Double loop algebras and elliptic root systems

Abstract

In this note, we describe an elliptic root system and elliptic Weyl group, due to Saito (Publ RIMS Kyoto Univ 21:75–179, 1985), from view point of double loop algebra and its group. A natural action of the double loop group will be introduced on a trivial $\mathbb {C}^*$-bundle over the space of $\overline{\partial }$-connections on a $C^\infty$-trivial principal bundle over an elliptic curve that would be constructed from 2-dimensional central extension of a double loop algebra. The invariant theory of the elliptic Weyl group will be also discussed.

1 Introduction

Saito [14] arrived at the notions of elliptic root system and elliptic Weyl group in the course of his study on simply elliptic singularities [12]. An elliptic root system is a root system defined on a real vector space with positive semi-definite bilinear form whose radical is of dimension 2, extending finite and affine root system. The existence of imaginary roots which generate 2-dimensional radical indicates its relation to an elliptic curve.

It is natural to consider the construction of a Lie algebra with given elliptic root system as the next step, and there were several attempts to this problem by Wakimoto [19], Slodowy [18], Yamada (one of the authors of this article) [21] and Saito with Yoshii [16] etc. For recent developments, see e.g., [9]. Among these constructions, one has the ‘maximal’ one that is the universal central extension of a 2-toroidal Lie algebra $\mathfrak {g}\otimes \mathbb {C}[ s^{\pm 1}, t^{\pm 1}]$ for some simple finite-dimensional Lie algebra $\mathfrak {g}$ (cf. [5]). Notice that the kernel of the universal central extension of such an algebra is of infinite dimension, whereas that of the Lie algebra $\mathfrak {g}\otimes \mathbb {C}[t^{\pm 1}]$ is 1-dimensional.

A relation between a simple singularity and the simple Lie algebra of the same type was clarified by the Grothendieck-Brieskorn-Slodowy theory [17]. This description in terms of Lie algebras was a key to construct the period map, in particular the primitive form [13], of the semi-universal deformation of a simple singularity [22]. Helmke and Slodowy [2] constructed the space of semi-universal deformation of an isolated hypersurface simply elliptic singularity in terms of a holomorphic affine Lie group, where the relation between holomorphic principal G-bundles, where G is the connected and simply connected simple Lie group over $\mathbb {C}$ whose Lie algebra is $\mathfrak {g}$, and the affine Lie group associated with G played an important role. Our project is to describe the semi-universal deformation of an isolated simply elliptic singularity in terms of Lie algebras that would be 2-toroidal Lie algebras, as one can see in [2]. This construction may allow us to describe the primitive form and hence the period map for an isolated simply elliptic singularity.

In this article, we explain the algebraic structure of 2-toroidal Lie algebras, i.e., in view of elliptic root systems etc. Next, we show that all of these concepts have some natural meaning in terms of the space $\mathcal {C}(\mathfrak {g})$ of $\overline{\partial }$-connections on a topologically trivial principal G-bundle over an elliptic curve. This note is organized as follows.

In Sect. 2, we explain the so-called elliptic root system, elliptic Weyl group $W_{ell}$ and their hyperbolic extensions in view of 2-toroidal Lie algebras. In Sect. 3, we show that the elliptic Weyl group can be regarded as the quotient of the normalizer of its Cartan subalgebra of the $C^\infty$-completion $\mathcal {E}(\mathfrak {g})$ of a 2-toroidal Lie algebras by its centralizer in the corresponding group $\mathcal {E}(G)$. An $\mathcal {E}(G)$-action on the space $\mathcal {C}(\mathfrak {g})$ will be studied and its relation to the invariant theory of $W_{ell}$ will be explained in Sect. 4. In Sect. 5, an action of $SL(2,\mathbb {Z})$ will be studied.

2 Elliptic root systems and 2-toroidal Lie algebras

In this section, we will recall some facts about what is called an elliptic root system, its Weyl groups, and their hyperbolic extension, introduced by Saito [14]. We also describe them in terms of 2-toroidal Lie algebras and their central extensions.

2.1 Elliptic root systems and their Weyl groups

In this subsection, we recall the notion of an elliptic root system and an elliptic Weyl group. Let us generalize the classical notion of root system [14]:

Definition 2.1

Let F be a finite-dimensional vector space over $\mathbb {R}$, $(\cdot , \cdot )$ be a symmetric bilinear form of signature $(l_+, l_0, l_-)$, i.e., $l_+,l_0$ and $l_-$ signify the number of positive, zero and negative eigenvalues, respectively. We call a non-empty subset $R \subset F$, the root system associated with $(\cdot , \cdot )$ if it satisfies the next five axioms:

• 1. Let Q(R) be the $\mathbb {Z}$-submodule of F generated by R. Then, $\dim (Q(R) \otimes \mathbb {R})=\dim F$, i.e., Q(R) is a full lattice in F.

• 2. For any $\alpha \in R$, $(\alpha , \alpha )\ne 0$.

• 3. For $\alpha \in R$, let $w_\alpha \in GL(F)$ be the reflection defined by

  $$\begin{aligned} w_\alpha (\lambda )=\lambda -\frac{2(\lambda , \alpha )}{(\alpha ,\alpha )}\alpha \quad \lambda \in F. \end{aligned}$$

  Then, $w_\alpha (R)=R$ for any $\alpha \in R$.

• 4. For any $\alpha , \beta \in R$, $\dfrac{2(\alpha ,\beta )}{(\alpha ,\alpha )} \in \mathbb {Z}$.

• 5. (Irreducibility) There is no non-empty subsets $R_1$ and $R_2$ such that $R=R_1 \cup R_2$ and $R_1 \perp R_2$.

As is well known, when the signature $(l_+, l_0, l_-)$ of R is either of the form (l, 0, 0) or (l, 1, 0), R is a finite root system or an affine root system, respectively. In case when the signature of R is of the form (l, 2, 0), R is called an elliptic root system. This is the root system introduced by K. Saito for his study on simply elliptic singularities.

Here and after, we will study the structure of an elliptic root system, say R. Set

$$\begin{aligned} \mathrm {Rad}\,(\cdot , \cdot )=\{\lambda \in F \vert (\lambda , \gamma )=0 \quad \forall \, \gamma \in F\}. \end{aligned}$$

It follows that the vector space $\mathrm {Rad}\, (\cdot , \cdot )$ is a 2-dimensional subspace of F defined over $\mathbb {Q}$, i.e., $\mathrm {Rad}\, (\cdot , \cdot ) \cap Q(R)$ is a full sublattice of Q(R). In this sense, fixing a 1-dimensional subspace E of $\mathrm {Rad}\, (\cdot , \cdot )$ defined over $\mathbb {Q}$ is called a marking for R, and the pair (R, E) is called a marked elliptic root system.

Remark 2.1

Two non-isomorphic marked elliptic root systems can be isomorphic as elliptic root systems. $G_2^{(1,3)}$ and $G_2^{(3,1)}$ are such examples. See [14] for detail.

For a marked elliptic root system (R, E), set

$$\begin{aligned}&F_f=F/\mathrm {Rad}\,(\cdot , \cdot ), \, \,\qquad \quad F_a=F/E, \\&R_f=R/R \cap \mathrm {Rad}\,(\cdot , \cdot ), \quad R_a=R/R \cap E. \end{aligned}$$

Denote the symmetric bilinear form on $F_f$ and $F_a$, induced from $(\cdot , \cdot )$, by $(\cdot , \cdot )_f$ and $(\cdot , \cdot )_a$. It is clear that $R_f$ and $R_a$ are, respectively, the finite and affine root systems associated with $(F_f, (\cdot , \cdot )_f)$ and $(F_a, (\cdot , \cdot )_a)$.

Remark 2.2

$R_a$ is the real root system of the corresponding affine Lie algebra.

In the sequel, we regard $F_f$ and $F_a$ as vector subspaces of F and $R_f \subset R_a \subset R$, for simplicity.

Let $\{\alpha _1,{\ldots }, \alpha _l\}$ and $\{\alpha _0,\alpha _1,{\ldots }, \alpha _l\}$ be root basis of $R_f$ and $R_a$, respectively, i.e.,

$$\begin{aligned} F_f:=\bigoplus _{i=1}^l \mathbb {R}\alpha _i \quad F_a:=\bigoplus _{i=0}^l \mathbb {R}\alpha _i, \end{aligned}$$

and each root is a $\mathbb {Z}$-linear combination of $\alpha _i$’s where all nonzero coefficients are either all positive or all negative. We express the fundamental imaginary root of the affine root system $R_a$ as

$$\begin{aligned} \delta _1=\sum _{i=0}^l a_i \alpha _i, \end{aligned}$$

that is, $a_i$ are coprime positive integers such that $\delta _1$ is an imaginary root of the corresponding affine Lie algebra. We let

$$\begin{aligned} A=\left( \frac{2(\alpha _i,\alpha _j)}{(\alpha _i,\alpha _i)}\right) _{0\le i,j\le l} \end{aligned}$$

be the generalized Cartan matrix of $R_a$.

Now, following Wakimoto [19], we study the structure of the marked elliptic root system (R, E) in view of the generalized Cartan matrix A. An $(l+1)$-tuple of positive integers $(k_0,k_1,{\ldots }, k_l)$ is called counting weight, if the diagonal matrix $K=\mathrm {diag}(k_0,k_1,{\ldots }, k_l)$ satisfies the next two conditions:

• 1. $KAK^{-1}$ is a generalized Cartan matrix (cf. [3]), and

• 2. $\mathrm {G.C.D.}(k_0,k_1,{\ldots }, k_l)=1$.

Example 2.1

• 1. If A is the generalized Cartan matrix of type $X_l^{(1)}$ with $X=A,D,E$, a counting weight is uniquely determined as

  $$\begin{aligned} (k_0,k_1,{\ldots }, k_l)=(1,1,{\ldots }, 1). \end{aligned}$$

• 2. Let

  $$\begin{aligned} A=\begin{pmatrix} 2 &{}\quad -1 &{}\quad 0 \\ -1 &{}\quad 2 &{}\quad -1 \\ 0 &{}\quad -3 &{}\quad 2 \end{pmatrix} \end{aligned}$$

  be the generalized Cartan matrix of type $G_2^{(1)}$. There are two possible counting weights: (1, 1, 1) and (1, 3, 1). Namely, in general, a counting weight is not uniquely determined from a generalized Cartan matrix of affine type.

Let $W_a$ be the affine Weyl group associated with the affine root system $R_a$. It is well known that $R_a$ decomposes into a finite number of $W_a$-orbits (cf. Remark 2.2) and that each $W_a$-orbit of $R_a$ contains a simple root.

Lemma 2.1

If two simple roots $\alpha _i$ and $\alpha _j$ lie in the same $W_a$-orbit, then we have $k_i=k_j$.

Proof

We recall that $\alpha _i$ and $\alpha _j$ lie in the same $W_a$-orbit if and only if the vertices corresponding to these simple roots are connected by a subdiagram consisting of simple edges of the Dynkin diagram of $R_a$. Hence, since $R_a$ is irreducible, it is sufficient to show this lemma in the case when the vertices corresponding to $\alpha _i$ and $\alpha _j$ are connected by one simple edge.

Set $A=(a_{i,j})$ and $B=(b_{i,j})=KAK^{-1}$. As $b_{i,j}=a_{i,j}k_ik_j^{-1}$ and $a_{i,j}=a_{j,i}=-1$ by assumption, this implies that

$$\begin{aligned} b_{i,j}b_{j,i}=(a_{i,j}k_ik_j^{-1})(a_{j,i}k_jk_i^{-1})=a_{i,j}a_{j,i}=1. \end{aligned}$$

from which it follows that $b_{i,j}=b_{j,i}=-1$, since B is a generalized Cartan matrix. Thus, we conclude that $k_i=k_j$. $\square$

Now, for any $\alpha \in R_a$, there exists $w \in W_a$ and a simple root $\alpha _i$ such that $\alpha =w(\alpha _i)$. The above lemma assures that we can define the counting weight of the root $\alpha$ by

$$\begin{aligned} k(\alpha )=k_i. \end{aligned}$$

Let $\delta _2$ be a $\mathbb {Z}$-basis of the lattice $Q(R) \cap E$. The number $k(\alpha )$ is the smallest positive integer such that $\alpha +k(\alpha )\delta _2 \in R$. Indeed, we have

Proposition 2.2

(cf. [14, 19]) $R=\{\alpha +mk(\alpha )\delta _2 \vert \, \alpha \in R_a,\, m \in \mathbb {Z}\}.$

Remark 2.3

Let us state the relation between a counting weight and the elliptic Dynkin diagram due to Saito [14]. Let $a^\vee =(a_0^\vee , a_1^\vee , {\ldots }, a_l^\vee ) \in (\mathbb {Z}_{>0})^{l+1}$ be the vector satisfying

• 1. $a^\vee A=0$, and

• 2. $\mathrm {G.C.D.}(a_0^\vee , a_1^\vee , {\ldots }, a_l^\vee )=1$.

Set

$$\begin{aligned} I=\left\{ j \in \{0,1,{\ldots }, l\} \left| \frac{a_j^\vee }{k_j}=\max \left\{ \frac{a_0^\vee }{k_0}, \frac{a_1^\vee }{k_1}, {\ldots }, \frac{a_l^\vee }{k_l} \right\} \right\} . \right. \end{aligned}$$

For $j \in I$, we set $\alpha _j^*=\alpha _j+\delta _2$ and consider the set of vertices parametrized by

$$\begin{aligned} \{\alpha _0,\alpha _1,{\ldots }, \alpha _l\} \cup \{\alpha _j^*\vert j \in I\}. \end{aligned}$$

The elliptic Dynkin diagram for the root system R is the graph with the vertices given by the above set where the vertices $\alpha$ and $\beta$ are connected following the usual rules for Dynkin diagram depending on the values $\dfrac{2(\alpha ,\beta )}{(\alpha ,\alpha )}$ and $\dfrac{2(\beta ,\alpha )}{(\beta ,\beta )}$. When these values are 2, we connect these vertices with two dashed edges.

In the rest of this article, we are only interested in the elliptic root systems of type $X_l^{(1,1)}$ ($X=A,B,C,D,E,F,G$) in which case one always has $(k_0,k_1,{\ldots }, k_l)=(1,1,{\ldots }, 1)$. Hence, the above proposition in this case implies

Corollary 2.3

The root system of type $X_l^{(1,1)}$ is given by

$$\begin{aligned} R=\{\alpha _f+m\delta _1+n\delta _2 \vert \,\alpha _f \in R_f, \, m,n \in \mathbb {Z}\}. \end{aligned}$$

Now, we discuss on the structure of the Weyl group associated with the elliptic root system of type $X_l^{(1,1)}$. Recall that, for any $\alpha \in R$, the reflection $w_\alpha$ with respect to $\alpha$ is, by definition, given by

$$\begin{aligned} w_\alpha (\lambda )=\lambda -\frac{2(\lambda ,\alpha )}{(\alpha ,\alpha )}\alpha \quad \forall \, \lambda \in F. \end{aligned}$$

The subgroup of GL(F) generated by $\{w_\alpha \}_{\alpha \in R}$ is called elliptic Weyl group and will be denoted by $W_{ell}$.

