Rapoport and Zink constructed some rigid analytic analogues of Shimura varieties as moduli spaces of p-divisible groups in a given isogeny class. Subsequently, Kottwitz conjectured that their l-adic cohomology realizes some instances of the local Langlands correspondences. In view of Langlands' functoriality principle, Kottwitz's conjecture implies that only in the case when the given isogeny class is isoclinic (called the basic case) the supercuspidal component of the cohomology of the corresponding Rapoport-Zink spaces is non-zero. Following an analogy with the global theory, Harris proposed a conjectural formula computing the cohomology of non-basic Rapoport-Zink spaces in terms of that of basic spaces which also implies the absence of supercuspidal representations in the non-basic cases. Harris' conjectural formula (which is compatible with Kottwitz's predictions) can be regarded as a geometric realization of Langlands' functoriality.
The goal of this talk is to explain a proof of some cases of Harris' conjecture.