Abstract
The CR geometry is applied to the representation theory of the group $\mathrm{SU}(p, q)$. We prove that the kernel of the CR Yamabe operator on a CR manifold $M$ is a representation of the conformal CR automorphism group of M. So we can construct a representations of $\mathrm{SU}(p, q)$ on the kernel of the CR Yamabe operator on the projective hyperquadric $\Bar{Q}_{p,q}$. This is a complex version of Kobayashi-Orsted's model of the minimal irreducible unitary representation $\varpi _{p,q}$ of $\mathrm{SO}(p, q)$ on $S^{p-1} \times S^{q-1}$.
Citation
Wei Wang. "Representations of ${\rm SU}(p,q)$ and CR geometry I." J. Math. Kyoto Univ. 45 (4) 759 - 780, 2005. https://doi.org/10.1215/kjm/1250281656
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