The stack of spherical Langlands parameters

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.