## Translation for finite $W$-algebras

HTML articles powered by AMS MathViewer

- by Simon M. Goodwin PDF
- Represent. Theory
**15**(2011), 307-346 Request permission

## 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.Β 2, 219β320. 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.Β 3, 485β516. MR**1255424**, DOI 10.1007/BF02096800 - Jonathan Brown,
*Twisted Yangians and finite $W$-algebras*, Transform. Groups**14**(2009), no.Β 1, 87β114. 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.Β 1, 155β180. 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.Β 1, 136β195. 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.Β 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.Β 1, 1β57. 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. DβAndrea, 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.Β 1, 137β261. 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.Β 85β104. MR**2140715**, DOI 10.1090/trans2/213/05 - Wee Liang Gan and Victor Ginzburg,
*Quantization of Slodowy slices*, Int. Math. Res. Not.**5**(2002), 243β255. MR**1876934**, DOI 10.1155/S107379280210609X - Victor Ginzburg,
*Harish-Chandra bimodules for quantized Slodowy slices*, Represent. Theory**13**(2009), 236β271. 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.Β 8, 2058β2063. MR**2678838**, DOI 10.1016/j.jalgebra.2010.06.027 - Simon M. Goodwin, Gerhard RΓΆhrle, 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), 357β369. 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.Β 2, 101β184. MR**507800**, DOI 10.1007/BF01390249 - Ivan Losev,
*Quantized symplectic actions and $W$-algebras*, J. Amer. Math. Soc.**23**(2010), no.Β 1, 35β59. MR**2552248**, DOI 10.1090/S0894-0347-09-00648-1 - β,
*Finite dimensional representations of W-algebras*, preprint, arXiv:0807.1023 (2008). - β,
*On the structure of the category $\mathcal O$ for $W$-algebras*, preprint, arXiv:0812.1584 (2008). - β,
*$1$-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.Β 1, 1β55. 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.Β 3, 487β543. MR**2314105**, DOI 10.4171/JEMS/86 - Alexander Premet,
*Primitive ideals, non-restricted representations and finite $W$-algebras*, Mosc. Math. J.**7**(2007), no.Β 4, 743β762, 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.Β 1, 269β306. 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.Β 4, 333β344. 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.Β 123β151. MR**1039836**, DOI 10.2969/aspm/01410123

## Additional 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