Highest weight orbits: PBW and toric degenerations

Let M be a cyclic representation of a nilpotent Lie algebra. We consider the closure F inside the projectivization of M of the orbit through the cylcic line of the corresponding unipotent group. Using Vinberg's filtration, we construct a degeneration of F into a variety that is both a toric variety and a $\bG_a^N$ variety. This degeneration factors through the PBW degeneration. Under certain combinatorial assumption, we describe the coordinate rings of the initial orbit closure F as well as of its PBW degenerate version and prove that the degenerations are flat. We give two applications of our construction. First, we describe new $\bG_a^N$--toric degenerations of the flag varieties for simple Lie groups and give their description in types A, C and $G_2$. Second, we introduce the notion of the parabolic PBW degeneration of flag varieties and study in detail the degenerate Grassmannians.