Whittaker modules for the affine Lie algebra $A_1 ^{(1)}$

TL;DR

This paper studies the structure and irreducibility of Whittaker modules for the affine Lie algebra f1b1b2, providing new classifications, explicit descriptions, and Wakimoto type constructions at both critical and noncritical levels.

Contribution

It introduces new irreducibility results for universal and degenerate Whittaker modules of f1b1b2, including explicit descriptions and Wakimoto type realizations.

Findings

Irreducibility of universal non-degenerate Whittaker modules at noncritical level.

Explicit description of simple quotients at critical level.

Wakimoto type constructions for modules at both critical and noncritical levels.

Abstract

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra $\widehat{sl_2}$ of type $A_1^{(1)}$ with noncritical level which are also irreducible Whittaker modules over $\widetilde{sl_2} =\widehat{sl_2} + {\Bbb C} d $ with the same Whittaker function and central charge. We have to modulo a central character for ${sl_2}$ to obtain irreducible degenerate Whittaker $\widehat{sl_2} $-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker $\widehat{sl_2}$--module by a submodule generated by a scalar action of central elements of the vertex algebra $V_{-2}(sl_2)$ is irreducible as $\widehat{sl_2}$--module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for $\widetilde{sl_2}$. Quite surprisingly, with the same Whittaker function and the same central character of $V_{-2}(sl_2)$, some irreducible $\widetilde{sl_2}$ Whittaker modules can have semisimple or free action of $d$. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of generalized Whittaker irreducible modules for $\widehat{sl_2}$ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for irreducible degenrate Whittaker modules for $\widehat{sl_2}$ at noncritical level.