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Whittaker modules for generalized Weyl algebras

Authors: Georgia Benkart and Matthew Ondrus
Journal: Represent. Theory 13 (2009), 141-164
MSC (2000): Primary 17B10; Secondary 16D60
Published electronically: April 16, 2009
MathSciNet review: 2497458
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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, 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)$.

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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Matthew Ondrus
Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408

Received by editor(s): March 25, 2008
Received by editor(s) in revised form: February 9, 2009
Published electronically: April 16, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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