Explicit realization of bounded modules for symplectic Lie algebras: spinor versus oscillator
Vyacheslav Futorny, Dimitar Grantcharov, Luis Enrique Ramirez, Pablo Zadunaisky
TL;DR
The paper develops an explicit Gelfand–Tsetlin (GT) tableaux realization for all simple and injective (hence projective) objects in the category of bounded $\mathfrak{sp}(2n)$-modules by linking spinor-type $\mathfrak{so}(2n)$-structures to oscillator-type $\mathfrak{sp}(2n)$-structures. It establishes a spinor–oscillator correspondence through a combinatorial tableaux framework, and demonstrates a Gelfand–Graev-type continuation for generic and bounded $\mathfrak{sp}(2n)$-modules, contrasting with the $A$-type case. The work provides an explicit construction of GT tableaux bases for all simple and indecomposable injective (projective) bounded $\mathfrak{sp}(2n)$-modules, with simple bounded modules classified by data $(\overline{\nu},\Sigma)$ and central character $\chi_{}\,\sigma$, and injectives realized as twisted localizations or Weyl-algebra pullbacks. By connecting half-integer spinor-type $\mathfrak{so}(2n)$-modules to oscillator-type $\mathfrak{sp}(2n)$-modules, the paper illuminates a broader geometric–algebraic picture and provides explicit tools (GT formulas, central-character behavior, and localization techniques) for studying bounded weight modules in the symplectic setting.
Abstract
We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded $\mathfrak{sp}(2n)$-modules. This realization is defined via a natural tableaux correspondence between spinor-type modules of $\mathfrak{so}(2n)$ and oscillator-type modules of $\mathfrak{sp}(2n)$. In particular, we show that, in contrast with the $A$-type case, the generic and bounded $\mathfrak{sp}(2n)$-modules admit an analog of the Gelfand-Graev continuation from finite-dimensional representations.