Double coset decompositions and intertwining

Let F be a non-archimedean local field of residual characteristic different from 2. This paper is a first step in the description of the smooth representation theory of a unitary group G ? GL(N,F) via types. We intersect certain double cosets in GL(N,F) with G and hence obtain the intertwining of certain characters ?ß- of open compact subgroups of G, for ß ? Lie G. In the case when ß is elliptic regular "modulo &weierpF", we will then obtain supercuspidal representations of G.