On the cohomology of Rapoport-Zink spaces of EL-type

Abstract

In this article, we are interested in determining the $l$-adic cohomology of Rapoport-Zink spaces associated to ${\rm GL}_n$ over an unramified extension of ${\Bbb Q}_p$ (called unramified ``EL-type'') in connection with the local Langlands correspondence for ${\rm GL}_n$. In fact, we compute (the alternating sum of) certain representation-theoretic functors defined in terms of their cohomology on the level of Grothendieck groups. In case Rapoport-Zink spaces parametrize $p$-divisible groups of dimension one, the above alternating sum is determined by Harris and Taylor (without the unramifiedness assumption). For $p$-divisible groups of ${\rm dim}>1$, Fargues has obtained an answer for the supercuspidal part when the $p$-divisible groups are basic (i.e., their associated Newton polygons are straight lines). Our main result is an identity among the cohomology of Rapoport-Zink spaces that appear in the Newton stratification of the same unitary Shimura variety. Our result implies a theorem of Fargues, providing a second proof, and reveals new information about the cohomology of Rapoport-Zink spaces for non-basic $p$-divisible groups. We propose an inductive procedure to completely determine the above alternating sum, assuming a strengthening of our main result along with a conjecture of Harris.