Borel-Weil Theory

Abstract

In this chapter we want to develop enough of the representation theory for compact semisimple Lie groups to show that Kirillov’s method applies to this case. This development requires somewhat of a review of the classical theory of representations for Lie groups and algebras. One major result is the Borel–Weil theory for geometric realization of the representations. This is important, for as we show in the following chapters the geometric quantization of certain mechanical systems is embodied in the Borel–Weil Theorem, which allows us to calculate the multiplicities of the eigenvalues given by quantization.