Differential operators and immersions of a Riemann surface into a Grassmannian (Q1602442)

The author studies holomorphic immersions \(f:\widetilde{X}\to G\) of the universal cover \(\widetilde X\) of a compact Riemann surface \(X\) into the Grassmannian \(G = G(n,\mathbb{C}^{2n})\). The mapping \(f\) is assumed to satisfy a nondegeneracy condition, and to be equivariant with respect to a homomorphism \(\rho:\Gamma\to \text{GL}(\mathbb{C}^{2n})\), where \(\Gamma\) is the decktransformation group acting on \(\widetilde{X}\), \(X = \widetilde{X}/\Gamma\), and \(\rho\Gamma\) acts on \(G\). The author proves that such immersions \(f\) (up to equivalence) are in bijective correspondence with the holomorphic differential operators (up to equivalence) of order 2 on a rank \(n\) vector bundle over \(X\) such that the symbol of the operator is an isomorphism.