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 | 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 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 well-known results on Whittaker modules and their associated annihilators for .

**[AP]**D. Arnal and G. Pinczon,*On algebraically irreducible representations of the Lie algebra 𝑠𝑙(2)*, J. Mathematical Phys.**15**(1974), 350–359. MR**0357527****[B1]**Vladimir Bavula,*Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules*, Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 83–107. MR**1265277****[B2]**V. V. Bavula,*Generalized Weyl algebras and their representations*, Algebra i Analiz**4**(1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J.**4**(1993), no. 1, 71–92. MR**1171955****[Be]**Georgia Benkart,*Down-up algebras and Witten’s deformations of the universal enveloping algebra of 𝔰𝔩₂*, Recent progress in algebra (Taejon/Seoul, 1997) Contemp. Math., vol. 224, Amer. Math. Soc., Providence, RI, 1999, pp. 29–45. MR**1653061**, 10.1090/conm/224/03190**[BR]**Georgia Benkart and Tom Roby,*Down-up algebras*, J. Algebra**209**(1998), no. 1, 305–344. MR**1652138**, 10.1006/jabr.1998.7511**[Bl]**Richard E. Block,*The irreducible representations of the Lie algebra 𝔰𝔩(2) and of the Weyl algebra*, Adv. in Math.**39**(1981), no. 1, 69–110. MR**605353**, 10.1016/0001-8708(81)90058-X**[BK]**Jonathan Brundan and Alexander Kleshchev,*Shifted Yangians and finite 𝑊-algebras*, Adv. Math.**200**(2006), no. 1, 136–195. MR**2199632**, 10.1016/j.aim.2004.11.004**[C]**K. Christodoulopoulou, Whittaker Modules for Heisenberg and Affine Lie Algebras, Ph.D. thesis, University of Wisconsin-Madison 2007.**[DGO]**Yuri A. Drozd, Boris L. Guzner, and Sergei A. Ovsienko,*Weight modules over generalized Weyl algebras*, J. Algebra**184**(1996), no. 2, 491–504. MR**1409224**, 10.1006/jabr.1996.0270**[E]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[JWZ]**Ji Qingzhong, Wang Dingguo, and Zhou Xiangquan,*Finite-dimensional representation of quantum group 𝑈_{𝑞}(𝑓(𝐾))*, East-West J. Math.**2**(2000), no. 2, 201–213. MR**1825457****[K]**Bertram Kostant,*On Whittaker vectors and representation theory*, Invent. Math.**48**(1978), no. 2, 101–184. MR**507800**, 10.1007/BF01390249**[Ku]**Rajesh S. Kulkarni,*Down-up algebras and their representations*, J. Algebra**245**(2001), no. 2, 431–462. MR**1863888**, 10.1006/jabr.2001.8892**[MS]**Dragan Miličić and Wolfgang Soergel,*The composition series of modules induced from Whittaker modules*, Comment. Math. Helv.**72**(1997), no. 4, 503–520. MR**1600134**, 10.1007/s000140050031**[O1]**M. Ondrus, Whittaker Modules, Central Characters, and Tensor Products for Quantum Enveloping Algebras, Ph.D. Thesis, University of Wisconsin-Madison, 2004.**[O2]**Matthew Ondrus,*Whittaker modules for 𝑈_{𝑞}(𝔰𝔩₂)*, J. Algebra**289**(2005), no. 1, 192–213. MR**2139098**, 10.1016/j.jalgebra.2005.03.018**[R]**Alexander L. Rosenberg,*Noncommutative algebraic geometry and representations of quantized algebras*, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR**1347919****[S]**S. P. Smith,*A class of algebras similar to the enveloping algebra of 𝑠𝑙(2)*, Trans. Amer. Math. Soc.**322**(1990), no. 1, 285–314. MR**972706**, 10.1090/S0002-9947-1990-0972706-5**[Sw]**Richard G. Swan,*𝐾-theory of finite groups and orders*, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR**0308195****[T1]**Xin Tang,*On Whittaker modules over a class of algebras similar to 𝑈(𝑠𝑙₂)*, Front. Math. China**2**(2007), no. 1, 127–142. MR**2289914**, 10.1007/s11464-007-0009-2**[T2]**Xin Tang,*Construct irreducible representations of quantum groups 𝑈_{𝑞}(𝑓_{𝑚}(𝐾))*, Front. Math. China**3**(2008), no. 3, 371–397. MR**2425161**, 10.1007/s11464-008-0027-8

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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/S1088-4165-09-00347-1

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.