# Deligne-Lusztig theory for unipotent groups and the local Langlands correspondence

# Deligne-Lusztig theory for unipotent groups and the local Langlands correspondence

(1) A (very) special case of Deligne-Lusztig theory yields a construction of cuspidal irreducible representations of the finite group GL_n(F_q) in the cohomology of an algebraic variety equipped with an action of GL_n(F_q). There is also a well known relationship between cuspidal representations of GL_n(F_q) and depth 0 supercuspidal representations of GL_n(F), where F is a local field with residue field F_q. (2) On the other hand, thanks to the work of Boyer, Carayol, Deligne, Harris, Henniart, Laumon, Rapoport, Stuhler, Taylor..., it is known that the local Langlands correspondence for GL_n(F) is realized in the cohomology of the Lubin-Tate tower of rigid analytic spaces over F. There is a direct geometric link between (1) and (2): the first level of the Lubin-Tate tower contains an open affinoid with good reduction, whose special fiber is isomorphic to a Deligne-Lusztig variety for GL_n(F_q). I will explain a similar picture for certain supercuspidal representations of GL_n(F) of positive depth. In particular, I will describe the construction of an open affinoid (with good reduction) in a higher level of the Lubin-Tate tower, which has the following properties. On the one hand, its cohomology gives an explicit geometric realization of the local Langlands correspondence for a certain class of positive depth supercuspidal representations of GL_n(F). On the other hand, its special fiber is related to a certain unipotent group over F_q in a way that is similar to one of the known approaches to Deligne-Lusztig theory for reductive groups over F_q.