In the early 1970s Drinfeld introduced a family of rigid analytic
spaces parameterizing deformations of certain formal groups with level
structure. This family is called the Lubin-Tate tower. He found an
open affinoid in the first level of this tower whose reduction is
isomorphic to what is now known as a Deligne-Lusztig variety for GL_n
over a finite field. This established a link between depth 0
supercuspidal representations of GL_n(K) (where K is a local field)
and cuspidal representations of GL_n(F_q) (where F_q is the residue
field of K). I will explain a similar construction at a higher level
of the tower, which leads to an analogue of the Deligne-Lusztig theory
for a class of unipotent groups over finite fields. This approach
yields a geometric construction of explicit local Langlands
correspondence for a certain class of positive depth supercuspidal
representations of GL_n(K). The talk is based on joint work with Jared
Weinstein (Boston University). A large portion of the talk will be
very elementary and will require no background apart from some
familiarity with algebraic groups over finite fields.