Spherical functions on p-adic homogeneous spaces.

设G是定义在p-ady域k上的代数群，X是G的齐次空间，X=X(K)，H(G，K)是G=G(K)关于G的极大紧子群K的Hecke代数。H(G，K)-X上的公共特征函数称为球函数，这是数论和表示论中感兴趣的对象。在X中抛物子群关于轨道的某些条件下，我们利用函数方程和群的球函数给出了球函数的显式公式。我们可以将它应用于对称形式和厄米特形式的空间，这是数论中有趣的对象。在这些空间中，球函数可以看作是局部表示密度的生成函数，因此它们是密切相关的。得到对称形式和厄米特形式表示的局部密度的显式公式是数论中的一个经典问题.对于非分支厄米特形式空间X，我们用上述方法确定了球函数的一个显式公式，并以Schwartz-Bruhat函数的空间结构S(K\X)为Hecke模.我们还利用X上的球函数得到了两种局域密度的显式公式。通过与F.Sato的共同研究，我们得到了一般尺寸的p≠2对称形式的局域密度的完全显式公式。该方法是通过Iwahori子群的作用对对称形式进行分类，并计算某些高斯和.将上述方法推广到Hermitina形式的情形，得到了一般尺寸的p≠2的厄米特形式的局部密度的显式公式.

Let G be an algebraic group defined over a p-adic field k, X a homogeneous space of G and X = X (k).Let H(G,K) be the Hecke algebra of G = G(k) with respect to a maximal compact subgroup K of G.An H(G,K)-common eigenfunction on X is called a spherical function, which is an intereted object both in Number Theory and Representation Theory.Under certain condition on orbits by a parabolic subgroup in X, we have given a method to obtain explicit formulas of spherical functions by means of functional equations and spherical functions of groups. We can apply it to the spaces of symmetric forms and hermitian forms, which are interesting objects in Number theory. In these spaces, spherical functions can be viewed as generating functions of local densities of representation, so they are closely related. It is a classical problem in Number Theory to obtain explicit formulas of local densities of representations for symmetric forms and hermitian forms.For the space X of unramified hermitian forms, we have determined an explicit formula for spherical functions by the above method and the structure of the space of Schwartz-Bruhat functions S(K＼X) as Hecke module. We have also obtained two kinds of explicit formulas of local densities by using spherical functions on X.By a joint research with F. Sato, we have obtained a complete explicit formula of local densities of symmetric forms for p ≠ 2, in general sizes. The method is to classify symmetric forms by the action of Iwahori subgroup and calculate certain Gaussian sums.Extending the above method to the the case of hermitina forms, we have obtained an explicit formula of local densities of hermitian forms for p ≠ 2, in general sizes.