by sending a monomial of degree $k$ in some $x_{\lambda}$’s to the product of $\lambda$’s involved. The map $\tilde{\theta}_{k}$ factors through the quotient $({\mathcal{I}}^{k}/{\mathcal{I}}^{k+1})/(\mathcal{T}_{k}/{\mathcal{I}}^{k+1})$, hence giving a well-defined map ${\theta}_{k}:{\mathcal{I}}^{k}/\mathcal{T}_{k}\to S_{R}^{k}(\Lambda)$. Equivalently, $\tilde{\theta}_{k}$ gives a well-defined map ${\theta}_{k}:\mathcal{I}_{F}^{k}/\mathcal{I}_{F}^{k+1}\to S_{R}^{k}(\Lambda)$, since ${\mathcal{I}}^{k}/\mathcal{T}_{k}=\mathcal{I}_{F}^{k}/\mathcal{I}_{F}^{k+1}$. The sum ${\theta}:=\oplus{\theta}_{k}$ is also well-defined. Since every element in $\mathcal{I}_{F}^{k}/\mathcal{I}_{F}^{k+1}$ is an $R$-linear combination of elements of the form $x_{I}+\mathcal{I}_{F}^{k+1}$, where $I\in\Upsilon_{k}$, we see that $\theta$ is the desired inverse map. ∎

Remark 6.2**.**

Since every element in $S_{R}^{k}(\Lambda)$, $k>0$, can be written uniquely as an $R$-linear combination of monomials of the form $\alpha_{I}$, where the $I\in\Upsilon_{k}$, it follows from the isomorphism $\psi$ of 6.1 that every element in $\mathcal{I}_{F}^{k}/\mathcal{I}_{F}^{k+1}$ can be written uniquely as an $R$-linear combination of elements of the form $x_{I}+\mathcal{I}_{F}^{k+1}$, where $I\in\Upsilon_{k}$.

Lemma 6.3**.**

(cf.* [7, Prop. 4.6 (1)]) For any root $\alpha\in\Sigma$, the operator $\Delta_{\alpha}$ sends $\mathcal{I}_{F}^{k}$ to $\mathcal{I}_{F}^{k-1}$.*

Proof.

This follows directly from (4) of 5.3. ∎

By 6.3, the operator $\Delta_{\alpha}$ induces an $R$-linear operator of degree $-1$ on the graded ring $\mathcal{G}r_{F}^{*}(\Lambda),$ denoted $\mathcal{G}r\Delta_{\alpha}$, for all $\alpha\in\Sigma$.

Proposition 6.4**.**

(see* [7, Prop. 4.4]) For each root $\alpha\in\Sigma$, the isomorphism of 6.1 exchanges the operator $\mathcal{G}r\Delta_{\alpha}$ on $\mathcal{G}r_{(R,F)}^{*}(\Lambda)$ with $\Delta_{\alpha}^{R}$ on the symmetric algebra $S_{R}^{*}(\Lambda)$.*

Proof.

By 6.2, every element in $\mathcal{I}_{F}^{k}/\mathcal{I}_{F}^{k+1}$, $k>0$, can be written uniquely as an $R$-linear combination of elements of the form $x_{{I}}+\mathcal{I}_{F}^{k+1}$, where $I\in\Upsilon_{k}$. Now the result follows by induction on the degree of monomials using (4) of 5.3.∎

We recall the following fact concerning topological modules.

Remark 6.5**.**

(cf. [1, §10]) Let $S$ be a topological ring in the $I$-adic topology for some ideal $I$, and suppose $M$ is an $S$-module with filtration

[TABLE]

such that $IM^{(i)}\subseteq M^{(i+1)}$ for all $i\in\mathbb{Z}$, and $M=\cup_{i}M^{(i)}$. There is a natural topology on $M$ induced by this filtration, turning $M$ into a topological $S$-module. This topology is generated by elements of the form $x+M^{(i)}$, where $x\in M$ and $i\in\mathbb{Z}$. The $S$-module $M$ is complete with respect to this topology if and only if the canonical $S$-module homomorphism $M\to\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{i}}}(M/M^{(i)})$ is an $S$-module isomorphism, and $M$ is Hausdorff with respect to this topology if and only if $\cap_{i}M^{(i)}=(0)$.

Definition 6.6**.**

(see [7, Def. 4.5]) Let $\mathcal{D}_{(R,F)}(\Lambda)$ be the algebra of $R$-linear endomorphisms of $R\llbracket\Lambda\rrbracket_{F}$ generated by the formal Demazure operators $\Delta_{\alpha}$ for all roots $\alpha$, and by multiplication by elements in $R\llbracket\Lambda\rrbracket_{F}$.

Let $\mathcal{D}_{(R,F)}(\Lambda)^{(i)}$ be the left $R\llbracket\Lambda\rrbracket_{F}$-submodule of $\mathcal{D}_{(R,F)}(\Lambda)$ generated by elements of the form $u\Delta_{\alpha_{i_{1}}}\dotsb\Delta_{\alpha_{i_{k}}}$, where $u\in\mathcal{I}_{F}^{j}$, $j-k\geq i$, and the $\alpha_{i_{r}}\in\Sigma$. This defines a filtration on $\mathcal{D}_{(R,F)}(\Lambda)$,

[TABLE]

with the property $\mathcal{D}_{(R,F)}(\Lambda)=\cup_{i}\mathcal{D}_{(R,F)}(\Lambda)^{(i)}$. We define the associated graded $R\llbracket\Lambda\rrbracket_{F}$-module,

[TABLE]

Proposition 6.7**.**

(see* [7, Prop. 4.6]) The filtration on $\mathcal{D}_{(R,F)}(\Lambda)$ above has the following properties:*

1. (1)

For any root $\alpha$, we have

[TABLE] 2. (2)

For any integer $j$, we have $\mathcal{I}_{F}^{j}\mathcal{D}_{(R,F)}(\Lambda)^{(i)}\subseteq\mathcal{D}_{(R,F)}(\Lambda)^{(i+j)}$. 3. (3)

For any operator $D\in\mathcal{D}_{(R,F)}(\Lambda)^{(i)}$, we have $D(\mathcal{I}_{F}^{j})\subseteq\mathcal{I}_{F}^{(i+j)}$. 4. (4)

$\cap_{i}\mathcal{D}_{(R,F)}(\Lambda)^{(i)}={0}$.

Proof.

Since (4) of 5.3 and 6.3 hold, and $R\llbracket\Lambda\rrbracket_{F}$ is Hausdorff, the proof of this result is the same as the proof of [7, Prop. 4.6]. ∎

Point (3) of 6.7 implies that the graded module $\mathcal{G}r^{*}\mathcal{D}_{(R,F)}(\Lambda)$ acts by graded endomorphisms on the graded ring $\mathcal{G}r^{*}_{(R,F)}(\Lambda)$, which is isomorphic to $S_{R}^{*}(\Lambda)$ by 6.1. Recall the algebra of Demazure operators $\mathcal{D}_{R}(\Lambda)$ on $S_{R}^{*}(\Lambda)$ considered in Section 5. By 6.4, the class $\left[\Delta_{\alpha}\right]$ of $\Delta_{\alpha}$ in the quotient $\mathcal{D}_{(R,F)}(\Lambda)^{(-1)}/\mathcal{D}_{(R,F)}(\Lambda)^{(0)}$ acts by $\Delta_{\alpha}^{R}$ on $S_{R}^{*}(\Lambda)$, and the class $[x_{\alpha}]$ of multiplication by $x_{\alpha}$ in $\mathcal{D}_{(R,F)}(\Lambda)^{(1)}/\mathcal{D}_{(R,F)}(\Lambda)^{(2)}$ acts by multiplication by $\alpha$ on $S_{R}^{*}(\Lambda)$. Moreover, $\mathcal{G}r^{*}\mathcal{D}_{(R,F)}(\Lambda)$ is generated as a graded $\mathcal{G}r_{(R,F)}^{*}(\Lambda)$-module by the $[x_{\alpha}]$ and the $\left[\Delta_{\alpha}\right]$ over all $\alpha\in\Sigma$. We have justified the following result:

Proposition 6.8**.**

(see* [7, Prop. 4.8]) The graded module $\mathcal{G}r^{*}\mathcal{D}_{(R,F)}(\Lambda)$ over the graded ring $\mathcal{G}r_{(R,F)}^{*}(\Lambda)\simeq S_{R}^{*}(\Lambda)$ is isomorphic to $\mathcal{D}_{R}(\Lambda)$.*

For each element $w\in W$, fix a reduced sequence of simple roots $I_{w}$.

