  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.
