Abstract. In this paper a geometric construction is given of all unitary highest weight modules of G = U(p, q). The construction is based on the unitary model of the kth tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over G/K, and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics. A major area of research in the representation theory of Lie groups is the explicit analysis of naturally occurring unitary representations. One aspect of this problem is that of producing geometric constructions of the unitary representations of a semisimple Lie group G. Here we consider unitary highest weigh
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