Irreducible components of the moduli space of Langlands parameters

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.