Representations of SU(p, q) and CR geometry I

Abstract

The CR geometry is applied to the representation theory of the group 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 SU(p, q) on the kernel of the CR Yamabe operator on the projective hyperquadric Q̄p,q. This is a complex version of Kobayashi-Orsted's model of the minimal irreducible unitary representation Wp,q of SO(p, q) on Sp-1 × Sq-1.