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 $W_H \to W_G$.

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    $\begingroup$ I am not sure I understand the question: What properties do you require from $W_H\to W$? You always have the trivial homonorphism, but that's not what you want (unless $H\to G$ is trivial). +1 for an interesting question though. $\endgroup$ – Moishe Kohan Jun 4 '14 at 4:07
  • $\begingroup$ I added some more detail to what I want. $\endgroup$ – Sven Jun 4 '14 at 17:10
  • $\begingroup$ Sorry, but the question remains unclear. What do you mean by "an associated inclusion"? The Weyl group of $G$ is not (in general) a subgroup of $G$. Only some finite central extension embeds. Just look at the case of $SL(2,C)$. The Weyl group embeds naturally in $PSL(2,C)$ but the embedding does not lift to $SL(2,C)$. $\endgroup$ – Moishe Kohan Jun 4 '14 at 17:23
  • $\begingroup$ See also mathoverflow.net/questions/13139/…. $\endgroup$ – Moishe Kohan Jun 4 '14 at 17:26
  • $\begingroup$ I want the weyl groups to have an inclusion. It would be awesome if there was some naturality to this whole thing and the inclusion may have choices so thats why I said "associated". Maybe Tits's result would be helpful in proving/disproving my question but the paper is in French and thus barred from me. $\endgroup$ – Sven Jun 4 '14 at 17:36

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 $W_H = \langle s_\alpha\rangle \subset W_G$.

  • $\begingroup$ I applogize, I was in a rush when I wrote the first question. I should have used the term parabolic subgroup. You're comment is what I was thinking of. The group you mentioned is the semisimple subgroup of the parabolic group associated to that subset of simple roots. Can we do better? $\endgroup$ – Sven Jun 4 '14 at 17:06
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    $\begingroup$ In this case, I must apologise for trying to teach the fish how to swim. One can give further examples, as follows: If $T$ is a maximal torus in $G$, and $H$ is a $T$-regular semi-simple subgroup of $G$, obtained by a suitable sequence removals of nodes and/or elementary transformations of the (extended) Dynkin diagram of $G$, then the Weyl group $W_H$ is again a subgroup of $W_G$ generated by reflections $s_\alpha \in W_G$ corresponding to roots in a system of simple roots of $H$. Unfortunately, I am not competent enough to expand this vague comment into a more detailed answer. $\endgroup$ – ivanpenev Jun 4 '14 at 21:30
  • $\begingroup$ On the other hand, the inclusions $\operatorname{SO}_{2n+1} \to \operatorname{SL}_{2n+1}$, $\operatorname{Sp}_{2n} \to \operatorname{SL}_{2n}$ and $\operatorname{SO}_{2n} \to \operatorname{SL}_{2n}$, for instance, yield descriptions of the Weyl groups of the embedded subgroups as certain groups of permutations of $2n$ letters, i.e. as subgroups of the Weyl group of the ambient special linear group, even though the embedded subgroups are not regular with respect to any maximal torus of the ambient group. $\endgroup$ – ivanpenev Jun 4 '14 at 21:38
  • $\begingroup$ Your comment was better than what I would have written. I will have a look at those examples, thanks. $\endgroup$ – Sven Jun 5 '14 at 1:13

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