  We generalizes the main result of Demailly \cite{D2} for the bundles
$E_{k,m}^{GG}(V^*)$ of jet differentials of order $k$ and weighted degree $m$
to the bundles $E_{k,m}(V^*)$ of the invariant jet differentials of order $k$
and weighted degree $m$. We formulate and prove the existence of an asymptotic
duality along the fibers of the Green-Griffiths jet bundles over projective
manifolds.
