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The orbit method and analysis of automorphic forms
Acta Mathematica ( IF 3.7 ) Pub Date : 2021-03-30 , DOI: 10.4310/acta.2021.v226.n1.a1
Paul D. Nelson 1 , Akshay Venkatesh 2
Affiliation  

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.

中文翻译:

自守形式的轨道方法与分析

我们沿着微观局部分析的路线,以定量形式开发了轨道方法,并将其应用于自同构形式的分析理论。我们的主要全局应用是任意秩的Gan-Gross-Prasad周期平均值的渐近公式。较大的一组上的自守形态保持不变,而较小的一组上的自守形态在一个大小族中变化,大约是相应的$ L $函数的导体的第四根。Ratner的度量分类结果为证明提供了重要的信息。我们的局部结果包括某些较高阶李群的表示所引起的某些特殊函数的渐近展开,例如像Ichino–Ikeda猜想中由矩阵系数积分定义的相对特征。
更新日期:2021-03-30
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