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 s (1) over tilde (2) of type A(1)((1)) with noncritical level. These modules can become simple Whittaker modules over s (1) over tilde (2) = s (1) over tilde (2) Cd with the same Whittaker function and central charge. We have to modulo a central character for s (1) over tilde (2) to obtain simple degenerate Whittaker 42-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker s (1) over tilde (2)-module by a submodule generated by a scalar action of central elements.of the vertex algebra V-2(s (1) over tilde (2) ) is simple as s (1) over tilde (2)-module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for s (1) over tilde (2). Quite surprisingly, with the same Whittaker function some simple degenerate s (1) over tilde (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 s (1) over tilde (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 s (1) over tilde (2) at noncritical level. (C) 2015 Elsevier Inc. All rights reserved.