We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation, Lorentz and conformal groups are SO(d − 1), SO(d − 1, 1) × SO(1, 1) and SO(d, 2) × SO(2, 1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d = 3, 4, 6, 10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. Their rotation, Lorentz and conformal groups are those that occur in the first three rows of the Magic Square. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra "cocycle" coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2, 4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F 4(4) , E 6(2) , E 7(−5) and E 8(−24) of the simple Jordan family were given in our earlier work hep-th/0409272.

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We study the minimal unitary representations of noncompact exceptional groups that arise as U-duality groups in extended supergravity theories. First we give the unitary realizations of the exceptional group E 8(−24) in SU * (8) as well as SU(6, 2) covariant bases. E 8(−24) has E7 ×SU(2) as its maximal compact subgroup and is the U-duality group of the exceptional supergravity theory in d = 3. For the corresponding U-duality group E 8(8) of the maximal supergravity theory the minimal realization was given in hep-th/0109005. The minimal unitary realizations of all the lower rank noncompact exceptional groups can be obtained by truncation of those of E 8(−24) and E 8(8) . By further truncation one can obtain the minimal unitary realizations of all the groups of the "Magic Triangle". We give explicitly the minimal unitary realizations of the exceptional subgroups of E 8(−24) as well as other physically interesting subgroups. These minimal unitary realizations correspond , in general, to the quantization of their geometric actions as quasi-conformal groups as defined in hep-th/0008063.

We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones defined by a quartic norm invariant and have maximal rank subgroups of the form H × SL(2, R) such that G/H × SL(2, R) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A 2 , G 2 , D 4 ,F 4 , E 6 , E 7 , E 8 and Sp (2n, R). The minimal unitary representations of Sp (2n, R) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU (m, n), SO (m, n) , SO * (2n) and SL (m, R) are also given explicitly.We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H × SL(2, R). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F (4) and OSp (n|2, R) (in the degenerate limit). The minimal unitary representations of OSp (n|2, R) with even subgroups SO(n) × SL(2, R) are the singleton representations. We also give the minimal realization of the one parameter family of Lie superalgebras D (2, 1; σ).