Summary: AN ASYMPTOTIC FORMULA FOR REAL GROUPS
§1 Distributions and multipliers
§2 Spherical functions
§3 Reduction to constant terms
§4 Estimation of the constant terms
§5 Weighted virtual characters and spherical transforms
§6 The asymptotic formula
* Supported in part by NSERC Operating Grant A3483.
This paper is the first of a pair of articles on real harmonic analysis. The objects we
study are tempered distributions on real groups that occur on the geometric side of the
trace formula. In this paper, we shall study the distributions attached to a fixed reductive
group G(R). We shall establish explicit formulas for their values at data that approach
infinity. In the next paper, we shall use the asymptotic formulas to establish identities
among the distributions attached to different groups.
Suppose for a moment that G is abelian. The trace formula is then just the Poisson