Square-integrable automorphic forms, local Langlands correspondence and Gross-Prasad conjecture

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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.