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 $\phi$ be an elliptic $L$-parameter. For every irreducible representation $\pi$ of $G(F)$ with Fargues--Scholze $L$-parameter $\phi$, 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 $\phi$ 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 $G(F)$.

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