Representation Theory for Reductive Groups Over Local Fields

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The investigator proposes to study a number of interrelated topics having to do with representation theory and harmonic analysis for reductive groups over local fields. There are four general topics, some of which are divided up further. The first topic concerns a strange new formula involving characters of discrete series representations of real groups in the Hermitian symmetric case, a formula that will no doubt be needed in order to compare the Lefschetz formula with the Arthur-Selberg trace formula. The second topic concerns transfer factors in various settings: inner forms, Shimura varieties, descent for twisted transfer factors. The third topic concerns bad reduction of Shimura varieties. Currently this has lower priority but that could change in the course of the next three years. The fourth topic consists of five distinct but related questions about orbital integrals for reductive groups over non-archimedean local fields. In less technical language the investigator proposes to study a number of topics that belong to the theory of automorphic forms, a beautiful area of mathematics with ties to all three main branches of mathematics: algebra, analysis and geometry. The last several decades have been exciting times for workers in this area, one of the highlights being the essential use of automorphic forms in Wiles's spectacular proof of Fermat's Last Theorem. The coming decades promise many further exciting developments, and the investigator hopes to contribute to these directly, by solving some of the questions raised in the proposal, as well as indirectly, by helping to train young workers in this technically demanding field.