  The main purpose of the paper is to establish new tools in the study of
$\mathcal{O}^\mathfrak{p}$. We introduce the Jantzen coefficients of
generalized Verma modules. It comes from the Jantzen's simplicity criteria for
generalized Verma modules and has a deep relation with the structure of
$\mathcal{O}^\mathfrak{p}$. We develop a reduction process to compute those
coefficients. For which we need to consider generalized Verma modules induced
from maximal parabolic subalgebras having maximal nontrivial singularity, so
called basic generalized Verma modules. The classification of such modules is
obtained in this paper. As the first application of our results, we give a
refinement of Jantzen's simplicity criteria.
