Last updated: October 15, 2005
Let G be a connected complex reductive algebraic group, and fix a real form of G, defined by an involution θ. Let K be the group of θ-fixed points in G, and let H be a θ-invariant Cartan subgroup of G. The real Weyl group associated to H is the group W(K,T) = NK(T)/(K∩T), which is also W(G(R),H(R)) = NG(R)(T)/T(R).
This group, which is rather delicate to compute, plays an important role in the representation theory of G(R). Even though it may always be realized as a subgroup of GL(t) (where t is the Lie algebra of T) generated by elements of order 2, it is not a Coxeter group in general. It has the following structure:
W(K,T)=(WC)θ.((A.Wic) × WR)
where the symbol “.” denotes a semidirect product, the symbol “×” denotes direct product, and the groups have the following meaning:
A subtle aspect is that the mα above, and therefore the group A, depend on the choice of covering group. Already in the simple case of type A1, for the fundamental Cartan, and the split real form, we have that mα is non-trivial (and therefore A is trivial) when G is SL(2), whereas mα is trivial (and therefore A is non-trivial) when G is PSL(2).
The “realweyl” command will output the real Weyl group for any given conjugacy class of Cartan subgroups, together with a set of generators, expressed in terms of the standard generators of the Weyl group W.