Deligne-Lusztig Theory and the Local Langlands Correspondence
Mitya Boyarchenko, University of Michigan
Let F be a local non-Archimedean field with residue field Fq. The Deligne-Lusztig variety for GLn can be obtained as the special fiber of a certain affinoid subspace of the Lubin-Tate tower for F. In this way one obtains a geometric link between Deligne-Lusztig theory and the local Langlands correspondence for depth zero supercuspidal representations of GLn(F). This picture can be extended and generalized in two different directions. On the one hand, J. Weinstein (Boston University) found another affinoid in the Lubin- Tate tower for F, whose special fiber can be viewed as an analogue of a Deligne-Lusztig variety for a certain unipotent group over Fq. In joint work with Weinstein we proved that the cohomology of this affinoid realizes the local Langlands and Jacquet-Langlands correspondences for a certain family of positive depth supercuspidal representations of GLn(F).
On the other hand, let G be an arbitrary semisimple group over F and let T be a maximal torus in G. In 1979, Lusztig formulated a cohomological construction of an irreducible supercuspidal representation of G(F) corresponding to a smooth character of T(F) (satisfying a certain nondegeneracy assumption), provided T is unramified and anisotropic. However, until now it has been very difficult to compare this geometric construction with the other constructions that are more commonly used in p-adic representation theory. It turns out that the methods developed for our joint work with Weinstein can also be used to analyze Lusztig's construction. In my talk I will describe the two stories mentioned above and explain the connection between them.