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

摘要

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) × SL(3, R), SL(6,R), E_(6(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 v. We also give their quadratic Casimir operators and determine their values in terms of v and the dimension n_V of the underlying Jordan algebras. For v= -(n_V + 2) + ip 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 v 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.