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

• Published in: arXiv.org
• Main Authors: Gunaydin, Murat, Pavlyk, Oleksandr
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.04.2009
• Subjects: Mathematical analysis; Mathematical functions; Operators (mathematics); Representations; Subgroups; Supergravity; Vectors (mathematics)
• ISSN: 2331-8422

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.