Summary: Society Math6matique de France
Asterisque 171-172 (1989), p.13-71.
UNIPOTENT AUTOMORPHIC REPRESENTATIONS: CONJECTURES
In these notes, we shall attempt to make sense of the notions of semisimple and unipotent
representations in the context of automorphic forms. Our goal is to formulate some conjectures,
both local and global, which were originally motivated by the trace formula. Some of these con-
jectures were stated less generally in lectures  at the University of Maryland. The present
paper is an update of these lectures. We have tried to incorporate subsequent mathematical
developments into a more comprehensive discussion of the conjectures. Even so, we have been
forced for several reasons to work at a level of generality at which there is yet little evidence.
The reader may prefer to regard the conjectures as hypotheses, to be modified if necessary in the
face of further developments.
We had originally intended to describe in detail how the conjectures are related to the spec-
tral side of the trace formula. However, we decided instead to discuss the examples of Adams
and Johnson (§5), and the applications of the conjectures to intertwining operators (§7) and to the
cohomology of Shimura varieties (§8). We shall leave the global motivation for another paper
I would like to thank Robert Kottwitz and Diana Shelstad for a number of very helpful