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Branching laws for Verma modules and applications in parabolic geometry. I. (English) Zbl 1327.53044
Summary: 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, Transform. Groups 17, No. 2, 523–546 (2012; Zbl 1257.22014)], 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-Hölder 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 A. Juhl’s conformally invariant differential operators [Families of conformally covariant differential operators, Q-curvature and holography. Basel: Birkhäuser (2009; Zbl 1177.53001)] and its generalizations to spinor bundles. 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.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A30 Conformal differential geometry (MSC2010)
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
58J70 Invariance and symmetry properties for PDEs on manifolds
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