Mathematical Sciences: Smooth Representations of Reductive P-Adic Groups via G-Types

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Morris 9623288 Admissible representations of reductive groups over local fields have been the object of study for some time by virtue of the Langlands program relating them to local and global number theory. Among these representations, the supercuspidal representations have always played a distinguished role as building blocks. One way to construct them is via induction from compact open subgroups, and this has led to the philosophy of studying admissible representations of a reductive group by restricting them to appropriate compact open subgroups. Bushnell and Kutzko have applied this philosophy with great success to the general or special linear groups; this has led them to formalize it via the very general notion of S-types. This approach has also been successful in studying the smooth representations of an arbitrary reductive group that contains a fixed vector under the prounipotent radical of some parahoric subgroup (the "level zero case"). Finally, preliminary results suggest this approach can work more generally for other groups. The PI will continue his investigations in this area, and in particular, he will consider the following topics: (a) The study of S-types for classical groups; in particular the classification of the admissible dual of classical groups, (b) Applications of (a) to the study of the reducibility of generalized principal series of classical groups and related arithmetic, (c) Further study of problems arising from the level zero case; in particular, the classification of the level zero part of the admissible dual in terms of Langlands parameters and applications of level zero S-types to the study of reducibility and R-groups. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. Th e Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.