Irreducible Components of the Moduli Space of Langlands Parameters

Let $F/\mathbb{Q}_{p}$ be finite, $G$ be an $L$-group, and let $\mathfrak{X}_{G}$ be the moduli space of Langlands parameters $W_{F} \to G$, 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 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 $[{\mathfrak{X}}_{M}/M^{\circ }] \to [{\mathfrak{X}}_{G}/G^{\circ }]$ is smooth on this set.