Lie groups, maps and the Weyl group

Question

If I have a map of simple Lie groups $H \to G$, do I get a map of Weyl groups $W_H \to W_G$? If $H$ is the semisimple component of a parabolic subgroup then we can clearly get this (see Ivan's answer for this), but is there anything I can say in general? I want the following statement: If $H \to G$ is an inclusion of semisimple Lie groups then there is an associated inclusion of the associated Weyl groups $W_H \to W_G$.

Answer 1

Probably you would like to fix Cartan subgroups in both $H$ and $G$, to make the question more precise. Below, I will point out a case of an embedding of simple complex Lie groups $H \to G$, which leads to an inclusion $W_H \to W_G$.

Let $\mathfrak{t}$ be a Cartan subalgebra of $\mathfrak{g} = \operatorname{Lie}(G)$, and let $\Delta \subset \mathfrak{t}^*$ be a system of simple roots for $\mathfrak{g}$. Let $\Theta \subset \Delta$ be a non-empty proper subset and consider the abelian subspace $\mathfrak{a} = \{x \in \mathfrak{t}\ | \ \alpha(x) = 0 \ \text{for} \ \alpha \in \Theta\}$. Let $L$ be the centraliser of $\mathfrak{a}$ in $G$. This is a connected reductive subgroup of $G$ containing the maximal torus $T \subset G$ with Lie algebra $\mathfrak{t}$. The group $L$ has a positive-dimensional centre with Lie algebra $\mathfrak{a}$, and, in general, several simple normal subgroups. The Weyl group $W_L = W(L,T)$ is naturally a subgroup of $W_G = W(G,T)$. In fact, $W_L$ is generated by the reflections $s_\alpha$ along the roots $\alpha \in \Theta$. Now, consider the Dynkin diagram of $L$, which can be obtained from the Dynkin diagram of $G$ by removing all nodes belonging to $\Delta\setminus\Theta$ and the edges incident with them. Let $\Gamma \subset \Theta$ be the set of nodes in some connected component of the Dynkin diagram of $L$. The root subspaces $\mathfrak{g}^{\pm\alpha}$ with $\alpha \in \Gamma$ generate a simple subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. This subalgebra is tangent to a connected simple subgroup $H$ of $G$, which is a simple normal subgroup of $L$, and thus is normalised by $T$. The Weyl group $W_H = W(H, H\cap T) \simeq W(HT,T)$ is a normal subgroup (in fact, a direct factor) of $W_L$, and thus a subgroup of $W_G$. Similarly to $W_L$, the group $W_H$ is generated by the reflections $s_\alpha$ with $\alpha \in \Gamma$. For instance, if $\Gamma = \Theta = \{\alpha\}$ for some $\alpha \in \Delta$, then $H \simeq \operatorname{SL}_2(\mathbb{C})$ or $\operatorname{PGL}_2(\mathbb{C})$, with Lie algebra $\mathfrak{h} = \mathfrak{g}^\alpha + \mathfrak{g}^{-\alpha} + [\mathfrak{g}^{\alpha},\mathfrak{g}^{-\alpha}]$ and Weyl group $W_H = \langle s_\alpha\rangle \subset W_G$.