Motivic Decomposition of Projective Pseudo-homogeneous Varieties

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions and relate them to the motives of corresponding homogeneous varieties.