The investigator proposes seven intertwined research projects, all on the interface between group theory and geometry. For the most part these proposals are geometrically motivated approaches to specific problems in modern harmonic analysis connected to the representation theory of semi- simple Lie groups. The first is an approach to construction and analysis of singular unitary representations of real reductive Lie groups, in two steps: (i) geometric construction of those representations, as Frechet space representations on cohomology of homogeneous vector bundles over flag domains, and (ii) transforming the representation space to a space of functions on a Stein manifold defined by a system of differential equations, by means of a double fibration transform (the complex Penrose transform is a particular case) from the flag domain to its linear cycle space. The second project is an approach to construction and analysis of unitary and other representations for a class of infinite dimensional Lie groups, the direct limits of finite dimensional Lie groups, especially strict direct limits of finite dimensional real and complex reductive Lie groups, both in the analytic category and in the algebraic category. The third project is to complete the investigator's work on the Harish-Chandra Schwartz space of a general semisimple Lie group, by synthesizing structural analyses of the relative Schwartz spaces. The investigator's fourth project is to extend the notion of Dirac cohomology to a notion of partial Dirac cohomology so that it applies to all the representations of a semisimple Lie group that appear in the Plancherel formula. The fifth project is to extend the investigator's isospectral group method from the setting of spherical space forms to the setting of locally symmetric Riemannian spaces. The sixth project is to investigate certain restrictions of discrete series representations to a class of subgroups of great geometric interest. And, seventh, the investigator will continue his development of a method for direct reading of the character and growth properties (asymptotics) of admissible representations of finite dimensional real reductive Lie groups from the basic data that specify their construction on cohomology spaces of homogeneous vector bundles over flag domains. One goal here is to do this in such a way that is directly applicable to the first and second projects.
These seven research projects all depend on the use of symmetry to clarify analytic (and in one case geometric) problems. Traditionally symmetry considerations are used to simplify matters by decreasing the number of variables, but here they are used to enable the use of insight, tools and results from geometry and analysis. The symmetries are embodied in group theory, which is the algebraic abstraction of the notion of symmetry. But modern Lie group theory incorporates classical analysis (calculus, differential equations...) and is closely tied to the geometry (Riemannian, symplectic, Kaehler, ...) of the configurations on which the groups act as symmetries. An important aspect of this synthesis of geometry and analysis is a geometric form of quantization that is particularly well suited to the sorts of finite dimensional groups considered in several of the projects. This geometric quantization, originally inspired by physics and developed in some detail by mathematicians, has in turn been very useful in a variety of settings in mathematics and physics. This is very much the case in six of the seven projects, where the geometric quantization is combined with modern differential geometry to understand various analytic problems. This is especially evident in the first and fourth projects, whose objectives can be viewed as a sort of dequantization.
