Symbolic calculus on the nilpotent orbits of SO0(1,2)

The coadjoint conical orbits in so(1, 2)* similar or equal to su(1, 1)* are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space-time. It contains also, as an open dense subset, the phase space for massless particles on one-dimensional Minkowski space-time when one identifies the Poincare group to a subgroup of the conformal group SO0(2, 2) similar or equal to SO0(1, 2) x SO0(1,2). On the other hand, the quantum representation associated to these systems is an indecomposable extension of the first term of the discrete series of representations of SO0(1,2) similar or equal to SU(1, 1)/Z(2) We present in this paper a symbol map linking this representation and this orbit. This calculus is invariant and behaves correctly in the classical limit. As a result we have obtained a conformally invariant symbolic calculus for massless particles on (anti-)de Sitter or Minkowski space-time in 1 + 1 dimension.