Connected components of the moduli space of L-parameters

Recently, in order to formulate a categorical version of the local Langlands correspondence, several authors have constructed moduli spaces of Z[1/p]-valued L-parameters for p-adic groups. The connected components of these spaces over various Z[1/p]-algebras R are conjecturally related to blocks in categories of R-representations of p-adic groups. Dat-Helm-Kurinczuk-Moss described the components when R is an algebraically closed field and gave a conjectural description when R = Z̄[1/p]. In this paper, we prove a strong form of this conjecture applicable to any integral domain R over Z̄[1/p].