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 $\varphi$ be an elliptic $L$-parameter. For every irreducible representation $\pi$ of $G(F)$ with Fargues--Scholze $L$-parameter $\varphi$, we prove that there exists a finite set of irreducible representations $\{\pi_i\}_{i \in I}$ containing $\pi$, such that $\pi_i$ has Fargues--Scholze $L$-parameter $\varphi$ for all $i \in I$ and a certain non-zero $\mathbb{Z}$-linear combination $\Theta_{\pi_0}$ of the Harish-Chandra characters of $\{\pi_i\}_{i \in I}$ 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, $\Theta_{\pi_0}$ is a non-zero stable distribution on $G(F)$.