Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds

Let N be a manifold of dimension m with a flat vector bundle given by a representation ρ:π₁(N) → GL(n, R) where π₁(N) is finitely generated. The holonomy group $ρ$ is a k-partially hyperbolic holonomy representation if the flat bundle pulled back over the unit tangent bundle of a sufficiently large compact submanifold of N splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are k-dimensional with $k < n/2$. Suppose that each element of ρ(π₁(N)) has an eigenvalue of norm $1$, or, alternatively, each element of it has a singular value that is uniformly bounded above and below. We show that $ρ$ is a P-Anosov representation for a parabolic subgroup P of GL(n, R) if and only if $ρ$ is a partially hyperbolic representation. We are going to primarily employ representation theory techniques. As an application, we will show this holds when N is a complete affine n-manifold and $ρ$ is a linear part of the holonomy representation. This had never been done over the full general linear group.