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We prove the locally differentiable rigidity of generic partially hyperbolicabelian algebraic high-rank actions on compact homogeneous spaces obtained fromsplit symplectic Lie groups. We also gave a non-generic action rigidity exampleon compact homogeneous spaces obtained from SL(2n,R) or SL(2n,C). Theconclusions are based on geometric Katok-Damjanovic way and progress towardscomputations of the generating relations in these groups.

New Cases of Differential Rigidity for Non-Generic Partially Hyperbolic Actions