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

TL;DR

This paper constructs unified quasiconformal realizations of exceptional and orthogonal groups relevant to supergravity, analyzes their spherical vectors, Casimir operators, and representations, and explores implications for U-duality in supergravity theories.

Contribution

It provides a unified quasiconformal framework for exceptional groups and SO(n+3,m+3), including spherical vectors, Casimir operators, and their role in supergravity U-duality groups.

Findings

Derived spherical vectors for all quasiconformal groups.

Computed quadratic Casimir operators in terms of parameters.

Linked quasiconformal actions to unitary representations in supergravity contexts.

Abstract

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.