The generating rank of the symplectic grassmannians: Hyperbolic and isotropic geometry

Exploiting the interplay between hyperbolic and isotropic geometry, we prove that the grassmannian of totally isotropic k -spaces of the polar space associated to the symplectic group Sp 2 n ( F ) has generating rank 2 n k − 2 n k − 2 when Char ( F ) ≠ 2 .