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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Alexander Stasinski

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

Author(s) from Durham

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.