In this project we will study irreducible unitary representations of semisimple Lie groups. We propose to give realizations of irreducible representations as holomorphic sections on a Stein extension of the Riemannian symmetric space G/K. The advantage of this approach is that it gives realizations of irreducible representations in a purely holomorphic setting, thus allowing methods from complex analysis such as Hardy space techniques. Let us call such a Stein extension M. The important properties M should possess are: (a) M is a Stein domain in a complexification of G/K, (b) M depends only on the group G, and (c) A large number of representations should occur as spaces of holomorphic sections on M. This project has two separate parts. First we will realize irreducible unitary representations using Szego kernels. The representations occur as spaces of holomorphic functions on M and the boundary values and inner product will be described in an explicit and natural way. We will also determine the proper space M. If G is of hermitian type then G/KxG/K is the correct choice for M . Otherwise there are some candidates for M, namely a parameter space of maximal compact subvarieties of an appropriate flag domain and the maximal domain in the complexification of G/K for which the Szego kernel extends holomorphically. These two candidates may turn out to be essentially the same (as in the known cases). We will need a simple explicit description of M and its boundary. Our second project is to study the indefinite quantization of elliptic coadjoint orbits. The goal is to explicitly give the unitary structure on the representations occurring in the Dolbeault cohomology of an elliptic coadjoint orbit. This is continuation of previous collaborations.
Symmetries play an fundamental role in mathematics and physics. Semisimple Lie groups provide a technical tool for studying such symmetries. For example a quantum mechanical system is, in part, a Hilbert space on which the group of symmetries acts. This is a representation of the group. Other representations occur very naturally in many areas of pure mathematics, such as number theory, classical analysis and probability. A fundamental problem is to classify all irreducible unitary representations. Along with this classification is the problem of constructing the representations. Our project is concerned with constructing unitary representations in a very natural way. Such an understanding of representations will lead to more applications of the theory to concrete physical problems as well as other areas of pure math.
