The primary goal of this project is to extend the Langlands-Shahidi method to non-generic cusp forms of certain classical groups. This is joint work with S. Friedberg. Our plan is to use Bessel models to play the role Whittaker models play in the generic case. This work should lead in the direction of establishing the automorphic transfer of cusp forms from classical to general linear groups, generalizing the generic case established by Cogdell - Kim - Piatetski-Shapiro - Shahidi. It should also allow for the completion of the work of Jiang - Soudry on the local Langlands correspondence for special orthogonal groups of odd dimension. Even establishing the transfer in low rank cases would give significant results. For example, we anticipate results on Hilbert-Modular forms with trivial central character from the most primitive example. In the course of this project we expect to address the local theory for the Archimedean case, as well as refining our earlier results in the non-Archimedean case. Additional goals of this project are to understand the poles of intertwining operators for maximal parabolic subgroups of quasi-split reductive groups over non-Archimedean fields (joint with F. Shahidi), construction of types, in the sense of Bushnell - Kutzko for the non-cuspidal spectrum of classical groups over non-Archimedean fields (joint with P. Kutzko and S. Stevens), and understanding reducibility of parabolically induced representations of Spin groups over non-Archimedean fields via Knapp - Stein theory.
Among the most powerful results in mathematics are those that display a connection between seemingly disparate ideas. These connections tend to improve our understanding of both ideas, and are interesting both as a matter of philosophy and application. For example, the power of Calculus lies not in our understanding of differential and integral calculus separately, but in the connection between the two via the Fundamental Theorem of Calculus. This is what lends the subject to the myriad applications which have impacted upon society, but also makes the subject a great achievement in human thought. This project is designed to address certain problems within what is known as the Langlands Program, which seeks to unify three major areas within mathematics. Namely, Number Theory, Harmonic Analysis, and Algebraic Geometry. Our work concentrates on one aspect of the conjectural connection between the first two areas. Tremendous progress has been made in recent years, and we hope to contribute to furthering our understanding of these subjects. This project will have broader impact not only through its contribution to mathematics (and more specifically number theory), but through raising the level of research at a public university with a diverse student population.
