When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis 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 $I=\{1,\cdots,n\}$. Let $\hat{\mathfrak{g}}$ be untwisted affine Lie algebra ,corresponding to $\mathfrak{g}$ , with index $\hat{I}=\{0,1\cdots,n\}$.
Let $P=\oplus _{i\in I}\mathbb{Z} \omega _i,Q=\oplus _{i\in I}\mathbb{Z} \alpha _i, P^{\lor}=\oplus _{i\in I}\mathbb{Z} \omega _{i}^{\lor}, Q^{\lor}=\oplus _{i\in I}\mathbb{Z} \alpha _{i}^{\lor}$ be weight lattices, root lattices, coweight lattices and coroots lattices, respectively.
Let $W$ and $\hat{W}$ be the Weyl groups of $\mathfrak{g}$ and $\hat{\mathfrak{g}}$. It is well-known that $W$ isgenerated 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 ?
$2:$ Hou could I understand the isomorphism $\widetilde{W}\cong \mathcal{T}\ltimes\hat{W}$?
Any help and references are greatly appreciated.
Thanks!
It has been answered at MO
