Branching laws for Verma modules and applications in parabolic geometry

We initiate a new study of dierential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, Transf. Groups (2012)], we are brought to natural settings of parabolic geome-tries for which there exist unique equivariant dierential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to nd singular vectors in generalized Verma modules, which signicantly simplies and generalizes many preceding works. In certain cases, it also determines the JordanHölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant dierential operators [Juhl, Progr. Math. 2009] and its generalizations to spinor bundles. This article is the rst in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries