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

TL;DR

This paper demonstrates the equivalence of quasi-conformal actions and quaternionic discrete series representations for SU(2,1) and G_2(2), providing explicit formulas and potential applications in physics and automorphic forms.

Contribution

It explicitly shows the equivalence of two constructions of representations for SU(2,1) and G_2(2), and derives formulas for principal series, discrete series, and minimal representations.

Findings

Explicit formulas for lowest K-types in various polarizations.

Demonstration of the equivalence between quasi-conformal actions and quaternionic discrete series.

Potential applications to topological strings, black hole micro-states, and automorphic forms.

Abstract

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. We expect our results to have applications to topological strings, black hole micro-state counting and to the theory of automorphic forms.