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) in five and four dimensions with symmetric scalar manifolds, 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)}, 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), SU*(6), E_{6(6)} 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. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic groundwork for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series. For rank 2 cases, SU(2,1) and G_{2(2)}, corresponding to simple N=2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries of matter coupled N=4 and N>4 supergravity theories. They lead to quasiconformal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms.