Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space

Let GrL⊂Gr(n,V) be the space of all Lagrangian subspaces of C2nC2n equipped with the standard symplectic form. Let X˜ be a universal cover of a compact connected Riemann surface XX. We consider all immersions f:X˜⟶GrL satisfying the following two conditions: (1) the map ff is equivariant with respect to some homomorphism into Sp(2n,C) of the Galois group of the covering X˜⟶X, and (2) the symmetric bilinear form on the pullback, to X˜, of the tautological vector bundle over GrL is fiberwise nondegenerate. Two such maps are called equivalent if they differ by the action of some fixed element of Sp(2n,C). We prove that the equivalence classes of all such maps are bijectively parametrized by pairs of the form (P,(F,∇))(P,(F,∇)), where PP is a projective structure on XX and (F,∇)(F,∇) is an equivalence class of flat O(n,C)-connection on XX. Two flat O(n,C)-bundles are equivalent if the corresponding flat PO(n,C)-bundles are isomorphic.