The objective of this project is to investigate various aspects of the relationship between spherical varieties and the Langlands program. It is motivated by recent conjectures relating periods of automorphic forms over spherical subgroups to Euler products of functionals arising from the Plancherel formula of a spherical variety, and expressing the support of Plancherel measure in terms of Arthur parameters. A major part of the project is devoted to creating trace formula-theoretic tools, both locally and globally, necessary to put the conjectures in the correct setting, to clarify some aspects and to prove particular instances of them. Other parts include extending previous harmonic-analytic results and developing a relative trace formula in specific cases in order to analyze the pertinent periods of automorphic forms.
L-functions are very central objects in various branches of number theory, and there are strong conjectures and results (in and around the Langlands program) relating most of the interesting types of L-functions to those L-functions which are called automorphic. Automorphic L-functions are studied by constructions of global harmonic analysis, that is explicit integrals of functions on the quotient of a Lie group by a discrete, arithmetic subgroup, but these constructions remain mysterious after many decades of use. A lot of them involve spherical ("large") subgroups, and a general theory has started to emerge which connects global harmonic analysis to Euler products (and hence L-functions) via local harmonic analysis on spherical varieties. According to this theory, which for now is mostly conjectural, the local Langlands conjecture admits a generalization to the harmonic analysis of a spherical variety over a local field, where a "dual group" is attached to the spherical variety and describes the representations distinguished by it; and spherical periods of automorphic forms, satisfying certain assumptions, are eulerian in a very explicit way and, hence, related to L-functions or special values of those. This project aims at improving these conjectures and investigating ways for their proof, mainly by trace formula-theoretic techniques.
