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. In particular, we show that all these line
bundles are tensor products of pullbacks of line bundles over $X(\C)$ by
evaluation maps.

For the special case where $\cA$ is a $C^*$-algebra, our results lead to a
complete classification of all irreducible involutive holomorphic
representations of $G(\cA)$ on Hilbert spaces.