Irreducible components of the moduli space of Langlands parameters

TL;DR

This paper characterizes the irreducible components of the moduli space of Langlands parameters over p-adic fields, analyzes local structures at tamely ramified points, and studies smoothness properties of related moduli stack maps.

Contribution

It determines the irreducible components of the moduli space, describes local structures at specific points, and identifies conditions for smoothness of moduli stack maps involving Levi subgroups.

Findings

Irreducible components of the moduli space are explicitly characterized.

Local structure around tamely ramified points is described in relation to Gelfand--Graev representations.

An open dense set where the moduli stack map is smooth is identified.

Abstract

Let $F/\mathbb{Q}_p$ be finite and let $\mathfrak{X}_G$ be the moduli space of Langlands parameters valued in $G$, in characteristic distinct from $p$. First, we determine the irreducible components of $\mathfrak{X}_G$. Then, we determine the local structure around tamely ramified points for which the image of `tame inertia' is regular. This local structure is related to the endomorphism rings of Gelfand--Graev representations, by work of Li. Lastly, we determine an open dense set in $\mathfrak{X}_M$, when $M$ is a Levi subgroup of $G$, such that the natural map of moduli stacks $[\mathfrak{X}_M/M] \to [\mathfrak{X}_G/G]$ is smooth on this set.