  This paper generalizes a theorem of Hida on the structure of ordinary
representations on unitary groups to $P$-ordinary representations, where $P$ is
a general parabolic subgroup of some general linear group. When $P$ is minimal,
we recover Hida's theorem which asserts that ordinary subspaces are
1-dimensional. While analogous $P$-ordinary subspaces are infinite-dimensional
in general, we use the theory of Bushnell-Kutzko types to canonically associate
a finite-dimensional type to the representation (under minor assumptions) that
has multiplicity one in its $P$-ordinary subspace. We simultaneous develop the
theory of modular forms on unitary groups with $P$-Iwahoric level structure
whose nebentypus is a type (instead of a character) and construct lattices of
$P$-ordinary modular forms inside $P$-ordinary automorphic representations. We
also obtain direct consequences for the dual notion of $P$-anti-ordinary forms
and representations.
