Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let G be a reductive algebraic group over a field k. Let S be a maximal split torus, Z its centraliser and N its normaliser. The Weyl group W is then defined to be the quotient N(k)/Z(k). Now we cannot hope for W to be realisable as a subgroup of G, but I would like to know how close we can get.

There is a classical result of Tits in the case where G is split which says that for each simple reflection s in the Weyl group, we can find a lift ws in G with the property that these lifts satisfy the braid relations. They do not however square to the identity, instead square to an order 2 element of S, and we get an extension of W by an elementary abelian 2-group embedding in G.

So my question ends up becoming, what generalisation of the above theorem of Tits exists when G is no longer assumed to be split? Ideally I'd get an answer for general reductive G, and if there happens to be a simpler formulation in the quasi-split case, I'd be interested in hearing that too.

share|improve this question
1  
Actually, the question is already answered in the paper you are referring to, "Normalisateurs de tores. I. Groupes de Coxeter étendus" (1966), where Tits gives a reference to the theorem 7.2 I mentioned. –  Guntram Oct 14 '10 at 8:19
    
By the way, the article Normalisateurs de tores by Tits (part I, with the promised second part never published) appears in J. Algebra 4 (1966), 96-116; the comment about the nonsplit case pointed out by Guntram occurs already in the first paragraph. Also, the Borel-Tits framework allows for reductive rather than just semisimple groups; but the essential arguments here involve the latter case. –  Jim Humphreys Oct 14 '10 at 11:54

1 Answer 1

up vote 7 down vote accepted

Via Theorem 7.2 in Borel-Tits, Groupes reductifs (1965), Tits's lifting result for the Weyl group also applies in the non-split case (for connected groups).

This theorem states that there exists a split subgroup $F$ of $G$ such that $F$ contains the maximal split torus $S$ of $G$ and intersects each relative root group of $G$ non-trivially. In particular, the Weyl groups of $F$ and $G$ are isomorphic.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.