## 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.

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