Theorem 6.9**.**

(see* [7, Lem. 4.11 (3)]) The operators $\Delta_{I_{w}}$ form a basis of $\mathcal{D}_{R,F}(\Lambda)$ as a left $R\llbracket\Lambda\rrbracket_{F}$-module.*

Proof.

By 5.5 and 5.10, the operators $\Delta_{I_{w}}^{R}$ form a basis of $\mathcal{D}_{R}(\Lambda)$ over $S_{R}^{*}(\Lambda)$. Therefore, by 6.8, the classes $\left[\Delta_{I_{w}}\right]$ form a basis of $\mathcal{G}r^{*}\mathcal{D}_{(R,F)}(\Lambda)$ over $\mathcal{G}r_{(R,F)}^{*}(\Lambda)$. By (4) of 6.7, $\mathcal{D}_{(R,F)}(\Lambda)$ is Hausdorff. Moreover, $R\llbracket\Lambda\rrbracket_{F}$ is complete. Therefore, the rest of this proof is the same as the proof of [7, Lem. 4.11]. ∎

Remark 6.10**.**

We can view $\mathcal{D}_{R}(\Lambda)$ as the subalgebra of the endomorphism algebra of $(S_{R}^{*}(\Lambda))^{\wedge}$ generated by the Demazure operators $\Delta_{\alpha}^{R}$ and by multiplication by elements in $S_{R}^{*}(\Lambda)$. If $(R,F)=(R,F_{a})$ is the additive formal group law over $R$, then, through the isomorphism $\phi:R\llbracket\Lambda\rrbracket_{F_{a}}\to(S_{R}^{*}(\Lambda))^{\wedge}$ of 4.14, we can view $\mathcal{D}_{(R,F_{a})}(\Lambda)$ as the completion of $\mathcal{D}_{R}(\Lambda)$ at the kernel of the map $\mathcal{D}_{R}(\Lambda)\to R$, sending $\alpha^{*}\mapsto 0$ for all $\alpha\in\Sigma$. Here, we use $\alpha^{*}$ to denote multiplication by $\alpha$.

7. The formal affine Demazure algebra

In this section, we define and discuss the formal affine Demazure algebra $\mathbf{D}_{F}$. All results discussed in this section are extensions of results in [9] and [17] to all real finite reflection groups.

By (2) of 4.10, the formal group ring $R\llbracket\Lambda\rrbracket_{F}$ is an integral domain. Thus, we let $\mathcal{Q}^{(R,F)}_{\Lambda}$ be the localization of $R\llbracket\Lambda\rrbracket_{F}$ at the multiplicative subset generated by the elements $x_{\alpha}$ for all $\alpha\in\Sigma$. The ring $\mathcal{Q}^{(R,F)}_{\Lambda}$ is an integral domain since $R\llbracket\Lambda\rrbracket_{F}$ is an integral domain, and we call $\mathcal{Q}^{(R,F)}_{\Lambda}$ the localized formal group ring. As $W$ acts on $R\llbracket\Lambda\rrbracket_{F}$ by permuting the roots $\alpha\in\Sigma,$ it also acts on $\mathcal{Q}^{(R,F)}_{\Lambda}$.

Lemma 7.1**.**

(see* [9, Lem. 3.2]) The localization map $R\llbracket\Lambda\rrbracket_{F}\to\mathcal{Q}^{(R,F)}_{\Lambda}$ is injective.*

Proof.

This follows from the fact that $R\llbracket\Lambda\rrbracket_{F}$ is an integral domain. ∎

Following [17, Def. 6.1] and [19, §4.1], we define the twisted formal group ring to be the $R$-module $\mathcal{Q}^{(R,F)}_{\Lambda,W}:=\mathcal{Q}^{(R,F)}_{\Lambda}\otimes_{R}R[W]$ with multiplication given by

[TABLE]

and extended by $R$-linearity. Here, $\delta_{w}$ is the element in $R[W]$ corresponding to $w\in W$ (so that $\delta_{w}\delta_{w^{\prime}}=\delta_{ww^{\prime}}$ for all $w,w^{\prime}\in W$). We will denote the identity $\mathbf{1}=\delta_{1}$. The set $\{\delta_{w}\}_{w\in W}$ is the canonical basis of the group ring $R[W]$, and, hence, it is a basis of $\mathcal{Q}^{(R,F)}_{\Lambda,W}$ as a left $\mathcal{Q}^{(R,F)}_{\Lambda}$-module.

Consider the $R$-subalgebra, $D_{R\llbracket\Lambda\rrbracket_{F}}$, of $\mathcal{Q}^{(R,F)}_{\Lambda,W}$ generated by the elements

[TABLE]

where $\delta_{\alpha}=\delta_{s_{\alpha}}$ for all $\alpha\in\Sigma$. Following [17, Def. 6.2 and Def. 6.3], we call the elements $X_{\alpha}^{(R,F)}$, $\alpha\in\Sigma$, formal Demazure elements and we call $D_{R\llbracket\Lambda\rrbracket_{F}}$ the formal Demazure algebra. We call the $R$-subalgebra of $\mathcal{Q}^{(R,F)}_{\Lambda,W}$ generated by $D_{R\llbracket\Lambda\rrbracket_{F}}$ and $R\llbracket\Lambda\rrbracket_{F}$ the formal affine Demazure algebra and denote it $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F}}$. Often, we denote the formal Demazure elements by $X_{\alpha}$, ommiting the superscript, and we denote the formal affine Demazure algebra by $\mathbf{D}_{F}$. For any simple root $\alpha_{i}\in\Delta$, we define $X_{i}:=X_{\alpha_{i}}$ and $\delta_{i}:=\delta_{s_{\alpha_{i}}}$. For any sequence of simple roots $I=(\alpha_{i_{1}},\dotsc,\alpha_{i_{r}})$, we will use the the notation $\delta_{I}:=\delta_{i_{1}}\dotsb\delta_{i_{r}}$ and $X_{I}:=X_{i_{1}}\dotsb X_{i_{r}}$.

Lemma 7.2**.**

(see* [9, Lem. 5.8]) The formal affine Demazure algebra $\mathbf{D}_{F}$ coincides with the $R$-subalgebra of $\mathcal{Q}_{W}^{(R,F)}$ generated by the elements of $R\llbracket\Lambda\rrbracket_{F}$ and the formal Demazure elements $X_{i}$, $i\in\{1,\dotsc,n\}$.*

Proof.

By 2.2, any root $\alpha\in\Sigma$ can be written $\alpha=w(\alpha_{i})$ for some $\alpha_{i}\in\Delta$ and $w\in W$. Now this result follows by the proof [9, Lem. 5.8]. ∎

Lemma 7.3**.**

(see* [9, Lem. 5.4]) Given a reduced sequence $I_{v}$ of $v\in W$ of length $l(I_{v}),$ let*

[TABLE]

for some $a_{v,w}\in\mathcal{Q}_{\Lambda}^{(R,F)}.$ Then

1. (a)

$a_{v,w}=0$* unless $w\leq v$ with respect to the Bruhat order on $W;$* 2. (b)

$a_{v,v}=(-1)^{l(I_{v})}\prod\limits_{\alpha\in v(\Sigma^{-})\cap\Sigma^{+}}x_{\alpha}^{-1}$.

Proof.

