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 $\left(π_{λ,ε}, \mathcal{H}_{λ,ε}\right)_{ (λ,ε)\in \mathbb {C}\times \{\pm\}}$ be the family of (smooth) representations of $G$ induced from the characters of $P$. For $(λ, ε), (μ, η)\in \mathbb{C} \times \{\pm\}$, a differential operator $\mathbf{D}_{(λ,ε), (μ,η)}^{reg}$ on $X\times X$, acting $G$-covariantly from $\mathcal{H}_{λ,ε} \otimes \mathcal{H}_{μ, η}$ into $\mathcal{H}_{λ+1,-ε} \otimes \mathcal{H}_{μ+1, -η}$ is constructed.