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Zeta Functions of Prehomogeneous Vector Spaces with Coefficients Related to Periods of Automorphic Forms

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Zeta Functions of Prehomogeneous Vector Spaces with Coefficients Related to Periods of Automorphic Forms Proc. Indian Acad. Sci. (Math. Sci.), Vol. 104, No. l, February 1994, pp. 99-135. Printed in India. Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms FUMIHIRO SATO Department of Mathematics, Rikkyo University, Nishi-Ikehukuro, Toshimaku, Tokyo 171, Japan Dedicated to the memory of Professor K G Ramanathan Abstract. The theory of zeta functions associated with prehomogeneousvector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. Howeverthe general theory does not includezeta functionsrelated to automorphic forms such as the Hecke L-functions and the standard L-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 of K=-typeand 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 coefficientsof irreducible unitary representations of compact groups. Keywords. Zeta function;prehomogeneous vector space; functional equation; automorphic form. Introduction The theory of zeta functions associated with prehomogeneous vector spaces was developed by Sato and Shintani in [S] and [SS] and generalized later in [S1] to multivariable zeta functions. The theory 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, even though they can naturally be considered to be associated with prehomogeneous vector spaces. Among those are the Hecke L-functions with Gr6ssencharacters (cf. [Heel], [Hec2]), the Epstein zeta functions with spherical functions (cf. [El), the Maass zeta functions attached to quadratic forms (cf. [M1]-[M4]), and the standard L-functions of automorphic forms on GL(n) (cf. [GJ]). Our ultimate purpose is to generalize the theory of zeta functions associated with prehomogeneous vector spaces ([SS], IS 1]) to zeta functions whose coeff~-'ients involve periods of automerphic forms, which include the zeta functions mentioned above. In [$6], we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. The simplest example in this case is the Epstein zeta functions with spherical functions (cf. w2.2, [$4]). In the present paper, we are concerned mainly with prehomogeneous vector spaces with symmetric structure and prove the functional equations and the analytic 99 100 FumihiroSato continuations of zeta functions with automorphic forms in the case where the infinitesimal character of an automorphic form is generic and the prehomogeneous vector space in question has a symmetric structure of K,-type. Our results can be applied, for example, to zeta ,functions considered in [M3] and [Hej] and some special cases of zeta functions in [M 1, 2, 4] as well as several new zeta functions (cf. w6). Recall that, in the theory of prehomogeneous vector spaces developed in [SS] and [S1], the proof of the existence of analytic continuations and functional equations of zeta functions, is based on the following four properties: 1. Integral representation of the product of zeta functions and local zeta functions at the infinite prime (= the zeta integral); 2. Analytic continuation and the functional equation of the zeta integral; 3. Functional equations satisfied by local zeta functions; 4. The existence of b-functions (the Bernstein-Sato polynomials), which control the singularities of zeta functions and the gamma-factor of functional equations. Moreover, by using the b-functions, one can eliminate the troublesome contribution of rational points in the singular set to the zeta integral. We extend these four properties to zeta functions with automorphic forms. In w1, we introduce zeta functions with automorphic forms and give their integral representation (the zeta integral) in a rather general setting. In w2, as examples of generalized zeta functions defined in w1, we examine zeta functions with Gr6ssencharacters and the Epstein zeta functions with spherical functions from our point of view. The easiest part of the generalization is the functional equation of the zeta integral, which we describe in w3. In w4, we define the notion of symmetric structure of prehomogeneous vector spaces and establish some elementary properties. In w5, we prove the functional equation of local zeta functions (Theorem 5.3) and that of global zeta functions (Theorem 5.4) under the condition that the