This proof is the same as the proof of [9, Lem. 5.4]; however, the proof of [9, Lem. 5.4] uses the results [13, Th. 1.1, III, (ii)] and [4, Ch. VI, §1, No 6, Cor. 2], which are properties of the Weyl group. Since we are working with real finite reflection groups, we provide updated references. The result [13, Th. 1.1, III, (ii)] can be replaced by 2.10, and instead of [4, Ch. VI, §1, No 6, Cor. 2], we use 2.7. ∎

Remark 7.4**.**

By definition, we have $X_{\alpha}=\tfrac{1}{x_{\alpha}}(1-\delta_{\alpha})$. Therefore, since $W$ permutes the roots, the $a_{v,w}$ of 7.3 are sums of products of elements of the form $\tfrac{1}{x_{\alpha}}$, where $\alpha\in\Sigma$.

Corollary 7.5**.**

(see* [9, Cor. 5.6]) Choose a reduced sequence $I_{w}$ for each $w\in W$. The elements $\{X_{I_{w}}\}_{w\in W}$ form a basis of $\mathcal{Q}^{(R,F)}_{\Lambda,W}$ as a left $\mathcal{Q}^{(R,F)}_{\Lambda}$-module, and the element $\delta_{v}$ decomposes as $\sum_{w\leq v}b_{w,v}X_{I_{w}}$ with $b_{w,v}\in\mathcal{Q}^{(R,F)}_{\Lambda}$. Furthermore,*

[TABLE]

Proof.

Since 7.3 holds, this proof is the same as the proof of [9, Cor. 5.6]. ∎

The proofs of 7.6 and 7.7 are direct computations:

Lemma 7.6**.**

(see* [17, Lemma 6.5]) For all $\alpha\in\Sigma$, we have the following commuting relation in $\mathcal{Q}^{(R,F)}_{\Lambda,W}$,*

[TABLE]

Here, $\Delta_{\alpha}(q)$ denotes the element $\tfrac{q-s_{\alpha}(q)}{x_{\alpha}}\in\mathcal{Q}^{(R,F)}_{\Lambda}$. By 5.1, if $q\in R\llbracket\Lambda\rrbracket_{F}$, then $\Delta_{\alpha}(q)\in R\llbracket\Lambda\rrbracket_{F}$. Hence, the formula above gives a relation in $\mathbf{D}_{F}$.

Lemma 7.7**.**

(see* [17, Eq. (6.1)]) For any $\alpha\in\Sigma$ we have*

[TABLE]

Here, $\kappa_{\alpha}\in R\llbracket\Lambda\rrbracket_{F}$ by the argument in [7, Def. 3.12].

Remark 7.8**.**

If $(R,F)$ is a formal group law of additive type, then the $\kappa_{\alpha}$ of 7.7 is [math] for each $\alpha\in\Sigma$.

The following Proposition says that all formal affine Demazure algebras over ample rings containing $\mathcal{R}$ are isomorphic. Its proof is the same as the proof of [17, Thm. 7.4]. We include the proof.

Proposition 7.9**.**

(see* [17, Thm. 7.4]) Assume $R$ is an ample ring with respect to a formal group law $(\mathbb{C},F)$, and that $R$ contains $\mathcal{R}$. There is an $R$-algebra isomorphism $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F}}\simeq\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{a}}}$.*

Proof.

The $W$-equivariant continuous ring isomorphism $\widetilde{\mathrm{exp}}_{F}^{*}:R\llbracket\Lambda\rrbracket_{F}\to R\llbracket\Lambda\rrbracket_{F_{a}}$ of formal group rings sends $x_{\alpha}\mapsto\mathrm{exp}_{F}(x_{\alpha})=x_{\alpha}g(x_{\alpha})$, where $g(u)\in R\llbracket u\rrbracket$ is the invertible power series defined in the proof of 4.4. Thus, $\widetilde{\mathrm{exp}}_{F}^{*}$ induces an isomorphism of twisted formal group rings $\widetilde{\mathrm{exp}}_{F}^{*}:\mathcal{Q}^{(R,F)}_{\Lambda}\to\mathcal{Q}^{(R,F_{a})}_{\Lambda}$, and $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F}}$ is isomorphic to its image $D^{\prime}:=\mathrm{exp}_{F}^{*}(\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{a}}})$ under this map. Now, $D^{\prime}$ is generated over $R\llbracket\Lambda\rrbracket_{F_{a}}$ by the elements

[TABLE]

where $\alpha_{i}\in\Delta$. Since $\tfrac{\mathrm{exp}_{F}(x_{\alpha_{i}})}{x_{\alpha_{i}}}\in R\llbracket\Lambda\rrbracket_{F_{a}}$ has constant term $1$, it is invertible in $R\llbracket\Lambda\rrbracket_{F_{a}}$. Therefore, $D^{\prime}$ is generated over $R\llbracket\Lambda\rrbracket_{F_{a}}$ by the elements $\Delta_{i}^{(R,F_{a})}$, $\alpha_{i}\in\Delta$, and is, thus, isomorphic to $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{a}}}$. ∎

8. Presentation in terms of generators and relations

In this section, we prove a presentation for the formal affine Demazure algebra in terms of formal Demazure elements. This section closely follows [7, §5.2], [9, §6 and §7], and [17, §6].

The arguments used from here up to and including the proof of 8.1 closely follow the arguments used in [7, §5.2]. Recall that we can view $S^{*}_{\mathbb{C}}(\Lambda)$ as the symmetric algebra of the complex vector space $V$ generated by the roots $\alpha\in\Sigma.$ Let $w_{0}$ be the longest element of $W$, and let $N=l(w_{0}).$ By [16, Ch. IV, Prop. 1.6 and Prop. 1.7] (see also the proof of [16, Ch. IV, Thm. 1.10]), the element $d_{0}:=\prod\limits_{\beta\in\Sigma^{+}}\beta\in S^{N}_{R}(\Lambda)$ satisfies $\Delta_{I_{w_{0}}}^{\mathbb{C}}(d_{0})=|W|$ for any reduced sequence $I_{w_{0}}$ of $w_{0}$. Since each root $\beta$ is an $\mathcal{R}$-linear combination of the simple roots, it follows from 5.4 that $\Delta_{I_{w_{0}}}^{R}(d_{0})=|W|$ as well.

Through the isomorphism $\psi\colon S_{R}^{*}(\Lambda)\to\mathcal{G}r_{(R,F)}^{*}(\Lambda)$ of 6.1, we choose an element $u_{0}\in\mathcal{I}_{F}^{N}/\mathcal{I}_{F}^{N+1}$ with $u_{0}=\psi(d)$. Therefore, by 6.4, for any reduced sequence $I_{w_{0}}$ of $w_{0}$, we have that $\Delta_{I_{w_{0}}}^{(R,F)}(u_{0})=\Delta_{I_{w_{0}}}^{R}(d)+\mathcal{I}_{F}$. Hence, $\epsilon\Delta_{I_{w_{0}}}^{(R,F)}(u_{0})=\Delta_{I_{w_{0}}}^{R}(d)=|W|$ for any reduced sequence $I_{w_{0}}$ of $w_{0}$.

The proof of 8.1 is the same as the proof of [7, Lemma 5.3 (3)]. We include the proof of 8.1 because it refers to several facts in [7] that we have shown hold for all real finite reflection groups as well. The element $u_{0}$ satisfies the following property:

Lemma 8.1**.**

(cf.* [7, Lemma 5.3 (3)]) If $I$ is any sequence of simple roots, and $1\leq l(I)\leq N$, then*

[TABLE]

Proof.

For $k>0$, 6.3 implies that the operator $\Delta_{I}$ sends $\mathcal{I}_{F}^{k}$ to $\mathcal{I}_{F}^{k-l(I)}$. Thus, if $l(I)<N$, then $\epsilon\Delta_{I}(u_{0})\in\epsilon\left(\mathcal{I}_{F}^{N-l(I)}\right)=\{0\}$.

If $l(I)=N$ and $I$ is not reduced, then $\epsilon\Delta_{I}(u_{0})=0$. To see this, it suffices to show that, in this case, $\Delta_{I}^{F}$ sends $\mathcal{I}_{F}^{k}$ to $\mathcal{I}_{F}^{k-l(I)+1}$. Thus, it is enough to show that $\mathcal{G}r\Delta_{I}=0$, which, by 6.4, is equivalent to the statement $\Delta_{I}^{R}=0$. The fact that $\Delta_{I}^{R}=0$ when $I$ is not reduced follows from 5.7.

