Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Abstract

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of $$SL(2,\mathbb {R})$$ . 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 Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9 ) 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.