Automorphic forms of several variables and spherical functions of homoge-neous spaces

齐次空间的球函数或Shintani函数对于研究自同构形式的周期的傅里叶展开和与自同构形式相关的L函数是重要的。我们主要研究经典群的Shintani函数。特别地，我们证明了Shintani函数的唯一性和存在性，并在某些情况下得到了它们的显式公式。应用它们研究了自同构L函数的解析性质和算术性质。(Muarase-Sugano)我们的下一个目标将是定义与球形子群相关的合适的几何对象，并根据它们的上同调群来解释Shintani函数。

The spherical function or the Shintani function of homogeneous spaces are important in order to study the Fourier expansion of period of automorphic forms and L-functions associated to automorphic forms.We studied mainly the Shintani function of the classical groups. in particular we proved uniqueness and existence of the Shintani function and in some cases we obtained explicit formulas for them. We applied them to examine analytic and arithmetic properties of automorphic L-functions. (Muarase-Sugano)It would be our next objective to define suitable geometrical objects associated to spherical subgroups and to interpret the Shintani function in terms of their cohomology groups.