Whittaker Modules for Generalized Weyl Algebras
Abstract
We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl2 and of Heisenberg Lie algebras, Smith's generalizations of U(sl2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(phi,t) are in bijection with the phi-stable left ideals of R. We determine the annihilator AnnA(w) of the cyclic generator w of V. We also describe the annihilator ideal AnnA(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U(sl2).