An averaging formula for the cohomology of PEL-type Rapoport--Zink spaces

TL;DR

This paper establishes an averaging formula for the cohomology of PEL-type Rapoport--Zink spaces, extending previous results and connecting with the Langlands correspondence, thereby advancing understanding of these spaces' cohomological structure.

Contribution

It generalizes Shin's cohomology formula to PEL-type Rapoport--Zink spaces and provides a conjectural description of their cohomology using the Langlands correspondence.

Findings

Derived an averaging formula for PEL-type Rapoport--Zink spaces.

Connected Igusa variety cohomology with the Langlands correspondence.

Extended previous EL-type results to more general PEL-type spaces.

Abstract

We prove under certain assumptions a formula for the cohomology of PEL-type Rapoport--Zink spaces that "averages" over the Kottwitz set. Our formula generalizes that of Shin's beyond the EL-type case and is proven by combining Mantovan's formula with descriptions of the cohomology of Shimura and Igusa varieties. We then use this averaging formula to derive a conjectural description of the cohomology of Rapoport--Zink spaces, generalizing our earlier work for EL-type spaces. Along the way, we give a description of the cohomology of Igusa varieties in terms of the Langlands correspondence, generalizing work in Shin's thesis.