# Can we realize Weyl group as a subgroup?

Given a semisimple Lie group G, let T be a maximal torus, W be the Weyl group defined as the quotient N(T)/C(T), where N(T) denotes the normalizer of T and C(T) denotes the centralizer.

Two questions are:

1. How many ways are there we can realize W as a subset of G?

2. Can we always realize W as a subgroup of G?

• This question (the second, more interesting question) will probably need some clarification before it can be cleanly answered. Is $G$ a semisimple real Lie group? Is $T$ a maximal split torus? – Marty Jan 27 '10 at 15:21
• It's worth noting that the questions here make equal sense and have mostly the same answers when the group is assumed to be a (connected) semisimple algebraic group over an arbitrary algebraically closed field. In any case, smaller fields need more discussion. – Jim Humphreys Apr 16 '10 at 18:45

In general it is not possible to embed the Weyl group $$W$$ in the group $$G$$: already you can see this for $$SL_2(\mathbb C)$$, where the Weyl group has order $$2$$: if the torus fixes the lines spanned by $$e_1$$ and $$e_2$$ respectively, you want to pick the linear map taking $$e_1$$ to $$e_2$$ and $$e_2$$ to $$e_1$$, but this has determinant $$-1$$. A lift of $$W$$ to $$N(T)$$ must be an element of order $$4$$ not $$2$$, say $$e_1 \mapsto -e_2$$ and $$e_2 \mapsto e_1$$.
In fact, Tits has shown (Normalisateurs de Tores I. Groupes de Coxeter Étendus (Journ. Alg. 4, 1966, pp. 96-116) that this is essentially the only obstruction: the Weyl group can always be lifted to a group $$\tilde{W}$$ inside $$G$$ which is an extension of $$W$$ by an elementary abelian $$2$$-group of order $$2^l$$ where $$l$$ is the number of simple roots. If I recall correctly, this lift is then unique up to conjugation.
• One should point out that Tits' result is for G simply connected. If we use a smaller Lie group, then the rank of the $2$-group drops. I think I recall that, if G is the adjoint form of its Lie algebra, then W always lifts. – David E Speyer Jan 27 '10 at 16:37
• You're right I meant to take $G$ simply connected, but are you sure about the adjoint form case? For $E_8$ for example, the adjoint form is simply connected, so the Tits group can't get any smaller. – Kevin McGerty Jan 27 '10 at 17:03