Recall that

$$\begin{aligned} F=F_f \oplus \mathbb {R}\delta _1 \oplus \mathbb {R}\delta _2, \quad \mathrm {Rad}\,(\cdot , \cdot )=\mathbb {R}\delta _1\oplus \mathbb {R}\delta _2. \end{aligned}$$

We define the subspace $F_f^*$ of $F^*$ by

$$\begin{aligned} F_f^*=\{h \in F^*\vert {h(\delta _1)}={h(\delta _2)}=0\}, \end{aligned}$$

where F is identified with the dual of $F^*$. Let $d_1,d_2$ be the elements of $F^*$ satisfying

$$\begin{aligned} d_i\vert _{F_f}=0, \quad d_i(\delta _j)=\delta _{i,j}\quad (i,j \in \{1,2\}). \end{aligned}$$

It is clear that

$$\begin{aligned} F^*=F_f^*\oplus \mathbb {R}d_1 \oplus \mathbb {R}d_2. \end{aligned}$$

Let $\mu : F \rightarrow F^*$ be the linear map satisfying

• 1. for any $\lambda \in F_f$, $\mu (\lambda ) \in F_f^*$ such that

  $$\begin{aligned} \mu (\lambda )(\kappa )=(\lambda ,\kappa ) \quad \forall ~\lambda , \kappa \in F_f, \end{aligned}$$

• 2. $\mu \vert _{\mathbb {R}\delta _1 \oplus \mathbb {R}\delta _2}=0$.

It follows that the restriction of the linear map $\mu$ to $F_f$ is injective. Hence, for $\alpha _f \in R_f$, we set

$$\begin{aligned} \alpha _f^\vee =\frac{2}{(\alpha _f,\alpha _f)}\mu (\alpha _f) \in F^*. \end{aligned}$$

The elliptic group $W_{ell}$ naturally acts on the dual $F^*$, and its action is explicitly given by

$$\begin{aligned} w_\alpha (h)=h -h(\alpha )\alpha _f^\vee \quad \forall \, h \in F^*, \end{aligned}$$

for $\alpha =\alpha _f+m\delta _1+n\delta _2 \in R$ with $\alpha _f \in R_f$. For $\alpha _f \in R_f$, we set

$$\begin{aligned} t_{\alpha _f^\vee }^i=w_{\delta _i-\alpha _f} w_{\alpha _f} \quad (i=1,2). \end{aligned}$$

By direct computation, one can check that, for any $h +\omega _1 d_1+\omega _2 d_2 \in F^*$ with $h \in F_f^*$, one has

$$\begin{aligned} t_{\alpha _f^\vee }^i(h+\omega _1d_1+\omega _2d_2)=\,h+\omega _1d_1+\omega _2d_2+\omega _i\alpha _f^\vee \quad (i=1,2). \end{aligned}$$

Let $Q_f^\vee$ be the coroot lattice of $R_f$, i.e.,

$$\begin{aligned} Q_f^\vee =\bigoplus _{i=1}^l \mathbb {Z}\alpha _i^\vee . \end{aligned}$$

The above computation implies the next proposition:

Proposition 2.4

(cf. [15]) $W_{ell} \cong W_f \ltimes (Q_f^\vee \times Q_f^\vee )$, where $W_f$ is the Weyl group associated with the finite root system $R_f$.

Remark 2.4

The elliptic Weyl group $W_{ell}$ is generated by $\{ w_{\alpha _0}, w_{\alpha _1}, {\ldots }, w_{\alpha _l}, w_{\alpha _j^*}~(j \in I)\}$.

2.2 Hyperbolic extension

As in the previous subsection, let F be an $(l+2)$-dimensional $\mathbb {R}$-vector space, $(\cdot , \cdot )$ be a symmetric bilinear form with signature (l, 2, 0), $E \subset \mathrm {Rad}\,(\cdot ,\cdot )$ be a marking and (R, E) be a marked elliptic root system in F. As we have seen before, the radical $\mathrm {Rad}\, (\cdot , \cdot )$ signifies the existence of the translation in 2 directions. But, the space $(F_f^*)_\mathbb {C}{:=\mathbb {C}\otimes _\mathbb {R}F_f^*}$ is too small to consider the invariant functions with respect to the action of the elliptic Weyl group since the only invariant holomorphic function on $(F_f^*)_\mathbb {C}$ is a constant. Following Saito [14], the notion of the hyperbolic extension will be introduced depending upon the marking E.

Recall that we have chosen a basis $\{\delta _1,\delta _2\}$ of $\mathrm {Rad}\,(\cdot , \cdot )$ satisfying

$$\begin{aligned} \mathrm {Rad}\,(\cdot , \cdot ) \cap Q(R)=\mathbb {Z}\delta _1 \oplus \mathbb {Z}\delta _2, \quad E \cap Q(R)=\mathbb {Z}\delta _2. \end{aligned}$$

We define the $(l+3)$-dimensional $\mathbb {R}$-vector space $\widehat{F}$ and a symmetric bilinear form $(\cdot , \cdot )_E$ as follows:

$$\begin{aligned}&\widehat{F}=F \oplus \mathbb {R}\Lambda _1=F_f \oplus \mathbb {R}\delta _1 \oplus \mathbb {R}\delta _2 \oplus \mathbb {R}\Lambda _1=F_a \oplus \mathbb {R}\delta _2 \oplus \mathbb {R}\Lambda _1, \\&(\cdot , \cdot )_E \vert _{F \times F}=(\cdot , \cdot ), \quad (\delta _i,\Lambda _1)_E=\delta _{i,1}, \quad (F_f,\Lambda _1)_E=\{0\}, \quad {(\Lambda _1, \Lambda _1)_E=0}. \end{aligned}$$

We call $(\widehat{F},(\cdot ,\cdot )_E)$ a hyperbolic extension of $(F,(\cdot ,\cdot ))$. By definition, we have

$$\begin{aligned} \mathrm {Rad}\,(\cdot , \cdot )_E=E{=\mathbb {R}\delta _2}, \end{aligned}$$

and the restriction of $(\cdot ,\cdot )_E$ to the $(l+2)$-dimensional subspace

$$\begin{aligned} \widehat{F}_a:=F_a \oplus \mathbb {R}\Lambda _1 \end{aligned}$$

is non-degenerate.

For the elliptic root system R and $\alpha \in R$, define $\hat{w}_\alpha \in GL(\widehat{F})$ by

$$\begin{aligned} \hat{w}_\alpha (\lambda )=\lambda -\frac{2(\lambda ,\alpha )_E}{(\alpha ,\alpha )_E}\alpha \quad \forall \, \lambda \in \widehat{F} \end{aligned}$$

and consider the subgroup $\widehat{W}_{ell}$ of $GL(\widehat{F})$ generated by $\{ \hat{w}_\alpha \}_{\alpha \in R}$ called the hyperbolic extension of the elliptic Weyl group $W_{ell}$.

Let us study the structure of the group $\widehat{W}_{ell}$. Let $d_1,d_2,c_1$ be the elements of the dual $\widehat{F}^*$ satisfying

$$\begin{aligned}&d_i(\Lambda _1)=0, \quad d_i(\delta _j)=\delta _{i,j},\quad d_i(F_f)=\{0\}, \\&c_1(\Lambda _1)=1, \quad \,c_1(\delta _i)=0, \quad ~~c_1(F_f)=\{0\}. \end{aligned}$$

By definition, one sees that the symmetric bilinear form $(\cdot , \cdot )_E$ is non-degenerate on $\widehat{F}_a$. Hence, introducing two subspaces of $\widehat{F}^*$ by

$$\begin{aligned} \widehat{F}_f^*:=(\mathbb {R}\delta _1 \oplus \mathbb {R}\delta _2 \oplus \mathbb {R}\Lambda _1)^\perp , \quad \widehat{F}_a^*:=(\mathbb {R}\delta _2)^\perp , \end{aligned}$$

one has

$$\begin{aligned}&\widehat{F}_a^*=\widehat{F}_f ^*\oplus \mathbb {R}d_1 \oplus \mathbb {R}c_1, \\&\widehat{F}^*=\widehat{F}_f^*\oplus \mathbb {R}d_1 \oplus \mathbb {R}d_2 \oplus \mathbb {R}c_1=\widehat{F}_a^*\oplus \mathbb {R}d_2. \end{aligned}$$

Let $\hat{\mu }: \widehat{F} \rightarrow \widehat{F}^*$ be the linear map defined by

$$\begin{aligned} \hat{\mu }(\lambda )(\kappa )=(\lambda ,\kappa )_E \quad \lambda , \kappa \in \widehat{F}. \end{aligned}$$

Since $\mathrm {Rad}\, (\cdot , \cdot )_E=E=\mathbb {R}\delta _2$, it follows that $\hat{\mu }(\delta _2)=0$. Nevertheless, the restriction

$$\begin{aligned} \hat{\mu }\vert _{\widehat{F}_a}: \widehat{F}_a \longrightarrow \widehat{F}_a^*, \end{aligned}$$

is a linear isomorphism. Hence, we set $E^*=\mathbb {R}d_2$ and define the symmetric bilinear form $(\cdot , \cdot )_{E^*}$ on $\widehat{F}^*$ by

$$\begin{aligned} (\hat{\mu }(\lambda ), \hat{\mu }(\kappa ))_{E^*}:=(\lambda , \kappa )_E \quad (\lambda , \kappa \in \widehat{F}_a), \quad (d_2,\widehat{F}^*)_{E^*}:=\{0\}. \end{aligned}$$

By definition, one has

$$\begin{aligned} \mathrm {Rad}\, (\cdot , \cdot )_{E^*}=E^*=\mathbb {R}d_2. \end{aligned}$$

For $\alpha \in R \subset \widehat{F}$, set

$$\begin{aligned} \alpha ^\vee =\frac{2}{(\alpha ,\alpha )_E}\hat{\mu }(\alpha ). \end{aligned}$$

By definition, one has

$$\begin{aligned} \alpha ^\vee (\lambda )=\frac{2(\alpha ,\lambda )_E}{(\alpha ,\alpha )_E}. \end{aligned}$$

Th equalities $\hat{\mu }(\delta _1)=c_1$ and $\hat{\mu }(\delta _2)=0$ imply that, for $\alpha =\alpha _f+m\delta _1+n\delta _2 \in R$ with $\alpha _f \in R_f$, one has

$$\begin{aligned} \alpha ^\vee =\alpha _f^\vee +\frac{2m}{(\alpha ,\alpha )_E}c_1= \alpha _f^\vee +m\frac{(\alpha _f^\vee ,\alpha _f^\vee )_{E^*}}{2}c_1. \end{aligned}$$

Hence, the natural action of the hyperbolic extension $\widehat{W}_{ell}$ on $\widehat{F}^*$ is given by

$$\begin{aligned} \hat{w}_\alpha (h)=h-h(\alpha )\alpha ^\vee \quad h \in \widehat{F}^*\end{aligned}$$

for any $\alpha \in R$. By direct computation, one obtains

Lemma 2.5

For $\alpha =\alpha _f+m\delta _1+n\delta _2 \in R$ with $\alpha _f \in R_f$ and $h=h_f+\omega _1d_1+\omega _2d_2+uc_1 \in \widehat{F}^*$ with $h_f \in F_f^*$, one has

$$\begin{aligned} \hat{w}_\alpha (h)&=h-(h_f(\alpha _f)+m\omega _1+n\omega _2)\alpha _f^\vee \\&\quad \ -m\left( (h_f,\alpha _f^\vee )_{E^*}+(m\omega _1+n\omega _2)\cdot \frac{(\alpha _f^\vee , \alpha _f^\vee )_{E^*}}{2}\right) c_1. \end{aligned}$$

For $\alpha _f \in R_f$, we set

$$\begin{aligned} \hat{t}_{\alpha _f^\vee }^i:=\hat{w}_{\delta _i-\alpha _f}\hat{w}_{\alpha _f} \quad (i=1,2). \end{aligned}$$

By Lemma 2.5, one has

Corollary 2.6

For $\alpha _f, \beta _f \in R_f$ and $h=h_f+\omega _1d_1+\omega _2d_2+uc_1 \in \widehat{F}^*$ with $h_f \in F_f^*$, one has

• 1. $\hat{t}_{\alpha _f^\vee }^1(h)=h+\omega _1\alpha _f^\vee -\left( (h_f,\alpha _f^\vee )_{E^*}+\omega _1 \cdot \dfrac{(\alpha _f^\vee ,\alpha _f^\vee )_{E^*}}{2}\right) c_1$,

  $\hat{t}_{\alpha _f^\vee }^2(h)=h+\omega _2 \alpha _f^\vee$,

• 2. $\hat{t}_{\alpha _f^\vee }^1\hat{t}_{\beta _f^\vee }^2(\hat{t}_{\beta _f^\vee }^2\hat{t}_{\alpha _f^\vee }^1)^{-1}(h)=h-\omega _2(\alpha _f^\vee , \beta _f^\vee )_{E^*}c_1$.

Remark 2.5

For any $\alpha _f^\vee \in Q_f^\vee$, set

$$\begin{aligned}&\hat{t}_{\alpha _f^\vee }^1(h)=h+\omega _1\alpha _f^\vee -\left( (h_f,\alpha _f^\vee )_{E^*}+\omega _1 \cdot \frac{(\alpha _f^\vee ,\alpha _f^\vee )_{E^*}}{2}\right) c_1, \\&\hat{t}_{\alpha _f^\vee }^2(h)=h+\omega _2 \alpha _f^\vee . \end{aligned}$$

It can be verified that for any $\beta _1^\vee , \beta _2^\vee \in Q_f^\vee$, one has

$$\begin{aligned} \hat{t}_{\beta _1^\vee }^i \hat{t}_{\beta _2^\vee }^i=\hat{t}_{\beta _1^\vee +\beta _2^\vee }^i \quad (i=1,2). \end{aligned}$$

Thus, setting

$$\begin{aligned} H(Q_f^\vee )=\langle \hat{t}_{\alpha _f^\vee }^1, \hat{t}_{\beta _f^\vee }^2 \vert \, \alpha _f^\vee , \beta _f^\vee \in Q_f^\vee \rangle , \end{aligned}$$

Lemma 2.6 and Remark 2.5 imply that this group is a discrete Heisenberg group. Moreover, the next isomorphism is well known:

Proposition 2.7

([14, 21]) $\widehat{W}_{ell} \cong W_f \ltimes H(Q_f^\vee )$.

Notice that Moody and Shi [7] obtained similar results for n-toroidal Lie algebras.

2.3 2-Toroidal Lie algebras

In this subsection, we describe elliptic root systems and their Weyl groups in view of 2-toroidal Lie algebras $\mathfrak {g}_{\mathrm {tor}}=\mathfrak {g}\otimes \mathbb {C}[s^{\pm 1}, t^{\pm 1}]$ where $\mathfrak {g}$ is a simple finite-dimensional Lie algebra over $\mathbb {C}$. Recall that a 2-toroidal Lie algebra is the Lie algebra whose bracket structure is given as follows: for $X, Y \in \mathfrak {g}$,

$$\begin{aligned}{}[X \otimes s^mt^n, Y \otimes s^kt^l]:=[X,Y] \otimes s^{m+k}t^{n+l}, \end{aligned}$$

where the bracket in the right-hand side is the Lie bracket in $\mathfrak {g}$. Here and after, we do not distinguish them.

