Stability of Elliptic Fargues-Scholze $L$-packets

Abstract

Let $F$ be a non-archimedean local field. Let $\overline{F}$ be an algebraic closure of $F$. Let $G$ be a connected reductive group over $F$. Let $φ$ be an elliptic $L$-parameter. For every irreducible representation $π$ of $G(F)$ with Fargues--Scholze $L$-parameter $φ$, we prove that there exists a finite set of irreducible representations $\{π_i\}_{i \in I}$ containing $π$, such that $π_i$ has Fargues--Scholze $L$-parameter $φ$ for all $i \in I$ and a certain non-zero $\mathbb{Z}$-linear combination $Θ_{π_0}$ of the Harish-Chandra characters of $\{π_i\}_{i \in I}$ is stable under $G(\overline{F})$ conjugation, as a function on the elliptic regular semisimple elements of $G(F)$. Moreover, if $F$ has characteristic zero, $Θ_{π_0}$ is a non-zero stable distribution on $G(F)$.