Split quaternionic analysis and separation of the series for SL(2,R) and SL(2,C)/SL(2,R)

We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions HR. The special role of the unit sphere in the classical quaternions H – identified with the group SU(2) – is now played by the group SL(2,R) realized by the unit quaternions in HR. As in the previous work, we use an analogue of the Cayley transform to relate the analysis on SL(2,R) to the analysis on the imaginary Lobachevski space SL(2,C)/SL(2,R) identified with the one-sheeted hyperboloid in the Minkowski space M. We study the counterparts of Cauchy–Fueter and Poisson formulas on HR and M and show that they solve the problem of separation of the discrete and continuous series. The continuous series component on HR gives rise to the minimal representation of the conformal group SL(4,R), while the discrete series on M provides its K-types realized in a natural polynomial basis. We also obtain a surprising formula for the Plancherel measure of SL(2,R) in terms of the Poisson-type integral on the split quaternions HR. Finally, we show that the massless singular functions of four-dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary Lobachevski space SL(2,C)/SL(2,R). Our results once again reveal the central role of the Minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four-dimensional quantum field theory.