Excursion operators and the stable Bernstein center

We prove that the Fargues-Scholze construction of elements in the Bernstein center via excursion operators always yields stable distributions. We also prove a strong quantitative compatibility of the Fargues-Scholze construction with transfer across extended pure inner forms. The proofs combine the character formulas from [HKW22], the commutation of Hecke operators with excursion operators, an averaging trick due to Fu [Fu24], and Arthur’s theory of elliptic tempered virtual characters. The arguments work uniformly for all connected reductive groups over p -adic local fields.