Researches on zeta functions of prehomogeneous vector spaces

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Theory of prehomogeneous vector spaces gives a systematic method of constructing zeta functions from polynomial invariants of prehomogeneous group actions. We investigated zeta functions of prehomogeneous vector spaces from the following 3 view points : (1) relations to Eisenstein-periods, (2) relations to the Koecher-Maass zeta functions of automorphic forms, (3) zeta functions of invariants of non-prehomogeneous group-actions. The most important result is the construction of zeta functions of non-degenerate quadratic mappings, which include zeta functions of certain non-prehomogeneous polynomials of degree 4.