Compact Lie Groups and Maximal Tori

Abstract

In this chapter and Chapter 12we develop the representation theory of a connected, compact matrix Lie group K.{\thinspace}The main result is a ``theorem of the highest weight,'' which is very similar to our main results for semisimple Lie algebras. If we let ?$$\mathfrak{k}$$be the Lie algebra of K and we let ?$$\mathfrak{g}$$be the complexification of ?,$$\mathfrak{k},$$then ?$$\mathfrak{g}$$is reductive, which means (Proposition 7.6) that ?$$\mathfrak{g}$$is the direct sum of a semisimple algebra and a commutative algebra. We can, therefore, draw on our structure results for semisimple Lie algebras to introduce the notions of roots, weights, and the Weyl group. We will, however, give a completely different proof of the theorem of the highest weight. In particular, our proof of the hard part of the theorem, the existence of a irreducible representation for each weight of the appropriate sort, will be based on decomposing the space of functions on K under the left and right action of K.{\thinspace}This argument is independent of the Lie-algebraic construction using Verma modules.