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Extended Deligne–Lusztig varieties for general and special linear groups

Stasinski, Alexander

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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 Mar 29, 2024
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.

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Copyright 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.




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