Studies on prehomogeneous vector space and micro-local analysis

从微局部分析的角度，得到了与预齐次向量空间相关的zeta函数的各种结果。特别是对交换抛物型预齐次向量空间的zeta函数进行了实验计算。其他一些关于微局部分析的研究是由合作研究者完成的。本研究设计、分析和确定了预齐次向量空间上的不变超函数(1)，将其应用于研究zeta函数的残数和泛函方程(2)，以及在预齐次向量空间上的不变微分方程(3)。这是预齐次向量空间中的一些基本问题。下面介绍了分步研究的结果及其意义。我们知道构造基本解是获得给定方程所有解的方法之一。求基本解的一种方法是利用二次多项式的复次幂。通过对多项式的复幂进行拉普拉斯变换我们得到b函数复幂乘以它。在b函数的零点处，复幂相对于参数有一个极点。将点处的复幂展开为洛朗展开式，一些奇异超函数作为洛朗展开式的系数出现在主部。这里出现的奇异超函数的支持包含在多项式的零点集合中。函数是最重要的奇异超函数之一，它的支持集中在原点上。最基本的解是超函数，它在对微分算子进行运算后变成了函数。通过构造基本解，我们可以用基本解的积分来计算任何解。我们有很多计算基本解的方法，但是我们可以直接从多项式的复次幂中计算基本解。为了达到这个目的，我们必须找到一个多项式，它改变了b函数和微分算子后复幂的乘积。佐藤美纪夫开始在更广泛的多项式类别中寻找这样的多项式。尽管他尝试了各种各样的多项式，但他在半年的时间里都没有得到很好的结果。经过一些试验，他发现拉普拉斯算子的成功归因于二次齐次多项式在旋转群下的不变性。然后我们必须注意群不变性，但什么样的群作用适合于基本解的构造。答案是“群体行为是预先同质的”。问题是什么样的群体行为是预先同质的。第一个问题是向量空间上的预齐次作用的分类。为了使问题过于复杂，我们假设群是一个半简单李群（复代数群），群作用是一个线性作用，我们必须寻找预齐次作用。Sato Mikio首先研究了这个问题，并在“半简单性”和“不可约性”条件下完成了预齐次向量空间的分类。后来Kimura Tatsuo解决了异常李群的问题，并给出了预齐次向量空间的完备表。现在我们已经知道了存在什么样的预齐次向量空间。接下来我们要做什么来计算基本解呢？佐藤美雄似乎对根本的解决方案感到失望。他首先研究了预齐次向量空间的分类问题，下一个吸引他的问题是ζ函数的问题。他非常注意用复次幂来计算黎曼ζ函数的泛函方程。对于黎曼ζ函数的泛函方程，我们使用一元次齐次多项式，预齐次向量空间的相对不变量对应于该多项式。假设正则性是相对不变量存在的充分条件，我们考虑了相对不变量对应的偏微分算子，并构造了微分算子的基本解。另一方面，计算泛函方程的问题等价于计算相对不变量复幂的傅里叶变换的问题。相对不变量的傅里叶变换的计算比基本解的计算更重要如果你认为函数更重要的话。当时黎曼zeta函数的对应对象还没有找到，但如果认为相对不变量的复幂就是“局部”zeta函数，这就为zeta函数在实场上的候选提供了很好的依据。期望的“全局”zeta函数可能具有相同的函数方程。然后我们要研究相对不变量的傅里叶变换的显式计算。Sato Mikio与Shintani Takuro一起研究了泛函方程，并证明了可以用Gamma函数和复参数指数函数的多项式来表示泛函方程。在与Takuro Shintani的合作研究中，成功定义了全局zeta函数。因此我们可以看到全局zeta函数在什么条件下。然后在相对不变量的傅里叶变换的基础上，清晰地表述了与预齐次向量空间相关的ζ函数的概念。从数论的角度阐明了ζ函数的意义。对于应用，zeta函数的作用仍然不清楚，但我们看到了表示理论对象（预齐次向量空间）和数论对象（zeta函数）之间的联系。在实际研究中，我们一直围绕齐次向量空间理论和微局部分析的应用进行研究。特别地，我们对基本预齐次向量空间和既没有约化群也没有不可约表示的预齐次向量空间进行了费力的计算。在实际计算中，许多问题仍未得到解决，但在不可约的预齐次向量空间上取得了一些进展。然而，它正在缓慢而稳定地进行。此外，我们还用计算机代数研究了微分方程的图形。合作研究人员正在各自的领域研究b函数和相关主题。我们还为空气大学的学生写了一本关于微分方程邀请的书。这当然是微分方程的基础教材。在这里，分析偏向于以常微分方程为主的传统微分方程的介绍，目的是拓展他们对偏微分方程的视角。自2011财年开始在空中播放讲座。少

Various results on zeta functions associated with prehomogeneous vector spaces are obtained by viewing from the point of micro-local analysis. Especially experimental calculation on zeta functions of prehomogeneous vector space of commutative parabolic type are carried out. Some other studies on micro-local analysis are done by collaborative researchers.It has been designed in this study, analysis and determination of the invariant hyperfunctions on the prehomogeneous vector space (1), application to the study of residues and the functional equation of the zeta function (2), and invariant differential equations on the prehomogeneous vector space (3). These were some basic problems in prehomogeneous vector spaces. The following describes the step-by-research results and their significance.We know that constructing fundamental solutions is one of the method to obtain all the solutions to the given equation. One method to obtain the fundamental solution is to use the complex power of the … More polynomial of degree two. By operating the Laplacian to the complex power of the polynomial we get the b-function and the complex power is multiplied by it. At the zero-point of the b-function, the complex power has a pole with respect to the parameter. Expanding the complex power at the point to the Laurent expansion, some singular hyperfunctions appears in the rincipal part as coefficients of the Laurent expansion. The support of the singular hyperfunctions appearing here are contained in the set of the zero points of the polynomial. One of the most important singular hyperfunctions is the delta function, whose support concentrates on the origin. The fundamental solution is the hyperfunction which becomes the delta function after operating the differential operator. By constructing the fundamental solution, we can calculate any solution using the integral of the fundamental solution.We have some various method to compute the fundamental solutions, but we can compute the fundamental solution directly from the complex power of the polynomial. For this purpose we have to find polynomials that changes the product of b-functions and the complex power after operating differential operators. Sato Mikio began to seek for finding such polynomials in a more wide class of polynomials. But his trials did not work out well for about half of the year though he tried in the various polynomials. After some trials he found that the success in the case of the Laplacian is attributed to the invariance of homogeneous polynomials of degree two under the rotation group. Then we have to be careful to the group-invariance but what kind of group-action is appropriate for