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On Koszul duality for Kac-Moody groups. (English) Zbl 1326.20051

Summary: For any Kac-Moody group \(G\) with Borel \(B\), we give a monoidal equivalence between the derived category of \(B\)-equivariant mixed complexes on the flag variety \(G/B\) and (a certain completion of) the derived category of \(G^\vee\)-monodromic mixed complexes on the enhanced flag variety \(G^\vee /U^\vee\), here \( G^\vee\) is the Langlands dual of \(G\). We also prove variants of this equivalence, one of which is the equivalence between the derived category of \(U\)-equivariant mixed complexes on the partial flag variety \(G/P\) and a certain “Whittaker model” category of mixed complexes on \(G^\vee /B^\vee\). In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups of A. Beilinson, V. Ginzburg, and W. Soergel [in J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)].

MSC:

20G44 Kac-Moody groups
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 0864.17006
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References:

[1] Sergey Arkhipov, Roman Bezrukavnikov, and Victor Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), no. 3, 595 – 678. · Zbl 1061.17013
[2] A. A. Beĭlinson, On the derived category of perverse sheaves, \?-theory, arithmetic and geometry (Moscow, 1984 – 1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27 – 41. · doi:10.1007/BFb0078365
[3] A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel\(^{\prime}\)fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1 – 50. · Zbl 0790.22007
[4] 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). · Zbl 0536.14011
[5] A. Beilinson, R. Bezrukavnikov, and I. Mirković, Tilting exercises, Mosc. Math. J. 4 (2004), no. 3, 547 – 557, 782 (English, with English and Russian summaries). · Zbl 1075.14015
[6] Alexander Beilinson and Victor Ginzburg, Wall-crossing functors and \?-modules, Represent. Theory 3 (1999), 1 – 31. · Zbl 0910.05068
[7] A. A. Beĭlinson, V. A. Ginsburg, and V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317 – 350. · Zbl 0695.14009 · doi:10.1016/0393-0440(88)90028-9
[8] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473 – 527. · Zbl 0864.17006
[9] Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. · Zbl 0808.14038
[10] Roman Bezrukavnikov, Cohomology of tilting modules over quantum groups and \?-structures on derived categories of coherent sheaves, Invent. Math. 166 (2006), no. 2, 327 – 357. · Zbl 1123.17002 · doi:10.1007/s00222-006-0514-z
[11] Roman Bezrukavnikov, Alexander Braverman, and Ivan Mirkovic, Some results about geometric Whittaker model, Adv. Math. 186 (2004), no. 1, 143 – 152. · Zbl 1071.20039 · doi:10.1016/j.aim.2003.07.011
[12] Roman Bezrukavnikov and Michael Finkelberg, Equivariant Satake category and Kostant-Whittaker reduction, Mosc. Math. J. 8 (2008), no. 1, 39 – 72, 183 (English, with English and Russian summaries). · Zbl 1205.19005
[13] Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137 – 252 (French). · Zbl 0456.14014
[14] Victor Ginsburg, Perverse sheaves and \?*-actions, J. Amer. Math. Soc. 4 (1991), no. 3, 483 – 490. · Zbl 0760.14008
[15] Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25 – 83. · Zbl 0897.22009 · doi:10.1007/s002220050197
[16] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[17] Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of \?/\? for a Kac-Moody group \?, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 6, 1543 – 1545. , https://doi.org/10.1073/pnas.83.6.1543 Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of \?/\? for a Kac-Moody group \?, Adv. in Math. 62 (1986), no. 3, 187 – 237. · Zbl 0641.17008 · doi:10.1016/0001-8708(86)90101-5
[18] Yves Laszlo and Martin Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109 – 168. , https://doi.org/10.1007/s10240-008-0011-6 Yves Laszlo and Martin Olsson, The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 169 – 210. · Zbl 1191.14003 · doi:10.1007/s10240-008-0012-5
[19] Olivier Mathieu, Construction d’un groupe de Kac-Moody et applications, Compositio Math. 69 (1989), no. 1, 37 – 60 (French). · Zbl 0678.17012
[20] J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34 – 73. · Zbl 1007.18012 · doi:10.1006/aima.2001.1995
[21] Dragan Miličić and Wolfgang Soergel, The composition series of modules induced from Whittaker modules, Comment. Math. Helv. 72 (1997), no. 4, 503 – 520. · Zbl 0956.17004 · doi:10.1007/s000140050031
[22] Martin Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55 – 112. · Zbl 1137.14004 · doi:10.1515/CRELLE.2007.012
[23] Olaf M. Schnürer, Equivariant sheaves on flag varieties, Math. Z. 267 (2011), no. 1-2, 27 – 80. · Zbl 1217.14034 · doi:10.1007/s00209-009-0609-5
[24] Wolfgang Soergel, Kategorie \?, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421 – 445 (German, with English summary). · Zbl 0747.17008
[25] T. A. Springer, A purity result for fixed point varieties in flag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 271 – 282. · Zbl 0581.20048
[26] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier.
[27] J.-L. Verdier, Spécialisation de faisceaux et monodromie modérée, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 332 – 364 (French). · Zbl 0532.14008
[28] Zhiwei Yun, Weights of mixed tilting sheaves and geometric Ringel duality, Selecta Math. (N.S.) 14 (2009), no. 2, 299 – 320. · Zbl 1197.14016 · doi:10.1007/s00029-008-0066-8
[29] Zhiwei Yun, Goresky-MacPherson calculus for the affine flag varieties, Canad. J. Math. 62 (2010), no. 2, 473 – 480. · Zbl 1223.14056 · doi:10.4153/CJM-2010-029-x
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