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In this article, we introduce (weakly) root graded Banach–Lie algebras and corresponding Lie groups as natural generalizations of group likefor a Banach algebra  A  or groups like  C ( X , K ) of continuous maps of a compact space  X  into a complex semisimple Lie group  K . We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups  P  to representations of  G  on holomorphic sections of homogeneous vector bundles over  G /  P . One of our main results is an algebraic characterization of the space of sections which is used to show that this space actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation of a (weakly) root graded Banach–Lie group  G  and show that all holomorphic functions on the spaces  G /  P  are constant.

Borel–Weil Theory for Root Graded Banach–Lie Groups