Topics in Representation Theory: Fundamental Representations and Highest

We’ll now start the study of arbitrary irreducible representations of higher rank compact Lie groups, beginning with the purely Lie-algebraic aspects of the story. A fundamental weakness is the Lie-algebraic approach is the lack of any analog of the explicit representation in terms of homogeneous polynomials that we were able to use in the SU(2) case. Working just with the Lie algebra and its commutation relations, one can derive many properties of irreducible represen-tations, but the only actual construction of the representations in this context are rather inexplicit, involving taking quotients of infinite dimensional modules of the enveloping algebra known as Verma modules. We will not construct rep-resentations this way, instead using the Borel-Weil construction, a geometric technique that works at the level of the group and generalizes the homogeneous polynomial construction of the SU(2) case. For now we will describe some of the Lie-algebraic techniques that are both computationally useful and allow the precise formulation of the general picture, postponing proofs that these actu-ally give all irreducible representations until we use geometric techniques to construct them. In this lecture we’ll be following [1] chapter 14 fairly closely. As a result these notes may be even sketchier than usual. 1 Co-roots and Fundamental Weights For each simple root α, we would like to identify a copy of sl(2,C) ⊂ gC, which we will call sl(2,C)α. The weights of our representations will then be classified by how they behave under the maximal abelian subalgebras of each of these sl(2,C)α. Recall that [gα, gβ] ⊂ gα+β and [gα, g−α] ⊂ tC We can choose a set of three generators X+α, X α,Hα satisfying the sl(2,C) commutation relations, with X+α ⊂ gα, X−α ⊂ g−α and Hα ⊂ tC [Hα, X+α] = α(Hα)X α = 2