If $l(I)=N$ and $I$ is reduced, then $\epsilon\Delta_{I}(u_{0})=|W|$ by definition. ∎

In [7, 9], the torsion index $\mathfrak{t}$ of [11] serves the same role as the element $|W|$ does in this paper. For each $w\in W$, let $I_{w}$ be a reduced sequence of $w$. With the torsion index $\mathfrak{t}$ replaced by the element $|W|$, the following result goes through:

Lemma 8.2**.**

(see* [9, Lem. 6.1]) Suppose $|W|$ is invertible in $R$. Then the matrix $\left(\Delta_{I_{v}}\Delta_{I_{w}}(u_{0})\right)_{(v,w)\in W\times W}$ with coefficients in $R\llbracket\Lambda\rrbracket_{F}$ is invertible.*

Proof.

By hypothesis, the element $|W|$ is invertible in $R$. In addition, 8.1 holds. Thus, this result follows by the proof of [7, Prop. 6.6]. ∎

The arguments used from here up to and including the proof of 8.9 closely follow the arguments used in [9, §6 and §7] and [17, §6]. Let $\phi:\mathcal{Q}_{\Lambda,W}^{(R,F)}\to\textrm{End}_{R}(\mathcal{Q}_{\Lambda}^{(R,F)})$ be the $R$-algebra homomorphism induced by the left action of $\mathcal{Q}_{\Lambda,W}^{(R,F)}$ on $\mathcal{Q}_{\Lambda}^{(R,F)}$. By definition, $R\llbracket\Lambda\rrbracket_{F}$ acts on the left on both algebras, $\phi$ is $R\llbracket\Lambda\rrbracket_{F}$-linear, and $\phi(X_{\alpha})=\Delta_{\alpha}$. Let $\phi_{\mathbf{D}_{F}}$ be the restriction of $\phi$ to $\mathbf{D}_{F}$. Note that the image of $\phi_{\mathbf{D}_{F}}$ is $\mathcal{D}_{(R,F)}(\Lambda)$, since the $R$-algebra generators of $\mathbf{D}_{F}$ map to the $R$-algebra generators of $\mathcal{D}_{(R,F)}(\Lambda)$.

Theorem 8.3**.**

(see* [9, Thm. 7.10]) The map $\phi_{\textbf{D}_{F}}:\mathbf{D}_{F}\to\mathcal{D}_{(R,F)}(\Lambda)$ is a left $R\llbracket\Lambda\rrbracket_{F}$-algebra isomorphism.*

Proof.

Since 7.5, and 8.2 hold, this proof is the same as the proof of [9, Thm. 7.10]. ∎

Corollary 8.4**.**

The $R$-algebra $\mathbf{D}_{F}$ is free as a left $R\llbracket\Lambda\rrbracket_{F}$-submodule of $\mathcal{Q}^{(R,F)}_{\Lambda,W}$ with basis $\{X_{I_{w}}\}_{w\in W}$.

Proof.

This follows from 6.9 and 8.3. ∎

Lemma 8.5**.**

(see* [9, Lem. 7.1]) Let $I$ and $I^{\prime}$ be reduced sequences of the same element $w\in W$. Then in the formal affine Demazure algebra $\mathbf{D}_{F},$ we have*

[TABLE]

where the ordering $v<w$ is with respect to the Bruhat order on $W$.

Proof.

By 8.4, the difference $X_{I}-X_{I^{\prime}}=\sum_{v\in W}c_{v}X_{I_{v}}$ for some $c_{v}\in R\llbracket\Lambda\rrbracket_{F}$. Since 7.1, 7.3, and 7.5 hold, the rest of this proof is the same as the proof of [9, Lem. 7.1]. ∎

Remark 8.6**.**

Let $I$ and $I^{\prime}$ be reduced sequences of the same element $u\in W$. Then 5.5 and 8.3 imply that, in $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{a}}}$, we have $X_{I}^{(R,F_{a})}=X_{I^{\prime}}^{(R,F_{a})}$.

Remark 8.7**.**

Let $I$ and $I^{\prime}$ be reduced sequences of the same element $u\in W$. Suppose $\Sigma$ is crystallographic. By [7, Thm. 3.10], in $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{m}}}$, we have $X_{I}^{(R,F_{m})}=X_{I^{\prime}}^{(R,F_{m})}$. Note that [7, Thm. 3.10] refers to the proof of [12, Thm. 2, pp. 86], which relies on the geometry of flag varieties. If $\Sigma$ is noncrystallographic, we leave it open to determine whether $X_{I}^{(R,F_{m})}=X_{I^{\prime}}^{(R,F_{m})}$ in $\mathbf{D}_{R\llbracket\Lambda\rrbracket_{F_{m}}}$.

Remark 8.8**.**

(cf. [9, Ex. 7.3]) Suppose $\alpha_{i}$ and $\alpha_{j}$ are simple roots, and $m_{i,j}$ is the order of $s_{i}s_{j}$ in $W$. Set w_{0}^{i,j}=\underbrace{s_{i}s_{j}s_{i}\dotsb}_{\text{m_{i,j} times}}=\underbrace{s_{j}s_{i}s_{j}\dotsb}_{\text{m_{i,j} times}}. If $\Sigma$ is a root system of rank $2$, then $w_{0}^{1,2}$ is the unique element in $W$ with more than one reduced decomposition; $w_{0}^{1,2}$ has exactly two reduced sequences $(\alpha_{1},\alpha_{2},\alpha_{1},\dotsc)$ and $(\alpha_{2},\alpha_{1},\alpha_{2},\dotsc)$, both of length $m_{1,2}$; and $w_{0}^{1,2}$ is the longest word in $W$.

Theorem 8.9**.**

(see* [9, Thm. 7.9] and [17, Thm. 6.14]) Let $\Sigma$ be a root system of a real finite reflection group $W$. Let $\mathcal{Q}^{(R,F)}_{\Lambda}$ denote the localization of $R\llbracket\Lambda\rrbracket_{F}$ at the elements $x_{\alpha},$ $\alpha\in\Sigma.$ Given a set of simple roots $\{\alpha_{1},\dotsc,\alpha_{n}\}$ with associated simple reflections $\{s_{1},\dotsc,s_{n}\}$, let $m_{i,j}$ denote the order of the product $s_{i}s_{j}$ in $W$. The elements $q\in\mathcal{Q}^{(R,F)}_{\Lambda}$ (resp. $q\in R\llbracket\Lambda\rrbracket_{F}$) and the formal Demazure elements $X_{i}$ satisfy the following relations for all $i,j=1,\dotsc,n$:*

1. (1)

$X_{i}q=\Delta_{i}(q)+s_{i}(q)X_{i};$** 2. (2)

$X_{i}^{2}=\kappa_{\alpha_{i}}X_{i}$, where $\kappa_{\alpha_{i}}=\tfrac{1}{x_{\alpha_{i}}}+s_{i}\left(\tfrac{1}{x_{\alpha_{i}}}\right);$ 3. (3)

\underbrace{X_{i}X_{j}X_{i}\dotsb}_{\text{m_{i,j} times}}-\underbrace{X_{j}X_{i}X_{j}\dotsb}_{\text{{m_{i,j} times}}}=\sum\limits_{w<w_{0}^{i,j}}\eta_{w}^{i,j}X_{I_{w}},    $\eta_{w}^{i,j}\in R\llbracket\Lambda\rrbracket_{F}$.

Here, $w_{0}^{i,j}$ is defined in 8.8, and the ordering $w<w_{0}^{i,j}$ is with respect to the Bruhat order on $W$. These relations, together with the ring law in $R\llbracket\Lambda\rrbracket_{F}$ and the fact that the $X_{i}$ are $R$-linear, form a complete set of relations in the localized twisted formal group ring $Q^{(R,F)}_{\Lambda,W}$ (resp. the formal affine Demazure algebra $\mathbf{D}_{F}$).

Proof.

