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We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra ofquantum 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 justits homogeneous subspaces. We give a coherence realization for all the positiveroot vectors under the actions of Lusztig's braid automorphisms of$U_q({\mathfrak {sp}}_{2n})$.

Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space