Topics in representation theory and number theory

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." Benedict Gross plans to do research on the boundary between representation theory and number theory, exploring the implications of the local and global Langlands correspondence. He expects to extend his conjectures with D. Prasad, on restriction from SO(n) to SO(n-1), to restriction problems for all classical groups. This will have implications for the arithmetic of Hermitian and orthogonal Shimura varieties. Gross also plans to investigate the simple supercuspidal representations he introduced with M. Reeder. He hopes to determine their wild Galois parameters in all cases, and to exploit their simple matrix coefficients in the trace formula.Benedict Gross plans to work on the boundary between questions in number theory, such as the representations of Galois groups, and questions in the representation theory of groups, such as the decomposition of the restriction of representations of the unitary group U(n) to the subgroup U(n-1). These problems are quite mysteriously related, via the conjectural Langlands correspondence, and Gross hopes to explore their connections in more detail.