## Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

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- by Roman Bezrukavnikov
- Represent. Theory
**7**(2003), 1-18 - DOI: https://doi.org/10.1090/S1088-4165-03-00158-4
- Published electronically: January 9, 2003
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## Abstract:

A certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group is introduced by the author in the paper*Perverse coherent sheaves*(the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means by the author in the paper

*On tensor categories attached to cells in affine Weyl groups*, to be published).

## References

- [AB]1 Arkhipov, S., Bezrukavnikov, R.,
- A. A. Beĭlinson, J. Bernstein, and P. Deligne,
*Faisceaux pervers*, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR**751966** - Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel,
*Koszul duality patterns in representation theory*, J. Amer. Math. Soc.**9**(1996), no. 2, 473–527. MR**1322847**, DOI 10.1090/S0894-0347-96-00192-0
[B1]B Bezrukavnikov, R., - A. I. Bondal and M. M. Kapranov,
*Representable functors, Serre functors, and reconstructions*, Izv. Akad. Nauk SSSR Ser. Mat.**53**(1989), no. 6, 1183–1205, 1337 (Russian); English transl., Math. USSR-Izv.**35**(1990), no. 3, 519–541. MR**1039961**, DOI 10.1070/IM1990v035n03ABEH000716 - 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 - Abraham Broer,
*A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles*, J. Reine Angew. Math.**493**(1997), 153–169. MR**1491811**, DOI 10.1515/crll.1997.493.153 - Abraham Broer,
*Decomposition varieties in semisimple Lie algebras*, Canad. J. Math.**50**(1998), no. 5, 929–971. MR**1650954**, DOI 10.4153/CJM-1998-048-6 - Neil Chriss and Victor Ginzburg,
*Representation theory and complex geometry*, Birkhäuser Boston, Inc., Boston, MA, 1997. MR**1433132** - Pierre Deligne,
*La conjecture de Weil. II*, Inst. Hautes Études Sci. Publ. Math.**52**(1980), 137–252 (French). MR**601520**, DOI 10.1007/BF02684780 - Michel Demazure,
*A very simple proof of Bott’s theorem*, Invent. Math.**33**(1976), no. 3, 271–272. MR**414569**, DOI 10.1007/BF01404206 - M. Gontcharoff,
*Sur quelques séries d’interpolation généralisant celles de Newton et de Stirling*, Uchenye Zapiski Moskov. Gos. Univ. Matematika**30**(1939), 17–48 (Russian, with French summary). MR**0002002**, DOI 10.1007/s002220100122
[Gi]Gi Ginzburg, V., - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157**, DOI 10.1007/978-1-4757-3849-0 - V. Hinich,
*On the singularities of nilpotent orbits*, Israel J. Math.**73**(1991), no. 3, 297–308. MR**1135219**, DOI 10.1007/BF02773843 - Bertram Kostant,
*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327–404. MR**158024**, DOI 10.2307/2373130 - George Lusztig,
*Cells in affine Weyl groups. IV*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**36**(1989), no. 2, 297–328. MR**1015001** - Ivan Mirković and Kari Vilonen,
*Perverse sheaves on affine Grassmannians and Langlands duality*, Math. Res. Lett.**7**(2000), no. 1, 13–24. MR**1748284**, DOI 10.4310/MRL.2000.v7.n1.a2
[O]O Ostrik, V., - D. I. Panyushev,
*Rationality of singularities and the Gorenstein property of nilpotent orbits*, Funktsional. Anal. i Prilozhen.**25**(1991), no. 3, 76–78 (Russian); English transl., Funct. Anal. Appl.**25**(1991), no. 3, 225–226 (1992). MR**1139878**, DOI 10.1007/BF01085494
[P]P Positselskii, L., private communication.
[PS]PS Parshall, B., Scott, L., - Jean-Louis Verdier,
*Des catégories dérivées des catégories abéliennes*, Astérisque**239**(1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR**1453167**

*Perverse sheaves on affine flags and Langlands dual group,*preprint, math.RT/0201073.

*On tensor categories attached to cells in affine Weyl groups,*preprint, math.RT/001008, to appear in “Representation Theory of Algebraic Groups and Quantum Groups,” Advanced Studies in Pure Math. [B2]izvrat Bezrukavnikov, R.,

*Perverse coherent sheaves,*preprint, math.AG/0005152. [B3]2 Bezrukavnikov, R.,

*Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group,*preprint, math.RT/0201256.

*Perverse sheaves on a Loop group and Langlands’ duality,*preprint, alg-geom/9511007.

*On the $K$-theory of the nilpotent cone,*preprint, math.AG/9911068.

*Derived categories, quasi-hereditary algebras, and algebraic groups,*preprint, 1987.

## Bibliographic Information

**Roman Bezrukavnikov**- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 347192
- Email: bezrukav@math.northwestern.edu
- Received by editor(s): February 15, 2002
- Received by editor(s) in revised form: July 31, 2002
- Published electronically: January 9, 2003
- Additional Notes: The author is supported by the NSF grant DMS0071967, and by the Clay Mathematical Institute
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory
**7**(2003), 1-18 - MSC (2000): Primary 20G99
- DOI: https://doi.org/10.1090/S1088-4165-03-00158-4
- MathSciNet review: 1973365