Research on prehomogeneous vector spaces and micro-local analysis

(a)（摘自“方阵空间和交变矩阵空间上的奇异不变超函数”一文中的摘要。）本文研究了n × n方阵空间和2n × 2n交替矩阵空间上奇异不变超函数的基本计算。通过将行列式函数或Pfaffian函数的复幂展开成关于复参数的Laurent级数，我们可以构造奇异不变超函数作为它们的Laurent展开系数。本文给出了复幂的极点的精确阶数，并确定了洛朗展开系数的精确支撑点。通过应用这些结果，我们证明了每一个拟相对不变超函数都可以表示为复幂的Laurent展开系数的线性组合，并且证明了每一个奇异拟相对不变超函数实际上在其支撑点的一般点上是相对不变的。在最后一节中，我们给出了奇异不变缓变分布的傅里叶变换公式。(b)（摘自论文《实对称矩阵空间上不变微分方程的不变超函数解》的摘要。）n次的实特殊线性群自然作用于n × n个实对称矩阵的向量空间。讨论了如何在n × n实对称矩阵的向量空间上确定多项式系数不变线性微分方程的不变超函数解。我们证明了每一个不变的超函数解都可以表示为行列式函数的复幂的洛朗展开系数关于幂参数的线性组合。然后将问题简化为洛朗膨胀系数的确定。

(a) (From the abstract of the paper "Singular invariant hyperfunctions on the square matrix space and the alternating matrix space".) Fundamental calculations on singular invariant hyperfunctions on the n × n square matrix space and on the 2n × 2n alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.(b) (From the abstract of the paper "Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices".) The real special linear group of degree n naturally acts on the vector space of n × n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n × n real symmtric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.