Stanton In previous joint work with Kroetz the proposer constructed a specific domain for which they showed that most matrix coefficients of irreducible unitary representations of semisimple Lie groups as well as related automorphic functions have a canonical holomorphic continuation. For those representations occuring in the Plancherel density of the associated Riemannian symmetric space they used their construction to identify a natural Kahler structure on these domains. We shall attempt to enlarge the class of the unitary representations so related to complex differential geometry by studying Kahler structures on vector bundles over the previously constructed domain. The success of such a complex differential geometric formulation of parts of the unitary dual should have interesting applications to harmonic analysis. Also following up on our previous work, we shall attempt to obtain estimates on the Fourier coefficients of automorphic functions by analyzing the boundary values of their holomorphic continuation on the distinguished boundary of this domain. A different project is joint work with Slupinski in which we propose to obtain very detailed structure of the topology and differential geometry of nilpotent co-adjoint orbits of semisimple Lie groups. We have identified a class of orbits associated to 5-gradings of the corresponding Lie algebra and have substantial progress towards relating this to an extended version of conformal geometry. Possible payoff of these investigations include a compactification of the moduli space of exceptional holonomy structures on low dimensional manifolds as well as a geometric construction of representations associated to these orbits. Any complex nxn matrix may be written as a sum of two matrices where one is diagonalizable and the other is nilpotent, i.e. the matrix times itself some number of times is the zero matrix. The group of invertible matrices acts via conjugation (i.e. pre-multiply by the matrix and post-multiply by the inverse) on the vector space of all complex nxn matrices. As the sets of diagonalizable and nilpotent matrices are preserved it is reasonable to seek for each matrix a representative from these classes which is in some uniformly recognizable form. For the diagonalizable class the diagonal matrices are a natural choice and are universally used. On the other hand, the nilpotent elements have only a finite number of possibilities leading to the familiar standard form called Jordan blocks in linear algebra. These finitely many possibilities of a fixed block seem to be richer in geometric structure that those of the semisimple classes but much less understood. The proposed research is give a detailed description of the differential geometry of these class of nilpotent matrices. Somewhat surprisingly, the geometry these classes lead to include ones of current interest to theoretical physicists in string theory as well as geometers. One of the applications of our work is to present a space of such geometries, and to examine possible degenerations in the geometry as one approaches the boundary of the space. Another possible application of our description is towards identifying new geometries associated with certain algebraic structures on the space of matrices. Our approach has many points of contact with a construction proposed many years ago in relativity theory by the physicist Penrose using an algebraic object called spinors. Indeed, the use of higher dimensional spinors is critical to our investigations.
