"Research on prehomogeneous vector spaces and micro-local analysis"

(a)讨论了n × n实对称矩阵空间上的奇异不变超函数。我们用行列式函数复幂的洛朗展开式的负阶系数构造奇异不变超函数，即其支撑点包含在集合S ={det(x) = O}中的不变超函数。特别地，我们给出了确定行列式函数复幂的极点阶的一种算法，并给出了复幂的洛朗展开式的主部中出现的奇异超函数的支持。(b)讨论了n × n复矩阵和四元数矩阵空间上的奇异不变超函数。给出了一种确定行列式函数复幂的极点的阶数的算法，并精确地确定了奇异不变超函数的支持度，即支持度包含在集合S: ={det(x) = O}中的不变超函数，其支持度由复幂的洛朗展开的负序系数得到。

(a) Singular invariant hyperfunctions on the space of n x n real symmetric matrices are discussed. We construct singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S ={det(x) = O}, in terms of negative order coefficients of the Laurent expansions of the complex powers of the determinant function. In particular, we give an algorithm to determine the orders of poles of the complex powers of the determinant functions and the support of the singular hyperfunctions appearing in the principal part of the Laurent expansions of the complex powers.(b) Singular invariant hyperfuncsions on the space of n x n complex and quaternion matrices are discussed. We give an algorithm to determine the orders of poles of the complex power of the determinant function and to determine exactly the support of singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set S : ={det(x) = O}, obtained as negative-order-coefficients of the Laurent expansions of the complex powers.