Fumihiro Sato

On Functional Equations of Zeta Distributions

Publisher Summary 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.

Local densities of alternating forms

Abstract We give an explicit formula for local densities of integral representations of non-degenerate alternating matrices with entries in the ring of p -adic integers in terms of elementary divisors. The proof of the formula is based on the local functional equation satisfied by the zeta function on the space of alternating forms and some properties of spherical functions.

MULTIPLICITY ONE PROPERTY AND THE DECOMPOSITION OF b-FUNCTIONS

Recently, extensive calculations have been made on b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions which seem to be correlated to the decomposition of representations. In the present paper, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Furthermore, we give some criteria for the multiplicity one property. Our method can be applied equally to non-regular prehomogeneous vector spaces.

Eisenstein series on reductive Symmetrie spaces and representations of Hecke algebras

0.1. Let G be a reductive algebraic group defined over Q and σ an involutive Qautomorphism of G. Denote by H the subgroup of G of elements fixed by σ. We consider the Symmetrie space X = G/H. A connected component XQ of the set X(i?) of real points is an affine Symmetrie space of the identity component of G (i?). If X0 is Riemannian, then for an arithmetic subgroup Γ of G, one can define the Selberg-Langlands Eisenstein series, which is a real analytic function on Γ\Χ0 (cf. [L], [HC]). In general, X(i?) may contain a connected component XQ that is not a Riemannian Symmetrie space. The action of an arithmetic subgroup Γ on such a component X0 is not properly discontinuous and we can no longer expect to define real analytic Eisenstein series on F\XQ.

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above.In this paper, we generalize the theory to p.v.’s with symmetric structure ofKε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups.

Maximal semigroup symmetry and discrete Riesz transforms

We raise a question if the Riesz transform on T n or Z n is characterized by the “maximal semigroup symmetry” that they satisfy? We prove that this is the case if and only if the dimension n = 1,2 or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform (i.e. the n = 1 case) on T and Z, and extends a theorem of Stein for the Riesz transform on R n . Unlike the R n case, we show that there exist infinitely many, linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on T n and Z n if n � 3 and is not a multiple of four.

The Hamburger Theorem for the Epstein Zeta Functions

Publisher Summary Hamburgers theorem, which characterizes the Riemann zeta function by means of its functional equation, has tempted a number of researchers to find an alternative proof and/or a generalization. A characterization of hyper-functions has been proved on Rn whose supports, as well as the supports of their Fourier transforms, are contained in Zn. Claiming the existence of functional equations of a certain family of L-functions with Grosencharacters, a formula is derieved similar to the Poisson summation formula. The chapter explains the method of Ehrenpreis, Kawai, and Yoshimoto to prove the Hamburger theorem for the Epstein zeta function, a typical example of zeta functions associated with prehomogeneous vector spaces.