The PI proposes to study various basic questions in representation theory and the theory of automorphic forms arising from the Langlands program, and their applications to questions of arithmetic interest. For a number of years, the PI has been pursuing the construction and classification of square-integrable automorphic forms as predicted by Arthur's conjecture, especially in the context of the exceptional groups. The PI hopes to complete this study in the next 3-year period. He also intends to study the local Langlands correspondence for certain classical and exceptional groups. In another direction, the PI hopes to establish certain cases of the Gross-Prasad conjecture regarding the restriction of representations of an orthogonal or unitary group to a smaller one, which has applications to special values of L-functions. Finally, the PI proposes to establish the second term identity for the regularized Siegel-Weil formula in the context of the theory of theta correspondences.
The Langlands program is an integral part of modern number theoretic research. Its initial goal is to understand certain groups which arise naturally in number theory and representation theory. In recent years, it has expanded beyond its traditional boundaries to connect with areas such as algebraic geometry and mathematical physics. It has already found unexpected applications in real life. Indeed, the Jacquet-Langlands correspondence, which is one of the first major results in the Langlands program, has been exploited to give a construction of the so-called Ramanujan graphs. These graphs are highly connected and serve as efficient networks. They have turned out to be very useful in communications theory. It is hoped that the questions investigated in this proposal can help to unearth more applications of this type.
This project is devoted to various aspects and applications of the so-called Langlands program, which consists of a series of deep conjectures (and results) linking two a priori unrelated but individually important areas of contemporary mathematical research. One of these is number theory, which is classically concerned with finding integer solutions to polynomial equations, and whose modern study has evolved to that of a class of objects known as Galois representations. The other is representation theory which is concerned with symmetries in nature and whose modern study is concerned with that of a class of objects known as automorphic forms. The deep insight of Langlands, forultaed some 40 years ago, is that these two classes of objects are the same, in very precise ways. In course of this project, the PI has established a basic conjecture in the Langlands program for some specific class of groups (the local Langlands conjecture for GSp(4) and its inner forms). This series of work has been widely used by researchers in related areas. The PI has also worked on extending the Langlands program beyond its classical realm, namely for the nonlinear covering groups and the so-called spherical varieties. In joint work with Gordan Savin, for example, the PI extended the well-known results of Waldspurger on the local Shimura correspondence to the general metaplectic groups. In joint work with R. Gomez, the PI has verified cases of a recent conjecture of Sakellaridis-Venkatesh on the spectrum of spherical varieties of low rank. Finally, the PI, in collaboration with B. Gross and D. Prasad, has considered a series of ``branching problems" in the representation theory of classical groups and formulated a series of conjectures. This series of conjectures has been very influential in the last few years and has stimulated a huge amount of research activities. Finally, the PI's research has led to several ramifications which were pursued by the PI's graduate students. During the course of this project, the PI has supervised 4 PhD students of diverse background (including a woman, an African American, a white American and a Mexican). The PI has also given a series of instructional lectures on the subject matter of this project in instructional conferences in Seville and Korea.
