Cohomological splitting criterion on Lagrangian Grassmannians

Develop a cohomological splitting criterion for vector bundles on the Lagrangian Grassmannian LG(k), valid for any k ≥ 1, that characterizes when a bundle splits as a direct sum of line bundles—potentially using vanishing conditions involving irreducible homogeneous bundles rather than only wedge powers.

Background

For Grassmannians Gr(P^k, P^n), Ottaviani (1989) proved a splitting criterion that characterizes sums of line bundles via cohomology vanishings involving Schur functors of the universal bundle. For LG(k), Oeding–Macias Marques (2009) established partial results: their vanishing condition implies splitting, and the converse holds for k ≤ 6, but fails for k = 7 for E = O.

The authors highlight that LG(k) remains an open case for finding a general splitting criterion analogous to Grassmannians, suggesting the likely need to use irreducible homogeneous bundles (maximal weights) rather than only wedge powers.