Whittaker modules for generalized Weyl algebras
Authors:
Georgia Benkart and Matthew Ondrus
Journal:
Represent. Theory 13 (2009), 141164
MSC (2000):
Primary 17B10; Secondary 16D60
Posted:
April 16, 2009
MathSciNet review:
2497458
Fulltext PDF Free Access
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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 and of Heisenberg Lie algebras, Smith's generalizations of , various quantum analogues of these algebras, and many others. We show that the Whittaker modules of the generalized Weyl algebra are in bijection with the stable left ideals of . We determine the annihilator of the cyclic generator of . We also describe the annihilator ideal under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's wellknown results on Whittaker modules and their associated annihilators for .
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 K. Christodoulopoulou, Whittaker Modules for Heisenberg and Affine Lie Algebras, Ph.D. thesis, University of WisconsinMadison 2007.
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 D. Eisenbud, Commutative Algebra With A View Toward Algebraic Geometry, Grad. Texts in Math. 150, SpringerVerlag, New York, 1995. MR 1322960 (97a:13001)
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 B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101184. MR 507800 (80b:22020)
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Additional Information
Georgia Benkart
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
benkart@math.wisc.edu
Matthew Ondrus
Affiliation:
Department of Mathematics, Weber State University, Ogden, Utah 84408
Email:
MattOndrus@weber.edu
DOI:
http://dx.doi.org/10.1090/S1088416509003471
PII:
S 10884165(09)003471
Received by editor(s):
March 25, 2008
Received by editor(s) in revised form:
February 9, 2009
Posted:
April 16, 2009
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
