Classification and representation theoretic study of the functional equation-spaces

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Let Cp(resp. Cq) be the Clifford algebra of a positive definite real quadratic form in p(resp. q) variables. For a representationρof the tensor product of Cp and Cq, we can associate a homogeneous polynomial P of degree 4 on the representation space ofρhaving the property "The Fourier transform of the complex power}}| P|^s coincides with|P*|^{-m/4-s}(m=the degree ofρ) with an explicit gamma factor. Owing to the theory of prehomogeneous vector spaces, the basic relative invariant of an irreducible regular prehomogeneous vector space satisfies the property above. However the polynomials P are not necessarily relative invariants of any prehomogeneous vector spaces. The polynomials P are relative invariants of prehomogeneous vector spaces only for quite few exceptional cases. In this study, we discuss the structure and the action of the group of linear transformations that leave P invariant.