  We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of
quantum differential operators on the quantum symplectic space
$\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a
$U_q({\mathfrak{sp}}_{2n})$-module algebra whose irreducible summands are just
its homogeneous subspaces. We give a coherence realization for all the positive
root vectors under the actions of Lusztig's braid automorphisms of
$U_q({\mathfrak {sp}}_{2n})$.
