Borel-Weil Theory for Root Graded Banach-Lie groups

TL;DR

This paper extends Borel-Weil theory to root graded Banach-Lie groups, establishing a framework for holomorphic representations and sections in infinite-dimensional settings, generalizing classical finite-dimensional results.

Contribution

It introduces root graded Banach-Lie groups and algebras, characterizes holomorphic sections, and provides a geometric realization of irreducible holomorphic representations.

Findings

Spaces of sections carry natural Banach structures

All holomorphic functions on G/P are constant

Provides a geometric realization of irreducible representations

Abstract

In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ 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.