Conformally Covariant Differential Operators for the Diagonal Action of \(O(p,\,q)\) on Real Quadrics


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.

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Correspondence to Jean-Louis Clerc.

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Clerc, J. Conformally Covariant Differential Operators for the Diagonal Action of \(O(p,\,q)\) on Real Quadrics. J Geom Anal 28, 3300–3311 (2018).

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  • Conformally covariant differential operator
  • Projective quadric

Mathematics Subject Classification

  • 43A85
  • 58J70