AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Dimensions of Affine Deligne–Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators
About this Title
Elizabeth Milićević, Department of Mathematics & Statistics, Haverford College, 370 Lancaster Avenue, Haverford, PA, USA, Petra Schwer, Department of Mathematics, Karlsruhe Institute of Technology, EnglerstraÃe 2, 76133 Karlsruhe, Germany and Anne Thomas, School of Mathematics & Statistics, Carslaw Building F07, University of Sydney NSW 2006, Australia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 261, Number 1260
ISBNs: 978-1-4704-3676-6 (print); 978-1-4704-5403-6 (online)
DOI: https://doi.org/10.1090/memo/1260
Published electronically: November 7, 2019
MSC: Primary 20G25; Secondary 05E10, 20F55, 51E24.
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries on Weyl groups, affine buildings, and related notions
- 3. Labelings and orientations, galleries, and alcove walks
- 4. Dimensions of galleries and root operators
- 5. Affine Deligne–Lusztig varieties and folded galleries
- 6. Explicit constructions of positively folded galleries
- 7. The varieties $X_x(1)$ in the shrunken dominant Weyl chamber
- 8. The varieties $X_x(1)$ and $X_x(b)$
- 9. Conjugating to other Weyl chambers
- 10. Diagram automorphisms
- 11. Applications to affine Hecke algebras and affine reflection length
Abstract
Let $G$ be a reductive group over the field $F=k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the (extended) affine Weyl group of $G$. The associated affine Deligne–Lusztig varieties $X_x(b)$, which are indexed by elements $b \in G(F)$ and $x \in W$, were introduced by Rapoport (2000). Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that $b$ is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman (2010). Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that $b$ is basic, and reveals new patterns. Since we work only in the standard apartment of the building for $G(F)$, our results also hold in the $p$-adic context, where we formulate a definition of the dimension of a $p$-adic Deligne–Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- E. Beazley. Codimensions of Newton strata for $\textrm {SL}_3(F)$ in the Iwahori case. Math. Z., 263(3):499–540, 2009. URL: http://dx.doi.org/10.1007/s00209-008-0429-z, doi:10.1007/s00209-008-0429-z.
- E. T. Beazley, Affine Deligne-Lusztig varieties associated to additive affine Weyl group elements, J. Algebra 349 (2012), 63–79. MR 2853626, DOI 10.1016/j.jalgebra.2011.10.020
- R. Bédard. The lowest two-sided cell for an affine Weyl group. Comm. Algebra, 16(6):1113–1132, 1988. URL: http://dx.doi.org/10.1080/00927878808823622, doi:10.1080/00927878808823622.
- Pierre Baumann and Stéphane Gaussent, On Mirković-Vilonen cycles and crystal combinatorics, Represent. Theory 12 (2008), 83–130. MR 2390669, DOI 10.1090/S1088-4165-08-00322-1
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
- F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
- R. Carter, Conjugacy classes in the Weyl group, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Springer, Berlin, 1970, pp. 297–318. MR 0269749
- R. Dąbrowski, Comparison of the Bruhat and the Iwahori decompositions of a $\mathfrak {p}$-adic Chevalley group, J. Algebra 167 (1994), no. 3, 704–723. MR 1287066, DOI 10.1006/jabr.1994.1208
- Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
- P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
- Matthew J. Dyer, On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2591–2595. MR 1838781, DOI 10.1090/S0002-9939-01-05876-2
- Jean-Marc Fontaine, Groupes $p$-divisibles sur les corps locaux, Société Mathématique de France, Paris, 1977 (French). Astérisque, No. 47-48. MR 0498610
- W. N. Franzsen. Automorphisms of Coxeter groups. PhD thesis, University of Sydney, 2001.
- Qëndrim R. Gashi, A vanishing result for toric varieties associated with root systems, Albanian J. Math. 1 (2007), no. 4, 235–244. MR 2367216
- Q. R. Gashi. Vanishing results for toric varieties associated to $\textrm {GL}_n$ and $\textrm {G}_2$. Transform. Groups, 13(1):149–171, 2008. URL: http://dx.doi.org/10.1007/s00031-008-9002-4, doi:10.1007/s00031-008-9002-4.
- Ulrich Görtz and Xuhua He, Dimensions of affine Deligne-Lusztig varieties in affine flag varieties, Doc. Math. 15 (2010), 1009–1028. MR 2745691
- U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman. Dimensions of some affine Deligne-Lusztig varieties. Ann. Sci. École Norm. Sup. (4), 39(3):467–511, 2006. URL: http://dx.doi.org/10.1016/j.ansens.2005.12.004, doi:10.1016/j.ansens.2005.12.004.
- U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman. Affine Deligne-Lusztig varieties in affine flag varieties. Compos. Math., 146(5):1339–1382, 2010. URL: http://dx.doi.org/10.1112/S0010437X10004823, doi:10.1112/S0010437X10004823.
- Ulrich Görtz, Xuhua He, and Sian Nie, $\bf P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 3, 647–665 (English, with English and French summaries). MR 3377055, DOI 10.24033/asens.2254
- S. Gaussent and P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (2005), no. 1, 35–88. MR 2126496, DOI 10.1215/S0012-7094-04-12712-5
- Meinolf Geck and Götz Pfeiffer, On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), no. 1, 79–94. MR 1250466, DOI 10.1006/aima.1993.1056
- Thomas J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583–642. MR 2192017
- Paul Hamacher, The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian, Int. Math. Res. Not. IMRN 23 (2015), 12804–12839. MR 3431637, DOI 10.1093/imrn/rnv081
- X. He. Note on affine Deligne-Lusztig varieties. arXiv:1309.0075 [math.AG], 2013.
