On Functional Equations of Zeta Distributions

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This chapter discusses the functional equations of zeta distributions. Recent development in the theory of prehomogeneous vector spaces—in particular the works of Gyoja–Kawanaka on prehomogeneous vector spaces defined over finite fields and of Igusa on prehomogeneous vector spaces defined over p-adic number fields—has revealed a striking resemblance between the theories over finite fields, p-adic number fields, real and complex number fields, and algebraic number fields, as is common in the theory of representations of algebraic groups. The chapter presents the fundamental theorem in the theory of prehomogeneous vector spaces. It explains the analogy between the theories over various fields. It focuses on the cases of p-adic number fields and the rational number field Q. The fundamental theorem over finite fields because of Gyoja–Kawanaka is explained in the chapter in connection with L-functions associated with prehomogeneous vector spaces.