  We generalize 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 $m$. Namely, Theorem 0.5 from \cite{D2} and Theorem 9.3
from \cite{D1} provide a lower bound $\frac{c^k}{k}m^{n+kr-1}$ on the number of
the linearly independent holomorphic global sections of $E_{k,m}^{GG} V^*
\bigotimes \mathcal{O}(-m \delta A)$ for some ample divisor $A$. The group
$G_k$ of local reparametrizations of $(\mathbb{C},0)$ acts on the $k$-jets by
orbits of dimension $k$, so that there is an automatic lower bound
$\frac{c^k}{k} m^{n+kr-1}$ on the number of the linearly independent
holomorphic global sections of $E_{k,m}V^* \bigotimes \mathcal{O}(-m \delta
A)$. We formulate and prove the existence of an asymptotic duality along the
fibers of the Green-Griffiths jet bundles over projective manifolds. We also
prove a Serre duality for asymptotic sections of jet bundles. An application is
also given for partial application to the Green-Griffiths conjecture.
