Cohomology of sheaves on the building and R -representations

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-Tits [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 symetry that we restaure (we need all vertices); in particular we describe how the groups vary along a geodesic starting from any point. In a discussion with Kutzko and with Schneider, it appeared that there are in the literature two different definitions for an Iwahori subgroup : the connected and the non connected. As I was not aware of this, and as there are may be other definitions, I include here the unique decomposition property for the “possible” Iwahori subgroups. This part is entirely geometric and is certainly trivial for someone who is familiar with the theory of Bruhat and Tits.