the construction of the fundamental solution. The answer is "the group-action is prehomogeneous".The problem is what kind of group-action is prehomogeneous. The first problem is the classification of prehomogeneous action on the vector space. In order to make the problem too complicated, we suppose that the group is a semi-simple Lie groups (complex algebraic group) and that the group action is a linear action and we have to search for the prehomogeneous action. Sato Mikio first works on this problem and rounded off the classification of prehomogeneous vector spaces under the conditions "semisimplicity" and "irreducibiliy". Later Kimura Tatsuo addressed the problem for the exceptional Lie group and make the complete table of prehomogeneous vector spaces.Now we have seen that what kind of prehomogeneous vector spaces exists. Then what we have to do next for the computation of the fundamental solution? Sato Mikio seemed to turn sour the fundamental solutions. He first worked on the problem of the classification of prehomogeneous vector spaces, and the next problem he was attracted was the problem of zeta functions. He gave considerable attention to the fact that the complex power is used to compute the functional equation of Riemann's zeta function. For the functional equation of Riemann's zeta function we used the homogeneous polynomial of one variable and of degree one, and the relative invariants of prehomogeneous vector spaces correspond to the polynomial. Supposing the regularity as the sufficient condition for the existence of relative invariant, we considered the partial differential operators corresponding to the relative invariants and we can construct the fundamental solutions to the differential operators. On the other hand, the problem of the computation of the functional equation is equivalent to that of the computation of the Fourier transform of the complex power of the relative invariant.The computation of the Fourier transform of relative invariant is more important than that of fundamental solution if you think that zeta functions are more important. The corresponding object of the Riemann's zeta functions was not found at that time, but if you think that the complex power of the relative invariant is the "local" zeta function, this has good grounds for the candidate of the zeta function on the real field. The "global" zeta function to be expected may have the same functional equation. Then we have to study on the explicit computation of the Fourier transform of the relative invariant. Sato Mikio studied the functional equation with Shintani Takuro and proved that the functional equation can be written by using Gamma function and the polynomial of exponential functions of complex parameters. Also, he succeeded to define the global zeta function in the collaborative study with Shintani Takuro. Consequently we can see global zeta function under what conditions. Then the concept of zeta functions associated with prehomogeneous vector spaces was clearly formulated on the basis of the Fourier transforms of relative invariants. The meaning of zeta functions from the point of view of number theory was also made clear. For the application, the role of the zeta function is still unclear, but we see the connection between the representation theoretic object (prehomogeneous vector spaces) and the numaber theoretic one (zeta functions).In the practical research, we have been studying around the theory of phomogeneous vector spaces and the application of micro-local analysis. In particular, we carried out laborious calculation on the basic prehomogeneous vector spaces and the prehomogeneous vector spaces with neither reductive group nor irreducible representation. In actual calculation, many problems remain unresolved but some progress has been made on irreducible prehomogeneous vector space . However, it is proceeding slowly but steadily. In addition, we have studied the graphics of differential equations using computer algebra. The collaborative researchers are studying b-functions and related topics in their own area. We also wrote a book on the invitation to differential equation for the students of the University of Air. This is of course an elementary textbook of differential equations. Here, the analysis was biased towards the mainly ordinary differential equations in the introduction of traditional differential equations, with the aim to expand their perspectives of partial differential equations. Since fiscal 2011 began broadcasting a lecture on the Air. Less