Geometric invariants and model theory for singular unitary representations

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In this research project, we investigated realization of singular irreducible unitary representations of Lie groups through geometric approach. First, we have established the Fock model version of Dvorsky-Sahi theory on an extension of the theta duality correspondence for singular unitary highest weight representations of reductive Lie groups, by decomposing tensor products of fundamental representations in terms of geometric invariants for representations in question. Second, the singular orbits in prehomogeneous vector spaces arising from quaternionic structure of exceptional simple Lie groups of real rank 4 have been described by using data on root systems, and we have proved that the singular quarternionic unitary representations, due to Gross and Wallach, can be realized by geometric quantization of the corresponding quarternionic nilpotent K-orbits.