Moduli of Langlands parameters

Abstract

Let $F$ be a non-archimedean local field of residue characteristic $p$, let $\hat{G}$ be a split reductive group scheme over $\mathbb{Z}[\frac{1}{p}]$ with an action of $W_F$, and let ${}^{L}G$ denote the semidirect product $\hat{G}⋊W_F$. We construct a moduli space of Langlands parameters $W_F\to {}^{L}G$, and show that it is locally of finite type and flat over $\mathbb{Z}[\frac{1}{p}]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components and the irreducible components of the geometric fibers of this space, and parameterizations of the connected components of the total space over $\overline{\mathbb{Z}}[\frac{1}{p}]$ (under mild hypotheses) and over ${\overline{\mathbb{Z}}}_{\ell }$ for $\ell \ne p$. In each case, we show precisely how each connected component identifies with the “principal” connected component attached to a smaller split reductive group scheme. Finally, we study the GIT quotient of this space by $\hat{G}$ and give a description of its fibers up to homeomorphism, and a complete description of its ring of functions after inverting an explicit finite set of primes depending only on ${}^{L}G$.