Analyse sur les groupes de Lie et théorie des représentations
The book is based on lectures presented at the summer school on analysis on Lie groups and representation theory, held in 1999 in Kénitra, Morocco. It contains a written form of three series of lectures. The first one (given by M. Vergne, recorded by S. Paycha) contains a description of the equivariant cohomology of a manifold, which was introduced independently by N. Berline and M. Vergne, E. Witten and M. F. Atiyah and R. Bott in the 80’s. In the special case of S1-action with isolated fixed points, the Paradan formula is used for a description of the equivariant cohomology in terms of fixed points of the action. As an important application, it is shown how to obtain the Duistermaat-Heckman stationary phase formula for a U(1)-action on a symplectic manifold. The second series of lectures (given by F. Rouviére) was devoted to the Damek-Ricci spaces. A manifold is harmonic if the mean value theorem holds for harmonic functions. The Damek-Ricci spaces are harmonic manifolds but they are not symmetric spaces. Basic facts from geometry and harmonic analysis on the Damek-Ricci spaces are described in the course. The third series of lectures (given by J. Faraut) is devoted to Hilbert spaces of holomorphic functions invariant under the action of a group of automorphisms of a complex manifold. The set of such Hilbert spaces forms a convex cone and it is possible to use methods of the Choquet theory for a description of its structure and for an integral representation of their reproducing kernels. The last parts are devoted to the case of invariant domains in the complexification of a compact symmetric space. A part of the school’s program was reserved for lectures on the research work of participants, the list of lectures and their abstracts can be found in the book. The book brings nice reviews of very interesting areas of the field.