The stack of spherical Langlands parameters

TL;DR

This paper introduces a new geometric object called the stack of spherical Langlands parameters for reductive groups over nonarchimedean fields, generalizing unramified parameters and applying it to prove Eichler-Shimura relations for Hodge type Shimura varieties.

Contribution

It defines the stack of spherical Langlands parameters using inertia-invariants, extending the unramified case, and applies this to derive congruence relations without ramification restrictions.

Findings

Defined the stack of spherical Langlands parameters for reductive groups.

Generalized unramified Langlands parameters to the ramified case.

Proved Eichler--Shimura congruence relations for Hodge type Shimura varieties.

Abstract

For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.