## Whittaker modules for generalized Weyl algebras

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- by Georgia Benkart and Matthew Ondrus
- Represent. Theory
**13**(2009), 141-164 - DOI: https://doi.org/10.1090/S1088-4165-09-00347-1
- Published electronically: April 16, 2009
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## 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 $\mathfrak {sl}_2$ and of Heisenberg Lie algebras, Smithβs 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 Kostantβs well-known results on Whittaker modules and their associated annihilators for $U(\mathfrak {sl}_2)$.## References

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## Bibliographic Information

**Georgia Benkart**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 34650
- Email: benkart@math.wisc.edu
**Matthew Ondrus**- Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408
- Email: MattOndrus@weber.edu
- Received by editor(s): March 25, 2008
- Received by editor(s) in revised form: February 9, 2009
- Published electronically: April 16, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**13**(2009), 141-164 - MSC (2000): Primary 17B10; Secondary 16D60
- DOI: https://doi.org/10.1090/S1088-4165-09-00347-1
- MathSciNet review: 2497458