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

DOI:
https://doi.org/10.1090/S1088-4165-09-00347-1

Published electronically:
April 16, 2009

MathSciNet review:
2497458

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Abstract | References | Similar Articles | Additional Information

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 $U(\mathfrak {sl}_2)$.

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Additional 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

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.