Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple
complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie
algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic
subgroup $P(\cA)$ of $G(\cA)$ and we obtain generalizations $X(\cA) :=
G(\cA)/P(\cA)$ of the generalized flag manifolds. In this note we provide an
explicit description of all homogeneous holomorphic line bundles over $X(\cA)$
with non-zero holomorphic sections.