There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector bundle $G\times_H V\to G/P$ is a trivial as a vector bundle. A trivialization can be written down explicitly. It is induced by the map $G\times V\to (G/H)\times V$ defined by $(g,v)\mapsto (gH,g\cdot v)$. This is evidently $H$-invariant and factorizes to an isomorphism $G\times_H V\to (G/H)\times V$ of vector bundles.