Abstract

Let be a commutative unital Banach algebra, 𝔤 be a semisimple complex Lie algebra and be the 1-connected Banach–Lie group with Lie algebra . Then there is a natural concept of a parabolic subgroup of and we obtain generalizations of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over 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 is a C*-algebra, our results lead to a complete classification of all irreducible involutive holomorphic representations of on Hilbert spaces.