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

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(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.