  We study the minimal unitary representation (minrep) of SO(4,2) over an
Hilbert space of functions of three variables, obtained by quantizing its
quasiconformal action on a five dimensional space. The minrep of SO(4,2), which
coincides with the minrep of SU(2,2) similarly constructed, corresponds to a
massless conformal scalar in four spacetime dimensions. There exists a
one-parameter family of deformations of the minrep of SU(2,2). For positive
(negative) integer values of the deformation parameter \zeta one obtains
positive energy unitary irreducible representations corresponding to massless
conformal fields transforming in (0,\zeta/2) ((-\zeta/2,0)) representation of
the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep
of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary
supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the
massless N=4 Yang-Mills supermultiplet in four dimensions. For each given
non-zero integer value of \zeta, one obtains a unique supermultiplet of
massless conformal fields of higher spin. For SU(2,2|4) these supermultiplets
are simply the doubleton supermultiplets studied in arXiv:hep-th/9806042.
