Electronic Only Electronic Research Announcements
Representation Theory
ISSN 1088-4165
Previous volume | Table of contents | Next volume
Articles in press | Most recent volume | All volumes
Previous Article | Next Article
     

Harish-Chandra bimodules for quantized Slodowy slices

Author(s): Victor Ginzburg
Journal: Represent. Theory 13 (2009), 236-271.
MSC (2000): Primary 81R10
Posted: June 30, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice.

In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct ``noncommutative resolutions'' of Slodowy slices via translation functors.

References:

[ABO]
M.-J. Asensio, M. Van den Bergh, and F. Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings. Trans. Amer. Math. Soc. 316 (1989), 537-553. MR 958890 (90c:16001)

[BB1]
A. Beilinson, J. Bernstein, Localisation de $ \mathfrak{g}$-modules, C. R. Acad. Sci. Paris 292 (1981), no. 1, 15-18. MR 610137 (82k:14015)

[BB2]
-, A proof of Jantzen conjectures. I. M. Gelfand Seminar, 1-50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. MR 1237825 (95a:22022)

[BG]
A. Beilinson, V. Ginzburg, Wall-crossing functors and $ \mathscr D$-modules. Represent. Theory 3 (1999), 1-31. MR 1659527 (2000d:17007)

[BGe]
J. Bernstein, S. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. Compositio Math. 41 (1980), 245-285. MR 581584 (82c:17003)

[Bj]
J.-E. Bjork, The Auslander condition on Noetherian rings. Séminaire d'Algèbre Paul Dubreil et Marie-Paul Malliavin, (Paris, 1987/1988), 137-173, Lect. Notes in Math., 1404, Springer, Berlin, 1989. MR 1035224 (90m:16002)

[BoBr]
W. Borho, J.-L. Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules. Invent. Math. 69 (1982), 437-476. MR 679767 (84b:17007)

[BoK]
W. Borho, H. Kraft, Über die Gelfand-Kirillov-Dimension. Math. Ann. 220 (1976), 1-24. MR 0412240 (54:367)

[Bo]
M. Boyarchenko, Quantization of minimal resolutions of Kleinian singularities. Adv. Math. 211 (2007), 244-265. MR 2313534 (2008f:14001)

[Br]
B. Broer, Line bundles on the cotangent bundle of the flag variety. Invent. Math. 113 (1993), 1-20. MR 1223221 (94g:14027)

[BK1]
J. Brundan, A. Kleshchev, Shifted Yangians and finite $ W$-algebras. Adv. Math. 200 (2006), 136-195. MR 2199632 (2006m:17010)

[BK2]
-, Representation theory of shifted Yangians and finite $ W$-algebras. Mem. Amer. Math. Soc. 196 (2008), no. 91. MR 2456464

[BGK]
J. Brundan, S. Goodwin, and A. Kleshchev, Highest weight theory for finite W-algebras. Int. Math. Res. Not. IMRN 2008, no. 15, Art. ID rnn051.

MR 2438067 (2009f:17011)

[CBH]
W. Crawley-Boevey, M. Holland, Noncommutative deformations of Kleinian singularities. Duke Math. J. 92 (1998), 605-635. MR 1620538 (99f:14003)

[CE]
H. Cartan, S. Eilenberg, Homological algebra. Princeton University Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)

[CG]
N. Chriss, V. Ginzburg, Representation theory and complex geometry. Birkhäuser Boston, 1997. MR 1433132 (98i:22021)

[Ga]
O. Gabber, The integrability of the characteristic variety. Amer. J. Math. 103 (1981), 445-468. MR 618321 (82j:58104)

[GG]
W.-L. Gan, V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Notices. 2002, no. 5, 243-255. MR 1876934 (2002m:53129)

[GS]
I. Gordon, J.T. Stafford, Rational Cherednik algebras and Hilbert schemes. I, II. Adv. Math. 198 (2005), 222-274; Duke Math. J. 132 (2006), 73-135. MR 2183255 (2008i:14006)

[GuS]
V. Guillemin, S. Sternberg, Convexity properties of the moment mapping. Invent. Math. 67 (1982), 491-513. MR 664117 (83m:58037)

[Hod]
T. Hodges, Noncommutative deformations of type-$ A$ Kleinian singularities. J. Algebra 161 (1993), 271-290. MR 1247356 (94i:14038)

[Ho]
M. P. Holland, Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers, Ann. Sci. École Norm. Sup., 32 (1999), 813-834. MR 1717577 (2001a:16042)

[HTT]
R. Hotta, K. Takeuchi, T. Tanisaki, $ D$-modules, perverse sheaves, and representation theory. Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008. MR 2357361 (2008k:32022)

[Kas]
M. Kashiwara, $ D$-modules and microlocal calculus. Translations of Mathematical Monographs, 217. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2003. MR 1943036 (2003i:32018)

[KK]
M. Kashiwara, T. Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities. Publ. Res. Inst. Math. Sci. 17 (1981), 813-979. MR 650216 (83e:58085)

[Ka]
N. Kawanaka, Generalized Gelfand-Graev representations and Ennola duality, in Algebraic groups and related topics, 175-206, Adv. Stud. Pure Math., 6, North-Holland, 1985. MR 803335 (87e:20075)

[Ko]
B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101-184. MR 507800 (80b:22020)

[Lo1]
I. Losev, Quantized symplectic actions and W-algebras. Preprint 2007, arXiv:0707.3108.

[Lo2]
-, Finite dimensional representations of $ W$-algebras, Preprint 2008. arXiv:0807.1023.

[Ma]
H. Matumoto, Whittaker vectors and associated varieties. Invent. Math. 89 (1987), 219-224. MR 892192 (88k:17022)

[Mœ]
C. Mœglin, Modules de Whittaker et ideaux primitifs complètement premiers dans les algèbres enveloppantes, C. R. Acad. Sci. Paris 303 (1986), 845-848. MR 870903 (87m:17020)

[Mu]
I. Musson, Hilbert schemes and noncommutative deformations of type A Kleinian singularities. J. Algebra 293 (2005), 102-129. MR 2173968 (2006g:16066)

[P1]
A. Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), 1-55. MR 1929302 (2003k:17014)

[P2]
-, Enveloping algebras of Slodowy slices and the Joseph ideal. J. Eur. Math. Soc. 9 (2007), 487-543. MR 2314105 (2008c:17006)

[P3]
-, Primitive ideals, non-restricted representations and finite $ W$-algebras. Mosc. Math. J. 7 (2007), 743-762. MR 2372212 (2008k:17012)

[P4]
-, Commutative quotients of finite W-algebras., Preprint 2008. arXiv:0809.0663.

[Sk]
S. Skryabin, The category equivalence, Appendix to [P1].

[Sl]
P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math., 815, Springer-Verlag, 1980. MR 584445 (82g:14037)

[Spa]
N. Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups. Topology 16 (1977), 203-204. MR 0447423 (56:5735)

Similar Articles:

Retrieve articles in Representation Theory with MSC (2000): 81R10

Retrieve articles in all Journals with MSC (2000): 81R10

Additional Information:

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: ginzburg@math.uchicago.edu

DOI: 10.1090/S1088-4165-09-00355-0
PII: S 1088-4165(09)00355-0
Received by editor(s): November 10, 2008
Received by editor(s) in revised form: March 31, 2009
Posted: June 30, 2009
Dedicated: Dedicated to the memory of Peter Slodowy
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google