 
Summary: Annals of Mathematics, 114 (1981), 174
The trace formula in invariant form
By JAMES ARTHUR*
Introduction
The trace formula for GL2 has yielded a number of deep results on
automorphic forms. The same results ought to hold for general groups, but so far,
little progress has been made. One of the reasons has been the lack of a suitable
trace formula.
In [l(d)] and [l(e)]we presented a formula
or, as we wrote it in [l(e),$51,
G is a reductive group defined over Q, and f is any function in C;(G(A)l). The
left hand side of (I*) is the trace of the convolution operator off on the space of
cusp forms on G(Q) \ G(A)'. It is a distribution which is of great importance in
the study of automorphic representations. One would hope to study it through
the distributions o E O} and {1;: x E X \ X(G)}. Unfortunately, these
distributions depend on a number of unpleasant things. There is the parameter
T , as well as a choice of maximal compact subgroup of G(A)' and a choice of
minimal parabolic subgroup. What is worse, they are not invariant; their values
change when f is replaced by a conjugate of itself. In any generalization of the
applications of the trace formula for GL2,we would not be handed the function
