The Maximal Torus of a Compact Lie Group

Abstract

In this chapter we show that every connected compact Lie group G contains a maximal torus T. This maximal torus is unique up to conjugation, and its conjugates cover G. If N is the normalizer of T, then the Weyl group W = N/T is finite and operates effectively on T. Thus there is a one-to-one correspondence between functions on G which are invariant under conjugation and functions on T which are invariant under the action of W. In particular, the characters of G are the W-invariant characters of T. For this reason understanding the operation of the Weyl group on the maximal torus is important to representation theory. In the third section we compute the maximal tori and Weyl groups of the classical Lie groups, and in the last section we give a generalization which handles the case of nonconnected groups.