Quasiconformal Realizations of E 6 ( 6 ) , E 7 ( 7 ) , E 8 ( 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), SL(3,R) × SL(3,R), SL(6,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.