Irreducibility of Certain $\widehat{\mathfrak{sl}}_2$-Modules of Wakimoto Type

Abstract

We investigate the irreducible smooth $\widehat{\mathfrak{sl}}_{2}$-modules recently constructed in [Adv. Math. 481 (2025), 110559, 34 pages, 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 by Adamović. We also generalize some Wakimoto modules constructed in [Adv. Math. 289 (2016), 438-479, arXiv:1409.5354] and identify them as generalized Whittaker modules.