DIFFERENTIAL OPERATORS AND THE DOUBLING ARCHIMEDEAN ZETA INTEGRALS (Analytic, geometric and $p$-adic aspects of automorphic forms and $L$-functions)

In this survey, we explain Shimura's theory in [Shi90] on the differential operators and the Lie algebra action on the automorphic forms. We also explain how his theory is used in choosing the archimedean sections for constructing p--adic £-functions through the doubling method, and how the corresponding archimedean zeta integrals can be computed