Representations of ${\rm SU}(p,q)$ and CR geometry I

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 $S^{p-1} \times S^{q-1}$.