Springer motives. (English) Zbl 1505.14052
This paper shows that the motives of Springer fibers in the triangulated category of Voevodsky motives are pure Tate, that is, they are isomorphic to direct sums of finitely many Tate motives. The main tool is the existence of finite pavings (or some of their variant forms) of these varieties. The paper then use Soergel-Virk-Wendt’s equivariant motive theory [W. Soergel et al., “Equivariant motives and geometric representation theory (with an appendix by F. Hörmann and M. Wendt)”, Preprint (2018); arXiv:1809.05480] to construct certain category of equivariant Springer motives on the nilpotent cone and show that it is isomorphic to the derived category of graded modules over the graded affine Hecke algebra. In this part the convolution construction of the graded affine Hecke algebra plays an essential role.
Reviewer: Changlong Zhong (Albany)
MSC:
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 20C08 | Hecke algebras and their representations |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14C25 | Algebraic cycles |
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 20G05 | Representation theory for linear algebraic groups |
References:
| [1] | Ayoub, Joseph, A guide to (\'{e}tale) motivic sheaves. Proceedings of the International Congress of Mathematicians-Seoul 2014. Vol. II, 1101-1124 (2014), Kyung Moon Sa, Seoul · Zbl 1373.14009 |
| [2] | Bernstein, Joseph; Lunts, Valery, Equivariant sheaves and functors, Lecture Notes in Mathematics 1578, iv+139 pp. (1994), Springer-Verlag, Berlin · Zbl 0808.14038 · doi:10.1007/BFb0073549 |
| [3] | Brosnan, Patrick, On motivic decompositions arising from the method of Bia\l ynicki-Birula, Invent. Math., 161, 1, 91-111 (2005) · Zbl 1085.14045 · doi:10.1007/s00222-004-0419-7 |
| [4] | Chriss, Neil; Ginzburg, Victor, Representation theory and complex geometry, Modern Birkh\"{a}user Classics, x+495 pp. (2010), Birkh\"{a}user Boston, Ltd., Boston, MA · Zbl 1185.22001 · doi:10.1007/978-0-8176-4938-8 |
| [5] | De Concini, C.; Lusztig, G.; Procesi, C., Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc., 1, 1, 15-34 (1988) · Zbl 0646.14034 · doi:10.2307/1990965 |
| [6] | Jens Niklas Eberhardt, \(K\)-Motives and Koszul Duality, 2019. |
| [7] | Eberhardt, Jens Niklas; Kelly, Shane, Mixed motives and geometric representation theory in equal characteristic, Selecta Math. (N.S.), 25, 2, Paper No. 30, 54 pp. (2019) · Zbl 1444.14046 · doi:10.1007/s00029-019-0475-x |
| [8] | Huber, Annette; Kahn, Bruno, The slice filtration and mixed Tate motives, Compos. Math., 142, 4, 907-936 (2006) · Zbl 1105.14022 · doi:10.1112/S0010437X06002107 |
| [9] | Marc Hoyois, Cdh descent in equivariant homotopy \(K\)-theory, 2016. · Zbl 1453.14068 |
| [10] | Hoyois, Marc, The six operations in equivariant motivic homotopy theory, Adv. Math., 305, 197-279 (2017) · Zbl 1400.14065 · doi:10.1016/j.aim.2016.09.031 |
| [11] | Jantzen, Jens Carsten, Nilpotent orbits in representation theory. Lie theory, Progr. Math. 228, 1-211 (2004), Birkh\"{a}user Boston, Boston, MA · Zbl 1169.14319 |
| [12] | Kelly, Shane, Voevodsky motives and \(l\) dh-descent, Ast\'{e}risque, 391, 125 pp. (2017) · Zbl 1386.14003 |
| [13] | Lusztig, George, Affine Hecke algebras and their graded version, J. Amer. Math. Soc., 2, 3, 599-635 (1989) · Zbl 0715.22020 · doi:10.2307/1990945 |
| [14] | Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2, xiv+216 pp. (2006), American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA · Zbl 1115.14010 |
| [15] | Rider, Laura, Formality for the nilpotent cone and a derived Springer correspondence, Adv. Math., 235, 208-236 (2013) · Zbl 1278.17004 · doi:10.1016/j.aim.2012.12.001 |
| [16] | Rider, Laura; Russell, Amber, Perverse sheaves on the nilpotent cone and Lusztig’s generalized Springer correspondence. Lie algebras, Lie superalgebras, vertex algebras and related topics, Proc. Sympos. Pure Math. 92, 273-292 (2016), Amer. Math. Soc., Providence, RI · Zbl 1427.17009 |
| [17] | Laura Rider and Amber Russell, Formality and Lusztig’s generalized Springer correspondence, arXiv preprint 1708.07783 (2017). · Zbl 1427.17009 |
| [18] | Wolfgang Soergel, Rahbar Virk, and Matthias Wendt, Equivariant motives and geometric representation theory. (with an appendix by F. H\"ormann and M. Wendt), arXiv preprint 1809.05480 (2018). · Zbl 1436.14015 |
| [19] | Soergel, Wolfgang; Wendt, Matthias, Perverse motives and graded derived category \({\mathcal{O}} \), J. Inst. Math. Jussieu, 17, 2, 347-395 (2018) · Zbl 1436.14015 · doi:10.1017/S1474748016000013 |
| [20] | Zhao, Gufang; Zhong, Changlong, Geometric representations of the formal affine Hecke algebra, Adv. Math., 317, 50-90 (2017) · Zbl 1427.20010 · doi:10.1016/j.aim.2017.03.026 |
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