Subregular representations of $\mathfrak {sl}_n$ and simple singularities of type $A_{n-1}$. II
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- by Iain Gordon and Dmitriy Rumynin
- Represent. Theory 8 (2004), 328-345
- DOI: https://doi.org/10.1090/S1088-4165-04-00186-4
- Published electronically: July 20, 2004
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Abstract:
The aim of this paper is to show that the structures on $K$-theory used to formulate Lusztig’s conjecture for subregular nilpotent $\mathfrak {sl}_n$-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
- Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak {sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241. MR 1714141, DOI 10.1007/s000290050047 bmr R. Bezrukavnikov, I. Mirkovic, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, preprint, math.RT/0205144.
- Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math. 128 (1997), no. 1, 159–175 (French, with English summary). MR 1437497, DOI 10.1007/s002220050138
- Iain Gordon and Dmitriy Rumynin, Subregular representations of $\mathfrak {sl}_n$ and simple singularities of type $A_{n-1}$, Compositio Math. 138 (2003), no. 3, 337–360. MR 2019445, DOI 10.1023/A:1027381710548
- Robert Gordon and Edward L. Green, Graded Artin algebras, J. Algebra 76 (1982), no. 1, 111–137. MR 659212, DOI 10.1016/0021-8693(82)90240-X
- Yukari Ito and Hiraku Nakajima, McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000), no. 6, 1155–1191. MR 1783852, DOI 10.1016/S0040-9383(99)00003-8
- Yukari Ito and Hiraku Nakajima, McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000), no. 6, 1155–1191. MR 1783852, DOI 10.1016/S0040-9383(99)00003-8
- M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), no. 3, 565–576. MR 1752785, DOI 10.1007/s002080050344
- George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
- G. Lusztig, Bases in equivariant $K$-theory, Represent. Theory 2 (1998), 298–369. MR 1637973, DOI 10.1090/S1088-4165-98-00054-5
- G. Lusztig, Bases in equivariant $K$-theory. II, Represent. Theory 3 (1999), 281–353. MR 1714628, DOI 10.1090/S1088-4165-99-00083-7
- G. Lusztig, Subregular nilpotent elements and bases in $K$-theory, Canad. J. Math. 51 (1999), no. 6, 1194–1225. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. MR 1756878, DOI 10.4153/CJM-1999-053-1
- George Lusztig, Notes on affine Hecke algebras, Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999) Lecture Notes in Math., vol. 1804, Springer, Berlin, 2002, pp. 71–103. MR 1979925, DOI 10.1007/978-3-540-36205-0_{3}
- Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344, DOI 10.1090/ulect/018
- 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
- Jeremy Rickard, Translation functors and equivalences of derived categories for blocks of algebraic groups, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 255–264. MR 1308990, DOI 10.1007/978-94-017-1556-0_{1}3
- Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0
- Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445, DOI 10.1007/BFb0090294
- Amnon Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), no. 1, 41–84. MR 1195406, DOI 10.1016/0021-8693(92)90148-F
Bibliographic 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
- Received by editor(s): January 22, 2003
- Received by editor(s) in revised form: February 12, 2004
- Published electronically: 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 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 8 (2004), 328-345
- MSC (2000): Primary 17B50; Secondary 14J17, 16S35, 18F25, 20G05
- DOI: https://doi.org/10.1090/S1088-4165-04-00186-4
- MathSciNet review: 2077485