Study on automorphic forms on algebraic groups and associated zeta functions

1.酉群的亚辛表示：我们研究了局部域上酉群的亚辛表示，并给出了它们的“泛”分裂，这在研究θ提升时特别有用。作为这个结果的应用，我们给出了亚代数表示的一个显式特征公式.三次酉群上自守形式的Fourier-Jacobi展开式：我们用Adelic语言重新表述了Shintani关于三次酉群上自守形式的Fourier-Jacobi展开式的理论，并计算了Eisenstein级数的Fourier-Jacobi展开式的显式形式和椭圆模形式的Kudla提升、theta提升。作为应用，给出了Kudla提升不为零的一个判据. Siegel-Weil公式：研究了对偶约化对（U（2，2），U（2，1））情形下的非正则Siegel-Weil公式. Kudla提升的内积公式：利用3中的公式，我们给出了Kudla提升的Petersson范数关于自守L-函数的特殊值的显式公式。作为应用，我们给出了一个不同于文献[2]的Kudla提升不为零的判据。(The研究2- 4是与Takashi Sugano的联合工作）。支持数论暑期班：每年举办的数论暑期班都得到了我们的财政支持，其主题如下：2001年的“Zeta函数”、2002年的“预齐次向量空间”、2003年的“岩泽理论”和2004年的“基本群和伽罗瓦表示”。

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.