Twists of the Iwasawa-Tate zeta function

Abstract

The intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by Tate [14] and Iwasawa [9] in independent efforts to simplify Hecke's analysis of his L-functions attached to Gr6Bencharakters. Our "twists" of the Iwasawa-Tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups. Many of the basic problems of this general theory have already been solved by Yukie and the author (see [20] and [21-1) but still await appearance in final written form. Thus, it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper. Let k be a global field (i.e. algebraic number field or function field of a curve defined over a finite field). Let N and A • denote the ring of adeles and the group of ideles of k, respectively. Let co be a quasicharacter of the idele group trivial on the embedded group k • of nonzero elements of k. The group f2 = f2 k of all such quasicharacters co has a natural Riemann surface structure. Let L" I~ denote the usual idele norm on A • and for s~tl2 let co~ be the principal quasicharacter co~(t)=ltl~. Let Re(co) be the real part of co~f2, defined as the unique real number a such that l o I= co~. For any reductive linear algebraic group G defined over k, let G~ denote the "adelizat ion" of G over k as defined in [16], and let Gk be the group of k-rational points of G embedded as a discrete group in GA. For simplicity, we shall fix G as G1, now, although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k-rational character. Let p: G ~ G1 (V) be an irreducible k-rational representation of G in the finite-dimensional vector space V of dimension m. Let I be the identity matrix in G. By Schur's lemma, there is an integer d such that p(tI) =tdp(I) for all scalars t. We shall always assume that d>0 . Let dg be any convenient left-invariant measure on the quotient Ga/Gk. Let ~ ( V ~ be the Schwartz-Bruhat space of functions on V~. For the time being, let Vk' be any Gk-invariant subset of Vk. The zeta function associated to the representation p may now be defined: