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After reviewing the underlying algebraic structures we give a unifiedrealization of split exceptional groups F_{4(4)},E_{6(6)}, E_{7(7)}, E_{8(8)}and of SO(n+3,m+3) as quasiconformal groups that is covariant with respect totheir (Lorentz) subgroups SL(3,R), SL(3,R)XSL(3,R), SL(6,R), E_{6(6)} andSO(n,m)XSO(1,1), respectively. We determine the spherical vectors ofquasiconformal realizations of all these groups twisted by a unitary character$\nu$. We also give their quadratic Casimir operators and determine theirvalues in terms of $\nu$ and the dimension $n_V$ of the underlying Jordanalgebras. For $\nu= -(n_V+2)+i\rho$ the quasiconformal action induces unitaryrepresentations on the space of square integrable functions in $(2n_V+3)$variables, that belong to the principle series. For special discrete values of$\nu$ the quasiconformal action leads to unitary representations belonging tothe discrete series and their continuations. The manifolds that correspond to"quasiconformal compactifications" of the respective $(2n_V+3)$ dimensionalspaces are also given. We discuss the relevance of our results to N=8supergravity and to N=4 Maxwell-Einstein supergravity theories and, inparticular, to the proposal that three and four dimensional U-duality groupsact as spectrum generating quasiconformal and conformal groups of thecorresponding four and five dimensional supergravity theories, respectively.

Quasiconformal Realizations of E_{6(6)}, E_{7(7)}, E_{8(8)} and SO(n+3,m+3), N=4 and N>4 Supergravity and Spherical Vectors