Mathematical Sciences: Representation Theory and Differential Geometry

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Bertram Kostant will continue his research in the theory of Lie groups. The work will proceed in three directions 1) a plan for carrying out geometric quantization of certain nilpotent orbits by use of quasi-invariant differential operators; 2) an approach to determine generalized exponents by the use of special Kazhdan-Lusztig polynomials in the affine case and new information on the equivariant K-theory of the corresponding infinite dimensional flag manifold; 3) an approach to establish the positive definiteness of the Hermitian structure on the space of physical states. The determination of representations whose wave front set is the closure of the Joseph orbit is an important and difficult problem. These representations are generalizations of the oscillator representation for the two fold covering of the symplectic group. The study of the Kazhdan-Lusztig polynomials for Lie groups associated with affine Lie algebras using geometry of the corresponding flag variety and its relationship with K- theory is a very important aspect of this research.