Relation between automorphic forms and zeta functions associated with prehomogeneous vector spaces

我们所研究的主要问题是：(1)找出与准齐次向量空间相关的Zeta函数，这些Zeta函数是依附于自同构形式的某种Zeta函数；(2)构造一个不属于准齐次向量空间理论的(局部)函数方程理论。(1)根据Sato和Kimura的分类理论，将不可约正则准齐次向量空间分为5类经典空间和24类零星空间。我们用某些实解析Eisenstein级数的标准L函数或Koecher-Maass Zeta函数证明了与4个经典类型的准齐次向量空间相关的Zeta函数。证明这些结果所必需的结果之一是Siegel级数(=Eisenstein级数的傅里叶系数的p部分)的新的积分表示。作为新积分表示的另一种应用，我们证明了正交群的p元半单对称空间上Siegel级数与球函数之间的联系公式。(2)证明了，给定Ra上满足局部函数方程的一对齐次多项式和R^m到R^n的一对非退化对偶二次映射，则多项式的二次映射的拉回也满足局部函数方程。这推广了Faraut-Koranyi-Clerc的一个结果，并且我们可以构造许多不在非齐次向量空间理论中涵盖的函数方程的例子。我们还对二次空间上的非退化对偶二次映射进行了分类，并证明了这种二次映射与2个Clifford代数的张量积的表示一一对应。

The main problems we investigated in this research project are(1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms(2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :(1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.(2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras.