As mentioned at
RF, Gunaydin and Pavlyk have posted a new paper
arXiv:0904.0784v1 [hep-th] on non-compact exceptional groups as quasiconformal groups over the split cubic Jordan algebras. The exceptional groups E_{6(6)}, E_{7(7)}, E_{8(8)} are known to arise from toroidal compactifications of D=11 supergravity down to d dimensions, where the global non-compact symmetry group of the maximally extended supergravity is given by E_{11-d(11-d)} (
arXiv:hep-th/0409263v1). It is believed the discrete subgroups E_{6(6)}(Z), E_{7(7)}(Z), E_{8(8)}(Z) yield the symmetries of the non-perturbative spectra of toroidally compactified M-theory (
arXiv:hep-th/9410167v2).
Algebraically, E_{6(6)}, E_{7(7)}, E_{8(8)} arise as transformation groups of the Freudenthal triple system over the split octonions. E_{6(6)} is the subgroup of the automorphism group of the Freudenthal triple system, E_{7(7)}, which preserves the cubic form of the split exceptional Jordan algebra. Geometrically, E_{6(6)} is the collineation group of the split Moufang plane OP^2_s, the space of projectors of the split exceptional Jordan algebra. E_{7(7)} and E_{8(8)} act as conformal and quasiconformal groups, respectively, over this space.
Gunaydin and Pavlyk show in their new paper that E_{6(6)}, E_{7(7)}, E_{8(8)} individually act as quasiconformal groups over the split cubic Jordan algebras J(3,C_s), J(3,H_s) and J(3,O_s). This implies they also act as quasiconformal groups over the split projective planes CP^2_s, HP^2_s and OP^2_s, respectively. Physically, this yields a new type of duality, a duality of U-dualities, so to speak. For example, M-theory compactified on an 8-torus with D=3 U-duality group E_{6(6)} is dual to M-theory compactified on a 6-torus with D=5 U-duality group E_{6(6)} as the symmetries of the non-perturbative spectra are equivalent.