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 s l 2 \mathfrak {sl}_2 and of Heisenberg Lie algebras, Smith’s generalizations of U ( s l 2 ) U(\mathfrak {sl}_2) , various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = A w V = Aw of the generalized Weyl algebra A = R ( ϕ , t ) A = R(\phi ,t) are in bijection with the ϕ \phi -stable left ideals of R R . We determine the annihilator Ann A ⁡ ( w ) \operatorname {Ann}_A(w) of the cyclic generator w w of V V . We also describe the annihilator ideal Ann A ⁡ ( V ) \operatorname {Ann}_A(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 ( s l 2 ) U(\mathfrak {sl}_2) .