The three relations hold by 7.6, 7.7, and 8.5. Since 7.5 and 8.4 hold and $W$ is a Coxeter group (see 2.3), the proofs of the presentations of $\mathcal{Q}_{\Lambda,W}^{(R,F)}$ and $\mathbf{D}_{F}$ are similar to the proof of [17, Thm. 6.14].∎

9. Computations of structure coefficients

In the present section, we study the braid relation (3) of 8.9. In particular, we give general formulas for several structure coefficients $\eta_{w}^{i,j}$ that appear in this braid relation, thus generalizing several formulas obtained in [17, Prop. 6.8] to all real finite reflection groups.

Before we proceed, we will define some notation that will be used in this section and in Section 10.

Notation 9.1**.**

For any root $\gamma\in\Sigma,$ set $y_{\gamma}=\tfrac{1}{x_{\gamma}}$. Let $\{\alpha,\beta\}$ be distinct simple roots, and let $m$ be the order of $s_{\alpha}s_{\beta}$ in $W$. Write

[TABLE]

As in the proof of 2.8, we set

[TABLE]

and we will use the notation

[TABLE]

We define the following notation for products of $i$ formal Demazure elements, $\delta$’s, and simple reflections:

[TABLE]

So, for example,

[TABLE]

We define the following :

[TABLE]

Let $i=1,\dotsc,m-2$. We define:

[TABLE]

By 2.7, $\upsilon_{\alpha,\beta}^{(0)}=\upsilon_{\beta,\alpha}^{(0)}=y_{\Sigma^{+}_{\alpha,\beta}}$.

For $j>i\geq 0$, we define the operators $S_{\beta,\alpha}^{(i,j)}:=\sum\limits_{k=i}^{j}s_{\beta,\alpha,\ldots}^{(k)}$ and $S_{\alpha,\beta}^{(i,j)}:=\sum\limits_{k=i}^{j}s_{\alpha,\beta,\ldots}^{(k)}$, which act on a root $\gamma\in\Sigma$ as follows:

[TABLE]

[TABLE]

We have the following extension of [17, Prop. 6.8] to all real finite reflection groups. The proof is very similar to the proof of [17, Prop. 6.8]. We include it because we will refer to the proof several times in this section.

Lemma 9.2**.**

(cf.* [17, Prop. 6.8]) The difference $X_{\alpha,\beta,\ldots}^{(m)}-X_{\beta,\alpha,\ldots}^{(m)}$ can be written as a linear combination*

[TABLE]

where $\kappa_{\alpha,\beta}^{(i)}\in R\llbracket\Lambda\rrbracket_{F}$, $i=1,\dotsc,m-2$.

Proof.

We let $\kappa_{\alpha,\beta}^{(m-1)}$ and $\kappa_{\beta,\alpha}^{(m-1)}$ denote the coefficients of $X_{\alpha,\beta,\ldots}^{(m-1)}$ and $X_{\beta,\alpha,\ldots}^{(m-1)}$ on the right side of Eq. 5. Similarly, we let $\kappa_{\alpha,\beta}^{(m)}$ and $\kappa_{\beta,\alpha}^{(m)}$ denote the coefficients of $X_{\alpha\ldots}^{(m)}$ and $X_{\beta,\alpha,\ldots}^{(m)}$ on the right side of Eq. 5. We will show by direct computation that all four of these coefficients equal [math].

The fact that $\kappa_{\alpha,\beta}^{(m)}=\kappa_{\beta,\alpha}^{(m)}=0$ follows by the same reasoning as in the proof of [17, Prop. 6.8]. We will include the proof because we use it in 9.4 and 9.10. Alternatively, $\kappa_{\alpha,\beta}^{(m)}=\kappa_{\beta,\alpha}^{(m)}=0$ by 8.5.

In the expansion of the product

[TABLE]

and in the expansion of the product $X_{\beta,\alpha,\ldots}^{(m)},$ the coefficient of $\delta_{w_{0}^{\alpha,\beta}}=\delta_{\alpha,\beta,\ldots}^{(m)}=\delta_{\beta,\alpha,\ldots}^{(m)}$ is

[TABLE]

where we use ‘$+$’ when $m$ is even and we use ‘$-$’ when $m$ is odd. Hence, in the difference $X_{\alpha,\beta,\ldots}^{(m)}-X_{\beta,\alpha,\ldots}^{(m)}$, the $\delta_{w_{0}^{\alpha,\beta}}$-term is

[TABLE]

So $\kappa_{\alpha,\beta}^{(m)}=\kappa_{\beta,\alpha}^{(m)}=0$. Next we will show that $\kappa_{\alpha,\beta}^{(m-1)}=\kappa_{\beta,\alpha}^{(m-1)}=0$.

Suppose $m$ is odd. To obtain a $\delta_{\alpha,\beta,\ldots}^{(m-1)}$-term in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m)}$, we must choose the $\delta$-term in each factor, except for the last $X_{\alpha}$, where we must choose the constant term. If we choose the constant term in an $X_{\beta}$ factor, we would get cancellations of $\delta_{\alpha}$, resulting in a word in $\delta_{\alpha}$ and $\delta_{\beta}$ of length less than $m-1.$ If we choose the constant term in the first $X_{\alpha}$, then the word would begin with $\delta_{\beta},$ and if we choose a constant term in an $X_{\alpha}$ factor that is not the first or last factor, we would again get cancellations of $\delta$’s and obtain a word of length less than $m-1$. Hence, the $\delta_{\alpha,\beta,\ldots}^{(m-1)}$-term is

[TABLE]

By a similar argument, the $\delta_{\beta,\alpha,\ldots}^{(m-1)}$-term in $X_{\alpha,\beta,\ldots}^{(m)}$ is

[TABLE]

By 2.8, $\alpha=s_{\beta,\alpha,\ldots}^{(m-1)}(\beta)$. Thus, 2.7 implies that $c=y_{\Sigma^{+}_{\alpha,\beta}}$. Therefore, in the difference $X_{\alpha,\beta,\ldots}^{(m)}-X_{\beta,\alpha,\ldots}^{(m)},$ the coefficients of $\delta_{\alpha,\beta,\ldots}^{(m-1)}$ and $\delta_{\beta,\alpha,\ldots}^{(m-1)}$ are zero, which implies that $\kappa_{\alpha,\beta}^{(m-1)}=\kappa_{\beta,\alpha}^{(m-1)}=0$.

Now suppose $m$ is even. One can show by a similar argument that the $\delta_{\alpha,\beta}^{(m-1)}$-term and the $\delta_{\beta,\alpha}^{(m-1)}$-term in $X_{\alpha,\beta,\ldots}^{(m)}$ are both $-y_{\Sigma_{\alpha,\beta}^{+}}$. Thus, $\kappa_{\alpha,\beta}^{(m-1)}=\kappa_{\beta,\alpha}^{(m-1)}=0$ when $m$ is even as well.

Now we consider the constant terms. By the discussion just before the statement of 8.3, there is a natural action of $\mathcal{Q}_{\Lambda,W}^{(R,F)}$ on $\mathcal{Q}_{\Lambda}^{(R,F)}$, where $X_{\alpha}(r)=\Delta_{\alpha}(r)=0$ and $X_{\beta}(r)=\Delta_{\beta}(r)=0$ for all $r\in R$. Therefore, the constant term on the right side of Eq. 5 is zero.

Finally, it follows from 8.5 that all coefficients on the right hand side of Eq. 5 are in $R\llbracket\Lambda\rrbracket_{F}$. ∎

Remark 9.3**.**

If $\Sigma$ is crystallographic, then the structure coefficients $\kappa_{\alpha,\beta}^{(i)}$ and $\kappa_{\beta,\alpha}^{(i)}$ can be explicitly computed from the formulas in [17, Prop. 6.8] by applying 7.6 and 7.7.

We assume that $m\geq 3$ is odd for the rest of this section. We believe that results similar to those proven in the rest of this section hold when $m$ is even. However, for simplicity of our exposition, we will only consider the case when $m$ is odd.

We use the formulas given in the following Corollary to prove 9.10.

Corollary 9.4**.**

Assume $m\geq 3$ is odd. The $\kappa_{\beta,\alpha}^{(1)},\dotsc,\kappa_{\beta,\alpha}^{(m-2)}$ of 9.2 satisfy

[TABLE]

Proof.