Two derivations $d_s=s\dfrac{\partial }{\partial s}$ and $d_t=t\dfrac{\partial }{\partial t}$ on $\mathbb {C}[s^{\pm 1}, t^{\pm 1}]$ form a commutative Lie algebra $\mathfrak {d}=\mathbb {C}d_s \oplus \mathbb {C}d_t$ which also acts naturally on $\mathfrak {g}_{\mathrm {tor}}$;    for $A \otimes s^mt^n \in \mathfrak {g}_{\mathrm {tor}}$,

$$\begin{aligned}{}[d_s, A \otimes s^mt^n]:=m A \otimes s^mt^n, \quad [d_t, A \otimes s^mt^n]:=nA \otimes s^mt^n. \end{aligned}$$

Set $\mathfrak {g}_\mathrm {tor}^\mathfrak {d}=\mathfrak {d}\ltimes \mathfrak {g}_\mathrm {tor}$. With the aide of a non-degenerate symmetric invariant bilinear form $(\cdot , \cdot )$ on $\mathfrak {g}$, we define the symmetric bilinear form $(\,\cdot \,\vert \, \cdot \,)$ on $\mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}$ as follows:   for $X\otimes s^mt^n, Y \otimes s^kt^l \in \mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}$,

$$\begin{aligned}&(X \otimes s^mt^n \vert Y \otimes s^kt^l)=(X,Y)\delta _{m+k,0}\delta _{n+l,0}, \\&(d_\sharp \vert X \otimes s^mt^n)=(d_s \vert d_t)=0 \quad (\sharp \in \{s,t\}). \end{aligned}$$

Unfortunately, this symmetric bilinear form is degenerate and is only $\mathfrak {g}_{\mathrm {tor}}$-invariant. Moreover, the restriction of $(\,\cdot \, \vert \,\cdot \,)$ to the commutative subalgebra $\mathfrak {h}^\mathfrak {d}:=\mathfrak {d}\oplus \mathfrak {h}$ satisfies

$$\begin{aligned} \mathrm {Rad}\, (\,\cdot \,\vert \,\cdot \,)\vert _{\mathfrak {h}^\mathfrak {d}\times \mathfrak {h}^\mathfrak {d}}=\mathfrak {d}, \end{aligned}$$

where $\mathfrak {h}$ is a Cartan subalgebra of $\mathfrak {g}$.

The Lie algebra $\mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}$ admits the simultaneous eigenspace decomposition with respect to $\mathfrak {h}^\mathfrak {d}$:

$$\begin{aligned} \mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}=\mathfrak {h}^\mathfrak {d}\oplus \left( \bigoplus _{\alpha \in (\mathfrak {h}^\mathfrak {d})^*}\mathfrak {g}_{\alpha }^\mathfrak {d}\right) , \quad \mathfrak {g}_{\alpha }^\mathfrak {d}=\{A \in \mathfrak {g}_{\mathrm {tor}} \vert [h,A]=\alpha (h)A \quad (h \in \mathfrak {h}^\mathfrak {d})\}. \end{aligned}$$

The set $\Delta _{ell}:=\{\alpha \in (\mathfrak {h}^\mathfrak {d})^*\vert \mathfrak {g}_{\alpha }^\mathfrak {d}\ne \{0\} \}$ is called a double affine root system. Let us identify $\mathfrak {h}^*$ with a subspace of $(\mathfrak {h}^\mathfrak {d})^*$ as follows:   for $\alpha \in \mathfrak {h}^*$, we set $\alpha (d_\sharp )=0~(\sharp \in \{s,t\})$. Let $\delta _s, \delta _t$ be the elements of $(\mathfrak {h}^\mathfrak {d})^*$ satisfying

$$\begin{aligned} \delta _\sharp \vert _{\mathfrak {h}}=0, \quad \delta _\sharp (d_\flat )=\delta _{\sharp , \flat } \quad (\sharp , \flat \in \{s,t\}). \end{aligned}$$

We have the next decomposition

$$\begin{aligned} (\mathfrak {h}^\mathfrak {d})^*=\mathfrak {h}^*\oplus \mathbb {C}\delta _s \oplus \mathbb {C}\delta _t. \end{aligned}$$

We extend the symmetric bilinear form $(\,\cdot \,\vert \,\cdot \,)$ on $\mathfrak {h}^*$ to $(\mathfrak {h}^\mathfrak {d})^*$ by

$$\begin{aligned} (\delta _\sharp \vert (\mathfrak {h}^\mathfrak {d})^*)=\{0\}\; \hbox {for}\; \sharp \in \{s,t\}. \end{aligned}$$

The next proposition is clear:

Proposition 2.8

$$\begin{aligned} \Delta _{ell}=\{ \alpha _f+m\delta _s+n\delta _t \vert \alpha _f \in \Delta _f \cup \{0\},~m,n \in \mathbb {Z}\} \setminus \{0\}, \end{aligned}$$

where $\Delta _f$ is the root system of $\mathfrak {g}$ with respect to $\mathfrak {h}$.

Set

$$\begin{aligned} \Delta _{ell}^{re}=\{\alpha \in \Delta _{ell} \vert (\alpha \vert \alpha )\ne 0\}, \quad \Delta _{ell}^{im}=\{\alpha \in \Delta _{ell} \vert (\alpha \vert \alpha )=0\}. \end{aligned}$$

The set $\Delta _{ell}^{re}$ (resp. $\Delta _{ell}^{im}$) is called real root system of $\Delta _{ell}$ (resp. imaginary root system of $\Delta _{ell}$). We have

$$\begin{aligned} \Delta _{ell}^{re}=\{\alpha _f +m\delta _s+n \delta _t \vert \alpha _f \in \Delta _f,~m,n \in \mathbb {Z}\}, \quad \Delta _{ell}^{im}=\{m\delta _s+n\delta _t \vert (m,n) \in \mathbb {Z}^2{\setminus } \{(0,0)\}\}, \end{aligned}$$

and

$$\begin{aligned} \mathfrak {g}_{\alpha _f+m\delta _s+n\delta _t}^\mathfrak {d}=\mathfrak {g}_{\alpha _f} \otimes \mathbb {C}s^mt^n, \quad \mathfrak {g}_{m\delta _s+n\delta _t}^\mathfrak {d}=\mathfrak {h}\otimes \mathbb {C}s^mt^n. \end{aligned}$$

Remark 2.6

For $\mathfrak {g}$ of type $X_l$, the set $\Delta _{ell}^{re}$ is the elliptic root system of type $X_l^{(1,1)}$ by Corollary 2.3.

Now, we consider the Weyl group of $\mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}$. Define the linear isomorphism $\nu _f: \mathfrak {h}\longrightarrow \mathfrak {h}^*$ by

$$\begin{aligned} \nu _f(h_f)(h_f')=(h_f\vert h_f') \quad h_f,h_f' \in \mathfrak {h}, \end{aligned}$$

and set

$$\begin{aligned} \alpha _f^\vee =\frac{2}{(\alpha _f\vert \alpha _f)}\nu _f^{-1}(\alpha _f) \quad (\alpha _f \in \Delta _f). \end{aligned}$$

Let $e_{\alpha _f}$ ($\alpha _f \in \Delta _f$) be a root vector of root $\alpha _f$ in $\mathfrak {g}$ normalized by the relations $[e_{\alpha _f}, e_{-\alpha _f}]=\alpha _f^\vee$ for any $\alpha _f$. For $\alpha =\alpha _f+m\delta _x+n\delta _y \in \Delta _{ell}^{re}$ with $\alpha _f \in \Delta _f$, we set $e_{\alpha }=e_{\alpha _f} \otimes s^mt^n$. The affine automorphism $s_{\alpha }$ of $\mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}$ defined by

$$\begin{aligned} s_{\alpha }=\exp (\mathrm {ad}(e_{\alpha })) \exp (-\mathrm {ad}(e_{-\alpha }))\exp (\mathrm {ad}(e_{\alpha })). \end{aligned}$$

stabilizes $\mathfrak {h}^\mathfrak {d}$, i.e.,

$$\begin{aligned} s_{\alpha }(h)=h-\alpha (h)\alpha _f^\vee \quad h \in \mathfrak {h}^\mathfrak {d}. \end{aligned}$$

Notice that this is an isometry on $\mathfrak {h}$ but is only an affine transformation on $\mathfrak {h}^\mathfrak {d}$. We denote the restriction of $s_{\alpha }$ ($\alpha \in \Delta _{ell}^{re}$) to $\mathfrak {h}^\mathfrak {d}$ by $w_{\alpha }$. The double affine Weyl group $W_{daf}$ is, by definition, the subgroup of the group of affine transformations on $\mathfrak {h}^\mathfrak {d}$ generated by $w_{\alpha }$ ($\alpha \in \Delta _{ell}^{re}$).

For any $\alpha _f \in \Delta _f$, we set

$$\begin{aligned} t_{\alpha _f^\vee }^\sharp =w_{\delta _\sharp -\alpha _f}\cdot w_{\alpha _f} \quad (\sharp \in \{s,t\}). \end{aligned}$$

It can be checked that, for any $h=h_f+\omega _sd_s+\omega _td_t\in \mathfrak {h}^\mathfrak {d}$ with $h_f \in \mathfrak {h}$, we have

$$\begin{aligned} t_{\alpha _f^\vee }^\sharp (h)=h+\omega _\sharp \alpha _f^\vee \quad (\sharp \in \{s,t\}). \end{aligned}$$

It follows that

Proposition 2.9

One has the isomorphism

$$\begin{aligned} W_{daf}\cong W_f \ltimes (Q_f^\vee \times Q_f^\vee ). \end{aligned}$$

This proposition shows that $W_{daf} \cong W_{ell}$, that is, we obtained a description of the elliptic Weyl group $W_{ell}$ in terms of the 2-toroidal Lie algebra $\mathfrak {g}_\mathrm {tor}$.

Next, we consider the hyperbolic extension from view point of 2-toroidal Lie algebras. For this purpose, we consider a 2-dimensional central extension $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$ of $\mathfrak {g}_\mathrm {tor}^\mathfrak {d}$ to obtain a non-degenerate symmetric invariant bilinear form on it. Namely, it is the vector space

$$\begin{aligned} \widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}:=\mathfrak {g}_{\mathrm {tor}}^\mathfrak {d}\oplus \mathbb {C}c_s \oplus \mathbb {C}c_t, \end{aligned}$$

enjoying the next commutation relations

$$\begin{aligned}&[\mathfrak {g}_\mathrm {tor}^\mathfrak {d}, c_s]=[\mathfrak {g}_\mathrm {tor}^\mathfrak {d}, c_t]=\{0\}, \\&[X \otimes s^mt^n, Y \otimes s^kt^l]=[X,Y]\otimes s^{m+k}t^{n+l}+ (X,Y)\delta _{m+k,0}\delta _{n+l,0}(mc_s+nc_t), \\&[d_s, X \otimes s^mt^n]=mX \otimes s^mt^n, \quad [d_t, X \otimes s^mt^n]=nX \otimes s^mt^n, \\&[d_s, d_t]=0, \end{aligned}$$

where $X,Y \in \mathfrak {g}$ and $k,l,m,n \in \mathbb {Z}$.

The Lie algebra $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$ possesses a non-degenerate symmetric $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$-invariant bilinear form $\langle \, \cdot \, \vert \, \cdot \, \rangle$ whose non-trivial pairings are given by

$$\begin{aligned}&\langle X \otimes s^mt^n \vert Y \otimes s^kt^l \rangle =(X,Y)\delta _{m+k,0}\delta _{n+l,0}, \\&\langle \partial _\sharp \vert d\flat \rangle =\delta _{\sharp , \flat } \quad (\sharp , \flat \in \{s,t\}). \end{aligned}$$

We remark that the restriction of this symmetric $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$-invariant bilinear form to the $(\mathrm {rk}\mathfrak {g}+4)$-dimensional commutative subalgebra

$$\begin{aligned} \widetilde{\mathfrak {h}}:=\mathfrak {h}^\mathfrak {d}\oplus \mathbb {C}c_s \oplus \mathbb {C}c_t, \end{aligned}$$

is again non-degenerate.

We identify $(\mathfrak {h}^\mathfrak {d})^*$ with a subspace of $\widetilde{\mathfrak {h}}^*$ as follows: for $\alpha \in (\mathfrak {h}^\mathfrak {d})^*$, we set $\alpha (c_\sharp )=0~(\sharp \in \{s,t\})$. Since $c_s$ and $c_t$ are central in $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$, the root space decomposition of $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$ with respect to $\widetilde{\mathfrak {h}}$ looks as follows:

$$\begin{aligned} \widetilde{\mathfrak {g}}_{\mathrm {tor}}^\mathfrak {d}=\widetilde{\mathfrak {h}} \oplus \left( \bigoplus _{\alpha \in \Delta _{ell}} \widetilde{\mathfrak {g}}_{\alpha } \right) , \quad \widetilde{\mathfrak {g}}_{\alpha }=\mathfrak {g}_{\alpha }^\mathfrak {d}. \end{aligned}$$

Let $\Lambda _s, \Lambda _t$ be the elements of $\widetilde{\mathfrak {h}}^*$ satisfying

$$\begin{aligned} \Lambda _\sharp (\mathfrak {h}^\mathfrak {d})=0, \quad \Lambda _{\sharp }(c_\flat )=\delta _{\sharp , \flat } \quad (\sharp , \flat \in \{s,t\}). \end{aligned}$$

We have

$$\begin{aligned} \widetilde{\mathfrak {h}}^*= (\mathfrak {h}^\mathfrak {d})^*\oplus \mathbb {C}\Lambda _s \oplus \mathbb {C}\Lambda _t = \mathfrak {h}^*\oplus \mathbb {C}\delta _s \oplus \mathbb {C}\delta _t \oplus \mathbb {C}\Lambda _s \oplus \mathbb {C}\Lambda _t. \end{aligned}$$

Define the linear isomorphism $\nu :\widetilde{\mathfrak {h}} \longrightarrow \widetilde{\mathfrak {h}}^*$ by

$$\begin{aligned} \nu (\tilde{h})(\tilde{h'})=\langle \tilde{h} \vert \tilde{h'} \rangle \quad \tilde{h}, \tilde{h'} \in \widetilde{\mathfrak {h}}. \end{aligned}$$

Via this isomorphism, we define a non-degenerate symmetric bilinear form on $\widetilde{\mathfrak {h}}^*$ by

$$\begin{aligned} \langle \tilde{\lambda } \vert \tilde{\kappa } \rangle = \langle \nu ^{-1}(\tilde{\lambda }) \vert \nu ^{-1}(\tilde{\kappa })\rangle \quad \tilde{\lambda }, \tilde{\kappa } \in \widetilde{\mathfrak {h}}^*. \end{aligned}$$

For $\alpha \in \Delta _{ell}^{re} \subset \widetilde{\mathfrak {h}}^*$, we set

$$\begin{aligned} \alpha ^\vee =\frac{2}{\langle \alpha \vert \alpha \rangle }\nu ^{-1}(\alpha ) \in \widetilde{\mathfrak {h}}. \end{aligned}$$

Notice that we have

$$\begin{aligned} \nu ^{-1}(\delta _\sharp )=c_\sharp , \quad \nu ^{-1}(\Lambda _\sharp )=d_\sharp \quad (\sharp \in \{s,t\}), \end{aligned}$$

and for $\alpha =\alpha _f+m\delta _x+n\delta _y \in \Delta _{ell}^{re}$ with $\alpha _f \in \Delta _f$, we also have

$$\begin{aligned} \alpha ^\vee =\alpha _f^\vee + \frac{2}{\langle \alpha _f\vert \alpha _f \rangle }(mc_s+nc_t). \end{aligned}$$

In the same way as $s_{\alpha }$ ($\alpha \in \Delta _{ell}^{re}$), we define the automorphism $\tilde{s}_{\alpha }$ of $\widetilde{\mathfrak {g}}_{\mathrm {tor}}^\mathfrak {d}$ by

$$\begin{aligned} \tilde{s}_{\alpha }=\exp (\mathrm {ad}(e_{\alpha }))exp(-\mathrm {ad}(e_{-\alpha }))\exp (\mathrm {ad}(e_{\alpha })). \end{aligned}$$

By direct computation, one can check that

$$\begin{aligned} \tilde{s}_{\alpha }(\tilde{h})=\tilde{h}-\alpha (\tilde{h})\alpha ^\vee , \end{aligned}$$

for any $\tilde{h} \in \widetilde{\mathfrak {h}}$, namely, $\tilde{s}_{\alpha }$ stabilizes $\widetilde{\mathfrak {h}}$. We denote the restriction of $\tilde{s}_{\alpha }$ to $\widetilde{\mathfrak {h}}$ by $\tilde{w}_{\alpha }$ and define $\widetilde{W}_{daf}$ as the subgroup of $O(\widetilde{\mathfrak {h}}, \langle \cdot \vert \cdot \rangle )$ generated by $\tilde{w}_{\alpha }$ ($\alpha \in \Delta _{ell}^{re}$). The group $\widetilde{W}_{daf}$ is too big for our purpose, and we need to reduce the space $\widetilde{\mathfrak {h}}$ to a smaller space which we explain below.

