Algorithmic Construction of Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

Masakazu Muro

Abstract: This is the second paper on invariant hyperfunction solutions of invariant linear differential equations on the vector space of $n \times n$ real symmetric matrices. In the preceding paper [Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices, J. Funct. Anal., 193 (2002), 346--384], we proved 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. Fundamental properties of the complex power have been investigated in [Tôhoku Math. J. (2) {\bf51} (1999), 329--364]. In this paper, we give algorithms to determine the space of invariant hyperfunction solutions and apply the algorithms to some examples. These algorithms enable us to compute in a fully constructive way all the invariant hyperfunction solutions for all the invariant differential operators in terms of Laurent expansion coefficients of the complex power of the determinant function.

Keywords: invariant hyperfunctions, symmetric matrix spaces, linear differential equations

Classification (MSC2000): 22E45, 58J15; 35A27

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