Borel--Weil Theory for Groups over Commutative Banach Algebras

TL;DR

This paper extends Borel--Weil theory to groups over commutative Banach algebras, providing explicit descriptions of line bundles and classifying certain holomorphic representations for these infinite-dimensional groups.

Contribution

It introduces a framework for understanding homogeneous line bundles and representations of Banach--Lie groups over commutative Banach algebras, generalizing classical finite-dimensional results.

Findings

Explicit description of holomorphic line bundles over generalized flag manifolds

All such line bundles are tensor products of pullbacks from classical cases

Complete classification of irreducible involutive holomorphic representations for C*-algebra cases

Abstract

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