Harmonic superspace, minimal unitary representations and quasiconformal groups

We show that there is a remarkable connection between the harmonic superspace (HSS) formulation of N ≤ 2, d ≤ 4 supersymmetric quaternionic Kähler sigma models that couple to N ≤ 2 supergravity and the minimal unitary representations of their isometry groups. In particular, for N ≤ 2 sigma models with quaternionic symmetric target spaces of the form G/H × SU(2) we establish a one-to-one mapping between the Killing potentials that generate the isometry group G under Poisson brackets in the HSS formulation and the generators of the minimal unitary representation of G obtained by quantization of its geometric realization as a quasiconformal group. Quasiconformal extensions of U-duality groups of four dimensional N ≤ 2, d ≤ 4 Maxwell-Einstein supergravity theories (MESGT) had been proposed as spectrum generating symmetry groups earlier. We discuss some of the implications of our results, in particular, for the BPS black hole spectra of 4d N ≤ 2 MESGTs.