Borel–Weil Theory for Root Graded Banach–Lie Groups

Abstract

In this paper we introduce (weakly) root graded Banach‐Lie algebras and corresponding Lie groups as natural generalizations of group like GLn(A) for 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.