Extended Deligne-Lusztig varieties for general and special linear groups

Stasinski, Alexander (2009) Extended Deligne-Lusztig varieties for general and special linear groups. Preprint, 29pp. (Submitted)

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 such 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 SL_2(F_q[[\varpi]]/(\varpi^2)), with odd q, the extended Deligne-Lusztig varieties do indeed afford all the irreducible representations

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