Mean square asymptotics and oscillatory integrals for maximal flat submanifolds of locally symmetric spaces

Abstract: Given a compact locally symmetric space of non-compact type, we present a mean square asymptotic for integrals of eigenfunctions along maximal flat submanifolds, constrained to eigenfunctions with suitably generic spectral parameter. This is motivated by questions concerning the maximal size of automorphic periods. The proof uses the pre-trace formula. The analysis of orbital integrals requires knowledge about the geometry of maximal flat submanifolds of the globally symmetric space S. When S is the hyperbolic plane, modeled by the upper half plane, the maximal flat submanifolds are geodesics, and they are lines or half-circles orthogonal to the real axis. The midpoints of the half-circles play a critical role, as do their analogues in higher rank spaces, and one is led to generalize their properties as well as other facts about maximal flat submanifolds.

analysis of PDEsnumber theoryrepresentation theory

Audience: researchers in the topic

Harmonic Analysis and Symmetric Spaces 2021