Cohomology of sheaves on the building and R-representations

摘要

This article contains four parts. The subject of the first one is to compare open compact subgroups of a reductive connected group G over a local non archimedean field, associated to certain natural concave functions by the theory of Bruhat-Tist [BT1]. We compare the groups which appear in the theory of unrefined minimal types by Moy and Prasad [MP1, MP2] with those which appear in the theory of complex sheaves on the building X by Schneider and Stuhler [SS]. Schneider and Stuhler give to special vertices a predominant role hiding in this way some symmetry that we restaure (we need all vertices); in particular we describe how the groups vary along a geodesic starting from any point.