ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES

If G is a connected reductive p-adic group, P is a parabolic subgroup of G, and M is a Levi factor of P, and if A is an Artinian local ring having a finite residue field of characteristic p, then we define a functor Ord(P) from the category of admissible smooth P-representations over A to the category of admissible smooth M-representations of A, which we call the functor of ordinary parts. We show that this functor is right adjoint to the functor of parabolic induction IndG/P, where (P) over bar is an opposite parabolic to P.