For a connected reductive group G over a non-archimedean local field and an elliptic Langlands parameter phi, this work proves a stability property for the packet of irreducible smooth representations whose Fargues-Scholze L-parameter is phi. It shows that any such representation lies in a finite subset whose members all have parameter phi and for which a suitable non-zero integer linear combination of their Harish-Chandra characters is invariant under G(F)-conjugation on elliptic regular semisimple elements; in characteristic zero this combination is a non-zero stable distribution. The result advances the expectation that Fargues-Scholze L-packets behave like classical (endoscopic) L-packets.
This paper studies stability for L-packets defined through the Fargues-Scholze local Langlands correspondence. Let F be a non-archimedean local field and G a connected reductive group over F, and let…