group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $(X, \omega)$ a symplectic manifold a metaplectic structure on $X$ is a G-structure for $G$ the metaplectic group, hence a lift of structure groups of the tangent bundle from the symplectic group to the metaplectic group through the double cover map $Mp(2n, \mathbb{R}) \to Sp(2n, \mathbb{R})$:
Analogously for the Mp^c-group one considers $Mp^c$-structures.
Let $(X,\omega)$ be a symplectic manifold and $L \subset T X$ a subbundle of Lagrangian subspaces of the tangent bundle. Then $T X$ admits a metaplectic structure precisely if $L$ admits a metalinear structure.
(Bates-Weinstein, theorem 7.16)
Every Sp-principal bundle has a lift to an Mp^c-principal bundle.
(Robinson-Rawnsley 89, theorem (6.2))
For more details, see at metaplectic group – (Non-)Triviality of Extensions.
Michael Forger, Harald Hess, Universal metaplectic structures and geometric quantization, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. (EUCLID)
R. J. Plymen, The Weyl bundle, Journal of Functional Analysis 49, 186-197 (1982) (journal)
P. L. Robinson, John Rawnsley, The metaplectic representation, $Mp^c$-structures and geometric quantization, 1989
Michel Cahen, Simone Gutt, John Rawnsley, Symplectic Dirac Operators and $Mp^c$-structures (arXiv:1106.0588)
Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, (pdf)