The stack of spherical Langlands parameters

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.