Relative invariants, difference equations, and the Picard-Vessiot theory

Acknowledgements First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master's thesis. I can not give enough thanks to his heartwarming encouragement all over the period of this program. I thank Professor T. Kogiso and Professor K. Sugiyama for being close advisers on prehomogeneous vector spaces. or July 2003, I had a chance to know the existence of the Professor Takeuchi's paper [2] when I was trying to understand [1]. Since I once attended to an introductory part of his lecture on Hopf algebras at 2002, the paper seemed like just what I wanted. Then I was absorbed in it and came to think that the results can be extended to involve [1]. But I could not have developed it in the presented form without the collaboration with Professor Masuoka. He suggested the viewpoint from relative Hopf modules as is described in Section 3.2, added many interesting results, and helped me to complete several proofs. Finally I would like to thank my parents for supporting my decision to study mathematics as the lifework.