  The adjacency matrix of a symplectic dual polar graph restricted to the
eigenspaces of an abelian automorphism subgroup is shown to act as the
adjacency matrix of a weighted subspace lattice. The connection between the
latter and $U_q(sl_2)$ is used to find the irreducible components of the
standard module of the Terwilliger algebra of symplectic dual polar graphs. The
multiplicities of the isomorphic submodules are given.
