×
Achar, Pramod N.; Riche, Simon

Modular perverse sheaves on flag varieties. II: Koszul duality and formality. (English) Zbl 1375.14162

Duke Math. J. 165, No. 1, 161-215 (2016).
The paper continues the study of modular perverse sheaves on flag varieties begun in [the authors, Ann. Sci. Éc. Norm. Supér. (4) 49, No. 2, 325–370 (2016; Zbl 1386.14178)] (with a joint appendix with Geordie Williamson). For varieties equipped with a stratification by affine spaces, the authors develop a formalism of “mixed modular perverse sheaves”. Two applications are given: (1) a “Koszul-type” derived equivalence relating a given flag variety to the Langlands dual flag variety, (2) a formality theorem for the modular derived category of a flag variety.
Reviewer: Vladimir L. Popov (Moskva)

Cited in 3 Reviews
Cited in 14 Documents

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
20G40 Linear algebraic groups over finite fields

Keywords:

modular perverse sheaves; Koszul duality; flag varieties; self-duality and formality

Citations:

Zbl 1386.14178
PDF BibTeX XML Cite
\textit{P. N. Achar} and \textit{S. Riche}, Duke Math. J. 165, No. 1, 161--215 (2016; Zbl 1375.14162)
Full Text: DOI arXiv Euclid
  • logo of hbz
  • logo of WorldCat

References:

[1] P. Achar and S. Riche, Koszul duality and semisimplicity of Frobenius , Ann. Inst. Fourier 63 (2013), 1511-1612. · Zbl 1348.14041
[2] P. Achar and S. Riche, Modular perverse sheaves on flag varieties, I: Tilting and parity sheaves (with a joint appendix with G. Williamson), to appear in Ann. Sci. Éc. Norm. Supér. · Zbl 1386.14178
[3] P. Achar and S. Riche, Modular perverse sheaves on flag varieties, III: Positivity conditions , [math.RT]. arXiv:1408.4189v1 · Zbl 1386.14178
[4] S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution , J. Amer. Math. Soc. 17 (2004), 595-678. · Zbl 1061.17013
[5] A. Beĭlinson, “On the derived category of perverse sheaves” in \(K\)-theory, Arithmetic and Geometry (Moscow, 1984-1986) , Lecture Notes in Math. 1289 , Springer, Berlin, 1987, 27-41.
[6] A. Beĭlinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” in Analyse et Topologie sur les Espaces Singuliers, I (Luminy, 1981) , Astérisque 100 , Soc. Math. France, Paris, 1982, 5-171.
[7] A. Beĭlinson, R. Bezrukavnikov, and I. Mirković, Tilting exercises , Mosc. Math. J. 4 (2004), 547-557, 782. · Zbl 1075.14015
[8] A. Beĭlinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory , J. Amer. Math. Soc. 9 (1996), 473-527. · Zbl 0864.17006
[9] J. Bernstein and V. Lunts, Equivariant Sheaves and Functors , Lecture Notes in Math. 1578 , Springer, Berlin, 1994. · Zbl 0808.14038
[10] R. Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone , Represent. Theory 7 (2003), 1-18. · Zbl 1065.20055
[11] R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and \(t\)-structures on derived categories of coherent sheaves , Invent. Math. 166 (2006), 327-357. · Zbl 1123.17002
[12] R. Bezrukavnikov and Z. Yun, On Koszul duality for Kac-Moody groups , Represent. Theory 17 (2013), 1-98. · Zbl 1326.20051
[13] P. Deligne, La conjecture de Weil, II , Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137-252. · Zbl 0456.14014
[14] D. Juteau, Decomposition numbers for perverse sheaves , Ann. Inst. Fourier 59 (2009), 1177-1229. · Zbl 1187.14022
[15] D. Juteau, C. Mautner, and G. Williamson, Parity sheaves , J. Amer. Math. Soc. 27 (2014), 1169-1212. · Zbl 1344.14017
[16] I. Mirković and S. Riche, Linear Koszul duality , Compos. Math. 146 (2010), 233-258. · Zbl 1275.14014
[17] I. Mirković and S. Riche, Linear Koszul duality, II, Coherent sheaves on perfect sheaves , preprint, [math.RT]. arXiv:1301.3924v2 · Zbl 1353.18014
[18] S. Riche, W. Soergel, and G. Williamson, Modular Koszul duality , Compos. Math. 150 (2014), 273-332. · Zbl 1426.20017
[19] C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences , Math. Z. 208 (1991), 209-223. · Zbl 0725.16011
[20] R. Rouquier, “Categorification of \(\mathfrak{sl}_{2}\) and braid groups” in Trends in Representation Theory of Algebras and Related Topics , Contemp. Math. 406 , Amer. Math. Soc., 2006, 137-167. · Zbl 1162.20301
[21] W. Soergel, “On the relation between intersection cohomology and representation theory in positive characteristic” in Commutative Algebra, Homological Algebra and Representation Theory (Catania/Genoa/Rome, 1998) , J. Pure Appl. Algebra 152 (2000), 311-335. · Zbl 1101.14302
[22] G. Williamson, Schubert calculus and torsion explosion , preprint, [math.RT]. arXiv:1309.5055v2
[23] G. Williamson and T. Braden, Modular intersection cohomology complexes on flag varieties , Math. Z. 272 (2012), 697-727. · Zbl 1284.14031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.