  We obtain BGG-type formulas for transfer matrices of irreducible
finite-dimensional representations of the classical Lie algebras
$\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and
which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$.
These transfer matrices are expressed in terms of transfer matrices of certain
infinite-dimensional highest weight representations (such as parabolic Verma
modules and their generalizations) in the auxiliary space. We further factorise
the corresponding infinite-dimensional transfer matrices into the products of
two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929,
arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially
based on the new BGG-type resolutions of the finite-dimensional
$\mathfrak{g}$-modules, which naturally arise geometrically as the restricted
duals of the Cousin complexes of relative local cohomology groups of ample line
bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.
