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Subregular representations of  and simple singularities of type  . II
Author(s):
Iain
Gordon;
Dmitriy
Rumynin
Journal:
Represent. Theory
8
(2004),
328-345.
MSC (2000):
Primary 17B50;
Secondary 14J17, 16S35, 18F25, 20G05
Posted:
July 20, 2004
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Abstract:
The aim of this paper is to show that the structures on  -theory used to formulate Lusztig's conjecture for subregular nilpotent  -representations are, in fact, natural in the McKay correspondence. The main result is a categorification of these structures. The no-cycle algebra plays the special role of a bridge between complex geometry and representation theory in positive characteristic.
References:
-
- 1.
- J. Bernstein, I. Frenkel and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of
 via projective and Zuckerman functors, Selecta Math. (N.S.) 5 no.2 (1999), 199-241. MR 1714141 (2000i:17009) - 2.
- R. Bezrukavnikov, I. Mirkovic, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, preprint, math.RT/0205144.
- 3.
- T. Bridgeland, A. King and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535-554. MR 1824990 (2002f:14023)
- 4.
- N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser, Berlin, 1997. MR 1433132 (98i:22021)
- 5.
- P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math. 128 (1997), 159-175. MR 1437497 (98b:20061)
- 6.
- I. Gordon and D.Rumynin, Subregular representations of
 and simple singularities of type  , Compositio Math. 138, (2003), 337-360. MR 2019445 - 7.
- R. Gordon and E.L. Green, Graded Artin algebras, J. Algebra 76, (1982) 111-137. MR 0659212 (83m:16028a)
- 8.
- Y. Ito and H. Nakajima, McKay correspondence and Hilbert schemes in dimension three, Topology 39 no. 6 (2000), 1155-1191. MR 1783852 (2001h:14004)
- 9.
- J.C. Jantzen, Subregular nilpotent representations of Lie algebras in prime characteristic, Represent. Theory 3 (1999), 153-222. MR 1783852 (2001h:14004)
- 10.
- M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316 (2000) 565-576. MR 1752785 (2001h:14012)
- 11.
- G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635. MR 0991016 (90e:16049)
- 12.
- G. Lusztig, Bases in equivariant K-theory, Represent. Theory 2 (1998), 298-369. MR 1637973 (99i:19005)
- 13.
- G. Lusztig, Bases in equivariant K-theory, II, Represent. Theory 3 (1999), 281-353. MR 1714628 (2000h:20085)
- 14.
- G. Lusztig, Subregular nilpotent elements and bases in K-theory, Canad. J. Math. 51 (1999), 1194-1225. MR 1756878 (2001g:19006)
- 15.
- G. Lusztig, Notes on affine Hecke algebras, in Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999), 71-103, Lecture Notes in Math. 1804, Springer, 2002. MR 1979925 (2004d:20006)
- 16.
- H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, AMS, 1999. MR 1711344 (2001b:14007)
- 17.
- A. Premet, Special transverse slices and enveloping algebras, Adv. Math., 170 (2002), 1-55. MR 1929302 (2003k:17014)
- 18.
- J. Rickard, Translation functors and equivalences of derived categories for blocks of algebraic groups, NATO Sci. Ser. C Math. Phys. Sci. 424, Finite-dimensional algebras and related topics (Ottawa, ON, 1992), 255-264, Kluwer, 1994. MR 1308990 (95k:20068)
- 19.
- P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37-108. MR 1831820 (2002e:14030)
- 20.
- P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes Math. 815, Springer, 1980. MR 0584445 (82g:14037)
- 21.
- A. Yekutieli, Dualizing Complexes over non-commutative graded algebras, J. Algebra 153 (1992), 41-84. MR 1195406 (94a:16077)
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Additional Information:
Iain
Gordon
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, United Kingdom
Email:
ig@maths.gla.ac.uk
Dmitriy
Rumynin
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email:
rumynin@maths.warwick.ac.uk
DOI:
10.1090/S1088-4165-04-00186-4
PII:
S 1088-4165(04)00186-4
Received by editor(s):
January 22, 2003
Received by editor(s) in revised form:
February 12, 2004
Posted:
July 20, 2004
Additional Notes:
Both authors were visiting and partially supported by MSRI while this research was begun and extend their thanks to that institution. Much of the research for this paper was undertaken while the first author was supported by TMR grant ERB FMRX-CT97-0100 at the University of Bielefeld and Nuffield grant NAL/00625/G
The second author was partially supported by EPSRC
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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