Borel--Weil Theory for Groups over Commutative Banach Algebras

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.