We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded sp(2n)-modules. This realization is defined via a natural tableaux correspondence between spinor-type modules of so(2n) and oscillator-type modules of sp(2n). In particular, we show that, in contrast with the A-type case, the generic and bounded sp(2n)-modules admit an analog of the Gelfand-Graev continuation from finite-dimensional representations.
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Futorny et al. (2024) studied this question.