The most famous examples of covariant bi-differential operators are the Rankin-Cohen operators. They appear in the theory of automorphic forms as constant coefficients bilinear differential operators on the upper half-plane in $\mathbb C$. They are covariant with respect to the holomorphic discrete series of $SL(2,\mathbb R)$.

In this talk I will report on joint works with J.-L. Clerc and K. Koufany where we address the general problem of constructing such intertwining operators in the setting of the degenerate principle series of representations of conformal groups (say $G$) of real simple Jordan algebras. In the case where $G = SL(2,\mathbb R)$, the degenerate principle series are indexed by pairs $(\lambda, \epsilon) \in \mathbb C \times \{\pm\}$. Given two such representations $\pi_{\lambda,\epsilon}$ and $\pi_{\mu,\eta}$, and a positive integer $k$, there exists a family of constant coefficients bi-differential operators on $\mathbb R$ which are covariant with respect to $(\pi_{\lambda,\epsilon} \otimes \pi_{\mu,\eta}, \pi_{\lambda+\mu+2k,\epsilon\eta})$. For special values of the parameters $\lambda$ and $\mu$, they coincide with the Rankin-Cohen operators. Our approach uses Bernstein-Sato type identities and Zeta functional equations for real simple Jordan algebras. We close the talk by mentioning the case of bi-differential operators on differential forms and $G = SO(1, n + 1)$.