2014/09/30 by Drazen Adamovic, Rencai Lu, Kaiming Zhao · 2 citations
Mathematics · Physics and Astronomy · #math.RT #math-ph #math.MP #math.QA #msc:17B69 #msc:17B67 #msc:17B68
paper · pdf · doi:10.1016/j.aim.2015.11.020
published as Advances in Mathematics 289 (2016) 438-479 · 34 pages, to appear in Advances in Mathematics
arxiv created 2015/11/25 · arxiv updated 2015/12/11
We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra \widehatsl2 of type A1(1) with noncritical level which are also irreducible Whittaker modules over \widetildesl2 =\widehatsl2 + \Bbb C d with the same Whittaker function and central charge. We have to modulo a central character for sl2 to obtain irreducible degenerate Whittaker \widehatsl2 -modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker \widehatsl2--module by a submodule generated by a scalar action of central elements of the vertex algebra V-2(sl2) is irreducible as \widehatsl2--module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for \widetildesl2. Quite surprisingly, with the same Whittaker function and the same central character of V-2(sl2), some irreducible \widetildesl2 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 \widehatsl2 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 \widehatsl2 at noncritical level.