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The affine Grassmannian and the Springer resolution in positive characteristic

Part of: Lie groups (Co)homology theory

Published online by Cambridge University Press:  28 November 2016

Pramod N. Achar  and
Laura Rider
Pramod N. Achar
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA email pramod@math.lsu.edu
Laura Rider
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email laurajoy@mit.edu
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Abstract

An important result of Arkhipov–Bezrukavnikov–Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of ‘mixed modular sheaves’ recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov’s ‘exotic t-structure’ on the Springer resolution.

Keywords

affine GrassmannianSpringer resolutionLanglands duality

MSC classification

Primary: 22E57: Geometric Langlands program
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2627 - 2677
Copyright
© The Authors 2016 

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References

Achar, P., Perverse coherent sheaves on the nilpotent cone in good characteristic , in Recent developments in Lie algebras, groups and representation theory, Proceedings of Symposia in Pure Mathematics, vol. 86 (American Mathematical Society, Providence, RI, 2012), 1–23.Google Scholar
Achar, P. and Riche, S., Koszul duality and semisimplicity of Frobenius , Ann. Inst. Fourier 63 (2013), 1511–1612.Google Scholar
Achar, P. and Riche, S., Modular perverse sheaves on flag varieties III: positivity conditions, Preprint (2014), arXiv:1408.4189.Google Scholar
Achar, P. and Riche, S., Modular perverse sheaves on flag varieties II: Koszul duality and formality , Duke Math. J. 165 (2016), 161–215.Google Scholar
Achar, P. and Rider, L., Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture , Acta Math. 215 (2015), 183–216.Google Scholar
Arkhipov, S. and Bezrukavnikov, R., Perverse sheaves on affine flags and Langlands dual group , Israel J. Math. 170 (2009), 135–183.Google Scholar
Arkhipov, S., Bezrukavnikov, R. and Ginzburg, V., Quantum groups, the loop Grassmannian, and the Springer resolution , J. Amer. Math. Soc. 17 (2004), 595–678.Google Scholar
Beĭlinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers , in Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque, vol. 100 (Société Mathématique de France, Paris, 1982), 5–171.Google Scholar
Beĭlinson, A., Bezrukavnikov, R. and Mirković, I., Tilting exercises , Mosc. Math. J. 4 (2004), 547–557.Google Scholar
Beĭlinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory , J. Amer. Math. Soc. 9 (1996), 473–527.Google Scholar
Bernstein, J. and Lunts, V., Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578 (Springer, Berlin, 1994).Google Scholar
Bezrukavnikov, R., Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone , Represent. Theory 7 (2003), 1–18.Google Scholar
Bezrukavnikov, R., Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves , Invent. Math. 166 (2006), 327–357.Google Scholar
Bezrukavnikov, R., Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group , Israel J. Math. 170 (2009), 185–206.Google Scholar
Bezrukavnikov, R. and Finkelberg, M., Equivariant Satake category and Kostant–Whittaker reduction , Mosc. Math. J. 8 (2008), 39–72.Google Scholar
Bezrukavnikov, R. and Mirković, I., Representations of semi-simple Lie algebras in prime characteristic and noncommutative Springer resolution , Ann. of Math. (2) 178 (2013), 835–919.Google Scholar
Bondal, A. and Kapranov, M., Representable functors, Serre functors, and reconstructions , Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183–1205.Google Scholar
Brion, M. and Kumar, S., Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231 (Birkhäuser, Boston, MA, 2005).Google Scholar
Deligne, P., La conjecture de Weil. II , Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.Google Scholar
Geiss, C., Leclerc, B. and Schröer, J., Preprojective algebras and cluster algebras , in Trends in representation theory of algebras and related topics, EMS Series of Congress Reports, vol. 1 (European Mathematical Society, Zürich, 2008), 253–283.Google Scholar
Finkelberg, M. and Mirković, I., Semi-infinite flags I. Case of global curve P 1 , in Differential topology, infinite-dimensional Lie algebras, and applications, American Mathematical Society Translations, series 2, vol. 194 (American Mathematical Society, Providence, RI, 1999), 81–112.Google Scholar
Jantzen, J. C., Representations of algebraic groups, Mathematical Surveys and Monographs, vol. 107, second edition (American Mathematical Society, Providence, RI, 2003).Google Scholar
Juteau, D., Mautner, C. and Williamson, G., Parity sheaves , J. Amer. Math. Soc. 27 (2014), 1169–1212.Google Scholar
Juteau, D., Mautner, C. and Williamson, G., Parity sheaves and tilting modules, Preprint (2014), arXiv:1403.1647.Google Scholar
Kumar, S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204 (Birkhäuser, Boston, MA, 2002).Google Scholar
Kumar, S., Lauritzen, N. and Thomsen, J. F., Frobenius splitting of cotangent bundles of flag varieties , Invent. Math. 136 (1999), 603–621.Google Scholar
Lakshimbai, V. and Gonciulea, N., Flag varieties, Travaux en Cours, no. 63 (Hermann, Paris, 2001).Google Scholar
Mautner, C. and Riche, S., Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture, Preprint (2015), arXiv:1501.07369.Google Scholar
Minn-Thu-Aye, M., Multiplicity formulas for perverse coherent sheaves on the nilpotent cone, PhD thesis, Louisiana State University (2013). Available athttp://etd.lsu.edu/docs/available/etd-07082013-113917/.Google Scholar
Mumford, D., Stability of projective varieties , Enseign. Math. (2) 23 (1977), 39–110.Google Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar