Full exceptional collections on the Lagrangian Grassmannians LG(4,8)LG(4,8) and LG(5,10)LG(5,10)

We construct full exceptional collections in the derived categories of coherent sheaves on the Lagrangian Grassmannians LG(4,8)LG(4,8) and LG(5,10)LG(5,10). The construction is radically different from all the previously considered homogeneous spaces in that one has to use homogeneous bundles associated with reducible representations of the parabolic subgroup.