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

We initiate a new study of differential 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, http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan--H\"older 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 differential operators [Juhl, http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and its generalizations. This article is the first 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

An extension problem related to the fractional Branson-Gover operators

The Branson-Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson-Gover operators. For Euclidean spaces we show that the fractional Branson-Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli-Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of its properties.Comment: 25 page