Abstract

This chapter deals with highest-weight modules, holomorphic representations of semigroups, the holomorphic discrete series, and Hardy spaces on compactly causal symmetric spaces. Holomorphic representations of the semigroup S(D) = G exp iD were introduced by Ol'shanskii. There is an extensive literature on highest-weight modules, and the classification of highest-weight modules. Harish-Chandra constructed the holomorphic discrete series of the group where the starting point of the analysis was on the bounded symmetric domains, unitary highest-weight modules, and the discrete series of the group. The analytic continuation was achieved by Wallach. The construction of Ψ (π,v) was by using the dual representation πv. The introduction of the Poisson kernel is new.

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