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 F₄₍₄₎,E₆₍₂₎,E_7(-5),E_8(-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), SUz.ast;(6), E_6(-26) and SO(n - 1, 1) × SO(1, 1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character ?. We present their quadratic Casimir operators and determine their values in terms of μ and the number n_V of vector fields of the respective 5D supergravity. For μ = -(nV + 2) + iρ the quasiconformal action induces unitary representations belonging to the principal series. For special discrete values of μ 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 G₂₍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.