Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces

TL;DR

This paper proves the Harris-Viehmann conjecture for a class of Rapoport-Zink spaces that are Hodge type and Hodge-Newton reducible, confirming their cohomology is parabolically induced, thus advancing understanding of local Shimura varieties and the local Langlands correspondence.

Contribution

It verifies the Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces by embedding them into EL type spaces where the conjecture is known.

Findings

Confirmed the conjecture for Hodge-Newton reducible spaces

Embedded spaces into EL type to leverage existing results

Supported the geometric realization of the local Langlands correspondence

Abstract

Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan.