Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-18T07:22:44.756Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic Newton points and the Newton poset in Iwahori-double cosets

Part of: Special aspects of infinite or finite groups Algebraic groups Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 November 2020

Elizabeth Milićević  and
Eva Viehmann
Elizabeth Milićević
Affiliation:
Haverford College, Department of Mathematics & Statistics, 370 Lancaster Avenue, Haverford, PA, 19041, USA; E-mail: [email protected]
Eva Viehmann
Affiliation:
Technische Universität München, Fakultät für Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei München, Germany; E-mail: [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.

Keywords

affine Deligne-Lusztig varietyNewton strataaffine flag variety

MSC classification

Primary: 11G25: Varieties over finite and local fields
Secondary: 14L05: Formal groups, $p$-divisible groups 20F55: Reflection and Coxeter groups
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e50
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Björner, A. and Brenti, B., Combinatorics of Coxeter Groups, Graduate Texts in Mathematics vol. 231 (Springer, New York, 2005).Google Scholar
Baumeister, B., Dyer, M., Stump, C., and Wegener, P., ‘A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements’, Proc. Amer. Math. Soc. Ser. B 1 (2014), 149–154.CrossRefGoogle Scholar
Beazley, E., ‘Codimensions of Newton strata for ${\mathrm{SL}}_3(F)$ in the Iwahori case’, Math. Z. 263(3) (2009), 499–540.CrossRefGoogle Scholar
Brenti, F., Fomin, S., and Postnikov, A., ‘Mixed Bruhat operators and Yang-Baxter equations for Weyl groups’, Internat. Math. Res. Notices (8) (1999), 419–441.CrossRefGoogle Scholar
Bhatt, B. and Scholze, P., ‘Projectivity of the Witt vector affine Grassmannian’, Invent. Math. 209(2) (2017), 329–423.CrossRefGoogle Scholar
Chai, C.-L., ‘Newton polygons as lattice points’, Amer. J. Math. 122(5) (2000), 967–990.CrossRefGoogle Scholar
Deligne, P. and Lusztig, G., ‘Representations of reductive groups over finite fields’, Ann. of Math. (2) 103(1) (1976), 103–161.CrossRefGoogle Scholar
Fan, C. K., ‘Schubert varieties and short braidedness’, Transform. Groups 3(1) (1998), 51–56.CrossRefGoogle Scholar
Fomin, S., Gelfand, S., and Postnikov, A., ‘Quantum Schubert polynomials’, J. Amer. Math. Soc. 10(3) (1997), 565–596.CrossRefGoogle Scholar
Gashi, Q. R., ‘On a conjecture of Kottwitz and Rapoport’, Ann. Sci. Éc. Norm. Supér. (4) 43(6) (2010), 1017–1038.CrossRefGoogle Scholar
Görtz, U. and He, X., ‘Dimensions of affine Deligne-Lusztig varieties in affine flag varieties’, Doc. Math. 15 (2010), 1009–1028.Google Scholar
Görtz, U., Haines, T. J., Kottwitz, R. E., and Reuman, D. C., ‘Affine Deligne-Lusztig varieties in affine flag varieties’, Compos. Math. 146(5) (2010), 1339–1382.CrossRefGoogle Scholar
Görtz, U., He, X., and Nie, S., $\textbf{P}$ -alcoves and nonemptiness of affine Deligne-Lusztig varieties’, Ann. Sci. Éc. Norm. Supér. (4) 48(3) (2015), 647–665.CrossRefGoogle Scholar
Hamacher, P., ‘The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type’, Duke Math. J. 164(15) (2015), 2809–2895.CrossRefGoogle Scholar
Hamacher, P., ‘The almost product structure of Newton strata in the deformation space of a Barsotti-Tate group with crystalline Tate tensors’, Math. Z. 287(3–4) (2017), 1255–1277.CrossRefGoogle Scholar
He, X., ‘Geometric and homological properties of affine Deligne-Lusztig varieties’, Ann. of Math. (2) 179(1) (2014), 367–404.CrossRefGoogle Scholar
He, X., ‘Hecke algebras and $p$ -adic groups’, in: Current Developments in Mathematics 2015 (Int. Press, Somerville, MA, 2016) 73–135.Google Scholar
