Abstract Li The ``classical'' theory of theta series and reductive dual pairs in the symplectic group has become a standard part of representation theory and automorphic forms, and continues to attract a great amount of interest and activity. The analogous theory for exceptional groups was begun just a few years ago, and is going through rapid development. In previous NSF sponsored projects, the proposer fruitfully studied some basic questions in the theory, such as the correspondences of infinitesimal characters, and developed new methods to determine the discrete spectrum of exceptional dual pairs. Here he proposes to continue his study of the remaining fundamental problems, and to apply the results to specific problems such as cohomology of arithmetic manifolds and unipotent representations. He plans to combine his own methods with those of Burger and Sarnak, and hopes to establish close relations between discrete series of certain non-symmetric homogeneous spaces and the spectrum of reductive dual pairs. The theory of Lie group representations is concerned with the study of certain kinds of symmetries. When two groups are embedded in a larger ambient group in a interesting way, one could ask how the symmetries of the larger group would behave upon restriction to the two embedded groups. Frequently this leads to deep information which plays important roles in both representation theory and number theory. Building upon earlier work by the proposer and other mathematicians, this project aims to settle several fundamental questions in the study of such restrictions of symmetries, and apply the results to the study of geometry and arithmetic of various interesting spaces associated to the groups in question.
