Deligne-Lusztig constructions for unipotent and p-adic groups

In 1979 Lusztig proposed a conjectural construction of supercuspidal representations of reductive p-adic groups, which is similar to the well known construction of Deligne and Lusztig in the setting of finite reductive groups. We present a general method for explicitly calculating the representations arising from Lusztig’s construction and illustrate it with several examples. The techniques we develop also provide background for the author’s joint work with Weinstein on a purely local and explicit proof of the local Langlands correspondence.