The Cohomology of Unramified Rapoport-Zink Spaces of EL-type and Harris's Conjecture

TL;DR

This paper investigates the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type, proposing a conjectural formula for certain representation morphisms and proving it for a class of representations, advancing understanding in the local Langlands program.

Contribution

It provides a conjectural formula for the morphisms $ ext{Mant}_{b, ta}$ for all representations and proves it for essentially square integrable cases, extending previous work.

Findings

Conjectural formula for $ ext{Mant}_{b, ta}( ho)$ for all $ ho$

Proof of the formula for essentially square integrable $ ho$

Agreement with Harris's conjecture for parabolic inductions

Abstract

We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, \mu}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, \mu}(\rho)$ for $\rho$ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm{Mant}_{b, \mu}(\rho)$ for all $\rho$ and prove it when $\rho$ is essentially square integrable. Our proof works for general $\rho$ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.