Topics in Representation Theory: Roots and Weights

Last time we defined the maximal torus T and Weyl group W (G,T ) for a compact, connected Lie group G and explained that our goal is to relate the representation theory of T to that of G. One aspect of the representation theory of T and of G for which there is a simple relation is that of their representation rings. Consider representations π of a compact Lie group G on complex vector spaces and recall that a representation is characterized up to isomorphism by its character χπ : g ∈ G→ Tr(π(g)) ∈ C