The orbit method and analysis of automorphic forms
Recent developments in Analytic Number Theory May 01, 2017 - May 05, 2017
Location: MSRI: Simons Auditorium
Automorphic forms
Kirillov orbit model
Gross-Prasad families
Ratner's Theorem
spherical harmonics
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11F22 - Relationship to Lie algebras and finite simple groups
11F66 - Langlands $L$-functions; one variable Dirichlet series and functional equations
Venkatesh
In the analytic theory of automorphic forms, especially in higher rank, one encounters complicated integrals over Lie groups which must be either evaluated or estimated. I will discuss how Kirillov's orbit method allows one to do this, at least heuristically. These ideas can often be made rigorous; I will apply it to evaluate the average value of L-functions over certain (Gross-Prasad) families, in any rank. This evaluation also uses Ratner's theorem on measure rigidity. Joint work with Paul Nelson.
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Venkatesh
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