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

Abstract

After reviewing the underlying algebraic structures we give a unified realization of split exceptional groups F 4(4) , E 6(6) , E 7( 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 in terms of and the dimension n V of the underlying Jordan algebras. For = -(n V + 2) + i the quasiconformal action induces unitary representations on the space of square integrable