DMS-9424053 PI: Collingwood The proposed research will investigate the connection between the singularity theory of Harish-Chandra modules and the matrix coefficient asymptotics of Harish-Chandra modules. The first step is to link the existence of global Whittaker models, the structure of cell representations of the Weyl group, the theory of nilpotent orbits and the structure of Jacquet modules. Next one uses recursive algorithm to compute the weight filtration of any degenerate series. Finally, relating the PI's earlier work on smooth embeddings with the embeddings of Matsuki-Oshima, one obtains better understanding of the embeddings of irreducible representations into principal series representations. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.
