## 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

## Bibliographic Information

**Ulrich Görtz**- Affiliation: Mathematisches Institut, Beringstr. 1, 53115 Bonn, Germany
- Email: ugoertz@math.uni-bonn.de
- Received by editor(s): September 19, 2008
- Received by editor(s) in revised form: December 8, 2008
- Published electronically: January 21, 2009
- Additional Notes: The author was partially supported by a Heisenberg grant and by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG (German Research Foundation)
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**13**(2009), 1-7 - MSC (2000): Primary 14L35; Secondary 20G40
- DOI: https://doi.org/10.1090/S1088-4165-09-00344-6
- MathSciNet review: 2471197