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Journal of Lie Theory 14 (2004), No. 1, 111--140
Copyright Heldermann Verlag 2004
Algorithmic Construction of Hyperfunction Solutions to
Invariant Differential Equations on the Space of Real Symmetric Matrices
Masakazu Muro
Gifu University, Yanagito 1-1, Gifu, 501-1193, Japan,
muro@cc.gifu-u.ac.jp
This is the second paper on invariant hyperfunction solutions of invariant
linear differential equations on the vector space of n x n real symmetric
matrices. In a preceding paper [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 by the author in the paper
"Singular invariant hyperfunctions on the space of real symmetric matrices"
[Tohoku Math. J. 51 (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.
MSC 2000: 22E45, 58J15, 35A27.
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