Study on spherical functions on Lie groups and automorphic L-functions

我们研究的目的是通过它们的积分表示(Zeta积分)建立关于自同构形式(自同构L函数)的Zeta函数的基本结果，例如解析延拓和函数方程。利用广义球函数的显式公式，即出现在自同构形的傅里叶展开式中的特殊函数，我们计算了阿基米德Zeta积分。特别计算了SO(2n+1)×GL(M)上的L函数和GL(3)上的标准L函数的阿基米德Zeta积分，得到了局部函数方程和整体函数方程。

The aim of our study is to establish fundamental results on zeta functions for automorphic forms (automorphic L-functions) such as the analytic continuations and the functional equations, through their integral representations (zeta integrals). Using explicit formulas for generalized spherical functions, that is, special functions appearing in the Fourier expansions of automorphic forms, we have computed archimedean zeta integrals. Especially we calculate the archimedean zeta integrals for the L-functions on SO(2n+1)×GL(m) and the standard L-function on GL(3) and obtained the local and the global functional equations.