Conformally covariant differential operators for the diagonal action of O(p, q) on real quadrics

Let $X=G\mathrm{/}P$ be a real projective quadric, where $G=O(p,q)$ and $P$ is a parabolic subgroup of $G$. Let $\left(\pi_{\lambda,\epsilon}, $\mathcal{H}${\lambda,\epsilon}\right){ (\lambda,\epsilon)\in \mathbb {C}\times {\pm}}$bethefamilyof(smooth)representationsof$G$ induced from the characters of $P$. For $(\lambda, \epsilon), (\mu, \eta)\in $\mathbb{C}$ \times {\pm}$,adifferentialoperator$$\mathbf{D}${(\lambda,\epsilon), (\mu,\eta)}^{reg}$on$X\times X$,acting$G$-covariantlyfrom$$\mathcal{H}${\lambda,\epsilon} \otimes $\mathcal{H}${\mu, \eta}$into$$\mathcal{H}${\lambda+1,-\epsilon} \otimes $\mathcal{H}$_{\mu+1, -\eta}$ is constructed.