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

TL;DR

This work investigates irreducible smooth $\widehat{\mathfrak{sl}}_2$-modules of Wakimoto type, showing that they admit Wakimoto realizations at both non-critical and critical levels. It connects the FGXZ universal modules $\widehat{M}(\varphi)$ to Wakimoto constructions, proving irreducibility in the non-critical case and identifying explicit simple quotients at the critical level with known irreducible Wakimoto modules. The authors also generalize Wakimoto modules to include generalized Whittaker structures, demonstrating irreducibility phenomena in the non-critical regime and illuminating the critical-level quotients via Schur-polynomial criteria and $L(\bm{\chi})$-type subquotients. Overall, the paper clarifies how Wakimoto realizations model and classify a broad class of irreducible $\widehat{\mathfrak{sl}}_2$-modules, enriching the interface between vertex algebras, Whittaker theory, and representation theory at both generic and critical levels.

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.

Theorems & Definitions (18)

• Conjecture 1.1
• Proposition 2.1
• Theorem 3.1: FGXZ
• Theorem 3.2: FGXZ
• Lemma 3.3
• Corollary 3.4
• Proposition 4.1
• Theorem 5.1
• proof
• Theorem 6.1