  Let K be a maximal unramified extension of a nonarchimedean local field with
arbitrary residual characteristic p. Let G be a reductive group over K which
splits over a tamely ramified extension of K. We show that the associated
Moy-Prasad filtration representations are in a certain sense independent of p.
We also establish descriptions of these representations in terms of explicit
Weyl modules and as representations occurring in a generalized Vinberg-Levy
theory.
  As an application, we use these results to provide necessary and sufficient
conditions for the existence of stable vectors in Moy-Prasad filtration
representations, which extend earlier results by Reeder and Yu (which required
p to be large) and by Romano and the author (which required G to be absolutely
simple and split). This yields new supercuspidal representations.
  We also treat reductive groups G that are not necessarily split over a tamely
ramified field extension.
