Mathematical Sciences: Representation Theory of p-adic Groups

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This research concerns the classification of the category of smooth representations of a p-adic group via restriction to compact, open subgroups. This classification is of interest both intrinsically and because of the fundamental role that the representation theory of p-adic groups plays in the program outlined by R.P. Langlands for tackling certain basic questions in the theory of numbers. In a series of papers with C.J. Bushnell, the principal investigator has singled out certain pairs, called types, consisting of a compact, open subgroup and an irreducible representation of this subgroup. The existence of such a type leads to the possibility of analyzing representations via Hecke algebra methods; the existence of a complete set of such types in the case of the groups GL(N) has already paid substantial dividends. This project extends the theory of types to others p-adic groups. The local Langlands correspondence is also considered in light of the theory of types, as is the possibility of using types to compute local factors. This research falls into the broad category of p-adic algebraic groups. Historically, algebraic groups arose in an effort to describe all the transformations or symmetries of an n-dimensional space. The spaces under consideration here, however, are not the familiar real spaces like the line and the plane, but related objects which are important in number theory and pure algebra.