## A stable trace formula. I: General expansions.(English)Zbl 1040.11038

Stabilization of the Arthur-Selberg trace formula is an important problem and has major applications to the functorial principle in Langlands program. This paper is the first in a series of three papers which establish stabilization of the trace formula under the assumption of the fundamental lemma, which has been proved for some low dimensional cases. This paper establishes some general expansion results that are parallel to expansions on both sides of the trace formula. The main results on stabilization of the trace formula are stated in this paper but will be proved in later papers. The Introduction gives a very good introduction to the history and applications of the stabilization of the trace formula.

### MSC:

 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11R39 Langlands-Weil conjectures, nonabelian class field theory 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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