7.3 and 2.10 imply that the elements $X_{\alpha,\beta,\ldots}^{(m)}$ and $X_{\beta,\alpha,\ldots}^{(m)}$ are $\mathcal{Q}_{\Lambda}^{(R,F)}$-linear combinations of the elements in

[TABLE]

So there must exist coefficients $p_{\mathbf{1},j}$, $p_{\alpha,j}^{(i)}$, $p_{\beta,j}^{(i)}\in\mathcal{Q}_{\Lambda}^{(R,F)}$; $i=1,\dotsc,m$; $j=1,2$, that satisfy

[TABLE]

[TABLE]

By 9.2, together with the fact that $\delta_{\alpha,\beta,\ldots}^{(m)}=\delta_{\beta,\alpha,\ldots}^{(m)}$ and the fact that the $\delta_{w}$, $w\in W$, are $\mathcal{Q}_{\Lambda}^{(R,F)}$-linearly independent, we find that

[TABLE]

Now, in the proof of 9.2, we showed that, for $j=1,2$, the coefficients

[TABLE]

Furthermore, since $m$ is odd, in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m)}$, the coefficient of $\delta_{\alpha,\beta,\ldots}^{(i)}$ equals the coefficient of $-\delta_{\alpha,\beta,\ldots}^{(i-1)}$ when $i$ is odd, and the coefficient of $\delta_{\beta,\alpha,\ldots}^{(i)}$ equals the coefficient of $-\delta_{\beta,\alpha,\ldots}^{(i-1)}$ when $i$ is even. This follows from the definition of the formal Demazure element, $X_{\alpha}=y_{\alpha}(\mathbf{1}-\delta_{\alpha})$, and from the fact that $\delta_{\alpha}^{2}=\delta_{\beta}^{2}=\mathbf{1}$. Switching $\alpha$ with $\beta$ gives a similar result for $X_{\beta,\alpha,\ldots}^{(m)}$. See the computations in 11.1 for concrete examples that display this property. This tells us that

[TABLE]

Combining this with Eq. 8 gives us

[TABLE]

Now we continue recursively to obtain the formulas for the coefficients that appear in the statement of this Corollary. ∎

Remark 9.5**.**

Assume $m\geq 3$ is odd. Suppose $(R,F)$ is the additive formal group law over $R$; or $(R,F)$ is the multiplication formal group law over $R$ and $\Sigma$ is crystallographic. In the first case, 8.6 implies that all the $\kappa_{\alpha,\beta}^{(i)}=0$. In the second case, 8.7 implies that all the $\kappa_{\alpha,\beta}^{(i)}=0$. Thus, in either case, 9.4 implies that

[TABLE]

We will now motivate 9.7 with the following example.

Example 9.6**.**

Suppose $m=5$. The coefficient of $\delta_{\alpha,\beta,\ldots}^{(3)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(5)}$ is

[TABLE]

To see this, note that in the expansion of

[TABLE]

all $\delta_{\alpha,\beta,\ldots}^{(3)}$ summands are obtained by choosing a $\delta$-term in each factor, except for two adjacent factors. In the adjacent factors, one should instead choose the constant term. This way, one obtains $4$ summands:

[TABLE]

The first summand is

[TABLE]

The second summand is

[TABLE]

The third summand is

[TABLE]

The fourth summand is

[TABLE]

Combining these summands, we see that

[TABLE]

Lemma 9.7**.**

Assume $m\geq 3$ is odd. The coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-2)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m)}$ is

[TABLE]

By symmetry, the coefficient of $\delta_{\beta,\alpha,\ldots}^{(m-2)}$ in the expansion of $X_{\beta,\alpha,\ldots}^{(m)}$ is $c_{\beta,\alpha}^{(m-2)}$.

Proof.

A $\delta_{\alpha,\beta,\ldots}^{(m-2)}$-summand in the expansion of $X_{\alpha,\beta,\ldots}^{(m)}$ is obtained by choosing $\delta$-terms in all factors except of two adjacent factors, $X_{\alpha}X_{\beta}$ or $X_{\beta}X_{\alpha}$, where one chooses constant terms (if one doesn’t choose adjacent factors, then there is cancellation of $\delta$’s, resulting in a word of length less than $m-2$). So we obtain $(m-1)$ summands,

[TABLE]

We use the multiplication in $\mathcal{Q}_{\Lambda,W}^{(R,F)}$ to obtain a formula for $c_{\alpha,\beta}^{(m-2)}$. Let $i=0,\ldots,m-2$. Then the $(i+1)^{st}$ summand above is

[TABLE]

Note that we can replace $s_{\alpha,\beta,\ldots}^{(i)}(y_{\omega_{i}}y_{\omega_{i+1}})$ by $s_{\alpha,\beta,\ldots}^{(i)}(y_{\alpha}y_{\beta})$. Combining these summands, we obtain the formula for the coefficient $c_{\alpha,\beta}^{(m-2)}$ that appears in the statement of the Lemma. ∎

We will now motivate 9.9 with the following example.

Example 9.8**.**

Suppose $m=5$. The coefficient of $\delta_{\beta,\alpha,\ldots}^{(2)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(5)}$ is

[TABLE]

To see this, note that in the expansion of

[TABLE]

all $\delta_{\beta,\alpha,\ldots}^{(2)}$ summands are obtained in one of two ways. In the first way: one chooses the constant term in the first factor, followed by the $\delta$-term in all remaining factors, except for two adjacent factors. In the two adjacent factors, one should instead choose the constant term. Here, one obtains $3$ summands. In the second way: one chooses the constant term in the second factor, and a $\delta$-term in the remaining factors. Here, the $\delta$’s in the first and third factors will cancel each other, giving a summand of $\delta_{\beta,\alpha,\ldots}^{(2)}$. So we get $4$ summands:

[TABLE]

The first summand is

[TABLE]

The second summand is

[TABLE]

The third summand is

[TABLE]

The fourth summand is

[TABLE]

Combining these summands, we see that

[TABLE]

Lemma 9.9**.**

Assume $m\geq 5$ is odd. Then the coefficient of $\delta_{\beta,\alpha,\ldots}^{(m-3)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m)}$ is

[TABLE]

By symmetry, the coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-3)}$ in the expansion of $X_{\beta,\alpha,\ldots}^{(m)}$ is $c_{\alpha,\beta}^{(m-3)}$.

Proof.

A $\delta_{\beta,\alpha,\ldots}^{(m-3)}$-summand in the expansion of $X_{\alpha,\beta,\ldots}^{(m)}$ is obtained in one of two ways. In the first way, one chooses the constant term in the first $X_{\alpha}$, followed by $\delta$-terms in the remaining factors, except for two adjacent factors. In the two adjacent factors, one should instead choose the constant terms (if one doesn’t choose adjacent factors, then there is a cancellation of $\delta$’s, resulting in a word of length less than $m-3$; if one chooses the $\delta$-term in the first factor, the product of $\delta$’s will begin with $\delta_{\alpha}$). This gives $m-2$ summands. In the second way, we obtain one additional summand by choosing the $\delta$-term in the first factor, followed by the constant term in the second factor, followed by the $\delta$-term in each of the remaining factors (so that the first and third $\delta_{\alpha}$’s will cancel). So we have $m-1$ summands in total:

[TABLE]

We use the multiplication in $\mathcal{Q}_{\Lambda,W}^{(R,F)}$ to obtain a formula for $c^{(m-3)}_{\beta,\alpha}$. The formula for the first summand above is

[TABLE]

Now, let $i=0,\ldots,m-3$. The $(i+2)^{nd}$ summand above is

[TABLE]

Note that we can replace $s_{\beta,\alpha,\ldots}^{(i)}(y_{\omega_{i+1}}y_{\omega_{i}})$ by $s_{\beta,\alpha,\ldots}^{(i)}(y_{\alpha}y_{\beta})$. Combining these summands, we obtain the general formula for the coefficient $c_{\beta,\alpha}^{(m-3)}$ that appears in the statement of the Lemma. ∎

Theorem 9.10**.**

Suppose $m\geq 3$ is odd. Below is an explicit formula for the structure coefficient $\kappa_{\beta,\alpha}^{(m-2)}$:

[TABLE]

Suppose $m\geq 5$ is odd. Below is an explicit formula for the structure coefficient $\kappa_{\alpha,\beta}^{(m-3)}$:

[TABLE]

Proof.

