Stability of elliptic Fargues-Scholze L-packets

The local Langlands correspondence conjecturally partitions the irreducible representations of a p-adic group into the so-called L-packets. Such a partition is conjecturally to be characterized by the stability condition, which is proven in many cases (when a construction of the local Langlands correspondence for certain representations is available) using the theory of endoscopy. In this talk, we will show that for elliptic L-parameters, the construction of Fargues-Scholze satisfies the stability condition. Using a formula of Hansen--Kaletha--Weinstein, we will reduce the problem of stability to showing equi-distribution properties of the weight multiplicities of highest weight representations of an algebraic group. Our proof of equi-distribution properties might be of independent interest.