The infinite dimensional discrete series representations of Sp(N,\({\mathbb{R}})\) are discussed. It is shown that, up to a certain equivalence, such representations labelled by \(<\sigma (\lambda)>\), where \(\sigma\) is either an integer or half an odd integer and \(\lambda =(\lambda_ 1,\lambda_ 2,...,\lambda_ N)\) is a partition, are unitary if and only if \({\tilde \lambda}{}_ 1\leq N-1\) and \({\tilde \lambda}{}_ 1+{\tilde \lambda}_ 2\leq 2\sigma\) where \({\tilde \lambda}=({\tilde \lambda}_ 1,{\tilde \lambda}_ 2,....)\) is the partition conjugate to \(\lambda\). The authors exploit the duality between Sp(N,\({\mathbb{R}})\) and O(2\(\sigma)\) arising from a consideration of the metaplectic or harmonic oscillator or Segal-\(Shale\)-\(Weil\) representation of Sp(2\(\sigma\) N,\({\mathbb{R}})\) [M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772-1780 (1971; Zbl 0247.20051), M. Kashiwara and M. Vergne, Invent. Math. 44, 1-47 (1978; Zbl 0375.22009)]. The application of S-function methods [D. E. Littlewood, The theory of group characters and matrix representations of groups (1940; Zbl 0025.00901)] in this context leads to a complete specification of the characters of Sp(N,\({\mathbb{R}})\) in terms of those of U(N). Matrix elements are then constructed explicitly for the particular representations \(<\sigma (1^{\alpha})>\) for \(0\leq \alpha \leq N\).