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.