The orbit method and analysis of automorphic forms

Abstract

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms.Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner's results on measure classification provide an important input to the proof.Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.

The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

“…dilated) or high center often exclude these narrow classes and thus, does not usually become fruitful to yield subconvex bound of an L-function which has conductor drop, see e.g. [2,26,25].…”

“…In general, for a pair of groups H ≤ G and their representations π and Π respectively, it is an interesting question to asymptotically evaluate moments of the central L-values of the Rankin-Selberg (if defined) product Π ⊗ π. Previously, in [26] Nelson-Venkatesh asymptotically evaluated the first moment keeping Π fixed and letting π vary over a dilated Plancherel ball when (G, H) are Gan-Gross-Prasad pairs, more interestingly, allowing arbitrary weights in the spectral side. More recently in [25] Nelson proved a Lindelöf-consistent upper bound of the second moment for the groups (G, H) = (U(n + 1), U(n)) in the split case keeping π fixed and letting Π vary over a Plancherel ball with high center.…”