infinitesimal character of an automorphic form is generic and its symmetric structure is of K,-type. In w6, we present explicit formulas for the functional equations of zeta functions for some concrete examples of prehomogeneous vector spaces with symmetric structure. Notation. As usual, 7/, Q, • and C stand for the ring of rational integers, the field of rational numbers, the field of real numbers and the field of complex numbers, respectively. For a complex number z, Re(z) is the real part of z. Let X be an affine algebraic variety defined over a field K. Then the set of K-rational points of X is denoted by Xr. For a linear algebraic group G defined over a field K, G O= the identity component of G with respect to the Zariski topology, Ru(G) = the unipotent radical of G, X(G) = the group of rational characters of G, X(G) r = the subgroup of X(G) of rational characters defined over K. For a real Lie group G, GO = the identity component of G with respect to the usual Hausdorfftopology. Let X be a C~~ Then C~(X) stands for the space of smooth functions on X with compact support. Let Va be a finite-dimensional real vector space. Then ff'(Va) stands for the space of rapidly decreasing functions on Va. Let V o be a finite-dimensional Q-vector space. A C-valued function f on V o is called a Schwartz- Zeta functions with automorphic forms 101 Bruhat function, if there exist lattices L t, L 2 in V~ such that the support of f is contained in L~ and the value of f(x) is determined by the residue class x (mod L2). We denote by 6e(V~) the space of Schwartz-Bruhat functions on V~, For m,n >i !, we put M(m, n) = the space of m by n complex matrices, M(n) = M(n, n) = the ring of complex square matrices of size n, I,eM(n) = the identity matrix of size n, 0"~'"~ = the zero matrix in M(m, n). For a subring R of C, denote by M(m, n; R) the subset of M(m, n) of matrices with entries in R. For a real Lie algebra 9, we denote by U(9) the universal enveloping algebra of the complexification 9c = g | ~ C. 1 Definition of zeta functions and their integral representations 1.1 Let (G,p,V) be a prehomogeneous vector spaces defined over a field K of characteristic 0 and denote its singular set by S. Then, by definition, Vg-Sg is a single Gg-orbit, where/( is the algebraic closure of K. Let St ..... S, be the K-irreducible hypersurfaces contained in S and take K-irreducible polynomials P t .... , Pn defining S~ ..... Sn, respectively. It is known that the polynomial Pi is unique up to a non-zero constant factor in K. For each i = 1,..., n, there exists a K-rational character Xi satisfying Pi(p(g)x)= ~i(g)Pi(x) (geG, xeV). We call P~ ..... P, as the basic relative invariants over K. Any relative invariant of (G,p, V) with coefficients in K can be expressed as a product of Pt ..... Pn, negative power being allowed. Denote by Xp(G)r the subgroup of X(G)K generated by Zt,..., Z,, which is a free abelian group of rank n. (For basic definitions and properties in the theory of prehomogeneous vector spaces, refer to [S1, w 1] and [SK].) 1.2 In the following, we consider a prehomogeneous vector space (G, p, V) defined over the rational number field Q. For an x~V, put G~ -" {g~GIp(g)x = x}. We assume that (A- 1) for any x E V o - So, the isotropy subgroup Gx is reductive and X ((Gx)~ = { 1}; (A-2) G has a semidirect product decomposition G = LU, where L is a connected reductive Q-subgroup and U is a connected normal Q-subgroup with X(U) = {1). The group G always has a semi-direct product decomposition satisfying (A-2). Namely G is a semi-direct product of U = R~(G), the unipotent radical, and a Levi subgroup L. In the following we fix a decomposition G = LU satisfying (A-2) once for all, which may not be the Levi decomposition (for concrete examples, see w2, w4.1 and w6). 102 Fumihiro Sato One of the consequences of assumption (A-l) is the following: Lemma I. I The singular set S is a hypersurface. Proof Since Gx is reductive, the open orbit V- S is an affine variety. This implies that S is a hypersurface. Let P1 ..... P. be the basic relative invariants over Q and X1 ..... X. the rational characters of G corresponding to Pa ..... P,, respectively. Let Go be the identity component of (~ kerx~ with respect to the Zariski topology. Put Lo = LnGo. Then i=1 Lo is connected and we have Go = Lo U (semi-direct product). Lemma 1.2 The oroups X(Lo) ~ and X(Go)~ are trivial.
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