Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors

After reviewing the algebraic structures that underlie the geometries of N = 2 Maxwell-Einstein supergravity theories (MESGT) with symmetric scalar manifolds in five and four dimensions, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F-4(4), E-6(2), E7(-5), E8(-24) and SO(n+2, 4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3, R), SL(3, C), SU*(6), E6(-26) and SO(n - 1, 1) x SO(1, 1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character nu. We present their quadratic Casimir operators and determine their values in terms of nu and the number n(V) of vector fields of the respective 5D supergravity. For nu = -(n(V) + 2) + i rho the quasiconformal action induces unitary representations belonging to the principal series. For special discrete values of nu it leads to unitary representations belonging to the quaternionic discrete series. Our results lay the algebraic groundwork for constructing explicitly the quaternionic discrete series unitary representations. For rank 2 cases, SU(2, 1) and G(2(2)), corresponding to simple N = 2 supergravity in four and five dimensions, respectively, this program was carried out in arXiv:0707.1669 and applied to quantum attractor flows.