Set

$$\begin{aligned} \mathcal {H}^{\pm }=\left\{ (\omega _s,\omega _t) \in \mathbb {C}\times \mathbb {C}^*\; \left| \; \pm \mathrm {Im}\left( \frac{\omega _s}{\omega _t}\right) >0 \; \right\} , \right. \end{aligned}$$

and

$$\begin{aligned} \widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }}=\{ \tilde{h}=h_f+\omega _sd_s+\omega _td_t+u_sc_s+u_tc_t \in \widetilde{\mathfrak {h}}\; \vert \; \,h_f \in \mathfrak {h},~(\omega _s,\omega _t)\in \mathcal {H}^{\pm } \; \}. \end{aligned}$$

Notice that we treat $d_s$ and $d_t$ unequally. For an elliptic root system R, the choice of a marking E causes unequal treatment, and it is natural to choose $\mathbb {C}d_t$ as $E^*$. Let

$$\begin{aligned} X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }})=\{\widetilde{h} \in \widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }}\; \vert \; \langle \widetilde{h} \vert \widetilde{h} \rangle =0 \; \} \end{aligned}$$

be a complex submanifold of $\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }}$. Since the symmetric bilinear form $\langle \, \cdot \, \vert \, \cdot \, \rangle$ is $\widetilde{W}_{daf}$-invariant, $\widetilde{W}_{daf}$ acts on $X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }})$. By definition, this action commutes with the natural $\mathbb {C}^*$-action. Hence, $\widetilde{W}_{daf}$ acts on the projectified space

$$\begin{aligned} \mathbb {P}(X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }})):=X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }})/{\mathbb {C}^*}. \end{aligned}$$

Before studying the $\widetilde{W}_{daf}$-action on $\mathbb {P}(X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }}))$, let us describe this latter space explicitly.

We remark that $\widetilde{h}=h_f+\omega _s d_s + \omega _t d_t+u_sc_s+u_tc_t \in X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{\pm }})$ with $h_f \in \mathfrak {h}$ implies the next equalities:

$$\begin{aligned} \langle h_f\vert h_f\rangle +2(\omega _su_s+\omega _tu_t)=0 \quad \text {i.e.,} \quad u_t=-\frac{1}{\omega _t}\left( \omega _su_s+\frac{1}{2}\langle h_f\vert h_f\rangle \right) . \end{aligned}$$

Hence, setting $\widehat{\mathfrak {h}}_\mathbb {H}=\mathbb {H}\times \mathfrak {h}\times \mathbb {C}$, where $\mathbb {H}=\{\tau \in \mathbb {C}\vert \mathrm {Im}\,\tau >0\}$, we have

Proposition 2.10

The holomorphic mapping

$$\begin{aligned} \varphi :\mathbb {P}(X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{-}})) \longrightarrow \widehat{\mathfrak {h}}_\mathbb {H}\end{aligned}$$

defined by

$$\begin{aligned}{}[h_f+\omega _s d_s + \omega _t d_t+u_sc_s+u_tc_t] \longmapsto \left( -\frac{\omega _s}{\omega _t},-\frac{h_f}{\omega _t},-\frac{u_s}{\omega _t} \right) \end{aligned}$$

is an isomorphism of complex manifolds.

Set $\tau =-\dfrac{\omega _s}{\omega _t} \in \mathbb {H}$. Let us compute the action of $\widetilde{W}_{daf}$ on $\widehat{\mathfrak {h}}_\mathbb {H}\cong \mathbb {P}(X(\widetilde{\mathfrak {h}}_{\mathcal {H}^{-}}))$.

For $\alpha \in \Delta _{ell}^{re}$, we denote by $\hat{w}_{\alpha }$ the element of $\mathrm {Aut}(\widehat{\mathfrak {h}}_\mathbb {H})$ induced from $\tilde{w}_{\alpha } \in \widetilde{W}_{daf}$.

Lemma 2.11

For $(\tau , h_f, u) \in \widehat{\mathfrak {h}}_{\mathbb {H}}$ and $\alpha =\alpha _f+m\delta _s+n\delta _t \in \Delta _{ell}^{re}$ with $\alpha _f \in \Delta _f$, one has

$$\begin{aligned}&\hat{w}_{\alpha }(\tau ,h_f,u) \\&\quad = \left( \tau , h_f-(\alpha _f(h_f)+m\tau -n)\alpha _f^\vee , u-m\left( \langle \alpha _f^\vee \vert h_f \rangle +(m\tau -n)\frac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) \right) . \end{aligned}$$

For $\alpha _f, \beta _f \in \Delta _f$, we set

$$\begin{aligned} \hat{t}_{\alpha _f^\vee }^s:=\hat{w}_{\delta _s-\alpha _f}\cdot \hat{w}_{\alpha _f}, \quad \hat{t}_{\beta _f^\vee }^t:=\hat{w}_{\delta _t+\beta _f}\cdot \hat{w}_{\beta _f}. \end{aligned}$$

Notice that the difference of the sign in two formulas occurs because of the definition $\tau =-\dfrac{\omega _s}{\omega _t}$. By Lemma 2.11, we have

Corollary 2.12

For any $(\tau ,h_f,u) \in \widehat{\mathfrak {h}}_\mathbb {H}$ and $\alpha _f, \beta _f \in \Delta _f$, we have

• 1. $\hat{t}_{\alpha _f^\vee }^s(\tau ,h_f,u)= \left( \tau , h_f+\tau \alpha _f^\vee ,u-\left( \langle \alpha _f^\vee \vert h_f \rangle +\tau \cdot \dfrac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) \right)$.

• 2. $\hat{t}_{\beta _f^\vee }^t(\tau ,h_f,u)=(\tau ,h_f+\beta _f^\vee , u)$.

• 3. $\hat{t}_{\alpha _f^\vee }^s\hat{t}_{\beta _f^\vee }^t(\hat{t}_{\beta _f^\vee }^t\hat{t}_{\alpha _f^\vee }^s)^{-1}(\tau ,h_f,u)=(\tau ,h_f,u-\langle \alpha _f^\vee \vert \beta _f^\vee \rangle ).$

Remark 2.7

Motivated by Corollary 2.12 1. and 2., we introduce the automorphisms $\hat{t}_{\alpha _f^\vee }^s$ and $\hat{t}_{\alpha _f^\vee }^t$ of $\widehat{\mathfrak {h}}_\mathbb {H}$ for $\alpha _f^\vee \in Q_f^\vee$ by

$$\begin{aligned}&\hat{t}_{\alpha _f^\vee }^s(\tau ,h_f,u)= \left( \tau , h_f+\tau \alpha _f^\vee ,u-\left( \langle \alpha _f^\vee \vert h_f \rangle +\tau \cdot \dfrac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) \right) , \\&\hat{t}_{\alpha _f^\vee }^t(\tau ,h_f,u)=(\tau ,h_f+\alpha _f^\vee , u). \end{aligned}$$

It can be checked that, for any $\beta _1^\vee , \beta _2^\vee \in Q_f^\vee$, one has

$$\begin{aligned} \hat{t}_{\beta _1^\vee }^s \hat{t}_{\beta _2^\vee }^s=\hat{t}_{\beta _1^\vee +\beta _2^\vee }^s, \quad \hat{t}_{\beta _1^\vee }^t \hat{t}_{\beta _2^\vee }^t=\hat{t}_{\beta _1^\vee +\beta _2^\vee }^t. \end{aligned}$$

Set

$$\begin{aligned} \widehat{W}_{daf}=\langle \hat{w}_\alpha \vert \alpha \in \Delta _{ell}^{re}\rangle . \end{aligned}$$

By Corollary 2.12 and Remark 2.7, the group generated by $\hat{t}_{\alpha _f^\vee }^s$ and $\hat{t}_{\beta _f^\vee }^t$ $(\alpha _f^\vee , \beta _f^\vee \in Q_f^\vee )$ is a discrete Heisenberg group (cf. Corollary 2.12 (3)) isomorphic to $H(Q_f^\vee )$. Indeed, we have

Proposition 2.13

$\widehat{W}_{daf} \cong W_f \ltimes H(Q_f^\vee ).$ In particular, we have $\widehat{W}_{daf} \cong \widehat{W}_{ell}$.

Thus, we conclude that the group $\widehat{W}_{daf}$ realizes the hyperbolic extension of $W_{ell}$. We call the space $\widehat{\mathfrak {h}}_\mathbb {H}\cong \mathbb {P}(X(\widetilde{\mathfrak {h}}_{\mathcal {H}}))$ the hyperbolic extension of $\mathbb {H}\times \mathfrak {h}$. Let us end up this subsection with a remark which would be related to invariant theory of the Weyl group $W_{ell}$ that will be discussed in Sect. 4.3:

Remark 2.8

By definition, we have

$$\begin{aligned} \langle \alpha _f^\vee \vert \beta _f^\vee \rangle =(\alpha _f^\vee , \beta _f^\vee ) \in \mathbb {Z}\end{aligned}$$

for any $\alpha _f^\vee , \beta _f^\vee \in Q_f^\vee$. This means that the $\widehat{W}_{ell}$-action on $\widehat{\mathfrak {h}}_\mathbb {H}$ induces a $W_{ell}$-action on $\mathbb {H}\times \mathfrak {h}\times \mathbb {C}^*$ via the exponential map $\mathbb {C}\longrightarrow \mathbb {C}^*; z \mapsto \exp (2\pi \sqrt{-1}z)$.

3 Double loop groups and elliptic Weyl groups

In this section, we will show that the elliptic Weyl group, recalled in Sect. 2.1 and related to 2-toroidal Lie algebras in Sect. 2.3, can be obtained naturally from double loop groups associated with a connected and simply connected simple Lie group G over $\mathbb {C}$. Here, a double loop group signifies the Fréchet Lie group

$$\begin{aligned} \mathcal {E}(G):=C^\infty (S^1 \times S^1,G) \end{aligned}$$

with its Lie algebra

$$\begin{aligned} \mathcal {E}(\mathfrak {g}):=C^\infty (S^1 \times S^1,\mathfrak {g}) \end{aligned}$$

called a double loop algebra, where $\mathfrak {g}$ signifies the Lie algebra of G. As usual, their structures are defined by pointwise operations. In Sect. 2.3, we used the terminology ‘toroidal’, but in this section, as we work on $C^\infty$-class, we use ‘double loop’ to distinguish with the former. We realize $S^1$ as $\sqrt{-1}\mathbb {R}/2\pi \sqrt{-1}\mathbb {R}$ and set

$$\begin{aligned} \partial _x=\frac{\partial }{\partial x}=s \frac{\partial }{\partial s}=d_s, \quad \partial _y=\frac{\partial }{\partial y}=t \frac{\partial }{\partial t}=d_t \quad (s=e^{x},\, t=e^{y}). \end{aligned}$$

As in Sect. 2.3, $\mathfrak {d}:=\mathbb {C}\partial _x \oplus \mathbb {C}\partial _y$ acts naturally on $\mathcal {E}(\mathfrak {g})$:

$$\begin{aligned}{}[\partial _\sharp , X \otimes f]:=X \otimes \partial _\sharp f \quad X \in \mathfrak {g}, f \in C^\infty (\mathbb {T}, \mathbb {C}), \sharp \in \{x,y\}. \end{aligned}$$

Via this action, we introduce a Lie algebra structure on $\mathcal {E}(\mathfrak {g})^\mathfrak {d}:=\mathfrak {d}\ltimes \mathcal {E}(\mathfrak {g})$.

Remark 3.1

Notice that the group $\mathcal {E}(G)$ is not only a regular F Lie group in the sense of [10] but also locally exponential in the sense of [8], where the exponential map $\exp : \mathcal {E}(\mathfrak {g}) \longrightarrow \mathcal {E}(G)$ is a local diffeomorphism between neighborhoods of $0 \in \mathcal {E}(\mathfrak {g})$ and of $e \in \mathcal {E}(G)$.

We define a left action of $\mathcal {E}(G)$ on $\mathcal {E}(\mathfrak {g})^\mathfrak {d}$ as follows:

$$\begin{aligned} \begin{aligned}&L: \mathcal {E}(G) \times \mathcal {E}(\mathfrak {g})^\mathfrak {d}\longrightarrow \mathcal {E}(\mathfrak {g})^\mathfrak {d}; \\&(g,(\xi , A)) ~\longmapsto ~ (\xi , \mathrm {Ad}(g)A-dg\cdot g^{-1}(\xi )). \end{aligned} \end{aligned}$$

(1)

We sometimes denote $L(g,(\xi , A))$ by $L_g(\xi ,A)$.

We choose a commutative $\mathrm {ad}$-diagonalizable subalgebra of $\mathcal {E}(\mathfrak {g})^\mathfrak {d}$ by

$$\begin{aligned} \mathfrak {h}^\mathfrak {d}=\mathfrak {d}\oplus \mathfrak {h}, \end{aligned}$$

and set

$$\begin{aligned}&N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})=\{g \in \mathcal {E}(G)\vert L_g(\mathfrak {h}^\mathfrak {d})=\mathfrak {h}^\mathfrak {d}\}, \\&Z_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})=\{g \in \mathcal {E}(G)\vert L_g(\xi ,h)=(\xi ,h)~\text {for}~(\xi ,h) \in \mathfrak {h}^\mathfrak {d}\}. \end{aligned}$$

Our purpose of this subsection is to show that the group

$$\begin{aligned} W_{ell}':=N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})/Z_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d}), \end{aligned}$$

is isomorphic to $W_{ell}$, i.e., we have the next theorem:

Theorem 3.1

$W_{ell}' \cong W_f \ltimes (Q_f^\vee \times Q_f^\vee ).$

Proof

For $g \in N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})$, $(\xi ,h) \in \mathfrak {h}^\mathfrak {d}$, we have

$$\begin{aligned} L_g(\xi , h)=(\xi , \mathrm {Ad}(g)(h)-dg\cdot g^{-1}(\xi )) \in \mathfrak {h}^\mathfrak {d}\end{aligned}$$

by definition. In particular, letting $h=0$, we obtain $t:=dg\cdot g^{-1}(\xi ) \in \mathfrak {h}$. $\square$

Now, suppose that $\xi =\omega _x\partial _x+\omega _y\partial _y\in \mathfrak {d}$ satisfies $-\dfrac{\omega _x}{\omega _y} \in \mathbb {H}$. For $(a,b) \in \mathbb {R}^2$, setting

$$\begin{aligned} K_{a,b}(x,y)=g(x,y)^{-1}g(x+a,y+b) \in \mathcal {E}(G), \end{aligned}$$

we see that

$$\begin{aligned} \xi K_{a,b}(x,y)&= \xi (g(x,y)^{-1})g(x+a,y+b)+g(x,y)^{-1}\xi (g(x+a,y+b)) \\&= -g(x,y)^{-1}(\xi g(x,y)g(x,y)^{-1})g(x+a,y+b) \\&\ \quad + g(x,y)^{-1}(\xi g(x+a,y+b)g(x+a,y+b)^{-1})g(x+a,y+b)\\&= -g(x,y)^{-1}tg(x+a,y+b) +g(x,y)^{-1}tg(x+a,y+b) =0, \end{aligned}$$

that is, $K_{a,b}(x,y)$ is $\xi$-holomorphic. Since $\mathbb {T}=S^1 \times S^1$ is compact, $K_{a,b}$ has to be a constant, say $A(a,b) \in G$. We regard A as a $C^\infty$-function $\mathbb {T}\longrightarrow G$. By definition, one has $A(a,b)=g(0,0)^{-1}g(a,b)$. Moreover, we have

Lemma 3.2

$$\begin{aligned} A(a,b) \in Z_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})=H, \end{aligned}$$

where H signifies the Cartan subgroup of G whose Lie algebra is $\mathfrak {h}$.

