Harish-Chandra bimodules for quantized Slodowy slices

Ginzburg, Victor

doi:10.1090/S1088-4165-09-00355-0

Harish-Chandra bimodules for quantized Slodowy slices
HTML articles powered by AMS MathViewer

by Victor Ginzburg

Represent. Theory 13 (2009), 236-271

DOI: https://doi.org/10.1090/S1088-4165-09-00355-0

Published electronically: June 30, 2009

PDF | Request permission

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

Maria Jesus Asensio, Michel Van den Bergh, and Freddy Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), no. 2, 537–553. MR 958890, DOI 10.1090/S0002-9947-1989-0958890-X
Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1–50. MR 1237825
Alexander Beilinson and Victor Ginzburg, Wall-crossing functors and $\scr D$-modules, Represent. Theory 3 (1999), 1–31. MR 1659527, DOI 10.1090/S1088-4165-99-00063-1
J. N. Bernstein and S. I. Gel′fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245–285. MR 581584
Jan-Erik Björk, The Auslander condition on Noetherian rings, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 137–173. MR 1035224, DOI 10.1007/BFb0084075
Walter Borho and Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, 437–476. MR 679767, DOI 10.1007/BF01389364
Walter Borho and Hanspeter Kraft, Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), no. 1, 1–24. MR 412240, DOI 10.1007/BF01354525
Mitya Boyarchenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math. 211 (2007), no. 1, 244–265. MR 2313534, DOI 10.1016/j.aim.2006.08.003
Bram Broer, Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), no. 1, 1–20. MR 1223221, DOI 10.1007/BF01244299
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, 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
William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
Ofer Gabber, The integrability of the characteristic variety, Amer. J. Math. 103 (1981), no. 3, 445–468. MR 618321, DOI 10.2307/2374101
Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255. MR 1876934, DOI 10.1155/S107379280210609X
I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), no. 1, 222–274. MR 2183255, DOI 10.1016/j.aim.2004.12.005
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117, DOI 10.1007/BF01398933
Timothy J. Hodges, Noncommutative deformations of type-$A$ Kleinian singularities, J. Algebra 161 (1993), no. 2, 271–290. MR 1247356, DOI 10.1006/jabr.1993.1219
Martin P. Holland, Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 6, 813–834 (English, with English and French summaries). MR 1717577, DOI 10.1016/S0012-9593(00)87719-8
Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
Masaki Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Mutsumi Saito; Iwanami Series in Modern Mathematics. MR 1943036, DOI 10.1090/mmono/217
Masaki Kashiwara and Takahiro Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), no. 3, 813–979. MR 650216, DOI 10.2977/prims/1195184396
N. Kawanaka, Generalized Gel′fand-Graev representations and Ennola duality, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 175–206. MR 803335, DOI 10.2969/aspm/00610175
Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
I. Losev, Quantized symplectic actions and W-algebras. Preprint 2007, arXiv:0707.3108.
—, Finite dimensional representations of $W$-algebras, Preprint 2008. arXiv:0807.1023.
Hisayosi Matumoto, Whittaker vectors and associated varieties, Invent. Math. 89 (1987), no. 1, 219–224. MR 892192, DOI 10.1007/BF01404678
Colette Mœglin, Modèles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 17, 845–848 (French, with English summary). MR 870903
Ian M. Musson, Hilbert schemes and noncommutative deformations of type A Kleinian singularities, J. Algebra 293 (2005), no. 1, 102–129. MR 2173968, DOI 10.1016/j.jalgebra.2005.07.021
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
—, Commutative quotients of finite W-algebras., Preprint 2008. arXiv:0809.0663.
S. Skryabin, The category equivalence, Appendix to \cite{premet}.
Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445, DOI 10.1007/BFb0090294
Nicolas Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology 16 (1977), no. 2, 203–204. MR 447423, DOI 10.1016/0040-9383(77)90022-2