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 symmetric 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.