Quaternionic Analysis and the Schrodinger Model for the Minimal Representation of O(3,3)

TL;DR

This paper links quaternionic analysis with the representation theory of O(3,3), introducing operators that clarify the structure of minimal representations and verifying key coefficients through the Schrödinger model.

Contribution

It introduces a new operator d/dR Pl_R on quaternionic spaces and uses the Schrödinger model to compute its effects, providing independent verification of Plancherel measure coefficients.

Findings

Operators on quaternionic spaces can be analyzed via the Schrödinger model.

The effect of the operator d/dR Pl_R on series components is explicitly computed.

The approach confirms and extends previous results on minimal representations of O(3,3).

Abstract

In the series of papers [FL,FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(4,C) and its real forms. This approach has proven very fruitful and pushed further the parallel with complex analysis and develop a rich theory. In [FL2] we study the counterparts of Cauchy-Fueter and Poisson formulas on the spaces of split quaternions H_R and Minkowski space M and show that they solve the problem of separation of the discrete and continuous series on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R). In particular, we introduce an operator Pl_R, compute its effect on the discrete and continuous series components of the space of functions H(H_R) and obtain a surprising formula for the Plancherel measure of SL(2,R). The proof is based on a transition to the Minkowski space M and some pretty lengthy computations. In this paper we introduce an operator d/dR Pl_R on H(H_R) and show that its effect on the discrete and continuous series components can be easily computed using the Schrodinger model for the minimal representation of O(p,q) (with p=q=3) and the results of Kobayashi-Mano from [KM], particularly their computation of the integral expression for the operator F_C. This provides an independent verification of the coefficients involved in the formula for Pl_R. This paper once again demonstrates a close connection between quaternionic analysis and representation theory of various O(p,q)'s.