Whittaker modules for the affine Lie algebra [formula omitted

Abstract

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra sl2ˆ of type A1(1) with noncritical level. These modules can become simple Whittaker modules over sl2˜=sl2ˆ+Cd with the same Whittaker function and central charge. We have to modulo a central character for sl2 to obtain simple degenerate Whittaker sl2ˆ-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker sl2ˆ-module by a submodule generated by a scalar action of central elements of the vertex algebra V−2(sl2) is simple as sl2ˆ-module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for sl2˜. Quite surprisingly, with the same Whittaker function some simple degenerate sl2˜ 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 sl2ˆ 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 sl2ˆ at noncritical level.