Hecke Algebras, Buldings and Harmonic Analysis on p-adic Groups

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9970454This project concerns problems in representation theory. Recent progress in the theory of "Types" suggests that the Harmonic analysis on p-adic groups can be completely understood via harmonic analysis on the associated Hecke algebras. Moreover these Hecke algebras are expected to be generalized affine Hecke algebras and so should be easier to analyze. Dr. Kim will try to establish the fact that many of the classical groups defined over p-adic fields give rise to algebras that are generalized affine Hecke algebras. Further she intends to study the relation between buildings of the classical groups and buildings of the general linear group. Finally she intends to consider how base changes relate to the representations of Hecke algebras.Representation theory is a major mathematical technique for exploiting the presence of symmetry. For example, the structure of the hydrogen atom, one of the fundamental computations in quantum mechanics, is controlled by representation theory. Another way of thinking about representation theory is as a generalization of the theory of eigenvalues and matrices that many people see in a college course on basic linear algebra. This project addresses two important questions in representation theory. One potentially has important applications to number theory. The other concerns a generalization of the theory of angular momentum in quantum mechanics. It will help describe how pair of interacting quantum mechanical systems of a rather general type breaks up into simpler systems.