Tuesday, 15 March 2011

Can we realize Weyl group as a subgroup?

In general it is not possible to embed the Weyl group W in the group G: already you can see this for SL2(mathbbC), where the Weyl group has order 2: if the torus fixes the lines spanned by e1 and e2 respectively, you want to pick the linear map taking e1 to e2 and e2 to e1, but this has determinant 1. A lift of W to N(T) must be an element of order 4 not 2, say e1mapstoe2 and e2mapstoe1.



In fact, Tits has shown that this is essentially the only obstruction: the Weyl group can always be lifted to a group tildeW inside G which is an extension of W by an elementary abelian 2-group of order 2l where l is the number of simple roots. If I recall correctly, this lift is then unique up to conjugation.

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