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

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. These modules can become simple Whittaker modules over $\widetilde{sl_2}=\widehat{sl_2}+\mathbb{C}d$ with the same Whittaker function and central charge. We have to modulo a central character for sl2sl2 to obtain simple 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 $V_{\widehat{sl_2}}$ is simple 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 some simple degenerate $\widetilde{sl_2}$ Whittaker modules at the critical level have semisimple action of d and others have free action of d . At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker 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 degenerate Whittaker modules for $\widehat{sl_2}$ at noncritical level.

D. Adamović, R. Lu, K. Zhao, Whittaker modules for the affine Lie algebra $A_1 ^{(1)}$ , Advances in Mathematics 289 (2016) 438-479.

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