Quasiconformal realizations of E₆₍₆₎, E₇₍₇₎, E₈₍₈₎ and SO(n + 3, m + 3), N ≥ 4 supergravity and spherical vectors

After reviewing the underlying algebraic structures we give a unified realization of split exceptional groups F₄₍₄₎, E₆₍₆₎, E₇₍₇₎, E₈₍₈₎ and of SO(n + 3,m + 3) as quasiconformal groups that is covariant with respect to their (Lorentz) subgroups SL(3,R{double-struck}), SL(3,R{double-struck}) × SL(3,R{double-struck}), SL(6,{double-struck}R), E₆₍₆₎ and SO(n,m) × SO(1, 1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character ν. We also give their quadratic Casimir operators and determine their values in terms of ν and the dimension nV of the underlying Jordan algebras. For ν = -(nV + 2) + iρ the quasiconformal action induces unitary representations on the space of square integrable functions in (2nV + 3) variables, that belong to the principle series. For special discrete values of ν the quasiconformal action leads to unitary representations belonging to the discrete series and their continuations. The manifolds that correspond to "quasiconformal compactifications" of the respective (2nV + 3) dimensional spaces are also given. We discuss the relevance of our results to N = 8 supergravity and to N = 4 Maxwell- Einstein supergravity theories and, in particular, to the proposal that three and four dimensional U-duality groups act as spectrum generating quasiconformal and conformal groups of the corresponding four and five dimensional supergravity theories, respectively.