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

TL;DR

This paper constructs explicit combinatorial models for all simple and injective bounded modules over symplectic Lie algebras, linking spinor and oscillator modules via tableaux, and extends Gelfand-Graev continuation to this setting.

Contribution

It provides a new combinatorial realization of bounded modules for symplectic Lie algebras, connecting spinor and oscillator modules through tableaux correspondence.

Findings

Explicit combinatorial realization of modules

Tableaux correspondence between spinor and oscillator modules

Extension of Gelfand-Graev continuation to symplectic modules

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.