Highest-Weight Theory

Abstract

In this chapter we study the regular representations of a classical group G by the same method used for the adjoint representation. When G is a connected classical group, an irreducible regular G-module decomposes into a direct sum of weight spaces relative to the action of a maximal torus of G. The theorem of the highest weight asserts that among the weights that occur in the decomposition, there is a unique maximal element, relative to a partial order coming from a choice of positive roots for G. We prove that every dominant integral weight of a semisimple Lie algebra g is the highest weight of an irreducible finite-dimensional representation of g. When g is the Lie algebra of a classical group G, the corresponding regular representations of G are constructed in Chapters 5 and 6 and studied in greater detail in Chapters 9 and 10. A crucial property of a classical group is the complete reducibility of its regular representations. We give two (independent) proofs of this: one algebraic using the Casimir operator, and one analytic using Weyl’s unitarian trick.