The Selberg Trace Formula. V: Questions of Trace Class
M. Scott Osborne, Garth Warner · Transactions of the American Mathematical Society · 1984
The purpose of this paper is to develop criteria which will ensure that the K-finite elements of C^°(G) are represented on L¿ia(G/T) by trace class operators. Introduction.This is the fifth in a projected series of papers in which we plan to come to grips with the Selberg trace formula, the ultimate objective being a reasonably explicit expression.Let G, therefore, be a reductive Lie group and T a lattice in G, both subject to the usual conditions.Consider, for the moment, the special case when rank(r) = 1.Given a If-type 6, suppose that the poles of the c-functions c¿,o in the interval [-|p|,0[ are bounded uniformly away from 0 in the orbit types 0. Then Vq £C?(G) st x"*a = a, the operator Lça,r(a) is trace class.In the present paper, this assertion will be established.Actually, its proof turns out to be quite easy, so our primary goal will be to formulate and prove the requisite generalization for T of any rank, a task offering additional difficulties.During our consideration of this problem, we discovered a weaker condition, sufficient, nevertheless, to allow one to draw the same conclusion, the point being that the poles in the rank-one case, say, can be allowed to approach the origin provided they do so in a "tempered" way.The issue of trace class on the full discrete spectrum is a vexing question of longtime standing.In this connection, recall that L2dia(G/T) = L2ua(G/T)®L2ea(G/r).Now, on the cuspidal spectrum, the relevant operators are, as is well known, trace class.Consequently, it is the residual spectrum where all the problems lie.The conditions behind our hypotheses serve to guarantee trace class, but, of course, their validity has yet to be shown (except in familiar, elementary situations).Various preliminaries of a general nature are dealt with in §2, while §3 is devoted to a compilation of certain facts about the eigenfunctions in L2es(G/T).The residue hypothesis (and its generalization) is formulated in §4, where there is also to be found an ad-hoc discussion when rank(r) = 1.The proofs that the hypotheses lead to trace class are given in §7 and §8, the demonstrations depending on an important uniformity possessed by the degrees of the constant term polynomials (obtained in §5), in conjunction with the Laplace transform theory of §6.Certain