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We extend a theorem of Ottaviani on cohomological splitting criterion forvector bundles over the Grassmannian to the case of the symplectic isotropicGrassmanian. We find necessary and sufficient conditions for the case of theGrassmanian of symplectic isotropic lines. For the general case thegeneralization of Ottaviani's conditions are sufficient for vector bundles overthe symplectic isotropic Grassmannian. By a calculation in the program LiE, wefind that Ottaviani's conditions are necessary for Lagrangian Grassmannian ofisotropic $k$-planes for $k\leq 6$, but they fail to be necessary for the caseof the Lagrangian Grassmannian of isotropic 7-planes. Finally, we find arelated set of necessary and sufficient splitting criteria for the LagrangianGrassmannian.

Splitting criteria for vector bundles on the symplectic isotropic Grassmannian