Whittaker Modules for Generalized Weyl Algebras

Understand

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 sl_2 and of Heisenberg Lie algebras, Smith's generalizations of U(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 Ann_A(w) of the cyclic generator w of V.
• We also describe the annihilator ideal Ann_A(V) under certain assumptions that hold for most of the examples mentioned above.