Irreducible components of the moduli space of Langlands parameters

Let $F\mathrm{/}{\mathbb{Q}}_p$ 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 $[{\mathfrak{X}}_M\mathrm{/}M]\to [{\mathfrak{X}}_G\mathrm{/}G]$ is smooth on this set.