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

(A)(摘自“实对称矩阵空间上不变微分方程的超函数解”一文的摘要)N阶实特殊线性群自然作用于n阶实对称矩阵的向量空间。讨论了n阶实对称矩阵向量空间上系数多项式不变线性微分方程超函数不变解的确定问题。我们观察到，每个不变超函数解都表示为行列式函数的复幂关于幂的参数的Laurent展开系数的线性组合。然后将问题归结为需要表示的洛朗展开系数的确定。我们给出了一个确定它们的算法，并在一些例子中应用了该算法。(B)(摘自“一种通过准齐次向量空间上不变微分算子的Grobner基计算b_P-函数的算法”)本文考虑有限维向量空间上群不变微分算子P(x，∂)的b_P-函数的Grobner基计算问题。设(G，V)是正则的非齐次向量空间。通常观察到，微分方程P(x，∂)u(X)=v(X)的所有G-不变超函数解u(X)的空间由它的b_P-函数决定，b_P-函数是与G-不变微分算子P(x，∂)相关的多项式。本文证明了对于一类典型的非齐次向量空间，b_P-函数是利用V上Weyl代数的Grobner基的一个算法来计算的。

(a) (From the abstract of the paper "Hyperfunction solutions of 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 symmtric 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 observe 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 which is needed to express. We give an algorithm to determine them and apply the algorithm in some examples.(b) (From the abstract of the paper "An algorithm th compute the b_P-functions via Grobner bases of invariant differential operators on prehomogeneous vector spaces".) The calculation of b_P-function via Grobner basis for an group invariant differential operator P (x, ∂) on a finite dimensional vector space is considered in this paper. Let (G, V) be a regular prehomogeneous vector space. It is often observed that the space of all G-invariant hyperfunction solutions u (x) to the differential equation P (x, ∂) u (x)=v (x) is determined by its b_P-function, a polynomial associated with the G-invariant differential operator P (x, ∂). We prove in this paper that the b_P-function is computed by an algorithm using Grobner basis of the Weyl algebra on V for a typical class of prehomogeneous vector spaces.