A unified approach to the minimal unitary realizations of noncompact groups and supergroups

We study the minimal unitary representations of non-compact groups andsupergroups obtained by quantization of their geometric realizations asquasi-conformal groups and supergroups. The quasi-conformal groups G leavegeneralized light-cones, defined by a quartic norm, invariant and have maximalrank subgroups of the form H X SL(2,R) such that G/H X SL(2,R) arepara-quaternionic symmetric spaces. We give a unified formulation of theminimal unitary representations of simple non-compact groups of type A_2, G_2,D_4, F_4, E_6, E_7, E_8 and Sp(2n,R). The minimal UIRs of Sp(2n,R) are simplythe singleton representations and correspond to a degenerate limit of theunified construction. The minimal unitary representations of the othernoncompact groups SU(m,n), SO(m,n), SO*(2n) and SL(m,R) are also givenexplicitly. We extend our formalism to define and construct the corresponding minimalrepresentations of non-compact supergroups G whose even subgroups are of theform H X SL(2,R). If H is noncompact then the supergroup G does not admit anyunitary representations, in general. The unified construction with H simple orAbelian leads to the minimal representations of G(3), F(4) and OSp(n|2,R) (inthe degenerate limit). The minimal unitary representations of OSp(n|2,R) withthe even subgroup SO(n) X SL(2,R) are the singleton representations. We alsogive the minimal realization of one parameter family of Lie superalgebrasD(2,1;\sigma).Comment: Minor revision to improve the presentation. Typos corrected. Version to appear in JHEP. 35 pages; Latex fil