Representation Theory and Automorphic Forms

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AbstractSpeh/BarbaschThis proposal is concerned with the study of automorphic forms,representations of reductive groups and their applications. Dan Barbasch, together with various coworkers, is proposing tocontinue investigations of the unitary dual of real and p-adicgroups. In particular he will study necessary conditions forunitarity. He will also continue the study of unipotentrepresentations, in particular their associated cycles. Joint with various coworkers he will study occurences of representations in thedual reductive pairs correspondence, and for p-adic groups he willstudy the Bernstein center. Birgit Speh, together withvarious coworkers, will continue the study of cohomology of locally symmetricspaces. In particular she will study representation theoreticdescriptions of modular symbols. She will also work on proving uniform convergence of terms in the Arthur-Selberg trace formula. It is expected that the results of this proposal will contribute significantly to the understanding of the geometry and topology of locally symmetric spaces. Graduate and undergraduate students as well as postdoctoral faculty are expected to be involved in studying problems generated by this research. Many problems in number theory and mathematical physics are concernedwith functions that are solutions to differential equations which havecertain symmetry properties. These properties are expressed inmathematics as saying that the solutions form a unitary representationof a reductive group. A major part of this proposal is concerned with thedetermination of the building blocks which are calledunitary irreducible representations. The aforementioned functionsrelevant to number theory are called automorphic functions and haveexpansions in terms of unitary irreducible representations. Theseexpansions can be thought of as generalizations of classical Fourierseries. The problem of convergence of these expansions is importantfor applications to number theory, and forms an important componentof this proposal.