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

Abstract

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