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Extended Deligne–Lusztig varieties for general and special linear groups.

Stasinski, Alexander (2011) 'Extended Deligne–Lusztig varieties for general and special linear groups.', Advances in mathematics., 226 (3). pp. 2825-2853.


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.

Item Type:Article
Keywords:Deligne–Lusztig varieties, Representations, Linear groups over finite rings.
Full text:(AM) Accepted Manuscript
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Publisher statement:This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 226, 3, 2011, 10.1016/j.aim.2010.10.010.
Date accepted:No date available
Date deposited:07 May 2014
Date of first online publication:February 2011
Date first made open access:No date available

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