Representations in L2-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L2-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”