Nolan R . Wallach Symmetry , Representations , and Invariants Graduate Texts in Mathematics 255 Springer

In this appendix we give another algebraic proof of the Weyl character formula, using methods that have many other applications in Lie theory. We begin by setting up the machinery of Lie algebra cohomology (without assuming any previous background in homological algebra). We define the cohomology spaces for a Lie algebra representation in terms of a cochain complex and differential, with the cohomology in degree zero being the subspace of invariants. We show that the short left-exact sequence of invariants associated to a submodule and quotient module extends to a long exact sequence in cohomology. Then we determine the cohomology of the universal enveloping algebra of a Lie algebra. For a semisimple Lie algebra g the cohomology spaces associated with the nilradial n+ of a Borel subalgebra of g are particularly important (we already saw this in the classification of irreducible representations using the n+-invariant vectors in Chapter 3). These cohomology spaces are completely described by a theorem of Kostant, which we prove using an identity for the Casimir operator due to Casselman and Osborne. From Kostant’s theorem we obtain the Weyl character formula via the Euler-Poincaré principle. E.1 Lie algebra cohomology E.1.1 Cochain complex Let g be a Lie algebra over C, and let (ρ,V ) be a representation of g. Here we do not assume that g or V is finite dimensional. For each integer p = 0,1, . . . consider the space Cp(g,V ) of all p-multilinear maps ω : g×·· ·×g } {{ } p // V that are alternating in their arguments. We call such a map ω a p-cochain. For example, a 0-cochain is a constant map from g to V , which we identify with its value