Representation Theory of p-Adic Groups

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ABSTRACT Philip Kutzko University of Iowa DMS-97 32527 Professor Kutzko will study the smooth complex representation theory of a reductive p-adic group using the method of types. A type in a group is a compact open subgroup together with an irreducible representation in such a way that the pair carries information about all irreducible representations. This approach has been used successfully in the past to study other linear groups. Among other projects, the investigator will try to apply the method of types be applied in the context of split classical groups with the goal of understanding reducibility of parabolically induced representations. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and consider the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples above. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, the subject has been systematized into a branch of number theory called the Langlands program. This project is an important part of the Langlands program.