Study on automorphic forms on algebraic groups and associated zeta functions

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1. Metaplectic representations of unitary groups :We studied metaplectic representations of unitary groups over local fields and gave their "universal" splitting, which are in particular useful in the study of theta lifting. As an application of this result, we gave an explicit character formula for metaplectic representations.2. Fourier-Jacobi expansion of automorphic form on unitary groups of degree three :We reformulated Shintani' s theory on Fourier-Jacobi expansion of automorphic forms on unitary groups of degree three in adelic language, and calculated explicit form for Fourier-Jacobi expansion of Eisenstein series and Kudla lifts, theta lifts from elliptic modular forms. As an application, we gave a criterion for the non-vanishing of Kudla lifts.3. Siegel-Weil formula :We studied a non-regularized Siegel-Weil formula in the case of the dual reductive pair (U(2,2), U(2, 1)).4. Inner product formula for Kudla lifts :Using the formula stated in 3, we gave an explicit formula for the Petersson norms of Kudla lifts in term of special values of automorphic L-functions. As an application, we gave a criterion for the non-vanishing of Kudla lifts different from the one given in 2. (The studies 2- 4 are joint works with Takashi Sugano).5. Support for the Summer School of Number Theory :We supported financially the Summer School of Number Theory held annually.The themes were as follows : "Zeta functions" in 2001, "Prehomogeneous vector spaces" in 2002, "Iwasawa theory" in 2003 and "Fundamental groups and Galois representations" in 2004.