Explicit realization of bounded modules for symplectic Lie algebras: spinor versus oscillator

We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded $\mathfrak{s}\mathfrak{p}(2n)$-modules. This realization is defined via a natural tableaux correspondence between spinor-type modules of $\mathfrak{s}\mathfrak{o}(2n)$ and oscillator-type modules of $\mathfrak{s}\mathfrak{p}(2n)$. In particular, we show that, in contrast with the $A$-type case, the generic and bounded $\mathfrak{s}\mathfrak{p}(2n)$-modules admit an analog of the Gelfand-Graev continuation from finite-dimensional representations.