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 ζ, one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0, ζ/2)((−ζ/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 ζ, 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 hep-th/9806042.

We study the minimal unitary representation (minrep) of SO(6, 2) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO(6, 2), which coincides with the minrep of SO * (8) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of "deformations" of the minrep of SO * (8) labeled by the spin t of an SU (2) T subgroup of the little group SO(4) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO * (8) that describe massless conformal fields in six dimensions. The SU (2) T spin t is the six dimensional counterpart of U (1) deformations of the minrep of 4D conformal group SU (2, 2) labeled by helicity. We also construct the supersymmetric extensions of the minimal unitary representation of SO * (8) to minimal unitary representations of OSp(8 * |2N ) that describe massless six dimensional conformal supermultiplets. The minimal unitary supermultiplet of OSp(8 * |4) is the massless supermultiplet of (2, 0) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 .

Abstract.Let g denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let I) denote a Cartan subalgebra of g. In this paper we study finitely generated g-modules that decompose into direct sums of finite dimensional l)-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g is a simple Lie algebra that admits a torsion free module, then g is of type A or C.

We extend our earlier work on the minimal unitary representation of SO(d, 2) and its deformations for d = 4, 5 and 6 to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d, 2) and its deformations and massless conformal fields in Minkowskian spacetimes in d dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS (d+1) /CF T d higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d − 2) for massless representations.

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