There are two major themes in this proposal, the determination of the unitary dual, and the structure of unipotent representations. Barbasch will continue and extend his previous work to find necessary and sufficient conditions for unitarity for classes of Langlands parameters such as principal series in real and p-adic groups. The aim is to find proofs that are as uniform and conceptual as possible, and express the answers in a closed form. Unipotent representations are (conjectured to be) the building blocks of the unitary dual. Their character theory plays an important role in the theory of automorphic forms. Barbasch will study the character theory of unipotent representations in the real case, with an emphasis on relations to endoscopic groups, particularly twisted ones. Realizations of unipotent representations as local factors of automorphic forms, and in the context of the dual pairs correspondence are also envisioned. This research will provide a deeper understanding of representation theory of reductive Lie groups, in particular unitary representations, which play an essential role in the applications of representation theory to automorphic forms, mathematical physics analysis and geometry.
This research will generate problems suitable for Ph.D. theses as well as combinatorial problems for undergraduate research. Together with several faculty members at Cornell, Barbasch will run the ``Sophus Lie Days at Cornell'' an instructional conference designed to expose undergraduates and graduate students in mathematics and other sciences to the theory of Lie groups and their representations, one of the aims being to get them interested and involved in research at an early stage. The results of this research will be disseminated through research papers, talks at seminars and conferences, and various web sites.
