UNITARIZATION OF A SINGULAR REPRESENTATION OF SO(P,Q)

A geometric construction of a certain singular unitary representation of SO(e)(p,q), with p + q even is given. The representation is realized geometrically as the kernel of a SO(e)(p,q)-invariant operator on a space of sections over a homogeneous space for SO(e)(p,q). The K-structure of these representations is elucidated and we demonstrate their unitarity by explicitly writing down an SD(p,q)-invariant positive definite hermitian form. Finally, we demonstrate that the annihilator in U[g] of this representation is the Joseph ideal, which is the maximal primitive ideal associated with the minimal coadjoint orbit.