Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group

Question

When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.

In the paper, we assume that $\mathfrak{g}$ be a simple Lie algebra with rank $n$ and $I=\{1,\dotsc,n\}$. Let $\hat{\mathfrak{g}}$ be untwisted affine Lie algebra , corresponding to $\mathfrak{g}$ , with index $\hat{I}=\{0,1\dotsc,n\}$.

Let $P=\bigoplus _{i\in I}\mathbb{Z} \omega _i$, $Q=\bigoplus _{i\in I}\mathbb{Z} \alpha _i$, $P^{\vee}=\bigoplus _{i\in I}\mathbb{Z} \omega _{i}^{\vee}$, $Q^{\vee}=\bigoplus _{i\in I}\mathbb{Z} \alpha _{i}^{\vee}$ be the weight lattice, root lattice, coweight lattice and coroot lattice, respectively.

Let $W$ and $\hat{W}$ be the Weyl groups of $\mathfrak{g}$ and $\hat{\mathfrak{g}}$. It is well-known that $W$ is generated by simple reflections $s_i$ for $i\in I$ and $s_i$ for $i\in \hat{I}$, respectively. The Weyl group $W$ acts on the root lattice $Q$ by extending $s_i(\alpha_j)=\alpha_j−a_{ij}\alpha_i$. Then the author says $\hat{W}$ is isomorphic to the semi-direct product $W\ltimes Q$.

The author defines the extended Weyl group $\widetilde{W}\triangleq{} $the semi-direct product $W\ltimes P$. For any $w\in W$, we write $(w,0)\in \widetilde{W}$ and for $\omega\in P$, we write $t_\omega$ for the elements $(1,\omega)$.

The author says $\hat{W}$ is a normal subgroup of $\widetilde{W}$, then we have $\mathcal{T}=\widetilde{W}/\hat{W}$ is a finite group isomorphic to a subgroup of the group of diagram automorphisms of $\hat{\mathfrak{g}}$, i.e. the bijections $\tau: \hat{I}\to \hat{I}$ such that $a_{\tau(i)\tau(j)}=a_{ij}$ for $i,j\in \hat{I}$.

Moreover, there is an isomorphism of groups $\widetilde{W}=\mathcal{T}\ltimes\hat{W}$, where the semi-direct product is defined using the action of $\mathcal{T}$ in $\hat{W}$ given by $\tau.s_i = s_{\tau(i)}\tau$.

Here are my questions:

1: Hou could I get $\mathcal{T}=\widetilde{W}/\hat{W}$ is a finite group which is isomorphic to a subgroup of the group of diagram automorphisms ?

2: How could I understand the isomorphism $\widetilde{W}\cong \mathcal{T}\ltimes\hat{W}$?

Any help and references are greatly appreciated.

Thanks!

I also posted it on MSE Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group.

New explanations and thoughts:

I have also read details in Chapter $4$ of the book Reflection Groups and Coxeter Groups written by James E. Humphreys. In $4.5$, the author writes: where $A=$ connected components of $V^\circ:=V\setminus\bigcup_{H\in\mathcal{H}}H$, $\mathcal{H}$ the collection of all hyperplanes $H_{\alpha,k}$ and $A_0:=\{\lambda\in V|0<(\lambda,\alpha)<1\text{ for all }\alpha\in\Phi^+\}.$

I think this is the similar isomorphism from $W\ltimes P\to \mathcal{T}\ltimes > \hat{W}.$ Is it true?

Answer 1

This is discussed in [B]: Bourbaki - Lie groups and Lie algebras, Chapters 4–6, Ch. VI, §2, no. 3. (My numbering is from the French version, which is what I've got to hand, but I think it matches the English version.)

Once you have realised $\hat W$ as the subgroup $W \ltimes Q$ of $W \ltimes P = \tilde W$, you have that the inclusion of $P/Q$ in $\tilde W/\hat W$ is an isomorphism. This is finite, because $P$ and $Q$ are both rank-$n$ lattices.

1: We let $P/Q$ act on the extended Dynkin diagram by acting on an (affine) alcove $C$ in $P \otimes_{\mathbb Z} \mathbb R$. For definiteness, let's choose $C$ so that $0$ is contained in its closure. Then we can choose $x \in P$, regard it as an element of $\tilde W$, adjust by an element of $Q$ to arrange that the closure of $x C$ also contains $0$, and then adjust by an element of $W$ so that $C$ and $x C$ lie in the same (spherical) chamber, hence (since both contain $0$ in their closure) are equal. Then all that $x$ can do is permute the vertices of $C$, and this describes its permutation action on $\hat I$.

2: This also realises $\tilde W$ as a semi-direct product, since its quotient $\mathcal T$ can now be seen as the subgroup of $\tilde W$ that preserves $C$.

These actions are described concretely for the various root systems in parts (XII) of Plates I–IX in the back matter of [B].