Whittaker Modules for Generalized Weyl Algebras

TL;DR

This paper studies Whittaker modules within generalized Weyl algebras, establishing a bijection with phi-stable ideals and describing annihilators, thus extending classical results to a broad algebraic context.

Contribution

It provides a comprehensive analysis of Whittaker modules for generalized Weyl algebras, including classification and annihilator descriptions, generalizing Kostant's classical results.

Findings

Bijective correspondence between Whittaker modules and phi-stable ideals

Explicit description of annihilators of cyclic generators

Extension of Kostant's results to 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 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. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U(sl_2).