Non-abelian Lubin-Tate Theory Modulo $\ell$

Let p and l be two distinct prime numbers, and fix a positive integer d . I will explain how the F_l-cohomology complex of the Lubin-Tate tower of height d of a p-adic field K realizes mod l versions of both the semi-simple Langlands correspondence for GL_d(K) and the "Langlands-Jacquet" transfer from GL_d(K) to the central division K-algebra of invariant 1/d . Then I will give an explicit description of the supercuspidal part of the integral l-adic cohomology of this LT tower in terms of certain universal deformations. Finally, I will speculate about how to get a cohomological realization of the full Langlands correspondence mod l, including the mysterious nilpotent part of this correspondence. My current attempt involves a certain equivariant Lefschetz operator on the cohomology complex.