The Geometry and Cohomology of Some Simple Shimura Varieties.

This paper has twin aims. On the one hand we prove the local Langlands conjecture for GL n over a p-adic field. On the other hand in many cases we are able to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz in [Ko4]. These two problems go hand in hand. The local Langlands conjecture is one of those hydra like conjectures which seems to grow as it gets proved. However the generally accepted formulation seems to be the following (see [He2]). Let K be a finite extension of Q p . Fix a non-trivial additive character # : K