This is joint work with Tasho Kaletha.  The local Langlands correspondence predicts that representations of a reductive group G over a p-adic field are related to Galois representations into the Langlands dual of G.  A conjecture of Kottwitz (as generalized by Rapoport and Viehmann) asserts that this relationship appears in a precise way in the cohomology of "local Shimura varieties", which were shown to exist by Scholze.  We don't know how Galois acts on this cohomology yet, but we can verify much of the rest of the conjecture, in a large degree of generality, using a Lefschetz-Verdier fixed point formula.