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 conjectures were stated less generally in lectures [2] 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 spectral 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 [5].