Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

Abstract

Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored.