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

We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones defined by a quartic norm invariant and have maximal rank subgroups of the form H × SL(2, ℝ) such that G/H × SL(2, ℝ) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A ₂, G₂, D₄, F₄, E₆, E ₇, E₈ and Sp(2n, ℝ). The minimal unitary representations of Sp(2n, ℝ) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU(m, n), SO(m, n), SO*(2n) and SL(m, ℝ) are also given explicitly. We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H × SL(2, ℝ). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F(4) and O Sp(n|2, ℝ) (in the degenerate limit). The minimal unitary representations of O Sp(n|2, ℝ) with even subgroups SO(n) × SL(2, ℝ) are the singleton representations. We also give the minimal realization of the one parameter family of Lie superalgebras D(2, 1; σ).