The authors develop and analyse certain character reductions for the infinite-dimensional unitary discrete series representations of the non-compact symplectic group Sp(2n,R). The group reductions considered are Sp(2k,R) contains/implies Sp(2,R)*O(k) and the more general Sp(2nk,R) contains/implies Sp(2n,R)*O(k). They use Schur function techniques to derive succinct formulae involving certain infinite series of Schur functions. The results are relevant to the study of many-body systems with interactions of bilinear form and to the description of various quantum phenomena including collective behaviour.