Representation Theory of Reductive P-Adic Groups

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ABSTRACT Gopal Prasad Michigan 98 01262 Gopal Prasad will continue his work on the classification of admissible representations of reductive p-adic groups in terms of their restrictions to compact-open subgroups. He hopes to be able to refine the theory of ("unrefined")minimal K-types developed by him and A. Moy, and also find a suitable extension of the Kirillov theory for this purpose. In another direction, he proposes to find a proof of the centrality of the congruence subgroup kernel for simply connected anisotropic groups of type A-n and determine the normal subgroups of the group of rational points of these groups. A reductive p-adic group is the special collection of symmetries of an arithmetic or geometric object. This study is expected to unravel intricate and fundamental properties of the objects and patterns which are of interest in arithmetic and geometry through the study of these symmetries. One way to understand a group of symmetries is through a representation of its elements as matrices. Representation theory does this in a systematic way. Ultimately the goal is to be able to construct and classify all representations of the reductive p-adic groups, and from this settle important questions about the underlying geometric object. This is important because so many interesting arithmetic and geometric objects have interesting symmetries that form a reductive p-adic group.