Translation for finite $W$-algebras
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- by Simon M. Goodwin
- Represent. Theory 15 (2011), 307-346
- DOI: https://doi.org/10.1090/S1088-4165-2011-00388-5
- Published electronically: April 5, 2011
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Abstract:
A finite $W$-algebra $U(\mathfrak {g},e)$ is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element $e$ of a complex reductive Lie algebra $\mathfrak {g}$. It is possible to give the tensor product of a $U(\mathfrak {g},e)$-module with a finite dimensional $U(\mathfrak {g})$-module the structure of a $U(\mathfrak {g},e)$-module; we refer to such tensor products as translations. In this paper, we present a number of fundamental properties of these translations, which are expected to be of importance in understanding the representation theory of $U(\mathfrak {g},e)$.References
- Tomoyuki Arakawa, Representation theory of $\scr W$-algebras, Invent. Math. 169 (2007), no. 2, 219–320. MR 2318558, DOI 10.1007/s00222-007-0046-1
- Jan de Boer and Tjark Tjin, Quantization and representation theory of finite $W$ algebras, Comm. Math. Phys. 158 (1993), no. 3, 485–516. MR 1255424, DOI 10.1007/BF02096800
- Jonathan Brown, Twisted Yangians and finite $W$-algebras, Transform. Groups 14 (2009), no. 1, 87–114. MR 2480853, DOI 10.1007/s00031-008-9041-x
- Jonathan Brundan and Simon M. Goodwin, Good grading polytopes, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 155–180. MR 2293468, DOI 10.1112/plms/pdl009
- Jonathan Brundan, Simon M. Goodwin, and Alexander Kleshchev, Highest weight theory for finite $W$-algebras, Int. Math. Res. Not. IMRN 15 (2008), Art. ID rnn051, 53. MR 2438067, DOI 10.1093/imrn/rnn051
- 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
- Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite $W$-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918, viii+107. MR 2456464, DOI 10.1090/memo/0918
- Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008), no. 1, 1–57. MR 2480709, DOI 10.1007/s00029-008-0059-7
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- A. D’Andrea, C. De Concini, A. De Sole, R. Heluani and V. Kac, Three equivalent definitions of finite $W$-algebras, appendix to [DK].
- Alberto De Sole and Victor G. Kac, Finite vs affine $W$-algebras, Jpn. J. Math. 1 (2006), no. 1, 137–261. MR 2261064, DOI 10.1007/s11537-006-0505-2
- A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, vol. 213, Amer. Math. Soc., Providence, RI, 2005, pp. 85–104. MR 2140715, DOI 10.1090/trans2/213/05
- Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255. MR 1876934, DOI 10.1155/S107379280210609X
- Victor Ginzburg, Harish-Chandra bimodules for quantized Slodowy slices, Represent. Theory 13 (2009), 236–271. MR 2515934, DOI 10.1090/S1088-4165-09-00355-0
- Simon M. Goodwin, A note on Verma modules for finite $W$-algebras, J. Algebra 324 (2010), no. 8, 2058–2063. MR 2678838, DOI 10.1016/j.jalgebra.2010.06.027
- Simon M. Goodwin, Gerhard Röhrle, and Glenn Ubly, On 1-dimensional representations of finite $W$-algebras associated to simple Lie algebras of exceptional type, LMS J. Comput. Math. 13 (2010), 357–369. MR 2685130, DOI 10.1112/S1461157009000205
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
- Ivan Losev, Quantized symplectic actions and $W$-algebras, J. Amer. Math. Soc. 23 (2010), no. 1, 35–59. MR 2552248, DOI 10.1090/S0894-0347-09-00648-1
- —, Finite dimensional representations of W-algebras, preprint, arXiv:0807.1023 (2008).
- —, On the structure of the category $\mathcal O$ for $W$-algebras, preprint, arXiv:0812.1584 (2008).
- —, $1$-dimensional representations and parabolic induction for $W$-algebras, preprint, arXiv:0906.0157, (2009).
- T. E. Lynch, Generalized Whittaker vectors and representation theory, Ph.D. thesis, M.I.T., 1979.
- Alexander Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), no. 1, 1–55. With an appendix by Serge Skryabin. MR 1929302, DOI 10.1006/aima.2001.2063
- Alexander Premet, Enveloping algebras of Slodowy slices and the Joseph ideal, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 3, 487–543. MR 2314105, DOI 10.4171/JEMS/86
- Alexander Premet, Primitive ideals, non-restricted representations and finite $W$-algebras, Mosc. Math. J. 7 (2007), no. 4, 743–762, 768 (English, with English and Russian summaries). MR 2372212, DOI 10.17323/1609-4514-2007-7-4-743-762
- Alexander Premet, Commutative quotients of finite $W$-algebras, Adv. Math. 225 (2010), no. 1, 269–306. MR 2669353, DOI 10.1016/j.aim.2010.02.020
- S. Skryabin, A category equivalence, appendix to [Pr1].
- K. de Vos and P. van Driel, The Kazhdan-Lusztig conjecture for finite $W$-algebras, Lett. Math. Phys. 35 (1995), no. 4, 333–344. MR 1358298, DOI 10.1007/BF00750840
- Nolan R. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 123–151. MR 1039836, DOI 10.2969/aspm/01410123
Bibliographic Information
- Simon M. Goodwin
- Affiliation: School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
- MR Author ID: 734259
- Email: goodwin@for.mat.bham.ac.uk
- Received by editor(s): September 22, 2009
- Received by editor(s) in revised form: June 4, 2010
- Published electronically: April 5, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 15 (2011), 307-346
- MSC (2010): Primary 17B10, 17B35
- DOI: https://doi.org/10.1090/S1088-4165-2011-00388-5
- MathSciNet review: 2788896