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

Quasi-conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane, to Hermitian symmetric tube domains based on arbitrary Jordan algebras, and further to arbitrary Freudenthal triple systems. In the mathematics literature, quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic-Kähler spaces. These two constructions are essentially identical, as we show explicitly for the two rank 2 cases SU(2, 1) and G ₂₍₂₎. We obtain explicit results for certain principal series, quaternionic discrete series and minimal representations of these groups, including formulas for the lowest K-types in various polarizations. We expect our results to have applications to topological strings, black hole micro-state counting and to the theory of automorphic forms.