Proof

Since $g \in N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})$ implies $dg\cdot g^{-1}(\xi ) \in \mathfrak {h}$ as we have already shown, we have $\mathrm {Ad}(g)\mathfrak {h}\subset \mathfrak {h}$ which implies $A(a,b) \in N_G(\mathfrak {h})$ for each (a, b). By the connectivity of $\mathbb {T}$, it follows that $\mathrm {Im} A \subset N_G(\mathfrak {h})$ is connected since A is continuous. Now, $A(0,0)=e \in G$ implies $\mathrm {Im}\, A \subset H$ since G is simply-connected by assumption. $\square$

Next, we show

Lemma 3.3

$$\begin{aligned} A \in \mathrm {Hom}_{\mathrm {Grp}}(\mathbb {T}, H). \end{aligned}$$

Proof

By direct computation, one has

$$\begin{aligned} A(a+a',b+b')&= g(0,0)^{-1}g(a+a',b+b') \\&= g(0,0)^{-1}g(a,b) \cdot g(a,b)^{-1}g(a+a',b+b') \\&= A(a,b)K_{a',b'}(a,b)=A(a,b)A(a',b'). \end{aligned}$$

$\square$

We remark that $\mathrm {Hom}_{\mathrm {Grp}}(\mathbb {T}, H) \cong Q_f^\vee \times Q_f^\vee$. By definition and Lemma 3.3, we have

$$\begin{aligned} g(a,b)=g(0,0)A(a,b)\in N_G(\mathfrak {h}) \ltimes (Q_f^\vee \times Q_f^\vee ), \end{aligned}$$

which implies the existence of the surjection $N_G(\mathfrak {h}) \ltimes (Q_f^\vee \times Q_f^\vee ) \twoheadrightarrow N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})$. In particular, we obtain $Z_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})=Z_G(\mathfrak {h})=H$. As $W_f=N_G(\mathfrak {h})/Z_G(\mathfrak {h})$, we see that there is a surjection $W_f \ltimes (Q_f^\vee \times Q_f^\vee ) \twoheadrightarrow N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})/Z_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})$.

Now, for $\alpha \in \Delta _{ell}^{re}$, we define the element $s_{\alpha }'$ of $\mathcal {E}(G)$ by

$$\begin{aligned} s_{\alpha }'=\exp (e_{\alpha })\exp (-e_{-\alpha })\exp (e_{\alpha }). \end{aligned}$$

We denote the subgroup of $\mathcal {E}(G)$ generated by $\{ s_{\alpha }'\vert \alpha \in \Delta _{ell}^{re}\}$ by $\mathcal {W}_{ell}$. By direct calculation, we have

$$\begin{aligned} L_{s_{\alpha }'}(\xi ,h)=(\xi ,h-\alpha (\xi ,h)\alpha _f^\vee ) \qquad (\xi ,h) \in \mathfrak {h}^\mathfrak {d}, \end{aligned}$$

i.e., $s_{\alpha }' \in N_{\mathcal {E}(G)}(\mathfrak {h}^\mathfrak {d})$. Set

$$\begin{aligned} w_{\alpha }'=\left. L_{s_{\alpha }'}\right| _{\mathfrak {h}^\mathfrak {d}}. \end{aligned}$$

It follows that

Lemma 3.4

The group generated by $\{w_{\alpha }' \vert \alpha \in \Delta _{ell}^{re}\}$ is isomorphic to $W_f \ltimes (Q_f^\vee \times Q_f^\vee )$.

Proof

First of all, it can be checked easily that $\langle w_\alpha ' \vert \alpha \in \Delta _f\rangle =W_f$. For $\alpha \in \Delta _f$, we set

$$\begin{aligned} (t_{\alpha ^\vee }^\sharp )'=w_{\delta _\sharp -\alpha }' \circ w_{\alpha }' \quad \sharp \in \{x,y\}. \end{aligned}$$

It can be verified by direct calculation that these elements act on $\mathfrak {h}^\mathfrak {d}$ and satisfy

$$\begin{aligned} (t_{\alpha ^\vee }^\sharp )'(\xi ,h)=(\xi , h+\omega _\sharp \alpha ^\vee ), \end{aligned}$$

where we set $\xi =\omega _x \partial _x+\omega _y\partial _y \in \mathfrak {d}$. Hence the result follows. $\square$

By this lemma, Theorem 3.1 follows.

Let $P_f^\vee$ be the co-weight lattice of $\Delta _f$, i.e., the dual lattice of the root lattice $Q_f$, and for $\Lambda ^\vee \in P_f^\vee$, we define two elements of $C^\infty (\mathbb {R}^2, G)$ by

$$\begin{aligned} \phi _{\Lambda ^\vee }^x:=\exp (-x\Lambda ^\vee ), \quad \phi _{\Lambda ^\vee }^y:=\exp (-y\Lambda ^\vee ). \end{aligned}$$

(The group $C^\infty (\mathbb {R}^2, G)$ acts on the Lie algebra $C^\infty (\mathbb {R}^2, \mathfrak {g})$ via the gauge transformation as in (1)). Notice that these elements preserve $\mathcal {E}(\mathfrak {g}) \subset C^\infty (\mathbb {R}^2, \mathfrak {g})$ as can be seen from the next formulae: for $\sharp \in \{x,y\}$,

$$\begin{aligned} \begin{aligned}&L_{\phi _{\Lambda ^\vee }^\sharp }(\xi , h)=(\xi , h+\omega _\sharp \Lambda ^\vee ), \\&L_{\phi _{\Lambda ^\vee }^\sharp }(\xi , e_\alpha )=(\xi , e^{-\alpha (\Lambda ^\vee )\sharp }e_\alpha +\omega _\sharp \Lambda ^\vee ). \end{aligned} \end{aligned}$$

(2)

It turns out that $\phi _{\Lambda ^\vee }^x, \phi _{\Lambda ^\vee }^y \in \mathcal {E}(G)$ if and only if $\Lambda ^\vee \in Q_f^\vee$. Indeed, it can be checked that $\phi _{\alpha ^\vee }^\sharp =s_{\delta _\sharp -\alpha }' s_{\alpha }'$ for $\alpha \in \Delta _f$ and $\sharp \in \{x,y\}$ (cf. see, e.g., Sect. 1). In particular, we have $\left. L_{\phi _{\alpha ^\vee }^\sharp }\right| _{\mathfrak {h}^\mathfrak {d}}=(t_{\alpha ^\vee }^\sharp )'$ for $\alpha ^\vee \in Q_f^\vee$. Here and after, for $\Lambda ^\vee \in P_f^\vee$ and $\sharp \in \{x,y\}$, we set

$$\begin{aligned} (t_{\Lambda ^\vee }^\sharp )'=\left. L_{\phi _{\Lambda ^\vee }^\sharp }\right| _{\mathfrak {h}^\mathfrak {d}}. \end{aligned}$$

We call the subgroup of $GL(\mathfrak {h}^\mathfrak {d})$ generated by $W_f$ and $\langle (t_{\Lambda ^\vee }^x)', (t_{\Lambda ^\vee }^y)' \vert \Lambda ^\vee \in P_f^\vee \rangle$ the extended elliptic Weyl group and denote it by $W_{ell}^e$.

Remark 3.2

(cf. [14]) Similarly, one can show that the group $W_{ell}^e \cong W_f \ltimes (P_f^\vee \times P_f^\vee )$ is isomorphic to $N_{\mathcal {E}(G^{ad})}(\mathfrak {h}^\mathfrak {d})/Z_{\mathcal {E}(G^{ad})}(\mathfrak {h}^\mathfrak {d})$, i.e., the Weyl group of $\mathcal {E}(G^{ad})$, where $G^{ad}$ is the Chevalley group of adjoint type associated with the Lie algebra $\mathfrak {g}$.

4 Central extensions of $\mathcal {E}(\mathfrak {g})$ and $\mathcal {E}(G)$

Here, we reconstruct the hyperbolic extension of the elliptic Weyl group recalled in Sect. 2.3, from viewpoint of a central extension of the double loop group $\mathcal {E}(G)$.

4.1 $\widetilde{\mathcal {E}}(G)$-Action on $\widetilde{\mathcal {E}}(\mathfrak {g})^{\mathfrak {d}}$

Let $\widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}$ be the Fréchet space

$$\begin{aligned} \widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}:=\mathcal {E}(\mathfrak {g})^\mathfrak {d}\oplus (\mathbb {C}dx \oplus \mathbb {C}dy) \end{aligned}$$

with the smooth Lie bracket satisfying

$$\begin{aligned}&[{\widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}}, \mathbb {C}dx \oplus \mathbb {C}dy]=\{ 0\}, \\&[X \otimes f, Y \otimes g]=[X,Y] \otimes fg +(X,Y)\left[ \left( \int _{\mathbb {T}}(\partial _xf)g\omega \right) dx+ \left( \int _{\mathbb {T}}(\partial _yf)g\omega \right) dy \right] , \\&[\partial _x, X \otimes f]=X \otimes \partial _xf, \quad [\partial _y, X \otimes f]=X \otimes \partial _yf,\\&[\partial _x, \partial _y]=0. \end{aligned}$$

Here, $X,Y \in \mathfrak {g}$, $f,g \in C^\infty (\mathbb {T},\mathbb {C})$ with $\mathbb {T}=\left( \sqrt{-1}\mathbb {R}/2\pi \sqrt{-1}\mathbb {Z}\right) ^{2}$ and $\omega =-\dfrac{1}{4\pi ^{2}}dx\wedge dy$ is a volume form on $\mathbb {T}$.

The Lie algebra $\widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}$ possesses the non-degenerate symmetric invariant bilinear form $\langle \cdot \vert \cdot \rangle$ whose non-trivial pairings are given by

$$\begin{aligned} \begin{aligned}&\langle A \vert B \rangle =\int _{\mathbb {T}}(A,B)\omega \quad (A,B \in \mathcal {E}(\mathfrak {g})), \\&\langle \partial _\sharp \vert d\flat \rangle = \delta _{\sharp , \flat } \quad (\sharp , \flat \in \{ x,y\}). \end{aligned} \end{aligned}$$

(3)

Notice that this bilinear form is the smooth extension of the bilinear form defined on $\widetilde{\mathfrak {g}}_\mathrm {tor}^\mathfrak {d}$.

Next, we introduce a central extension of $\mathcal {E}(G)$ by $\mathbb {C}$, denoted by $\widehat{\mathcal {E}}(G)$, as follows.

Let $\Theta$ be the two-cocycle defined by

$$\begin{aligned} \Theta (g_1,g_2)=\frac{1}{8\pi ^2}\int _{\mathbb {T}}(g_1^{-1}dg_1 \wedge dg_2 \cdot g_2^{-1}), \end{aligned}$$

(4)

for $g_1, g_2 \in \mathcal {E}(G)$. Here, the two-cocyle condition means, for $g_i \in \mathcal {E}(G)$ ($1\le i\le 3$),

$$\begin{aligned} \Theta (g_1,g_2)+\Theta (g_1g_2,g_3)=\Theta (g_1,g_2g_3)+\Theta (g_2,g_3). \end{aligned}$$

(5)

The Fréchet Lie group $\widehat{\mathcal {E}}(G)$ is the central extension of $\mathcal {E}(G)$ by $\mathbb {C}$ with the two-cocyle $\Theta$, i.e., for $(g_i,c_i) \in \widehat{\mathcal {E}}(G)~(i=1,2)$, we define their product by

$$\begin{aligned} (g_1, c_1) \cdot (g_2, c_2):=(g_1g_2, c_1+c_2+ \Theta (g_1,g_2)). \end{aligned}$$

(6)

The next lemma describes a central extension of (1):

Lemma 4.1

Let $\widehat{L}: \widehat{\mathcal {E}}(G) \times \widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}\longrightarrow \widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}$ be the map defined by

$$\begin{aligned}&\widehat{L}((g,c), (\xi ,A,\alpha ))=\widehat{L}_{(g,c)}(\xi ,A,\alpha ) \\&\quad := \left( \xi , \mathrm {Ad}(g)A-dg\cdot g^{-1}(\xi ), \alpha -c\cdot \xi \lrcorner ~(-4\pi ^2\omega ) + \langle A \vert g^{-1}dg \rangle - \frac{1}{2}\langle dg\cdot g^{-1}(\xi ) \vert dg\cdot g^{-1}\rangle \right) . \end{aligned}$$

Here, $\xi \in \mathfrak {d}, A \in \mathcal {E}(\mathfrak {g})$ and $\alpha \in \mathbb {C}dx\oplus \mathbb {C}dy$.

• 1. $\widehat{L}$ defines a left $\widehat{\mathcal {E}}(G)$-action on $\widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}$.

• 2. This left $\widehat{\mathcal {E}}(G)$-action keeps the bilinear form $\langle \cdot \vert \cdot \rangle$ invariant.

Remark 4.1

• 1. Since $\mathbb {T}$ has no boundary, we have

  $$\begin{aligned} \Theta (g_1,g_2)=0 \quad g_i=exp(A_i\otimes f_i)\in \mathcal {E}(G)~(i=1,2). \end{aligned}$$

• 2. If $\mathrm {Im}\,\dfrac{\omega _x}{\omega _y}< 0$, $\xi =\omega _x\partial _x+\omega _y\partial _y \in \mathfrak {d}$ defines a holomorphic structure on $\mathbb {T}$.

We set

$$\begin{aligned} X&=\{ \widetilde{A}=(\xi , A,\alpha )\in \widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}\; \vert \; \langle \widetilde{A}\vert \widetilde{A}\rangle =0 \; \}, \\ X_{\mathcal {H}^{\pm }}&=\{(\omega _x \partial _x + \omega _y \partial _y, A,\alpha )\in X \; \vert \; (\omega _x,\omega _y)\in \mathcal {H}^{\pm } \; \}, \end{aligned}$$

where $\mathcal {H}^{\pm }$ is defined in Sect. 2.1. By Lemma 4.1, we have

Corollary 4.2

The group $\widehat{\mathcal {E}}(G)$ acts on $X_{\mathcal {H}^{\pm }}$.

By definition, this action commutes with the natural $\mathbb {C}^*$-action. Hence, $\widehat{\mathcal {E}}(G)$ acts on the projectified space

$$\begin{aligned} \mathbb {P}(X_{\mathcal {H}^{\pm }}):=X_{\mathcal {H}^{\pm }}/{\mathbb {C}^*}. \end{aligned}$$

We remark that $\widetilde{A}=(\omega _x \partial _x + \omega _y \partial _y,A, u_xdx+u_ydy) \in X_{\mathcal {H}^{\pm }}$ implies the next equalities:

$$\begin{aligned} \langle A\vert A\rangle +2(\omega _xu_x+\omega _yu_y)=0 \quad \text {i.e.,} \quad u_y=-\frac{1}{\omega _y}\left( \omega _xu_x+\frac{1}{2}\langle A\vert A\rangle \right) . \end{aligned}$$

Proposition 4.3

The map

$$\begin{aligned} \psi ^{\pm }:\mathbb {P}(X_{\mathcal {H}^{\pm }}) \longrightarrow \mathbb {H}\times \mathcal {E}(\mathfrak {g}) \times \mathbb {C}\end{aligned}$$

defined by

$$\begin{aligned}{}[\omega _x \partial _x + \omega _y \partial _y,A, u_xdx+u_ydy] \longmapsto \left( {\pm }\frac{\omega _x}{\omega _y},{\pm }\frac{A}{\omega _y},-\frac{u_x}{\omega _y} \right) \end{aligned}$$

is an isomorphism of complex Fréchet manifolds.