- Xuhua He, Geometric and homological properties of affine Deligne-Lusztig varieties, Ann. of Math. (2) 179 (2014), no. 1, 367–404. MR 3126571, DOI 10.4007/annals.2014.179.1.6
- Petra Hitzelberger, Kostant convexity for affine buildings, Forum Math. 22 (2010), no. 5, 959–971. MR 2719765, DOI 10.1515/FORUM.2010.051
- Petra Hitzelberger, Non-discrete affine buildings and convexity, Adv. Math. 227 (2011), no. 1, 210–244. MR 2782192, DOI 10.1016/j.aim.2011.01.019
- X. He and S. Nie. Minimal length elements of finite Coxeter groups. Duke Math. J., 161(15):2945–2967, 2012. URL: http://dx.doi.org/10.1215/00127094-1902382, doi:10.1215/00127094-1902382.
- J. E. Humphreys. Littelmann path operators for an arbitrary positive root. http://mathoverflow.net/a/84253, 2011.
- M. Kashiwara. Crystalizing the $q$-analogue of universal enveloping algebras. Comm. Math. Phys., 133(2):249–260, 1990. URL: http://projecteuclid.org/getRecord?id=euclid.cmp/1104201397.
- Nicholas M. Katz, Slope filtration of $F$-crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 63, Soc. Math. France, Paris, 1979, pp. 113–163. MR 563463
- Michael Kapovich and John J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory, Groups Geom. Dyn. 2 (2008), no. 3, 405–480. MR 2415306, DOI 10.4171/GGD/46
- Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220. MR 809866
- Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
- R. Kottwitz and M. Rapoport, On the existence of $F$-crystals, Comment. Math. Helv. 78 (2003), no. 1, 153–184. MR 1966756, DOI 10.1007/s000140300007
- P. Littelmann. A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math., 116(1-3):329–346, 1994. URL: http://dx.doi.org/10.1007/BF01231564, doi:10.1007/BF01231564.
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- V. Lakshmibai and C. S. Seshadri, Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 279–322. MR 1131317
- C. Lucarelli. A converse to Mazur’s inequality for split classical groups. J. Inst. Math. Jussieu, 3(2):165–183, 2004. URL: http://dx.doi.org/10.1017/S1474748004000064, doi:10.1017/S1474748004000064.
- George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8–12, 1977. MR 518617
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- I. G. Macdonald. Affine Hecke algebras and orthogonal polynomials, volume 157 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2003. URL: http://dx.doi.org/10.1017/CBO9780511542824, doi:10.1017/CBO9780511542824.
- B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972), 653–667. MR 330169, DOI 10.1090/S0002-9904-1972-12976-8
- Jon McCammond and T. Kyle Petersen, Bounding reflection length in an affine Coxeter group, J. Algebraic Combin. 34 (2011), no. 4, 711–719. MR 2842917, DOI 10.1007/s10801-011-0289-1
- Ivan Mirković and Kari Vilonen, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), no. 1, 13–24. MR 1748284, DOI 10.4310/MRL.2000.v7.n1.a2
- I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. MR 2342692, DOI 10.4007/annals.2007.166.95
- J. Parkinson, A. Ram, and C. Schwer. Combinatorics in affine flag varieties. J. Algebra, 321(11):3469–3493, 2009. URL: http://dx.doi.org/10.1016/j.jalgebra.2008.04.015, doi:10.1016/j.jalgebra.2008.04.015.
- Arun Ram, Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue: In honor of Robert D. MacPherson., 963–1013. MR 2282411, DOI 10.4310/PAMQ.2006.v2.n4.a4
- M. Rapoport. A positivity property of the Satake isomorphism. Manuscripta Math., 101(2):153–166, 2000. URL: http://dx.doi.org/10.1007/s002290050010, doi:10.1007/s002290050010.
- Michael Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR 2141705
- D. C. Reuman. Formulas for the dimensions of some affine Deligne-Lusztig varieties. Michigan Math. J., 52(2):435–451, 2004. URL: http://dx.doi.org/10.1307/mmj/1091112084, doi:10.1307/mmj/1091112084.
- M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure, Compositio Math. 103 (1996), no. 2, 153–181. MR 1411570
- Christoph Schwer, Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra, Int. Math. Res. Not. , posted on (2006), Art. ID 75395, 31. MR 2264725, DOI 10.1155/IMRN/2006/75395
- Petra Schwer, Root operators, root groups and retractions, J. Comb. Algebra 2 (2018), no. 3, 215–230. MR 3845717, DOI 10.4171/JCA/2-3-1
- Jian Yi Shi, A two-sided cell in an affine Weyl group, J. London Math. Soc. (2) 36 (1987), no. 3, 407–420. MR 918633, DOI 10.1112/jlms/s2-36.3.407
- E. Viehmann. The dimension of some affine Deligne-Lusztig varieties. Ann. Sci. École Norm. Sup. (4), 39(3):513–526, 2006. URL: http://dx.doi.org/10.1016/j.ansens.2006.04.001, doi:10.1016/j.ansens.2006.04.001.
- Zhongwei Yang, Class polynomials for some affine Hecke algebras, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–Hong Kong University of Science and Technology (Hong Kong). MR 3450176