Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ 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 $$(\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}$$ 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 $$\mathbf D_{(\mu ,\eta )}^\mathrm{reg}$$ 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.