For The metalinear group is a Lie group that is a -group extension of the general linear group .
Inside the symplectic group sits the general linear group
as the subgroup that preserves the standard Lagrangian submanifold . Restriction of the metaplectic group extension along this inclusion defines the metalinear group
Hence a metaplectic structure on a symplectic manifold induces a metalinear structure on its Lagrangian submanifolds.