By 2.8, we have $s_{\alpha,\beta,\ldots}^{(m-1)}(\alpha)=\beta$ when $m$ is odd. We use this property implicitly in this proof.

First, we will determine the coefficient $\kappa_{\beta,\alpha}^{(m-2)}.$ Replacing $m$ with $m-2$ in 9.2 and using the same method of proof, we deduce that the coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-2)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m-2)}$ is

[TABLE]

In 9.7, we found the coefficient of $\delta_{\beta,\alpha,\ldots}^{(m-2)}$ in the expansion of $X_{\alpha,\beta,\ldots}^{(m)}$, which we denoted $c_{\beta,\alpha}^{(m-2)}$. Therefore, by 9.4, we have the following formula:

[TABLE]

where

[TABLE]

One can check that, after cancellations and simplifications, this equation is the same as the equation given in the statement of this Theorem.

Now we will determine the coefficient $\kappa_{\alpha,\beta}^{(m-3)}.$ Replacing $m$ with $m-2$ in 9.2 and using the same method of proof, we deduce that the coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-3)}$ in the expansion of the product $X_{\beta,\alpha,\ldots}^{(m-2)}$ is

[TABLE]

Replacing $m$ with $m-3$ instead, we deduce that the coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-3)}$ in the expansion of the product $X_{\alpha,\beta,\ldots}^{(m-3)}$ is

[TABLE]

In 9.9, we found the coefficient of $\delta_{\alpha,\beta,\ldots}^{(m-3)}$ in the expansion of $X_{\beta,\alpha,\ldots}^{(m)}$, which we denoted $c_{\alpha,\beta}^{(m-3)}$. Therefore, by 9.4, we have the following formula:

[TABLE]

where

[TABLE]

One can check that, after cancellations and simplifications, this equation is the same as the equation given in the statement of this Theorem. ∎

Remark 9.11**.**

We leave the problem of computing general formulas for the remaining structure coefficients of 9.2 open. In Section 10, we compute all structure coefficients for the groups $I_{2}(5)$, $I_{2}(7)$, $H_{3}$, and $H_{4}$.

10. Applications

In this section, we specialize the structure coefficients derived in Section 9 to certain formal group laws. We also compute all structure coefficients for the dihedral groups $I_{2}(5)$ and $I_{2}(7)$, and the reflection groups $H_{3}$ and $H_{4}$.

We will use the following Lemma in 10.2.

Lemma 10.1**.**

Suppose $m\geq 3$ is odd. Let $i=1,\dotsc,m-1$. If $i$ is odd, we have $s_{\alpha,\beta,\ldots}^{(i)}(\beta)=s_{\beta,\alpha,\ldots}^{(m-i-1)}(\alpha)$, and if $i$ is even, we have $s_{\alpha,\beta,\ldots}^{(i)}(\alpha)=s_{\beta,\alpha,\ldots}^{(m-i-1)}(\beta).$

Proof.

It follows from 2.8 that $s_{\alpha,\beta,\ldots}^{(m)}(\beta)=-\alpha$ when $m$ is odd. Applying compositions of reflections $s_{\alpha}$ and $s_{\beta}$ to both sides of this equation to reduce the length of $s_{\alpha,\beta,\ldots}^{(m)}$ gives the desired equations. ∎

Example 10.2**.**

Suppose that $m\geq 3$ is odd. Let $(R,F)$ be any formal group law such that, for all $\alpha\in\Sigma$, the formula $x_{\alpha}+x_{-\alpha}-qx_{\alpha}x_{-\alpha}=0$ holds, where $q\in R$ is fixed. This includes the multiplicative formal group law over $R$, and any formal group law of additive type. Observe that, in $\mathcal{Q}_{\Lambda}^{(R,F)}$, the condition $x_{\alpha}+x_{-\alpha}-qx_{\alpha}x_{-\alpha}=0$ for all $\alpha\in\Sigma$ is equivalent to the condition $y_{-\alpha}=q-y_{\alpha}$ for all $\alpha\in\Sigma$. We show that under this condition, the structure coefficients of 9.10 satisfy

[TABLE]

(a) Using the formula of $\kappa_{\alpha,\beta}^{(m-2)}$ obtained in 9.10 and making the substitution $y_{-\alpha}=(q-y_{\alpha})$, we can write

[TABLE]

[TABLE]

By 10.1, we have the following relations:

[TABLE]

Using these relations and comparing $\kappa_{\beta,\alpha}^{(m-2)}$ and $\kappa_{\alpha,\beta}^{(m-2)}$, it is straightforward to verify that $\kappa_{\alpha,\beta}^{(m-2)}=\kappa_{\beta,\alpha}^{(m-2)}$.

(b) Using the formula for $\kappa_{\alpha,\beta}^{(m-3)}$ obtained in 9.10 and making the substitution $y_{-\alpha}=(q-y_{\alpha})$, we can write

[TABLE]

By 10.1, we have the following relations:

[TABLE]

These relations, together with the relations Eqs. 9, 10, 11 and 12, allow us to deduce that the terms of $\kappa_{\alpha,\beta}^{(m-3)}$ cancel in pairs. We conclude that $\kappa_{\alpha,\beta}^{(m-3)}=\kappa_{\beta,\alpha}^{(m-3)}=0$.

10.3 follows from 9.2 and 9.10:

Example 10.3**.**

1. (a)

Suppose $m=2$. In this case,

[TABLE] 2. (b)

Suppose $m=3$. In this case,

[TABLE]

where $\kappa_{\beta,\alpha}^{(1)}=y_{\alpha}y_{\beta}+s_{\beta}(y_{\alpha}y_{\beta})-y_{\alpha}s_{\beta}(y_{\alpha})$.

Example 10.4**.**

Suppose $m=5$. We provide explicit formulas for $\kappa_{\beta,\alpha}^{(1)},\kappa_{\alpha,\beta}^{(2)}$, and $\kappa_{\beta,\alpha}^{(3)}$:

[TABLE]

We use the results proven in Section 9 to justify these formulas.

The formulas for $\kappa_{\beta,\alpha}^{(3)}$ and $\kappa_{\alpha,\beta}^{(2)}$ are obtained from 9.10. Now we will find $\kappa_{\beta,\alpha}^{(1)}$. In 11.1, we provide formulas of products of up to seven formal Demazure elements. We use these formulas, together with the formulas for the coefficients $\kappa_{\beta,\alpha}^{(3)}$ and $\kappa_{\alpha,\beta}^{(2)}$, to determine $\kappa_{\beta,\alpha}^{(1)}$. We do this by subtracting $\kappa_{\beta,\alpha}^{(3)}X_{\beta}X_{\alpha}X_{\beta}-\kappa_{\alpha,\beta}^{(2)}X_{\alpha}X_{\beta}+y_{\Sigma^{+}_{\alpha,\beta}}(\mathbf{1}-\delta_{\beta})$ from $X_{\beta\ldots}^{(5)}$, and then using 9.4.

The coefficient of $(\mathbf{1}-\delta_{\beta})$ in the expansion of the product $X_{\beta,\alpha,\ldots}^{(5)}$ is

[TABLE]

The coefficient of $(\mathbf{1}-\delta_{\beta})$ in the expansion of the product $\kappa_{\beta,\alpha}^{(3)}X_{\beta}X_{\alpha}X_{\beta}$ is

[TABLE]

The coefficient of $(\mathbf{1}-\delta_{\beta})$ in the expansion of the product $-\kappa_{\alpha,\beta}^{(2)}X_{\alpha}X_{\beta}$ is

[TABLE]

After cancellations, the coefficient of $(\mathbf{1}-\delta_{\beta})$ in the term $X_{\beta\ldots}^{(5)}-\kappa_{\beta,\alpha}^{(3)}X_{\beta}X_{\alpha}X_{\beta}+\kappa_{\alpha,\beta}^{(2)}X_{\alpha}X_{\beta}$ is

[TABLE]

The coefficient of $(\mathbf{1}-\delta_{\beta})$ in $X_{\beta}$ is $y_{\beta}$. Therefore, by 9.4, the coefficient at $X_{\beta}$ in the braid relation is $\kappa_{\beta}^{1}=\tfrac{C-y_{\Sigma^{+}_{\alpha,\beta}}}{y_{\beta}}$. Finally, by 10.1, we have that $y_{\alpha}=s_{\beta,\alpha,\ldots}^{4}(y_{\beta})$, and that $y_{\Sigma^{+}_{\alpha,\beta}}=y_{\beta}s_{\beta}(y_{\alpha})s_{\beta}s_{\alpha}(y_{\beta})s_{\beta}s_{\alpha}s_{\beta}(y_{\alpha})s_{\beta,\alpha,\ldots}^{(4)}(y_{\beta})$.

