Topics in Representation Theory: Maximal Tori and the Weyl Group

The next part of this course will be concerned with compact, connected Lie groups and their representations. Our goal is to • Classify compact connected Lie groups • Classify all irreducible representations of such groups • Calculate the characters of these irreducible representations Recall that the Peter-Weyl theorem tells one that the matrix elements of irreducible representations form an orthonormal basis of L2(G) for G a compact Lie group. Equivalently, there is an isomorphism of Hilbert spaces L2(G) = ⊕pi∈ĜVpi ⊗ V ∗pi where the Hilbert space direct sum is over all irreducible representations of G. Under the left action of G, the infinite dimensional representation on L2(G) breaks up into finite dimensional representation spaces Vpi with multiplicity dim Vpi. What the Peter-Weyl theorem doesn’t do is tell us what the represen-tations are and how to tell them apart. We would like to know how to project from L2(G) onto each of the Vpi ⊗ V ∗pi. This requires classifying the irreducible representations and finding their characters. 1 Maximal Tori For the case of G = U(1), Fourier analysis of functions on the circle tells us that irreducible representations are all one dimensional, labelled by an integer n and have character χ(eiθ) = einθ A group consisting of a finite product of k copies of U(1) will be called a torus and its irreducible representations are labelled by k integers. For compact, connected G, we will approach the problem of determining its irreducible rep-resentations by considering a torus subgroup T of G that is as large as possible and relating the representation theory of G (which we don’t understand) and that of T (which we do understand). This will involve the space G/T of T cosets of G. The geometry and topology of G/T is quite interesting and intimately involved in the problem of finding the representations of G. The spaces G/T have several different characterizations that we will explore later. They are not only Kähler manifolds, but projective algebraic varieties and so can be studied with the methods of algebraic geometry. They are sometimes called “flag manifolds”, since in the case of G = U(n), G/T is the space of complex “flags ” in Cn, i.e chains of inclusion