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

In the series of papers [FL1, FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(2,HC) ' 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 HR 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 PlR, 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 PlR on H(H + R ) and show that its effect on the discrete and continuous series components can be easily computed using the Schrödinger 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 FC . This provides an independent verification of the coefficients involved in the formula for PlR. This paper once again demonstrates a close connection between quaternionic analysis and representation theory of various O(p, q)’s.