He, X., ‘Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties’, Ann. Sci. Éc. Norm. Supér. (4) 49(5) (2016), 1125–1141.CrossRefGoogle Scholar
He, X., ‘Note on affine Deligne-Lusztig varieties’, in: Proceedings of the Sixth International Congress of Chinese Mathematicians, vol. I, Adv. Lect. Math. (ALM) vol. 36 (Int. Press, Somerville, MA, 2017),297–307.Google Scholar
Hamacher, P. and Viehmann, E., ‘Finiteness properties of affine Deligne-Lusztig varieties’, Doc. Math. 25 (2020), 899–910.Google Scholar
Hartl, U. and Viehmann, E., ‘The Newton stratification on deformations of local $G$ -shtukas’, J. Reine Angew. Math. 656 (2011), 87–129.CrossRefGoogle Scholar
Hartl, U. and Viehmann, E., ‘Foliations in deformation spaces of local $G$ -shtukas’, Adv. Math. 229(1) (2012), 54–78.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Isocrystals with additional structure’, Compositio Math. 56(2) (1985), 201–220.Google Scholar
Kottwitz, R. E., ‘Isocrystals with additional structure. II’, Compositio Math. 109(3) (1997), 255–339.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Dimensions of Newton strata in the adjoint quotient of reductive groups’, Pure Appl. Math. Q. 2(3), special issue in honor of R. D. MacPherson, part 1 (2006), 817–836.CrossRefGoogle Scholar
Kottwitz, R. and Rapoport, M., ‘On the existence of $F$ -crystals’, Comment. Math. Helv. 78(1) (2003), 153–184.CrossRefGoogle Scholar
Lam, T. and Shimozono, M., ‘Quantum cohomology of $G/ P$ and homology of affine Grassmannian’, Acta Math. 204(1) (2010), 49–90.CrossRefGoogle Scholar
Lucarelli, C., ‘A converse to Mazur’s inequality for split classical groups’, J. Inst. Math. Jussieu 3(2) (2004), 165–183.CrossRefGoogle Scholar
Milićević, E., ‘Maximal Newton points and the quantum Bruhat graph’, Michigan Math. Journal. (2016), arXiv.org/1606.07478.Google Scholar
Milićević, E., Schwer, P., and Thomas, A., ‘Dimensions of affine Deligne–Lusztig varieties: a new approach via labeled folded alcove walks and root operators’, Mem. Amer. Math. Soc. 261(1260) (2019).Google Scholar
Nie, S., ‘Fundamental elements of an affine Weyl group’, Math. Ann. 362(1–2) (2015), 485–499.CrossRefGoogle Scholar
Postnikov, A., ‘Quantum Bruhat graph and Schubert polynomials’, Proc. Amer. Math. Soc. 133(3) (2005), 699–709 (electronic).CrossRefGoogle Scholar
Rapoport, M., ‘A guide to the reduction modulo $p$ of Shimura varieties’, Astérisque, (298) (2005), 271–318.Google Scholar
Rapoport, M. and Richartz, M., ‘On the classification and specialization of $F$ -isocrystals with additional structure’, Compositio Math. 103(2) (1996), 153–181.Google Scholar
Ragnarsson, K. and Tenner, B. E., ‘Homotopy type of the Boolean complex of a Coxeter system’, Adv. Math. 222(2) (2009), 409–430.CrossRefGoogle Scholar
Stembridge, J. R., ‘On the fully commutative elements of Coxeter groups’, J. Algebraic Combin. 5(4) (1996), 353–385.CrossRefGoogle Scholar
Viehmann, E., ‘Newton strata in the loop group of a reductive group’, Amer. J. Math. 135(2) (2013), 499–518.CrossRefGoogle Scholar
Viehmann, E., ‘Truncations of level 1 of elements in the loop group of a reductive group’, Ann. of Math. (2) 179(3) (2014), 1009–1040.CrossRefGoogle Scholar
Viehmann, E., ‘Minimal Newton strata in Iwahori double cosets’, Int. Math. Res. Not. IMRN, (2020).CrossRefGoogle Scholar
Viehmann, E., ‘On the geometry of the Newton stratification’, in: Shimura Varieties, London Math. Soc. Lecture Note Ser . vol. 457 (Cambridge Univ. Press, Cambridge, 2020), 192–208.Google Scholar
Viehmann, E. and Wu, H., ‘Central leaves in loop groups’, Math. Res. Lett. 25 (2018), 989–1008.CrossRefGoogle Scholar
Zhu, X., ‘Affine Grassmannians and the geometric Satake in mixed characteristic’, Ann. of Math. (2) 185(2) (2017), 403–492.CrossRefGoogle Scholar