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We initiate a new study of differential operators with symmetries and combinethis with the study of branching laws for Verma modules of reductive Liealgebras. By the criterion for discretely decomposable and multiplicity-freerestrictions of generalized Verma modules [T. Kobayashi,http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we arebrought to natural settings of parabolic geometries for which there existunique equivariant differential operators to submanifolds. Then we apply a newmethod (F-method) relying on the Fourier transform to find singular vectors ingeneralized Verma modules, which significantly simplifies and generalizes manypreceding works. In certain cases, it also determines the Jordan--H\"olderseries of the restriction for singular parameters. The F-method yields anexplicit formula of such unique operators, for example, giving an intrinsic andnew proof of Juhl's conformally invariant differential operators [Juhl,http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and itsgeneralizations. This article is the first in the series, and the next onesinclude their extension to curved cases together with more applications of theF-method to various settings in parabolic geometries.

Branching laws for Verma modules and applications in parabolic geometry. I