Whittaker modules for generalized Weyl algebras

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 $\mathfrak{sl}_2$ and of Heisenberg Lie algebras, Smith's generalizations of $U(\mathfrak{sl}_2)$, 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 $\operatorname{Ann}_A(w)$ of the cyclic generator $w$ of $V$. We also describe the annihilator ideal $\operatorname{Ann}_A(V)$ 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 $U(\mathfrak{sl}_2)$.