Irreducible components of the moduli space of Langlands parameters

Abstract

Let F/Qp be finite and let X G be the moduli space of Langlands parameters valued in G, in characteristic distinct from p. First, we determine the irreducible components of 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 X M , when M is a Levi subgroup of G, such that the natural map of moduli stacks [X M /M ] → [X G /G] is smooth on this set.