Maximal Torus - Weyl Group

Weyl Group

Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T). The representation theory of G is essentially determined by T and W.

• The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
• The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.
• The identity component of the normalizer of T is also equal T. The Weyl group is therefore equal to the component group of N(T).
• The normalizer of T is closed, so the Weyl group is finite
• Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, the conjugacy classes of G intersect T in a Weyl orbit.
• The space of conjugacy classes in G is diffeomorphic to the orbit space T/W.