We set $\tau =-\dfrac{\omega _x}{\omega _y} \in \mathbb {H}$ and $\overline{\partial }=\tau \partial _x-\partial _y$. We denote the elliptic curve $(\mathbb {T}, \overline{\partial })$ by $E_\tau$. Set

$$\begin{aligned} \mathcal {C}(\mathfrak {g})=\mathbb {H}\times \mathcal {E}(\mathfrak {g}), \quad \mathcal {C}(\mathfrak {g})_\tau =\{\tau \} \times \mathcal {E}(\mathfrak {g}). \end{aligned}$$

The latter space can be identified with the space of $\overline{\partial }$-connections on $C^\infty$-trivial principal G-bundles over $E_\tau$. By Remark 4.1.2, we consider the map $\psi ^-$ in the rest of this section.

Denoting the induced action of $\widehat{\mathcal {E}}(G)$ on $\mathcal {C}(\mathfrak {g}) \times \mathbb {C}$ via $\psi ^-$ by the same letter $\widehat{L}$, we obtain the next proposition:

Proposition 4.4

For $(g,c) \in \widehat{\mathcal {E}}(G)$ and $(\tau ,A,u) \in \mathcal {C}(\mathfrak {g})\times \mathbb {C}$, we have

$$\begin{aligned}&\widehat{L}_{(g,c)}(\tau ,A,u) \\&\quad = \left( \tau , \mathrm {Ad}(g)(A)-(\overline{\partial }g)g^{-1}, u-c+ \langle A \vert g^{-1}\partial _xg \rangle -\frac{1}{2}\langle (\overline{\partial }g)g^{-1} \vert (\partial _xg)g^{-1}\rangle \right) . \end{aligned}$$

Remark 4.2

The canonical projection $\mathcal {C}(\mathfrak {g})\times \mathbb {C}\twoheadrightarrow \mathcal {C}(\mathfrak {g})$ induces the left $\mathcal {E}(G)$-action on $\mathcal {C}(\mathfrak {g})$ which we denote by L.

Now, we reconstruct the hyperbolic extension of $W_{ell}$ in terms of the left $\widehat{\mathcal {E}}(\mathfrak {g})$-action on $\mathcal {C}(\mathfrak {g})\times \mathbb {C}$. For $\alpha \in \Delta _{ell}^{re}$, we define the element $\hat{s}_{\alpha }'$ of $\widehat{\mathcal {E}}(G)$ by

$$\begin{aligned} \hat{s}_{\alpha }'= (\exp (e_{\alpha }), 0)\cdot (\exp (-e_{-\alpha }),0)\cdot (\exp (e_{\alpha }),0). \end{aligned}$$

Lemma 4.5

For $\alpha \in \Delta _{ell}^{re}$, we have

• 1. $\Theta (\exp (e_{\alpha }), \exp (-e_{-\alpha }))= \Theta (\exp (e_{\alpha })\cdot \exp (-e_{-\alpha }),\exp (e_{\alpha }))=0$.

• 2. $\hat{s}_{\alpha }'=(s_{\alpha }',0)$.

Proof

• (1) For $\alpha =\alpha _f+m\delta _x+n\delta _y$ ($\alpha _f \in \Delta _f$),

  $$\begin{aligned} e_{\alpha }=e_{\alpha _f} \otimes s^mt^n, \quad e_{-\alpha }=e_{-{\alpha _f}} \otimes s^{-m}t^{-n}. \end{aligned}$$

  Hence, by Remark 4.1, we obtain the result.

• (2) This follows from (1) and the definition of $\hat{s}_{\alpha }'$.

$\square$

Set

$$\begin{aligned} \widehat{\mathfrak {h}}_\mathbb {H}'=\mathbb {H}\times \mathfrak {h}\times \mathbb {C}\subset \mathcal {C}(\mathfrak {g}) \times \mathbb {C}. \end{aligned}$$

By Proposition 4.4 and Lemma 4.5, we obtain

Corollary 4.6

For $(\tau ,h,u) \in \widehat{\mathfrak {h}}_\mathbb {H}'$ and $\alpha =\alpha _f+m\delta _x+n\delta _y \in \Delta _{ell}^{re}~(\alpha _f \in \Delta _f)$, we have

$$\begin{aligned}&\widehat{L}_{\hat{s}_{\alpha }'}(\tau ,h,u) \\&\quad = \left( \tau , h-(\alpha _f(h)+(m\tau -n))\alpha _f^\vee , u-m\left( \langle \alpha _f^\vee \vert h\rangle +(m\tau -n)\frac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) \right) . \end{aligned}$$

In particular, $\hat{s}_{\alpha }' \in \widehat{\mathcal {E}}(G)$ stabilizes $\widehat{\mathfrak {h}}_\mathbb {H}'$. Hence, we set

$$\begin{aligned} \hat{w}_{\alpha }'=\left. \hat{L}_{\hat{s}_{\alpha }'}\right| _{\widehat{\mathfrak {h}}_\mathbb {H}'}. \end{aligned}$$

For $\alpha _f, \beta _f \in \Delta _f$, we set

$$\begin{aligned} (\hat{t}_{\alpha _f^\vee }^x)'=\hat{w}_{\delta _x-\alpha _f}' \circ \hat{w}_{\alpha _f}', \quad (\hat{t}_{\beta _f^\vee }^y)'=\hat{w}_{\delta _y+\beta _f}'\circ \hat{w}_{\beta _f}'. \end{aligned}$$

Notice that we have

$$\begin{aligned} (\hat{t}_{\alpha _f^\vee }^x)'= \left. \hat{L}_{(s_{\delta _x-\alpha _f}'\cdot s_{\alpha _f}',0)}\right| _{\widehat{\mathfrak {h}}_\mathbb {H}'}, \quad (\hat{t}_{\beta _f^\vee }^y)'= \left. \hat{L}_{(s_{\delta _y+\beta _f}'\cdot s_{\beta _f}',0)}\right| _{\widehat{\mathfrak {h}}_\mathbb {H}'}. \end{aligned}$$

From Proposition 4.4 and Corollary 4.6, the next lemma follows:

Lemma 4.7

For any $(\tau ,h,u) \in \widehat{\mathfrak {h}}_\mathbb {H}'$ and any $\alpha _f, \beta _f \in \Delta _f$, one has

• 1. $(\hat{t}_{\alpha _f^\vee }^x)'(\tau ,h,u)= \left( \tau , h+\tau \alpha _f^\vee , u-\left( \langle \alpha _f^\vee \vert h \rangle +\tau \dfrac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) \right)$.

• 2. $(\hat{t}_{\beta _f^\vee }^y)'(\tau ,h,u)=(\tau ,h+\beta _f^\vee ,u)$.

• 3. $(\hat{t}_{\alpha _f^\vee }^x)'(\hat{t}_{\beta _f^\vee }^y)'(\tau ,h,u)=(\hat{t}_{\beta _f^\vee }^y)'(\hat{t}_{\alpha _f^\vee }^x)'(\tau ,h,u-\langle \alpha _f^\vee \vert \beta _f^\vee \rangle )$.

Let $\widehat{W}_{ell}'$ be the subgroup of $\mathrm {Aut}(\widehat{\mathfrak {h}}_\mathbb {H}')$ generated by $\hat{w}_{\alpha }'~(\alpha \in \Delta _{ell}^{re})$.

By Lemma 2.11, Corollary 2.12 and Lemma 4.7, $\widehat{W}_{ell}'$ is isomorphic to the hyperbolic extension $\widehat{W}_{ell}$ of $W_{ell}$. Namely, we obtain

Theorem 4.8

$\widehat{W}_{ell}' \cong W_f \ltimes H(Q_f^\vee ) \cong \widehat{W}_{ell}.$

4.2 $\mathcal {E}(G)$-Action on $\widetilde{\mathcal {C}}(\mathfrak {g})$

In the previous subsection, we have studied the $\widehat{\mathcal {E}}(G)$-action on $\mathcal {C}(\mathfrak {g}) \times \mathbb {C}$. Here, via the exponential map $\mathbb {C}\rightarrow \mathbb {C}^*; z \mapsto e^{2\pi \sqrt{-1}z}$, we will show that the group $\mathcal {E}(G)$ acts on $\widetilde{\mathcal {C}}(\mathfrak {g}):=\mathcal {C}(\mathfrak {g})\times \mathbb {C}^*$. In addition, we will study the $W_{ell}$-action on $\widetilde{\mathfrak {h}}_\mathbb {H}:=\mathbb {H}\times \mathfrak {h}\times \mathbb {C}^*\subset \widetilde{\mathcal {C}}(\mathfrak {g})$ which will play an important role in the invariant theory.

Let $\theta$ be the Maurer–Cartan form of G and define the 3-form $\sigma$ on G as follows:

$$\begin{aligned} \sigma =\frac{1}{24\pi ^2}(\theta \wedge d\theta ). \end{aligned}$$

(7)

Here and after, the invariant form $(\cdot , \cdot )$ is normalized so that the square length of the long root is 2. We normalize the exterior product as in Sect. 5.2.

The next lemma, called the Polyakov–Wiegmann identity, follows by direct computation:

Lemma 4.9

([11]) Let $f,g:D \times [0,1] \rightarrow G$ be a $C^\infty$-map, where $D=\{z \in \mathbb {C}\vert \, \vert z \vert \le 1\}$ is a closed disk. One has the next identity:

$$\begin{aligned} (f \cdot g)^*\sigma =f^*\sigma +g^*\sigma +\frac{1}{8\pi ^2}d(f^{-1}\cdot df \wedge dg \cdot g^{-1}). \end{aligned}$$

Let $S\mathbb {T}$ be the solid torus bounded by $\mathbb {T}$ and let $g \in \mathcal {E}(G)$. Regarding g as a loop on the loop group $L(G):=C^\infty (S^1,G)$, it follows that there exists $\overline{g} \in C^\infty (S\mathbb {T},G)$ whose restriction to $\mathbb {T}$ is g since $\pi _1(L(G)) \cong \pi _1(G) \times \pi _1(\Omega G) \cong 0$. Hence, we fix such element $\overline{g}$ and set

$$\begin{aligned} \lambda (g)=\int _{S\mathbb {T}}\overline{g}^*\sigma . \end{aligned}$$

(8)

It follows that $\lambda (g)~\mathrm {mod}~\mathbb {Z}$ does not depend on the choice of $\overline{g}$. In particular, the number $e^{2\pi \sqrt{-1}\lambda (g)}$ is well defined (see, e.g., [6] for detail). By Lemma 4.9, we obtain the next lemma:

Lemma 4.10

For any $g,g' \in \mathcal {E}(G)$, the next identity holds:

$$\begin{aligned} \lambda (g \cdot g') \equiv \lambda (g)+\lambda (g')+\Theta (g,g') \quad \mod \mathbb {Z}. \end{aligned}$$

We set

$$\begin{aligned} \widetilde{\mathcal {E}}(G)=\mathcal {E}(G) \times \mathbb {C}^*, \end{aligned}$$

and define the structure of the group as follows:

$$\begin{aligned} (g,u) \cdot (g',v)=(gg',uve^{-2\pi \sqrt{-1}\Theta (g,g')}). \end{aligned}$$

By Lemma 4.10, we have

Corollary 4.11

The central extension

$$\begin{aligned} 1 \longrightarrow \mathbb {C}^*\longrightarrow \widetilde{\mathcal {E}}(G) \longrightarrow \mathcal {E}(G) \longrightarrow 1, \end{aligned}$$

splits. Indeed, $\mathcal {E}(G) \hookrightarrow \widetilde{\mathcal {E}}(G);~g \mapsto (g,e^{-2\pi \sqrt{-1}\lambda (g)})$ is a section of this short exact sequence.

Thus, by setting

$$\begin{aligned} \widetilde{\mathcal {C}}(\mathfrak {g})=\mathcal {C}(\mathfrak {g}) \times \mathbb {C}^*, \quad \widetilde{\mathcal {C}}(\mathfrak {g})_\tau =\mathcal {C}(\mathfrak {g})_\tau \times \mathbb {C}^*, \end{aligned}$$

the $\widehat{\mathcal {E}}(G)$-action given in Proposition 4.4 lifts to an $\widetilde{\mathcal {E}}(G)$-action on $\widetilde{\mathcal {C}}(\mathfrak {g})$. In particular, by the splitting $\mathcal {E}(G) \hookrightarrow \widetilde{\mathcal {E}}(G)$ as above, it induces naturally the left action $\widetilde{L}$ of $\mathcal {E}(G)$ on $\widetilde{\mathcal {C}}(\mathfrak {g})$. This can be explicitly given as follows:

Proposition 4.12

For $g \in \mathcal {E}(G)$ and $(\tau , A,u) \in \widetilde{\mathcal {C}}(\mathfrak {g})$, we have

$$\begin{aligned}&\widetilde{L}_g(\tau ,A,u) \\&\quad = \left( \tau , \mathrm {Ad}(g)(A)-\overline{\partial }g\cdot g^{-1}, u\cdot e^{2\pi \sqrt{-1}\left\{ \langle A \vert g^{-1}\partial _xg \rangle -\frac{1}{2}\langle \overline{\partial }g\cdot g^{-1}\vert \partial _xg\cdot g^{-1}\rangle \right\} }\cdot e^{-2\pi \sqrt{-1}\lambda (g)}\right) . \end{aligned}$$

Remark 4.3

• 1. By Remark 4.2, the canonical projection $\widetilde{\mathcal {C}}(\mathfrak {g}) \twoheadrightarrow \mathcal {C}(\mathfrak {g})$ is $\mathcal {E}(G)$-equivariant.

• 2. For $\tau \in \mathbb {H}$, the embedding $\mathcal {C}(\mathfrak {g})_\tau \hookrightarrow \mathcal {E}(\mathfrak {g})^\mathfrak {d}; (\tau , A) \mapsto \overline{\partial }+A$ is $\mathcal {E}(G)$-equivariant.

