Bushnell-Kutzko types for $P$-ordinary automorphic representations on unitary groups

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.