This proposal is concerned with the study of automorphic forms, representations of reductive groups and their applications. Dan Barbasch, together with various coworkers, is proposing to continue investigations of the unitary dual of real and p-adic groups. In particular he will study necessary conditions for unitarity. He will also continue the study of unipotent representations, in particular their associated cycles. Joint with various coworkers he will study occurences of representations in the dual reductive pairs correspondence, and for p-adic groups he will study the Bernstein center. Birgit Speh, together with various coworkers, will continue the study of cohomology of locally symmetric spaces. In particular she will study representation theoretic descriptions 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 concerned with functions that are solutions to differential equations which have certain symmetry properties. These properties are expressed in mathematics as saying that the solutions form a unitary representation of a reductive group. A major part of this proposal is concerned with the determination of the building blocks which are called unitary irreducible representations. The aforementioned functions relevant to number theory are called automorphic functions and have expansions in terms of unitary irreducible representations. These expansions can be thought of as generalizations of classical Fourier series. The problem of convergence of these expansions is important for applications to number theory, and forms an important component of this proposal.
