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

Understand

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-Kahler 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.