Whittaker modules for generalized Weyl algebras

Benkart, Georgia; Ondrus, Matthew

doi:10.1090/S1088-4165-09-00347-1

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

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

References

D. Arnal and G. Pinczon, On algebraically irreducible representations of the Lie algebra $\textrm {sl}(2)$, J. Mathematical Phys. 15 (1974), 350–359. MR 357527, DOI 10.1063/1.1666651
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, DOI 10.1007/bf01058631
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
Georgia Benkart, Down-up algebras and Witten’s deformations of the universal enveloping algebra of $\mathfrak {s}\mathfrak {l}_2$, Recent progress in algebra (Taejon/Seoul, 1997) Contemp. Math., vol. 224, Amer. Math. Soc., Providence, RI, 1999, pp. 29–45. MR 1653061, DOI 10.1090/conm/224/03190
Georgia Benkart and Tom Roby, Down-up algebras, J. Algebra 209 (1998), no. 1, 305–344. MR 1652138, DOI 10.1006/jabr.1998.7511
Richard E. Block, The irreducible representations of the Lie algebra ${\mathfrak {s}}{\mathfrak {l}}(2)$ and of the Weyl algebra, Adv. in Math. 39 (1981), no. 1, 69–110. MR 605353, DOI 10.1016/0001-8708(81)90058-X
Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite $W$-algebras, Adv. Math. 200 (2006), no. 1, 136–195. MR 2199632, DOI 10.1016/j.aim.2004.11.004
K. Christodoulopoulou, Whittaker Modules for Heisenberg and Affine Lie Algebras, Ph.D. thesis, University of Wisconsin-Madison 2007.
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, DOI 10.1006/jabr.1996.0270
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
Rajesh S. Kulkarni, Down-up algebras and their representations, J. Algebra 245 (2001), no. 2, 431–462. MR 1863888, DOI 10.1006/jabr.2001.8892
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, DOI 10.1007/s000140050031
M. Ondrus, Whittaker Modules, Central Characters, and Tensor Products for Quantum Enveloping Algebras, Ph.D. Thesis, University of Wisconsin-Madison, 2004.
Matthew Ondrus, Whittaker modules for $U_q({\mathfrak {sl}}_2)$, J. Algebra 289 (2005), no. 1, 192–213. MR 2139098, DOI 10.1016/j.jalgebra.2005.03.018
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, DOI 10.1007/978-94-015-8430-2
S. P. Smith, A class of algebras similar to the enveloping algebra of $\textrm {sl}(2)$, Trans. Amer. Math. Soc. 322 (1990), no. 1, 285–314. MR 972706, DOI 10.1090/S0002-9947-1990-0972706-5
Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 0308195, DOI 10.1007/BFb0059150
Xin Tang, On Whittaker modules over a class of algebras similar to $U(\textrm {sl}_2)$, Front. Math. China 2 (2007), no. 1, 127–142. MR 2289914, DOI 10.1007/s11464-007-0009-2
Xin Tang, Construct irreducible representations of quantum groups $U_q(f_m(K))$, Front. Math. China 3 (2008), no. 3, 371–397. MR 2425161, DOI 10.1007/s11464-008-0027-8