Local and Global Problems in the Relative Langlands Program

This award concerns mathematical objects called reductive groups which are special kinds of topological groups characterized by abundant symmetries. These symmetries serve as key insights into understanding the intrinsic structures of objects in our universe. The study of reductive groups dates back to the late 19th century. Two crucial areas of this field are the representation theory of reductive groups and automorphic forms on reductive groups, which are specialized functions with additional symmetry on reductive groups. These two areas also have many connections to various other disciplines, including physics and computer science. This project aims to explore the restriction of representations of reductive groups to a spherical subgroup and to investigate the period integrals of automorphic forms. In the meantime, the PI will continue advising his current undergraduate and graduate students, as well as any potential students interested in studying the Langlands program. He will hold weekly meetings with them and assign suitable thesis problems. He will also continue organizing seminars and conferences in this area. Additionally, he will maintain his outreach efforts in K-12 education by mentoring high school students and coaching local kids in the Newark area for math competitions, among other activities. To be specific, the primary objective in the local theory is to use the trace formula method to study the multiplicity problem for spherical varieties. In recent years, the PI and his collaborators have examined the local multiplicity for some spherical varieties and have proposed a conjectural multiplicity formula for all spherical varieties. Additionally, they have formulated an epsilon dichotomy conjecture for all strongly tempered spherical varieties. The PI intends to prove these conjectures and investigate further structures and properties related to multiplicity. Additionally, the PI plans to study the multiplicity for varieties that are not necessarily spherical, as well as the relations between distribution characters and orbital integrals. In the global theory, the PI intends to use the relative trace formula method and some beyond endoscopic type comparison method to study various relations between period integrals and automorphic L-functions (in particular proving the Ichino-Ikeda type formula for period integrals in some cases). Moreover, Ben-Zvi—Sakellaridis—Venkatesh have recently developed a beautiful theory of relative Langlangs duality. The PI intends to use this theory to explain all the existing automorphic integrals and to explore some new integrals. The PI also hopes to extend the theory of relative Langlands duality beyond the current spherical setting. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.