Periods and harmonic analysis on spherical varieties/ Yiannis Sakellaridis & Akshay Venkatesh

Includes bibliography

1. Introduction -- Part I. The dual group of a spherical variety -- 2. Review of spherical varieties -- 3. Proofs of the results on the dual group -- Part II. Local theory and the Ichino-Ikeda conjecture -- 4. Geometry over a local field -- 5. Asymptotics -- 6. Strongly tempered varieties -- Part III. Spectral decomposition and scattering theory -- 7. Results -- 8. Two toy models: the global picture and semi-infinite matrices -- 9. The discrete spectrum -- 10. Preliminiaries to the Bernstein morphisms: 'linear algebra" -- 11. The Bernstein morphisms -- 12. Preliminaries to scattering (I): direct integrals and norms -- 13. Preliminaries to scattering (II): consequesnce of the conjecture on discrete series -- 14. Scattering theory -- 15. Explicit Plancherel formula -- Part IV. Conjectures -- 16. The local X-distinguished spectrum -- 17. Speculation on a global period formula -- 18. Examples -- A. Prime rank one spherical varieties -- Bibliography

This volume elaborates the idea that harmonic analysis on a spherical variety X is intimately connected to the Langlands program. In the local setting, the key conjecture is that the spectral decomposition of L2(X) is controlled by a dual group attached to X. Guided by this, the authors develop a Plancherel formula for L2(X), formulated in terms of simpler spherical varieties which model the geometry of X at infinity. This local study is then related to global conjectures—namely, conjectures about period integrals of automorphic forms over spherical subgroups.