Representations in \(L^{2}\)-spaces on infinite-dimensional symmetric cones
In a previous paper in [J. Funct. Anal. 156, 263--300 (1998; Zbl 0908.22015)], the same authors classified unitary highest weight representations of automorphism groups of classical infinite-dimensional tube domains. In this paper they construct their \(L^2\)-realizations, including the vector-valued case. The domains in question are all contained in the bounded operators on a Hilbert space. In the earlier paper, the realization was in certain Hilbert spaces of holomorphic functions on bounded domains, while in the present paper the authors use the (analogs of) Siegel domains. The two versions are related by a Cayley transform. Furthermore, the authors construct a projective representation of the full identity component of the affine automorphism group of the Hilbert-Schmidt version \(\Sigma_2\) of the tube domain, which is actually a representation of the subgroup corresponding to the trace class version \(\Sigma_1\) of the tube domain, but it is not a representation of the whole group. Topics considered include also infinite-dimensional measure theory, in particular Gaussian measures and (analogs of) Wishart distributions, Laplace transforms, holomorphic functions on infinite-dimensional domains, infinite-dimensional Jordan algebras and associated conformal Lie algebras. In the last two sections the authors prove ergodicity for the measures involved in the scalar case, while for the operator-valued case they obtain results about densities with respect to the Wishart distributions.
scientific article; zbMATH DE number 1763760
scientific article