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

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.

Differential symmetry breaking operators. II. Rankin--Cohen operators for symmetric pairs

Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae, for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs.
For this purpose we use a new method (F-method) developed in the first part of the series and based on the algebraic Fourier transform for generalized Verma modules.
The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order.
We discover explicit formulae, of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

Global uniqueness of small representations

We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.