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Differential symmetry breaking operators. I: General theory and F-method. (English) Zbl 1338.22006
Let $$W\to Y$$ and $$V\to X$$ be a pair of vector bundles with a smooth map $$p:Y\to X$$. Let, further, $$G'\subset G$$ be a pair of Lie groups such that $$G'$$ acts equivariantly on $$W\to Y$$, $$G$$ acts equivariantly on $$V\to X$$, and $$p$$ is $$G'$$-equivariant. The main result of this paper establishes a duality between $$G'$$-intertwining differential operators (called symmetry breaking operators) and certain homomorphisms between induced representations of the corresponding Lie algebras. The authors develop a new method for characterizing symmetry breaking operators by means of systems of partial differential equations based on algebraic Fourier transforms for generalized Verma modules. Continuous symmetry breaking operators for Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting and are characterized by second order differential equations.

##### MSC:
 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 22E46 Semisimple Lie groups and their representations 11F55 Other groups and their modular and automorphic forms (several variables) 53C10 $$G$$-structures
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