  In this paper the authors investigate infinite-dimensional representations
$L$ in blocks of the relative (parabolic) category ${\mathcal O}_S$ for a
complex simple Lie algebra, having the property that the cohomology of the
nilradical with coefficients in $L$ ``looks like'' the cohomology with
coefficients in a finite-dimensional module, as in Kostant's theorem. A
complete classification of these ``Kostant modules'' in regular blocks for
maximal parabolics in the simply laced types is given. A complete
classification is also given in arbitrary (singular) blocks for Hermitian
symmetric categories.
