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

TL;DR

This paper constructs conformally covariant differential operators on real quadrics associated with the orthogonal group O(p, q), advancing the understanding of symmetry-invariant operators in geometric representation theory.

Contribution

It introduces a family of explicit differential operators acting covariantly on representations induced from parabolic subgroups of O(p, q).

Findings

Explicit construction of covariant differential operators on real quadrics.

Operators act between tensor products of induced representations.

Enhances understanding of symmetry-invariant differential operators in geometric analysis.

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.