Example 10.5**.**

Suppose $m=7$. We provide explicit formuals for $\kappa_{\beta,\alpha}^{(1)}$, $\kappa_{\alpha,\beta}^{(2)}$, $\kappa_{\beta,\alpha}^{(3)}$, $\kappa_{\alpha,\beta}^{(4)}$, and $\kappa_{\beta,\alpha}^{(5)}$:

[TABLE]

These formulas are obtained using the method of 10.4.

Remark 10.6**.**

Let $(R,F)$ be a formal group law such that, for all $\alpha\in\Sigma$, $x_{\alpha}+x_{-\alpha}-qx_{\alpha}x_{-\alpha}=0$, where $q\in R$ is fixed. By direct computation and using the method of 10.2, we compute $\kappa_{\alpha,\beta}^{(1)}=\kappa_{\beta,\alpha}^{(1)}$ when $m=5$. We also compute $\kappa_{\alpha,\beta}^{(3)}=\kappa_{\beta,\alpha}^{(3)}$, $\kappa_{\alpha,\beta}^{(2)}=\kappa_{\beta,\alpha}^{(2)}=0$, and $\kappa_{\alpha,\beta}^{(1)}=\kappa_{\beta,\alpha}^{(1)}$ when $m=7$.

Remark 10.7**.**

Suppose $m\geq 3$ is odd. Let $(R,F)$ be a formal group law such that, for all $\alpha\in\Sigma$, $x_{\alpha}+x_{-\alpha}-qx_{\alpha}x_{-\alpha}=0$, where $q\in R$ is fixed. In light of 10.2 and 10.6, we conjecture that $\kappa_{\alpha,\beta}^{(i)}=\kappa_{\beta,\alpha}^{(i)}$ when $i$ is odd, and $\kappa_{\alpha,\beta}^{(i)}=\kappa_{\beta,\alpha}^{(i)}=0$ when $i$ is even.

Remark 10.8**.**

Let $(\Sigma,\Delta)$ be the geometric realization of $W=I_{2}(3)$ given in 2.16, and let $(R,F_{a})$ be the additive formal group law over $R$. In this case, $\Delta=\{\alpha,\beta\}$. Using the formulas for the roots given in 2.16 and the formula for $\kappa_{\beta,\alpha}^{(1)}$ given in 10.3 (b), we compute directly that $\kappa_{\beta,\alpha}^{(1)}=0$. Thus, we have given a combinatorial proof of 8.6 for this geometric realization of $I_{2}(3)$.

Remark 10.9**.**

Let $(\Sigma,\Delta)$ be the geometric realization of $W=I_{2}(5)$ given in 2.18, and let $(R,F_{a})$ be the additive formal group law over $R$. In this case, $\Delta=\{\alpha,\beta\}$. We already know that $\kappa_{\alpha,\beta}^{(2)}=0$ by 10.2. Moreover, using the formulas for the roots given in 2.18, the formulas for the structure coefficients given in 10.4, and the relation $\tau^{2}=\tau+1$, we compute directly that $\kappa_{\beta,\alpha}^{(1)}=\kappa_{\beta,\alpha}^{(3)}=0$ as well. Thus, we have given a combinatorial proof of 8.6 for this geometric realization of $I_{2}(5)$.

Remark 10.10**.**

Let $(\Sigma,\Delta)$ be the geometric realization of $W=I_{2}(7)$ given in 2.19, and let $(R,F_{a})$ be the additive formal group law over $R$. In this case, $\Delta=\{\alpha,\beta\}$. We already know that $\kappa_{\alpha,\beta}^{(4)}=0$ by 10.2, and that $\kappa_{\alpha,\beta}^{(2)}=0$ by 10.6. Moreover, using the formulas for the roots given in 2.19, the formulas for the structure coefficients given in 10.5, and the relation $\zeta^{3}=\zeta^{2}+2\zeta-1$, we compute directly that $\kappa_{\beta,\alpha}^{(1)}=\kappa_{\beta,\alpha}^{(3)}=\kappa_{\beta,\alpha}^{(5)}=0$ as well. Thus, we have given a combinatorial proof of 8.6 for this geometric realization of $I_{2}(7)$.

We will now consider the reflection groups $H_{3}$ and $H_{4}$.

Example 10.11**.**

Let $(\Sigma,\Delta)$ be the geometric realization of $W=H_{3}$ given in Section 2. This geometric realization has simple roots $\Delta=\{\alpha_{1},\alpha_{2},\alpha_{3}\}$, where

[TABLE]

are the orders of $s_{i}s_{j}$ in $W$. The coefficient $\kappa_{\alpha_{2},\alpha_{3}}^{(1)}$ was computed in 10.3 (b). In addition, $X_{1}X_{3}=X_{3}X_{1}$ by 10.3 (a). Moreover, the coefficients $\kappa_{\alpha_{1},\alpha_{2}}^{(1)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(2)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(3)}$ were computed in 10.4.

When $(R,F_{a})$ is the additive formal group law over $R$, one can verify directly, as was done in 10.8, that the coefficient $\kappa_{\alpha_{2},\alpha_{3}}^{(1)}=0$. One can also verify directly, as was done in 10.9, that the coefficients $\kappa_{\alpha_{1},\alpha_{2}}^{(1)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(2)}$, and $\kappa_{\alpha_{1},\alpha_{2}}^{(3)}$ all equal [math] in this case.

Example 10.12**.**

Let $(\Sigma,\Delta)$ be the geometric realization of $W=H_{4}$ given in Section 2. This geometric realization has simple roots $\Delta=\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}$, where

[TABLE]

are the orders of $s_{i}s_{j}$ in $W$. The coefficients $\kappa_{\alpha_{2},\alpha_{3}}^{(1)}$ and $\kappa_{\alpha_{3},\alpha_{4}}^{(1)}$ were computed in 10.3 (b). In addition, $X_{1}X_{3}=X_{3}X_{1}$, $X_{1}X_{4}=X_{4}X_{1}$, and $X_{2}X_{4}=X_{4}X_{2}$ by 10.3 (a). Moreover, the coefficients $\kappa_{\alpha_{1},\alpha_{2}}^{(1)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(2)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(3)}$ were computed in 10.4.

When $(R,F_{a})$ is the additive formal group law over $R$, one can verify directly, as was done in 10.8, that the coefficients $\kappa_{\alpha_{2},\alpha_{3}}^{(1)}$ and $\kappa_{\alpha_{3},\alpha_{4}}^{(1)}$ equal [math]. One can also verify directly, as was done in 10.9, that the coefficients $\kappa_{\alpha_{1},\alpha_{2}}^{(1)}$, $\kappa_{\alpha_{1},\alpha_{2}}^{(2)}$, and $\kappa_{\alpha_{1},\alpha_{2}}^{(3)}$ all equal [math] in this case.

11. Appendix

In the present section, we provide computations for products of up to seven formal Demazure elements for all root systems.

Calculation 11.1**.**

Let $W$ be a real finite reflection group, let $\Sigma$ be a root system of $W$, and let $\alpha,\beta\in\Sigma$. Below are explicit formulas for the products of $X_{\alpha}$ and $X_{\beta}$ up to seven elements. The formulas are written so that the coefficient to the left of a $\delta$-term in the expansion of the product appears after the colon.

[TABLE]

[TABLE]