It is well known that the restriction of a finite-dimensional irreducible representation of the symplectic Lie algebra sp(2n) to the subalgebra sp(2m) with m < n need not be multiplicity-free. Each multiplicity space turns out to be an irreducible representation of a quantized enveloping algebra Y(sp(2n-2m)) called the twisted Yangian. We give a description of this representation by calculating its highest weight and Drinfeld polynomials.