Mathematical Sciences: Discrete Spectrum of the Oscillator Representation and Applications

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Li will study the discrete spectrum of the restriction of the oscillator representation to reductive dual pairs. He will relate this investigation to the study of the discrete series of generalized Stiefel manifolds. The wave front sets and asymptotic distributions of discrete series will play a vital role. The results will be applied to the construction of unitary automorphic forms on classical groups, including those with non- zero cohomology. They will also be used to construct elements of the Rammanujan Dual. 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.