Now, we study the action of $W_{ell}$ on

$$\begin{aligned} \widetilde{\mathfrak {h}}_\mathbb {H}:=\mathbb {H}\times \mathfrak {h}\times \mathbb {C}^*\subset \widetilde{\mathcal {C}}(\mathfrak {g}). \end{aligned}$$

For $\alpha \in \Delta _{ell}^{re}$, we define the element $\tilde{s}_{\alpha }$ of $\widetilde{\mathcal {E}}(G)$ by

$$\begin{aligned} \tilde{s}_{\alpha }&= (\exp (e_{\alpha }),e^{-2\pi \sqrt{-1}\lambda (\exp (e_{\alpha }))})\cdot (\exp (-e_{-\alpha }),e^{-2\pi \sqrt{-1}\lambda (\exp (-e_{-\alpha }))})\\&\quad \cdot (\exp (e_{\alpha }),e^{-2\pi \sqrt{-1}\lambda (\exp (e_{\alpha }))}). \end{aligned}$$

By Corollary , we have

$$\begin{aligned} \tilde{s}_{\alpha }=(s_{\alpha }',e^{-2\pi \sqrt{-1}\lambda (s_{\alpha }')}). \end{aligned}$$

Lemma 4.13

For any $\alpha \in \Delta _{ell}^{re}$, one has

$$\begin{aligned} \lambda (s_{\alpha }')=0. \end{aligned}$$

Proof

By Lemmas 4.5 and 4.10, we have

$$\begin{aligned} \lambda (s_{\alpha }')=\lambda (\exp (e_{\alpha }))+\lambda (\exp (-e_{-\alpha }))+\lambda (\exp (e_{\alpha })). \end{aligned}$$

Suppose that $\alpha =\alpha _f+\,m\delta _x+\,n\delta _y$ with $\alpha _f \in \Delta _f$. We set $g(x,y)=\exp (e_{\alpha })$ and $\overline{g}(x,y,r)=\exp (r^{\vert m+n\vert +2}e_{\alpha })$ ($0\le r\le 1$). Then $\overline{g}$ is a extension of g to a $C^\infty$-function on $S\mathbb {T}$. By the Maurer-Cartan equation

$$\begin{aligned} d\theta =-\frac{1}{2}[\theta , \theta ], \end{aligned}$$

we have $\overline{g}^*\sigma =0$. $\square$

By this lemma, we have

$$\begin{aligned} \tilde{s}_{\alpha }=(s_{\alpha }',1) \quad (\alpha \in \Delta _{ell}^{re}). \end{aligned}$$

Hence, by Proposition 4.12, the next lemma follows:

Lemma 4.14

For $\alpha =\alpha _f+m\delta _x+n\delta _y~(\alpha _f \in \Delta _f)$ and $(\tau ,h,u) \in \widetilde{\mathfrak {h}}_\mathbb {H}$, one has

$$\begin{aligned}&\widetilde{L}_{s_{\alpha }'}(\tau ,h,u) \\&\quad = \left( \tau ,h-(\alpha _f(h)+(m\tau -n))\alpha _f^\vee , u \cdot e^{-2\pi \sqrt{-1}m\left( \langle \alpha _f^\vee \vert h\rangle +(m\tau -n)\frac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) }\right) . \end{aligned}$$

This lemma implies that $s_{\alpha }'$ stabilizes $\widetilde{\mathfrak {h}}_\mathbb {H}$. We set

$$\begin{aligned} \tilde{w}_{\alpha }=\left. \widetilde{L}_{s_{\alpha }'}\right| _{\widetilde{\mathfrak {h}}_\mathbb {H}}, \end{aligned}$$

and

$$\begin{aligned} \tilde{t}_{\alpha _f^\vee }^x=\tilde{w}_{\delta _x-\alpha _f} \cdot \tilde{w}_{\alpha _f}, \quad \tilde{t}_{\beta _f^\vee }^y=\tilde{w}_{\delta _y+\beta _f}\cdot \tilde{w}_{\beta _f}, \end{aligned}$$

for $\alpha \in \Delta _{ell}^{re}$ and $\alpha _f, \beta _f \in \Delta _f$. We denote by $\widetilde{W}_{ell}$ the subgroup of $\mathrm {Aut}(\widetilde{\mathfrak {h}}_\mathbb {H})$ generated by $\tilde{w}_{\alpha }~(\alpha \in \Delta _{ell}^{re})$. Since $\langle \alpha _f^\vee \vert \beta _f^\vee \rangle =(\alpha _f^\vee , \beta _f^\vee )$ is an integer (cf. Remark 2.8), by Lemma 4.7, it follows that $\widetilde{W}_{ell}$ is isomorphic to $W_{ell}$. Namely, we obtain the next result:

Theorem 4.15

$\widetilde{W}_{ell} \cong W_f \ltimes (Q_f^\vee \times Q_f^\vee ) \cong W_{ell}$.

Remark 4.4

It can be shown that the $\mathcal {E}(G)$-orbit of $\{\tau \}\times \mathfrak {h}\times \mathbb {C}^*\subset \widetilde{\mathcal {C}}(\mathfrak {g})_\tau$ is dense and

$$\begin{aligned} \mathcal {O}_{\mathcal {E}(G)}(\tau ,h)\cap (\{\tau \}\times \mathfrak {h})=\mathcal {O}_{W_{ell}}(\tau ,h)\; \textit{for any} \;h \in \mathfrak {h}, \end{aligned}$$

where $\mathcal {O}_{\mathcal {G}}(X)$ $(\mathcal {G}=\mathcal {E}(G),W_{ell})$ signifies the $\mathcal {G}$-orbit of X. Thus, the latter can be seen as an analogue of the Chevalley restriction theorem. The proof of these statements require some arguments on principal G-bundles over $E_\tau$ and will be discussed in our future publication.

4.3 Invariant theory of $W_{ell}$

In this subsection, we discuss on the structure of $W_{ell}$-invariants holomorphic functions on $\widetilde{\mathfrak {h}}_\mathbb {H}=\mathbb {H}\times \mathfrak {h}\times \mathbb {C}^*$. For simplicity, we denote the ring of holomorphic functions on $\mathbb {H}$ and $\widetilde{\mathfrak {h}}_\mathbb {H}$ by $\mathcal {O}_\mathbb {H}$ and $\mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}$, respectively.

For $\alpha _f^\vee \in Q_f^\vee$, we define $\tilde{t}_{\alpha _f^\vee }^x, \tilde{t}_{\alpha _f^\vee }^y$, as in Remark 2.7, as follows:

$$\begin{aligned} \tilde{t}_{\alpha _f^\vee }^x(\tau ,h,u)&= \left( \tau , h+\tau \alpha _f^\vee , ue^{-2\pi \sqrt{-1}\left( \langle \alpha _f^\vee \vert h \rangle + \tau \frac{\langle \alpha _f^\vee \vert \alpha _f^\vee \rangle }{2}\right) }\right) , \\ \tilde{t}_{\alpha _f^\vee }^y(\tau ,h,u)&= (\tau , h+\alpha _f^\vee , u). \end{aligned}$$

It turns out that, as in Corollary 2.12, this is compatible with the action given by Lemma 4.14 for $\alpha _f \in \Delta _f$. Moreover, for $\alpha _f^\vee , \beta _f^\vee \in Q_f^\vee$, one has

$$\begin{aligned} \tilde{t}_{\alpha _f^\vee }^x\tilde{t}_{\beta _f^\vee }^x=\tilde{t}_{\alpha _f^\vee +\beta _f^\vee }^x, \quad \tilde{t}_{\alpha _f^\vee }^y\tilde{t}_{\beta _f^\vee }^y=\tilde{t}_{\alpha _f^\vee +\beta _f^\vee }^y. \end{aligned}$$

Hence, $\tilde{t}_{\alpha _f^\vee }^x, \tilde{t}_{\alpha _f^\vee }^y$ are elements of $W_{ell}$, viewed as the subgroup of $\mathrm {Aut}(\widetilde{\mathfrak {h}}_\mathbb {H})$ generated by $\{\tilde{w}_\alpha \}_{\alpha \in \Delta _{ell}^{re}}$. We denote the subgroup of $W_{ell}$ generated by $\{\tilde{t}_{\alpha _f^\vee }^x\}_{\alpha _f^\vee \in Q_f^\vee }$, $\{\tilde{t}_{\alpha _f^\vee }^y\}_{\alpha _f^\vee \in Q_f^\vee }$ and $\{\tilde{t}_{\alpha _f^\vee }^x, \tilde{t}_{\alpha _f^\vee }^y\}_{\alpha _f^\vee \in Q_f^\vee }$ by $T_x, T_y$ and T, respectively.

Let $P_f\subset \mathfrak {h}^*$ be the weight lattice of $\mathfrak {g}$, i.e., the dual lattice of $Q_f^\vee$. A $T_y$-invariant function on $\widetilde{\mathfrak {h}}_\mathbb {H}$ is nothing but the function on $\tau , u$ and $e^{2\pi \sqrt{-1}\lambda }~(\lambda \in P_f)$, where $e^{2\pi \sqrt{-1}\lambda }$ is the function on $\mathfrak {h}$ defined by $e^{2\pi \sqrt{-1}\lambda }:\, h \mapsto e^{2\pi \sqrt{-1}\lambda (h)}$. For $K \in \mathbb {Z}$, we set

$$\begin{aligned} f_{\lambda , K}(\tau ,h,u)=u^{-K}e^{2\pi \sqrt{-1}\lambda (h)}. \end{aligned}$$

By computing $\sum _{\alpha _f^\vee \in Q_f^\vee }f_{\lambda ,K}(\tilde{t}_{\alpha _f^\vee }^x(\tau ,h,u))$, we see that the function

$$\begin{aligned} \theta _{\lambda ,K}(\tau ,h,u):=u^{-K}\sum _{\gamma \in \nu _f(Q_f^\vee )}q^{\frac{1}{2K}\vert \vert \lambda +K\gamma \vert \vert ^2}e^{2\pi \sqrt{-1}(\lambda +K\gamma )(h)} \end{aligned}$$

on $\widetilde{\mathfrak {h}}_\mathbb {H}$ where $q=e^{2\pi \sqrt{-1}\tau } \in \mathbb {C}^*$ and the linear map $\nu _f:\mathfrak {h}\rightarrow \mathfrak {h}^*$ is defined in Sect. 2.3, is T-invariant and is holomorphic on $\widetilde{\mathfrak {h}}_\mathbb {H}$ for $K \in \mathbb {Z}_{>0}$.

Fix a set of simple roots $\Pi _f$ and let $\Delta _f^+$ be the set of positive roots of $\Delta _f$ with respect to $\Pi _f$. We denote the highest root of $\Delta _f^+$ by $\theta _f$ and the half sum of roots in $\Delta _f^+$ by $\rho _f$. Let $P_f^+$ be the set of dominant weights with respect to $\Pi _f$, i.e., the set $\{ \lambda \in P_f\vert \lambda (\alpha _f^\vee ) \ge 0~(\forall \, \alpha _f \in \Pi _f)\}$. We remark that $\rho _f \in P_f^+$. The number $h^\vee :=1+\rho _f(\theta _f^\vee )$, called the dual Coxeter number, plays an important role.

For $K \in \mathbb {Z}_{\ge 0}$, we set

$$\begin{aligned} P_{K}^+=\{ (\lambda ,K) \vert \, \lambda \in P_f^+, ~\lambda (\theta _f^\vee ) \le K\}, \quad P_{af}^+:=\bigcup _{K \in \mathbb {Z}_{\ge 0}} P_K^+. \end{aligned}$$

It is clear that $P_0^+=\{(0,0)\}$. Now for $(\lambda ,K) \in P_{af}^+$, we set

$$\begin{aligned} \chi _{\lambda ,K}(\tau ,h,u)= \frac{\sum _{w \in W_f}\varepsilon (w)\theta _{w(\lambda +\rho _f), K+h^\vee }(\tau ,h,u)}{\sum _{w \in W_f}\varepsilon (w)\theta _{w(\rho _f),h^\vee }(\tau ,h,u)}, \end{aligned}$$

where $\varepsilon (w)=\det _\mathfrak {h}(w)$ stands for the signature of $w \in W_f$. It is well known that $\chi _\lambda \in \mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}^{W_{ell}}$ is the character of the integrable highest weight modules over the affine Lie algebra associated with $\mathfrak {g}$ (cf. [3, 4]). Indeed, the next stronger statement is known:

Proposition 4.16

(cf. [1]) $\mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}^{W_{ell}}=\bigoplus _{(\lambda ,K) \in P_{af}^+} \mathcal {O}_\mathbb {H}\chi _{\lambda ,K}$ as $\mathcal {O}_\mathbb {H}$-module.

As an $\mathcal {O}_\mathbb {H}$-algebra, the following description of $\mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}^{W_{ell}}$ is known.

Let $\{\Lambda _{\alpha _f}\}_{\alpha _f \in \Pi _f} \subset \mathfrak {h}^*$ be the dual basis of $\Pi _f^\vee :=\{\alpha _f^\vee \}_{\alpha _f \in \Pi _f} \subset \mathfrak {h}$, i.e., $\Lambda _{\alpha _f}(\beta _f^\vee )=\delta _{\alpha _f,\beta _f}$. Set $a_0^\vee =1$ and $a_{\alpha _f}^\vee =\Lambda _{\alpha _f}(\theta _f^\vee )$. One has

Theorem 4.17

(cf. [1, 20]) For each $\alpha _f \in \Pi _f$, there exists $\chi _{\alpha _f} \in \bigoplus _{(\lambda ,a_{\alpha _f^\vee }) \in P_{af}^+} \mathcal {O}_{\mathbb {H}}\chi _{\lambda ,a_{\alpha _f^\vee }}$ such that

• 1. $\chi _{0,1}$ and $\chi _{\alpha _f}~(\alpha _f \in \Pi _f)$ are algebraically independent, and

• 2. $\mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}^{W_{ell}}$ is generated by $\chi _{0,1}$ and $\chi _{\alpha _f}~(\alpha _f \in \Pi _f)$ as $\mathcal {O}_\mathbb {H}$-algebra.

Remark 4.5

The above theorem together with Remark 4.4 implies that there might exist $\mathcal {E}(G)$-invariant holomorphic functions on $\widetilde{\mathcal {C}}(\mathfrak {g})_\tau$ that provide us an analogue of the adjoint quotient map $\widetilde{\mathcal {C}}(\mathfrak {g})_\tau \twoheadrightarrow \mathfrak {h}\oplus \mathbb {C}$. We will discuss on the existence of such a map with its application in our future publication.

In 1984, Kac and Peterson [4] gave formulas of the Jacobian of the fundamental characters $\chi _{0,1}, \chi _{\Lambda _{\alpha _f}, a_{\alpha _f}^\vee }~(\alpha _f \in \Pi _f)$ for type $A_l^{(1)}, B_l^{(1)}, C_l^{(1)}, D_l^{(1)}, G_2^{(1)}$ and for some twisted cases, without proof. Their formulas suggest

Conjecture 4.18

$\mathcal {O}_{\widetilde{\mathfrak {h}}_\mathbb {H}}^{W_{ell}}=\mathcal {O}_\mathbb {H}[\chi _{0,1}, \chi _{\Lambda _{\alpha _f}, a_{\alpha _f}^\vee }~(\alpha _f \in \Pi _f)]$ as $\mathcal {O}_\mathbb {H}$-algebra. In particular, $\chi _{0,1}$ and $\chi _{\Lambda _{\alpha _f}, a_{\alpha _f}^\vee }~(\alpha _f \in \Pi _f)$ are algebraically independent.

5 $\mathrm {SL}_2(\mathbb {Z})$-Action on $\mathbb {H}\times \mathcal {E}(\mathfrak {g}) \times \mathbb {C}$

In the previous subsection, we have studied an $\mathcal {E}(G)$-action on $\widetilde{\mathcal {C}}(\mathfrak {g})$ with the aid of Proposition 4.3 via the isomorphism $\psi ^-$. Here, via the isomorphism $\psi ^+$ in the same proposition, we will study an $SL(2,\mathbb {Z})$-action on $\widetilde{\mathcal {C}}(\mathfrak {g})=\mathbb {H}\times \mathcal {E}(\mathfrak {g}) \times \mathbb {C}$.

A natural left $\mathrm {SL}_2(\mathbb {Z})$-action on $\mathbb {R}^2$ is denoted by $\varphi$, i.e., for $\gamma \in SL(2,\mathbb {Z})$, we set

$$\begin{aligned} {\varphi }_{\gamma }:\mathbb {R}^2 \longrightarrow \mathbb {R}^2; \quad \begin{pmatrix}x\\ y\end{pmatrix}\longmapsto \gamma \begin{pmatrix}x\\ y\end{pmatrix}. \end{aligned}$$

The next lemma is easy to check:

Lemma 5.1

For $\gamma \in SL(2,\mathbb {Z})$ and $(\xi , A,\alpha ) \in \widetilde{\mathcal {E}}(\mathfrak {g})^\mathfrak {d}$, we set

$$\begin{aligned} \gamma .(\xi , A,\alpha ):=(\varphi _{\gamma *}(\xi ), (\varphi _{\gamma ^{-1}})^*(A),(\varphi _{\gamma ^{-1}})^*(\alpha )). \end{aligned}$$

• 1. This is a left action and keeps the bilinear form $\langle \cdot \vert \cdot \rangle$ invariant.

