Oscillator-like unitary representations of non-compact groups with a jordan structure and the non-compact groups of supergravity

Abstract

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.