Publication details for Alexander StasinskiStasinski, Alexander (2011). Extended Deligne–Lusztig varieties for general and special linear groups. Advances in Mathematics 226(3): 2825–2853.
- Publication type: Journal Article
- ISSN/ISBN: 0001-8708, 1090-2082
- DOI: 10.1016/j.aim.2010.10.010
- Keywords: Deligne–Lusztig varieties, Representations, Linear groups over finite rings.
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
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.