Mathematical Sciences: Automorphic L-functions and Interwining Operators

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Shahidi In the next three years, the investigator will study the following problems in Automorphic Forms and Representation Theory. As his first problem, the investigator wants to show that the poles of the standard intertwining operators for parabolically induced representations of a quasisplit group over a local field are among those of certain Langlands L-functions. This will have important consequences in the theory of Eisenstein series and global liftings. In particular, he plans to prove a definitive reducibility criterion for representations induced from irreducible quasi-tempered generic representations of Levi subgroups of these groups in terms of L-functions. This is a consequence of another problem to be studied by him which extends a result of Vogan to p-adic groups. It states that the standard modules whose Langlands quotients are generic are irreducible. These are parts of a joint project with W. Casselman. As his second and third problems, he will continue his work on the tempered spectrum of classical groups (with D. Goldberg) and their residual spectrum (with H. Kim). He also plans to prove the equality of certain coefficients defined by two different methods (Rankin-Selberg and Langlands-Shahidi) as well as the study of different approaches to understanding poles of L-functions using the second method. He will also try to establish certain identities satisfied by normalized intertwining operators, extending his results from the tempered case to the non-tempered ones. Finally, he plans to further study the symmetric cube L-function of a cusp form on GL(2) and its twists with arbitrary cusp forms with the hope of better understanding the symmetric cube lift from GL(2) to GL(4). The research falls in the general mathematical area of the Langlands program.The Langlands program is part of Number Theory, which is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving forc e in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.