Rankin-Cohen bidifferential operators are the projectors onto irreducible summands in the decomposition of the tensor product of two representations of SL(2,R). We consider the general problem to find explicit formulæ for such projectors in the setting of multiplicity-free branching laws for reductive symmetric pairs.
For this purpose we develop a new method (F-method) based on an algebraic Fourier transform for generalized Verma modules, which enables us to characterize those projectors by means of certain systems of partial differential equations of second order.
As an application of the F-method we give a solution to the initial problem by constructing equivariant holomorphic differential operators explicitly for all symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of Jacobi polynomials appear in these operators including the classical Rankin-Cohen brackets as a special case.
© Toshiyuki Kobayashi