Connected components of the moduli space of L-parameters

TL;DR

This paper proves a conjecture relating the connected components of moduli spaces of L-parameters to representation blocks for p-adic groups, extending previous results to any integral domain over Z[1/p].

Contribution

It establishes a strong form of the conjecture connecting moduli space components to representation blocks for any integral domain over Z[1/p], broadening prior work.

Findings

Proves a strong form of the conjecture for all integral domains over Z[1/p]

Extends previous results from algebraically closed fields to more general rings

Provides a unified description of connected components of moduli spaces

Abstract

Recently, in order to formulate a categorical version of the local Langlands correspondence, several authors have constructed moduli spaces of $\mathbf{Z}[1/p]$-valued L-parameters for $p$-adic groups. The connected components of these spaces over various $\mathbf{Z}[1/p]$-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 $R = \overline{\mathbf{Z}}[1/p]$. In this paper, we prove a strong form of this conjecture applicable to any integral domain $R$ over $\overline{\mathbf{Z}}[1/p]$.