 
Summary: The Selberg trace formula for groups
of Frank one
Introduction
An important tool for the study of automorphic forms is a non
abelian analogue of the Poisson summation formula, generally known as
the Selberg trace formula. There have been a number of publications on
the subject following Selberg's original paper [lo], the most recent being
[2] and [7, 3 161. With the exception of Selberg's brief account [Ill, how
ever, most authors have restricted themselves to the groups SL(2) and
GL(2). In this paper we develop the formula for a wider class of groups.
We shall work in an ad& framework so our group G will be a reduc
tive algebraic group defined over a number field F . We require that the
Frank of the semisimple component of G be one. To simplify our intro
duction, let us assume that G itself is semisimple. If A is the adele ring of
F , let \ be the regular representation of GA on L^G^/Gd. It is important to
try to decompose \ into irreducible representations.
To begin with, \ splits into a sum of two representions \, and \ such
that kOis a direct sum of irreducible representations while hi decomposes
continuously. The theory of Eisenstein series provides us with a fairly
good understanding of the decomposition of \. However, virtually nothing
