On the connectedness of Deligne-Lusztig varieties

Görtz, Ulrich

doi:10.1090/S1088-4165-09-00344-6

On the connectedness of Deligne-Lusztig varieties
HTML articles powered by AMS MathViewer

by Ulrich Görtz

Represent. Theory 13 (2009), 1-7

DOI: https://doi.org/10.1090/S1088-4165-09-00344-6

Published electronically: January 21, 2009

PDF | Request permission

Abstract:

We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties due to Lusztig.

References

Cédric Bonnafé and Raphaël Rouquier, On the irreducibility of Deligne-Lusztig varieties, C. R. Math. Acad. Sci. Paris 343 (2006), no. 1, 37–39 (English, with English and French summaries). MR 2241956, DOI 10.1016/j.crma.2006.04.014
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
F. Digne and J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya Math. J. 183 (2006), 35–103. MR 2253886, DOI 10.1017/S0027763000009260
T. Ekedahl, G. van der Geer, Cycle classes of the E-O stratification on the moduli of abelian varieties, arXiv:math.AG/0412272v2.
Ulrich Görtz, On the flatness of local models for the symplectic group, Adv. Math. 176 (2003), no. 1, 89–115. MR 1978342, DOI 10.1016/S0001-8708(02)00062-2
U. Görtz, C.-F. Yu, Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties, arXiv:math/0802.3260v2.
U. Görtz, C.-F. Yu, The supersingular locus in Siegel modular varieties with Iwahori level structure, arXiv:0807.1229v2.
Burkhard Haastert, Die Quasiaffinität der Deligne-Lusztig-Varietäten, J. Algebra 102 (1986), no. 1, 186–193 (German). MR 853238, DOI 10.1016/0021-8693(86)90135-3
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
R. Kottwitz and M. Rapoport, Minuscule alcoves for $\textrm {GL}_n$ and $G\textrm {Sp}_{2n}$, Manuscripta Math. 102 (2000), no. 4, 403–428. MR 1785323, DOI 10.1007/s002290070034
G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101–159. MR 453885, DOI 10.1007/BF01408569
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, DOI 10.1090/cbms/039
G. Lusztig, A class of perverse sheaves on a partial flag manifold, Represent. Theory 11 (2007), 122–171. MR 2336607, DOI 10.1090/S1088-4165-07-00320-2
Frans Oort, A stratification of a moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 345–416. MR 1827027, DOI 10.1007/978-3-0348-8303-0_{1}3
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
M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
Eva Viehmann, Connected components of closed affine Deligne-Lusztig varieties, Math. Ann. 340 (2008), no. 2, 315–333. MR 2368982, DOI 10.1007/s00208-007-0153-8