I will explain how a generalized theory of canonical subgroups applies to exhibit a stratification of Lubin-Tate spaces outside sufficiently great compact subset of the open p-adic ball. To each of this stratum is associated a parabolic subgroup of $GL_n$ and the cohomology (in the limit) of this stratum is parabolicaly induced. This applies to a conjecture of Prasad on autodual representations of division algebras. I will discuss how this should apply to finish M. Strauch program on proving part of Harris Taylor results by purely local methods.