Extended Deligne–Lusztig varieties for general and special linear groups

Stasinski, Alexander

Abstract

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL2(Fq〚ϖ〛/(ϖ2))SL2(Fq〚ϖ〛/(ϖ2)), with odd q, the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.

Citation

Stasinski, A. (2011). Extended Deligne–Lusztig varieties for general and special linear groups. Advances in Mathematics, 226(3), 2825-2853. https://doi.org/10.1016/j.aim.2010.10.010

• Journal Article Type: Article
• Publication Date: Feb 15, 2011
• Deposit Date: Mar 13, 2012
• Publicly Available Date: May 7, 2014
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 226
• Issue: 3
• Pages: 2825-2853
• DOI: https://doi.org/10.1016/j.aim.2010.10.010
• Keywords: Deligne–Lusztig varieties, Representations, Linear groups over finite rings.