UNITARY HIGHEST WEIGHT REPRESENTATIONS IN HILBERT-SPACES OF HOLOMORPHIC-FUNCTIONS ON INFINITE-DIMENSIONAL DOMAINS

Automorphism groups of symmetric domains in Hilbert spaces form a natu ral class of infinite dimensional Lie algebras and corresponding Banac h Lie groups. We give a classification of the algebraic category of un itary highest weight modules for such Lie algebras and show that infin ite dimensional versions of the Lie algebras SD(2, n) have no unitary highest weight representations and thus do not meet the physical requi rement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbe rt spaces of vector valued holomorphic functions on the relevant domai ns in Hilbert spaces. The construction of such holomorphic representat ions of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Bana ch setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite d imensional settings is explored. (C) 1998 Academic Press.