Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups

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 E ₇×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 E₈₍₈₎ of the maximal supergravity theory the minimal realization was given in [1], The minimal unitary realizations of all the lower rank noncompact exceptional groups can be obtained by truncation of those of E_8(-24) and E₈₍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 [2].