A study on relations between the theory of prehomogeneous vector spaces, the theory of group representations and the theory of automorphic forms

本文从函数方程与一般线性群表示的关系、自同构L函数的关系、非正则拟齐次向量空间的推广等方面研究了拟齐次向量空间的Zeta函数.(1)证明了Zeta函数的函数方程与一般线性群的退化主级数表示之间的缠绕算子密切相关，并利用这种关系得到了函数方程的Gamma矩阵的欧拉型积分表达式.这使得我们能够将函数方程中的变量变化识别为一般线性群的Weyl群中的元素的作用，并将函数方程分解为更基本的函数方程的乘积。对于p-进局部Zeta函数也有类似的结果。作为p进位理论的一个应用，我们研究了傅里叶系数…Sp(N)和GL(N)的Elsenstein级数以及球面变换理论。(2)我们证明了实解析Siegel Eisenstein级数的Koecher-Maass级数，其中Zeta函数与SO(n，n)的Siegel抛物子群作用在某个准齐次向量空间上。这一结果很有可能推广到其他经典群。Zeta函数的显式计算已经取得了相当大的进展。对于70%以上的不可约正则约化拟齐次向量空间，我们得到了Zeta函数的Riemann Zeta函数和Cohen-Eisenstein级数的Mellin变换的显式表示。(3)对于非正则拟齐次向量空间，我们发展了积分表示的一般理论和Zeta积分的函数方程，它是正则拟齐次向量空间理论的形式推广。我们还给出了非正则空间的显式函数方程的第一个例子。较少

In this research project, we investigated zeta functions of prehomogeneous vector spaces from the view points(1)relations between functional equations and representations of general linear groups,(2)relations to automorphic L-functions,(3)generalization of the theory to non-regular prehomogeneous vector spaces.(1)We showed that the functional equations of zeta functions are closely related to intertwining operators between degenerate principal series representations of general linear groups, and, using the relation, we obtained an integral expression of Eulerian type of the gamma matrices of functional equations. This enables us to identify the variable change in functional equations as an action of an element in the Weyl group of a general linear group, and to decompose functional equations into a product of more elementary functional equations. There exists a similar results for p-adic local zeta functions. As an application of p-adic theory, we investigated the Fourier coefficients … More of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic subgroup of SO(n, n) acts. It is quite probable that this result can be extended to other classical groups. A considerable progress has been made in explicit calculation of zeta functions. We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a formal generalization of the theory for regular prehomogeneous vector spaces. We also gave the first example of explicit functional equations for non-regular spaces. Less