Whittaker modules for the affine Lie algebra A1(1)

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