Analysis, geometry and arithmetic of automorphic forms of many variables and higher dimensional modular varieties

关于半单李群的矩阵系数，我们得到了SU（2，2）的中间离散级数的渐近展开式的一个更精确的结果。本文研究了SU（3，1）矩阵系数的显式表达式（两者都是与T。Hayata和H. Koseki）。利用渐近展开式，我们得到了Sp（2，R）的某些P_J-主级数表示的c-函数的显式表达式（与M. Iida）。推进了亏格2的Siegel模群的基本整环的0-胞腔的研究。

About matrix coefficients of semi-simple Lie groups, we obtained a more precise result for the middle discrete series of SU(2, 2) about the asymptotic expansion. We investigated the explicit formula of the matrix coefficients of SU(3, 1)(both are joint works together with T. Hayata and H. Koseki). Utilizing the asymptotic expansion, we obtained the explicit formula of the c-functions of certain P_J-principal series representations of Sp(2, R) explicitly(joint work with M. Iida). We push forward the investigation of 0-cells of the fundamental domain of the Siegel modular group of genus 2.