Extended Deligne-Lusztig varieties for general and special linear groups

TL;DR

This paper extends Deligne-Lusztig varieties to include tamely ramified maximal tori for general and special linear groups over finite quotients, analyzing their structure and representation-theoretic completeness.

Contribution

It introduces a family of extended Deligne-Lusztig varieties for all tamely ramified maximal tori, broadening the scope beyond unramified cases and analyzing their representation-theoretic properties.

Findings

Unramified varieties do not produce all irreducible representations in general.

For SL_2 over certain finite quotients, extended varieties produce all irreducible representations.

The structure of generalized Deligne-Lusztig varieties is thoroughly analyzed.

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 $\SL_{2}(\mathbb{F}_{q}[[\varpi]]/(\varpi^{2}))$, with odd $q$, the extended Deligne-Lusztig varieties do indeed afford all the irreducible representations.