Inductive systems of representations with small highest weights for natural embeddings of symplectic groups

For natural embeddings of symplectic groups, inductive systems of irreducible representations where the maximum of the highest weight value on the maximal root is equal to $2$ are studied. For such embeddings of algebraic groups of type $C_n$ in characteristic $3$, the inductive system of representations generated by irreducible representations with highest weight $2\omega_n$ is determined. It is proved that any inductive system of representations of such groups consisting of representations with the value of the highest weight on the maximal root at most $2$ and containing representations with such value equal to $2$ contains the subsystem generated by the standard representations or the subsystem generated by the representations with highest weight $\omega_n$, For algebraic groups of type $C_n$ in characteristic $3$, the restrictions of certain irreducible modules to subsystem subgroups of type $C_{n-1}$ are described.