Study on prehomogeneous vector spaces and micro-local analysis

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In the period of the research, we mainly have been studying invariant differential equations on prehomogeneous vector spaces. The invariant differential equations on prehomogeneous vector spaces we are studying are linear and of constant coefficients, so they are the easiest differential operators to analyze. However, the concrete analysis for the specific differential operators does not seem to be so easy. Indeed, except for the wave operators, the concrete analysis for the specific hyperbolic differential operators with higher order is less well understood. We approached the analysis of invariant differential operators on prehomogeneous vector spaces through the problem of the determination of the support and the singularity spectra of the fundamental solution. In particular, the invariant differential operators on the prehomogeneous vector spaces of commutative parabolic type are hyperbolic differential operators. It is the most orthodox way for the calculation of the singularity propagation to determine the exact singularity spectra of the fundamental solution. We have succeeded to define the singularity propagation sets of the differential operators and determine them.The most remarkable result is that we have established the "Huygens principle" for the singularity propagation and discover the example for which the "Huygens principle" holds by the explicit computation. We verified that the "Huygens principle" for the singularity propagation holds for the invariant differential operators on prehomogeneous vector spaces. It is an interesting and challenging problem whether the "Huygens principle" is valid for other differential operators.