Summary: Proceedings of Symposia in Pure Mathematics
Vol. 33 (1979), part 1, pp. 253-274
EISENSTEIN SERIES AND THE TRACE FORMULA
The spectral theory of Eisenstein series was begun by Selberg. It was completed
by Langlands in a manuscript which was for a long time unpublished but which
recently has appeared [I].The main references are
1. Langlands, On thefunctional equations satisfied by Eisenstein series, Springer-
Verlag, Berlin, 1976.
2. , Eisenstein series, Algebraic Groups and Discontinuous Subgroups,
Summer Research Institute (Univ. Colorado, 1965), Proc. Sympos. Pure Math.,
vol. 9, Amer. Math. Soc., Providence, R. I. 1966.
3. Harish-Chandra, Automorphic forms on semi-simple Lie groups, Springer-
Verlag, Berlin, 1968.
In the first part of these notes we shall try to describe the main ideas in the theory.
Let G be a reductive algebraic matrix group over Q. Then G(A)is the restricted
direct product over all valuations v of the groups G(Qu).If v is finite, define Ku
to be G(uJ if this latter group is a special maximal compact subgroup of G(Qu).
This takes care of almost all v. For the remaining finite v, we let KObe any fixed
special maximal compact subgroup of G(Qu).We also fix a minimal parabolic sub-