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

Abstract

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra $\widehat{s{l}_2}$ of type $A_1^{(1)}$ with noncritical level which are also irreducible Whittaker modules over $\widetilde{s{l}_2}=\widehat{s{l}_2}+\mathbb{C}d$ with the same Whittaker function and central charge. We have to modulo a central character for $s{l}_2$ to obtain irreducible degenerate Whittaker $\widehat{s{l}_2}$-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker $\widehat{s{l}_2}$--module by a submodule generated by a scalar action of central elements of the vertex algebra $V_{-2}(s{l}_2)$ is irreducible as $\widehat{s{l}_2}$--module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for $\widetilde{s{l}_2}$. Quite surprisingly, with the same Whittaker function and the same central character of $V_{-2}(s{l}_2)$, some irreducible $\widetilde{s{l}_2}$ 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 $\widehat{s{l}_2}$ 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 $\widehat{s{l}_2}$ at noncritical level.

34 pages, to appear in Advances in Mathematics

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• A Construction of Representations of Loop Group and Affine Lie Algebra of ${\mathfrak{sl}}_n$
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