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Differential symmetry breaking operators: I. General theory and F-method

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Abstract

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.

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Acknowledgments

T. Kobayashi was partially supported by Institut des Hautes Études Scientifiques, France, and Grant-in-Aid for Scientific Research (B) (22340026) and (A) (25247006), Japan Society for the Promotion of Science. Both authors were partially supported by Max Planck Institute for Mathematics (Bonn) where a large part of this work was done.

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Authors and Affiliations

  1. Kavli IPMU and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Toshiyuki Kobayashi

  2. Laboratoire de Mathématiques de Reims, Université de Reims-Champagne-Ardenne, FR 3399 CNRS, 51687, Reims, France

    Michael Pevzner

Authors
  1. Toshiyuki Kobayashi
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  2. Michael Pevzner
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Correspondence to Toshiyuki Kobayashi.

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Kobayashi, T., Pevzner, M. Differential symmetry breaking operators: I. General theory and F-method. Sel. Math. New Ser. 22, 801–845 (2016). https://doi.org/10.1007/s00029-015-0207-9

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