The Cohomology of Weight Varieties

We describe the cohomology ring of the symplectic reductions by tori of coadjoint orbits, or weight varieties. Weight varieties arise from representation theory considerations, and are temed as such because they are the symplectic analogue of the T-isotypic components (weight spaces) of irreducible representations of G. Recently, Tolman and Weitsman expressed the (ordinary) cohomology ring for the abelian quotients of any Hamiltonian T space M in terms of other data, such as the equivariant cohomology of M and the restriction in equivariant cohomology to the fixed points. While the TolmanWeitsman result indicates that a finite number of calculations is necessary to find the cohomology ring of a weight variety, one must choose among an infinite number the relevant ones. Furthermore, there is no systematic way to carry out the calculations. Here we apply the Tolman-Weitsman result to the specific case of weight varieties. Using results of Kirwan, Goresky-KottwitzMacPherson, Atiyah-Bott, Bernstein-Gelfand-Gelfand, and Kostant, we find a simplification which allows an explicit description for the cohomology rings of weight varieties for SU(n) coadjoint orbits. Our method consists of two distinct contributions to the work of previous authors: an explicit list of a finite number of sufficient calculations, and a method for carrying out these calculations. Thesis Supervisor: Victor Guillemin Title: Professor of Mathematics