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

Igor Frenkel

¹

,

Matvei Libine

²

Abstract

We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions H R . 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 H R . 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 H R and M and show that they solve the problem of separation of the discrete and continuous series. The continuous series component on H R 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 H R . 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.

Magic identities for the conformal four-point integrals; the Minkowski metric case

“…In this section we establish further notations and state relevant results from quaternionic analysis. We mostly follow our previous papers [FL1,FL2,L1,L2].…”

“…As in Section 2 of [FL2], we consider the space H consisting of C-valued functions on H C (possibly with singularities) that are holomorphic with respect to the complex variables z 11 , z 12 , z 21 , z 22 and harmonic, i.e. annihilated by…”

“…By abuse of notation, we denote these Lie algebra actions by π 0 l and π 0 r respectively. They are described in Subsection 3.2 of [FL2].…”

The second order pole over split quaternions

“…This is an addition to a series of papers [FL1,FL2,FL3,FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. Since the paper [FL4] serves as a starting point for this article, we highlight the most relevant results.…”

Projections Separating Spectra for $\boldsymbol{L^2}$ on Pseudounitary Groups $\boldsymbol{{\mathrm U}(p,q)}$

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