Topics in Representation Theory: Roots and Complex Structures

To recap our story so far: we began by identifying an important abelian subgroup of G, the maximal torus T . By restriction any G representation π is a T representation π|T . In general π|T is a reducible T representation and the irreducible representations of T (which are all one-dimensional) that occur in π|T are called the weights of π. Irreducible representations of T , and thus weights, are labelled by an element of t∗, one that is integral on the integer lattice exp−1(e) ⊂ T . When we refer to “weights”, we will often be referring to these labels. If π is represented on a vector space V , the weight-space Vα corresponding to a weight α will be the sum of the one-dimensional subspaces of V that are irreducible representations of T with weight α. The following explanation of how the geometry of G/T is linked to representation theory is part of a much larger story. For more details, and much material on the relation of the cohomology of G/T to representation theory, see [2]. G acts by conjugation on itself