Excursion operators and the stable Bernstein center

Key Points

• The aim is to verify that the Fargues-Scholze construction yields stable distributions through excursion operators.
• Proved stability of distributions from the Fargues-Scholze construction.
• Establish compatibility with transfer across extended pure inner forms.
• Utilized character formulas and commutation properties of Hecke and excursion operators.
• Applied an averaging technique from Fu and Arthur’s theory.
• Confirmed the Fargues-Scholze construction always results in stable distributions.
• Demonstrated strong quantitative compatibility with transfer methods.
• Showed uniform applicability for all connected reductive groups over p-adic local fields.