Moduli of Langlands Parameters
Moduli of Langlands Parameters
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发表时间：
2020-09
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通讯作者：
Jean-François Dat;David Helm;R. Kurinczuk;Gilbert Moss
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作者：
Jean-François Dat;David Helm;R. Kurinczuk;Gilbert Moss
Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split connected reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct a moduli space of Langlands parameters $W_F \to {^LG}$, and show that it is locally of finite type and flat over $\mathbb{Z}[1/p]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components of this space over algebraically closed fields of characteristic zero and characteristic $\ell\neq p$, as well as of the components over $\overline{\mathbb{Z}}_{\ell}$ and (conjecturally) over $\overline{\mathbb{Z}}[1/p]$. Finally we study the functions on this space that are invariant under conjugation by $\hat{G}$ (or, equivalently, the GIT quotient by $\hat{G}$) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on $^LG$.
Let $F$ be a nonarchimedean local field of residue characteristic $p$, let $\hat{G}$ be a split connected reductive group over $\mathbb{Z}[1/p]$ with an action of $W_F$, and let $^LG$ denote the semidirect product $\hat{G}\rtimes W_F$. We construct a moduli space of Langlands parameters $W_F \to {^LG}$, and show that it is locally of finite type and flat over $\mathbb{Z}[1/p]$, and that it is a reduced local complete intersection. We give parameterizations of the connected components of this space over algebraically closed fields of characteristic zero and characteristic $\ell\neq p$, as well as of the components over $\overline{\mathbb{Z}}_{\ell}$ and (conjecturally) over $\overline{\mathbb{Z}}[1/p]$. Finally we study the functions on this space that are invariant under conjugation by $\hat{G}$ (or, equivalently, the GIT quotient by $\hat{G}$) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on $^LG$.