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Quasi-conformal actions were introduced in the physics literature as ageneralization of the familiar fractional linear action on the upper halfplane, to Hermitian symmetric tube domains based on arbitrary Jordan algebras,and further to arbitrary Freudenthal triple systems. In the mathematicsliterature, quaternionic discrete series unitary representations of realreductive groups in their quaternionic real form were constructed as degree 1cohomology on the twistor spaces of symmetric quaternionic-Kahler spaces. Thesetwo constructions are essentially identical, as we show explicitly for the tworank 2 cases SU(2,1) and G_{2(2)}. We obtain explicit results for certainprincipal series, quaternionic discrete series and minimal representations ofthese groups, including formulas for the lowest K-types in variouspolarizations. We expect our results to have applications to topologicalstrings, black hole micro-state counting and to the theory of automorphicforms.

Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G_2(2)