On Certain Small Representations of Indefinite Orthogonal Groups

For any n ∈ N such that 2n ≤ min(p, q), we construct a representation πn of O(p, q) with p+q even as the kernel of a commuting set of n(n+1) 2 number of O(p, q)-invariant differential operators in the space of C∞ functions on an isotropic cone with a distinguished GLn(R)-homogeneity degree. By identifying πn with a certain representation constructed via the formalism of the theta correspondence, we show (except when p = q = 2n) that the space of K-finite vectors of πn is the (g, K)-module of an irreducible unitary representation of O(p, q) with Gelfand-Kirillov dimension n(p + q − 2n − 1). Our construction generalizes the work of Binegar and Zierau (Unitarization of a singular representation of SOe(p, q), Commun. Math. Phys. 138 (1991), 245–258) for n = 1. 1. Construction of the representation πn Let V = R ' R ⊕ R be the real vector space equipped with the quadratic form