Elsevier

Advances in Mathematics

Volume 285, 5 November 2015, Pages 1796-1852
Advances in Mathematics

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

https://doi.org/10.1016/j.aim.2015.08.020Get rights and content
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Highlights

  • We study equivariant differential operators between sections of homogeneous bundles over different flag manifolds.

  • It corresponds dually to the study of branching laws for generalized Verma modules of reductive Lie algebras.

  • We formulate a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules.

  • The F-method yields explicit formulae for such operators.

Abstract

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 (2012) [22]), 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 Juhl's conformally invariant differential operators (Juhl (2009) [16]) 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

primary
53A30
secondary
22E47
33C45
58J70

Keywords

F-method
Branching law
Conformal geometry
Parabolic geometry
Equivariant differential operator
Verma module
Symmetric pairs

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