Irreduciblity of certain $\widehat{\mathfrak sl}_2$-modules of Wakimoto type

TL;DR

This paper studies certain irreducible modules of the affine Lie algebra rak{sl}_2, showing they can be realized as Wakimoto modules at various levels and identifying their structure and quotients.

Contribution

It demonstrates that specific rak{rak{sl}}_2-modules admit Wakimoto-type realizations at both critical and non-critical levels, and generalizes Wakimoto modules as generalized Whittaker modules.

Findings

Modules admit Wakimoto-type realization at critical and non-critical levels

Identification of simple quotients with known Wakimoto modules

Generalization of Wakimoto modules as generalized Whittaker modules

Abstract

We investigate the irreducible smooth $\widehat{\mathfrak{sl}}_{2}$-modules recently constructed in arXiv:2404.03855, and demonstrate that these modules admit a Wakimoto-type realization at both critical and non-critical levels. In the critical level case, we identify simple quotients of these modules with the Wakimoto modules whose irreducibility was already established in arXiv:math/0602181, arXiv:1402.6100. We also generalize some Wakimoto modules constructed in arXiv:1409.5354 and identify them as generalized Whittaker modules.