• 2. For $\gamma \in SL(2,\mathbb {Z})$ and $(g,c) \in \widehat{\mathcal {E}}(G)$, the next identity holds:

  $$\begin{aligned} \gamma \circ \hat{L}_{(g,c)} \circ \gamma ^{-1}=\hat{L}_{((\varphi _{\gamma ^{-1}})^*g,c)}. \end{aligned}$$

The induced action of $SL(2,\mathbb {Z})$ on $\mathrm {Im}\, \psi ^+$:

Proposition 5.2

For $(\tau , A, u) \in \widetilde{\mathcal {C}}(\mathfrak {g})$ and $\gamma =\begin{pmatrix} a &{}\quad b \\ c &{}\quad d \end{pmatrix} \in SL(2,\mathbb {Z})$, we have

$$\begin{aligned} \gamma .(\tau , A,u)=\left( \frac{a\tau +b}{c\tau +d}, \frac{(\varphi _{{\gamma }^{-1}})^*A}{c\tau +d}, u-\frac{c\langle A \vert A \rangle }{2(c\tau +d)} \right) . \end{aligned}$$

This action coincides exactly with the $SL(2,\mathbb {Z})$-action of the $\mathbb {C}$-span of the characters of integrable modules over an affine Lie algebra given in [4].

Appendices

Appendix 1: Lifting of a Weyl group

In Sect. 3, we have calculated a lifting of a reflection

$$\begin{aligned} s_\alpha '=\exp (e_\alpha )\exp (-e_{-\alpha })\exp (e_\alpha ) \in \mathcal {E}(G) \end{aligned}$$

and its action on $\mathfrak {h}^\delta$. Here, for the sake of leader’s convenience, we will show how one can calculate its action on $\mathcal {E}(\mathfrak {g})^\mathfrak {d}$.

1.1 On $\mathfrak {h}^\mathfrak {d}$

For $\alpha \in \Delta _{ell}^{re}$ and $t \in \mathbb {C}^*$, set

$$\begin{aligned} \sigma _\alpha (t)=\exp (te_\alpha )\exp (-t^{-1}e_{-\alpha })\exp (te_\alpha ). \end{aligned}$$

Let us compute rapidly $L_{\sigma _\alpha (t)}(\xi ,h)$ for $(\xi ,h) \in \mathfrak {h}^\mathfrak {d}$. By definition, we have

$$\begin{aligned} L_{\sigma _\alpha (t)}(\xi ,h)&= L_{\exp (te_\alpha )}\circ L_{\exp (-t^{-1}e_{-\alpha })} \circ L_{\exp (te_\alpha )}(\xi ,h) \\&= L_{\exp (te_\alpha )}\circ L_{\exp (-t^{-1}e_{-\alpha })}(\xi , h-t\alpha (\xi ,h)e_\alpha ) \\&= L_{\exp (te_\alpha )}(\xi ,h-\alpha (\xi ,h)\alpha ^\vee -t\alpha (\xi ,h)e_\alpha ) \\&= (\xi ,h-\alpha (\xi ,h)\alpha ^\vee )=w_\alpha '(\xi ,h). \end{aligned}$$

In particular, this action does not depend upon the choice of $t \in \mathbb {C}^*$.

1.2 On $\mathcal {E}(\mathfrak {g})^\mathfrak {d}$

Let $\alpha \in \Delta _{ell}^{re}$ and $\beta \in \Delta _{ell}$ such that $\beta \ne \pm \alpha$. Let $p,q \in \mathbb {Z}_{\ge 0}$ be integers satisfying

$$\begin{aligned} (\beta +\mathbb {Z}\alpha ) \cap \Delta =\{\beta +i\alpha \vert -q\le i\le p\}: ~\alpha \text {-string through} \beta . \end{aligned}$$

Since $\{e_\alpha ,e_{-\alpha },\alpha ^\vee \}$ forms an $\mathfrak {sl}_2$-triplet, i.e., $[\alpha ^\vee ,e_{\pm \alpha }]=\pm 2e_{\pm \alpha }$ and $[e_\alpha ,e_{-\alpha }]=\alpha ^\vee$, setting $e_{\beta +i\alpha }=\dfrac{1}{(p-i)!}(\mathrm {ad}e_{-\alpha })^{p-i}(e_{\beta +p\alpha })$ for $-q \le i\le p$, one can check

$$\begin{aligned} \left\{ {\begin{array}{l} \left[ \alpha ^\vee ,e_{\beta +i\alpha }\right] =(2i-p+q)e_{\beta +i\alpha }, \\ \left[ e_{\alpha },e_{\beta +i\alpha }\right] =(i+q+1)e_{\beta +(i+1)\alpha }, \\ \left[ e_{-\alpha },e_{\beta +i\alpha }\right] =(p+1-i)e_{\beta +(i-1)\alpha }, \end{array}}\right. \end{aligned}$$

where we set $e_{\beta -(q+1)\alpha }=0=e_{\beta +(p+1)\alpha }$ for simplicity. Let us compute $L_{\sigma _\alpha (t)}(\xi ,e_\beta )$. Since we have

$$\begin{aligned} L_{\sigma _\alpha (t)}(\xi ,e_\beta )=L_{\sigma _\alpha (t)}(\xi ,0)+L_{\sigma _\alpha (t)}({0},e_\beta ) \end{aligned}$$

by definition, and the first term can be computed using the result of the last subsection, we may assume that $\xi =0$. Let us calculate step by step. First, we have

$$\begin{aligned}&L_{\exp (te_\alpha )}(0,e_\beta ) \\&\quad = \left( 0, \sum _{i=0}^p \frac{1}{i!}t^i (\mathrm {ad}e_\alpha )^i(e_\beta ) \right) = \left( 0,\sum _{i=0}^p\frac{(q+1)(q+2)\cdots (q+i+1)}{i!}t^ie_{\beta +i\alpha }\right) \\&\quad = \left( 0, \sum _{i=0}^p (-1)^i \left( {\begin{array}{c}-q-1\\ i\end{array}}\right) t^ie_{\beta +i\alpha }\right) . \end{aligned}$$

Second, we have

$$\begin{aligned}&L_{\exp (-t^{-1}e_{-\alpha })}\left( 0, \sum _{i=0}^p \left( {\begin{array}{c}-q-1\\ i\end{array}}\right) (-t)^ie_{\beta +i\alpha }\right) \\&\quad = \left( 0,\sum _{i=0}^p \left( {\begin{array}{c}-q-1\\ i\end{array}}\right) (-t)^i\sum _{j\ge 0}\frac{1}{j!}(-t)^{-j}(\mathrm {ad}e_{-\alpha })^je_{\beta +i\alpha }\right) \\&\quad = \left( 0,\sum _{i=0}^p \left( {\begin{array}{c}-q-1\\ i\end{array}}\right) \sum _{j\ge 0}\left( {\begin{array}{c}p+j-i\\ j\end{array}}\right) (-t)^{i-j}e_{\beta +(i-j)\alpha }\right) . \end{aligned}$$

Here, the sum is taken over

$$\begin{aligned}&\{(i,j) \in \mathbb {Z}_{\ge 0}^2 \vert \, 0\le i\le p,~0\le j\le q+i\} \\&\quad = \{(i,j) \in \mathbb {Z}_{\ge 0}^2 \vert \,j\le i\le p\} \cup \{(i,j) \in \mathbb {Z}_{\ge 0}^2 \vert \, 0\le i\le p,\,0< j-i\le q\}. \end{aligned}$$

We consider the change of variables in each domain of the right-hand side as follows:

$\underline{m=i-j \hbox { and } n=i}$ on the first and $\underline{m=j-i \hbox { and } n=i}$ on the second. We have

$$\begin{aligned}&= \left( 0,\sum _{m=0}^p \left( \sum _{n=m}^p \left( {\begin{array}{c}p-m\\ p-n\end{array}}\right) \left( {\begin{array}{c}-q-1\\ n\end{array}}\right) \right) (-t)^me_{\beta +m\alpha } \right. \\&\left. \quad + \sum _{m=1}^q\left( \sum _{n=0}^p\left( {\begin{array}{c}-q-1\\ n\end{array}}\right) \left( {\begin{array}{c}p+m\\ p-n\end{array}}\right) \right) (-t)^{-m}e_{\beta -m\alpha } \right) \\&= \left( 0,\sum _{m=-q}^p \left( {\begin{array}{c}p-q-1-m\\ p\end{array}}\right) (-t)^me_{\beta +m\alpha }\right) \\&=\left( 0,\sum _{m=p-q}^p \left( {\begin{array}{c}p-q-1-m\\ p\end{array}}\right) (-t)^me_{\beta +m\alpha }\right) \\&= \left( 0,\sum _{m=0}^q \left( {\begin{array}{c}-m-1\\ p\end{array}}\right) (-t)^{p-q+m}e_{\beta +(p-q+m)\alpha }\right) . \end{aligned}$$

Here, in the second equality, we used the formula

$$\begin{aligned} \sum _{\begin{array}{c} i,j\ge 0 \\ i+j=N \end{array}} \left( {\begin{array}{c}\alpha \\ i\end{array}}\right) \left( {\begin{array}{c}\beta \\ j\end{array}}\right) =\left( {\begin{array}{c}\alpha +\beta \\ N\end{array}}\right) \end{aligned}$$

which can be proved by looking at the coefficient of $t^N$ in $(1+t)^\alpha \cdot (1+t)^\beta =(1+t)^{\alpha +\beta }$. Hence, we obtain

$$\begin{aligned} L_{\exp (-t^{-1}e_{-\alpha })}\circ L_{\exp (te_\alpha )}(0,e_\beta ) = \left( 0,(-1)^p\sum _{m=0}^q \left( {\begin{array}{c}p+m\\ p\end{array}}\right) (-t)^{p-q+m}e_{\beta +(p-q+m)\alpha }\right) . \end{aligned}$$

Finally, we compute $L_{\sigma _\alpha (t)}(0,e_\beta )=L_{\exp (te_\alpha )}\circ L_{\exp (-t^{-1}e_{-\alpha })}\circ L_{\exp (te_\alpha )}(0,e_\beta )$. We have

$$\begin{aligned}&L_{\sigma _\alpha (t)}(0,e_\beta ) \\&\quad = \left( 0,(-1)^p\sum _{m=0}^q \left( {\begin{array}{c}p+m\\ p\end{array}}\right) (-t)^{p-q+m} \sum _{n=0}^{q-m}\frac{1}{n!}t^n(\mathrm {ad}e_\alpha )^ne_{\beta +(p-q+m)\alpha }\right) \\&\quad = \left( 0,(-1)^p\sum _{m=0}^q \left( {\begin{array}{c}p+m\\ p\end{array}}\right) (-t)^{p-q+m} \sum _{n=0}^{q-m}\left( {\begin{array}{c}p+m+n\\ n\end{array}}\right) t^ne_{\beta +(p-q+m+n)\alpha }\right) . \end{aligned}$$

Now, we compute the summation

$$\begin{aligned} \sum _{m=0}^q \left( {\begin{array}{c}p+m\\ p\end{array}}\right) (-t)^{p-q+m} \sum _{n=0}^{q-m}\left( {\begin{array}{c}p+m+n\\ n\end{array}}\right) t^n. \end{aligned}$$

By the change of variable $k=m+n$, we obtain

$$\begin{aligned}&\sum _{m=0}^q \left( {\begin{array}{c}p+m\\ p\end{array}}\right) (-t)^{p-q+m} \sum _{n=0}^{q-m}\left( {\begin{array}{c}p+m+n\\ n\end{array}}\right) t^n \\&\quad = (-t)^{p-q}\sum _{0\le m\le k\le q}(-1)^m\left( {\begin{array}{c}p+m\\ m\end{array}}\right) \left( {\begin{array}{c}p+k\\ k-m\end{array}}\right) t^k\\&\quad = (-t)^{p-q}\sum _{k=0}^q \sum _{m=0}^{k}\left( {\begin{array}{c}-p-1\\ m\end{array}}\right) \left( {\begin{array}{c}p+k\\ k-m\end{array}}\right) t^k \\&\quad = (-t)^{p-q}. \end{aligned}$$

Hence, the only non-trivial contribution in the above summation comes from the term corresponding to $(m,n)=(0,0)$. Thus, we obtain

$$\begin{aligned} L_{\sigma _\alpha (t)}(0,e_\beta )=(0, (-1)^qt^{p-q}e_{\beta +(p-q)\alpha }) =(0,(-1)^qt^{-\beta (\alpha ^\vee )}e_{w_\alpha (\beta )}). \end{aligned}$$

In particular, we have

$$\begin{aligned} L_{s_{\delta _\sharp -\alpha _f}'} \circ L_{s_{\alpha _f}'}(0,e_\beta ) =L_{s_{\delta _\sharp -\alpha _f}'} (0,(-1)^qe_{\beta -\beta (\alpha _f^\vee )\alpha _f}) =(0,e_{\beta -\beta (\alpha _f^\vee )\delta _\sharp }) \end{aligned}$$

for $\alpha =\alpha _f \in \Delta _f$ and $\sharp \in \{x,y\}$, and for any $\alpha \in \Delta _{ell}^{re}$

$$\begin{aligned} L_{\sigma _\alpha (t)^2}(0,e_\beta ) =L_{\sigma _\alpha (t)}(0,(-1)^qt^{-\beta (\alpha ^\vee )}e_{w_\alpha (\beta )}) =(0,(-1)^{p+q}e_\beta ), \end{aligned}$$

i.e.,

$$\begin{aligned} \sigma _\alpha (t)^2=(-1)^{\alpha ^\vee }. \end{aligned}$$

(9)

Appendix 2: Normalization on exterior product

Here, for a differentiable manifold M, we fix a normalization on differential p-form on M. For $\omega _1,{\ldots }, \omega _r \in \Omega _M^1$ and $X_1,{\ldots }, X_r\in \Theta _M$, we set

$$\begin{aligned} (\omega _1\wedge {\cdots } \wedge \omega _r)(X_1,{\ldots }, X_r) :=\det (\omega _i(X_j))_{1\le i,j\le r}. \end{aligned}$$

Hence, for $\omega \in \Omega _M^p, \omega ' \in \Omega _M^q$ and $X_1, {\ldots }, X_{p+q} \in \Theta _M$, the exterior product $\omega \wedge \omega '$ is defined by

$$\begin{aligned}&(\omega \wedge \omega ')(X_1,{\ldots }, X_{p+q}) \\ :=&\frac{1}{p!q!}\sum _{\sigma \in \mathfrak {S}_{p+q}}(\mathrm {sgn}\sigma )\omega (X_{\sigma (1)},{\ldots }, X_{\sigma (p)})\omega '(X_{\sigma (p+1)},{\ldots }, X_{\sigma (p+q)}). \end{aligned}$$

With this normalization, it is natural to define the de Rham differential d as follows: for $\omega \in \Omega _M^r$ and $X_0,{\ldots }, X_r \in \Theta _M$,

$$\begin{aligned} (d\omega )(X_0,{\ldots }, X_r) =&\sum _{i=0}^r(-1)^iX_i(\omega (X_0,{\ldots }, \overset{\vee }{X_i},{\ldots }, X_r) \\&+ \sum _{i<j}(-1)^{i+j}\omega ([X_i,X_j], {\ldots }, \overset{\vee }{X_i},{\ldots },\overset{\vee }{X_j},{\ldots },X_r). \end{aligned}$$