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

Abstract

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.