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In this paper a geometric construction is given of all unitary highest weight modules of      G = U ⁡  ( p , q ) G = \operatorname {U} (p,q)      . The construction is based on the unitary model of the      k k      th 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 G/K      , and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.

An 𝐿²-cohomology construction of unitary highest weight modules for 𝑈(𝑝,𝑞)