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.

Representations in L2-Spaces on Infinite-Dimensional Symmetric Cones

“…W 1 , we first use the correspondence to positive definite kernels on the bounded domains explained in [NØ98] and then an appropriate Cayley transform to obtain a correspondence between the bounded and the unbounded picture.…”

“…The main objective of this paper is to provide an L 2 -realization of the unitary highest weight representations of the automorphism groups of infinite-dimensional Hilbert domains which have been classified in [NØ98]. In this section we explain how one associates to a unitary highest weight representation of such a group a positive definite function on the domain W 1 , resp.…”

“…The cocycle is trivial on the subgroup corresponding to the trace class version S 1 , but non-trivial on the large group. The structure of the cocycle gives a geometric explanation for some of the difficulties we encountered in [NØ98] (Section IV).…”

“…In a previous paper [NØ98]we gave an algebraic classification of the unitary highest weight representations for the infinite-dimensional analogs of the automorphism groups of the classical bounded symmetric domains. We also realized these representations globally in certain Hilbert spaces of holomorphic functions on certain corresponding domains in (infinitedimensional) Hilbert spaces.…”

Iwasawa decompositions of some infinite-dimensional Lie groups

“…As regards the representation theory, it is well known that decompositions of this kind are particularly important for instance in the construction of principal series representations, and there has been a continuous endeavor to extend the ideas of representation theory to the setting of infinite-dimensional Lie groups. Some references related in spirit to the present paper are [Se57], [Ki73], [SV75], [Ol78] [Bo80], [Ca85], [Ol88], [Pic90], [Bo93], [Nee98], [NØ98], [NRW01], [DPW02], [Nee04], [Gru05], [Wo05], [BR07]; however, this list is very far from being complete. In this connection we wish to highlight the paper [Wo05] devoted to an investigation of direct limits of (Iwasawa decompositions and) principal series representations of reductive Lie groups.…”