  After reviewing the underlying algebraic structures we give a unified
realization 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 to
their (Lorentz) subgroups SL(3,R), SL(3,R)XSL(3,R), SL(6,R), E_{6(6)} and
SO(n,m)XSO(1,1), respectively. We determine the spherical vectors of
quasiconformal realizations of all these groups twisted by a unitary character
$\nu$. We also give their quadratic Casimir operators and determine their
values in terms of $\nu$ and the dimension $n_V$ of the underlying Jordan
algebras. For $\nu= -(n_V+2)+i\rho$ the quasiconformal action induces unitary
representations 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 to
the discrete series and their continuations. The manifolds that correspond to
"quasiconformal compactifications" of the respective $(2n_V+3)$ 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.
