A variant of the idea of generalized complex geometry given by passing from generalization of complex geometry to generalization of exceptional geometry. Instead of by reduction of structure groups along inclusions like it is controled by inclusions of into split real forms of exceptional Lie groups.
One dimension down, compactification of 10-dimensional type II supergravity on a 6-manifold preserves supersymmetry precisely if the generalized tangent bundle in the NS-NS sector admits G-structure for the inclusion
This is reviewed in (GLSW, section 2).
Survey slides include
Daniel Persson, Arithmetic and Hyperbolic Structures in String Theory (arXiv:1001.3154)
Nassiba Tabti, Kac-Moody algebraic structures in supergravity theories (arXiv:0910.1444)
Original articles include
Christian Hillmann, Generalized E(7(7)) coset dynamics and D=11 supergravity, JHEP 0903 (2009) 135 (arXiv:0901.1581)
Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, Generalised geometry from the ground up (arXiv:1307.8295)
(see also at 3d supergravity – possible gaugings).
The E11-geometry of 11-dimensional supergravity compactified to the point is discussed in
The generalized-U-duality+diffeomorphsim invariance in 11d is discussed in