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Summary: ON A FAMILY OF DISTRIBUTIONS OBTAINED
FROM EISENSTEIN SERIES I: APPLICATION OF THE
PALEY-WIENER THEOREM
Introduction. This is the first of two papers aimed at finding ex-
plicit formulas for certain distributions. The distributions are obtained
from Eisenstein series and are important ingredients in the trace formula.
In this paper we shall resolve some analytic difficulties that center around
an interchange of two limits. The next paper will be devoted to the
calculations which will culminate in the formulas.
Let G be a reductive algebraic group defined over Q. As usual, we
will write G(A)' for the intersection of the kernels of the maps
in which ranges over the group X(G)o of characters of G defined over Q.
The trace formula is an identity
between distributions on G(A)'. The distributions on the left are
parametrized by semisimple conjugacy classes in G(Q) and are closely
related to weighted orbital integrals on G(A)'. Although they need to be
better understood for any general applications of the trace formula, the
remaining problems are primarily local. We will not discuss them here.
The distributions on the right are defined in terms of truncated Eisenstein
series. They are parametrized by the set X of Weyl group orbits of pairs
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