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Part I. We prove a one-to-one correspondence between differential symmetrybreaking operators for equivariant vector bundles over two homogeneous spacesand certain homomorphisms for representations of two Lie algebras, inconnection with branching problems of the restriction of representations. Wedevelop a new method (F-method) based on the algebraic Fourier transform forgeneralized Verma modules, which characterizes differential symmetry breakingoperators by means of certain systems of partial differential equations. Incontrast to the setting of real flag varieties, continuous symmetry breakingoperators of Hermitian symmetric spaces are proved to be differential operatorsin the holomorphic setting. In this case symmetry breaking operators arecharacterized by differential equations of second order via the F-method.  Part II. Rankin--Cohen brackets are symmetry breaking operators for thetensor product of two holomorphic discrete series representations of SL(2,R).We address a general problem to find explicit formulae for such intertwiningoperatorsin the setting of multiplicity-free branching laws for reductivesymmetric pairs. We discover explicit formulae of new differential symmetrybreaking operators for all the six different complex geometries arising fromsemisimple symmetric pairs of split rank one, and reveal an intrinsic reasonwhy the coefficients of orthogonal polynomials appear in these operators(Rankin--Cohen type) in the three geometries and why normal derivatives aresymmetry breaking operators in the other three cases. Further, we analyze a newphenomenon that the multiplicities in the branching laws of Verma modules mayjump up at singular parameters.

Differential symmetry breaking operators. I-Genreal theory and F-method. II-Rankin-Cohen Operators for Symmetric Pairs