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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Whittaker modules for generalized Weyl algebras
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by Georgia Benkart and Matthew Ondrus PDF
Represent. Theory 13 (2009), 141-164 Request permission


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
  • MR Author ID: 34650
  • Email:
  • Matthew Ondrus
  • Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408
  • Email:
  • 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:
  • MathSciNet review: 2497458