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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?

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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

1 Answer 1

up vote 13 down vote accepted

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 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.

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So is Tits' group just the $\mathbf{Z}$-points of the normalizer, after one chooses a model over $\mathbf{Z}$? –  moonface Jan 27 '10 at 16:33
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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 Speyer Jan 27 '10 at 16:37
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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
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My recollection was incorrect. As Kevin McGerty points out, the adjoint and simply connected forms match for E_8; Allen Knutson points out (via e-mail) that SO_5 is another counter-example. –  David Speyer Jan 28 '10 at 18:47
    
There is interesting later work on this kind of question for compact Lie groups and analogues, for example by Dwyer and Wilkerson. A follow-up to the paper by Tits ("part I" still searching for its "part II"), with useful references: MR2174268 (2006f:55015) Dwyer, W. G.; Wilkerson, C. W. Normalizers of tori. Geom. Topol. 9 (2005), 1337--1380 –  Jim Humphreys Mar 17 '10 at 14:43

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