Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n)

Abstract

We study the space P of all polynomial functions on the complex cone Kn={𝒵= (𝒵1⋅⋅⋅𝒵n) ε Cn, 𝒵2=𝒵21+ ⋅⋅⋅+𝒵2n=0} (n=3,4,⋅⋅⋅). Its subspaces Kl (=Kln) of homogeneous polynomials of degree l (=0,1,2,⋅⋅⋅) provide a convenient realization of the carrier spaces for the symmetric tensor representations of the real orthogonal group SO(n). The multiplication operator 𝒵μ (μ=1,...,n) maps Kl into Kl+1. We define its adjoint as an interior differential operator on Kn which maps Kl+1 into Kl and transforms as an n-vector. We show that the lowest order differential operator with this property is proportional to Dμ= (n/2−1+√∂) ∂μ −(1/2) 𝒵μΔ. We define a scalar product in P with respect to which the operators 𝒵μ and Dμ are Hermitian adjoint to each other and consider the Hilbert space completion Kn of P with respect to this scalar product. The spaces Kn are imbedded for all n (=3,4,⋅⋅⋅) in the Fock type spaces Bn, studied earlier by Bargmann. The space Kn possesses a reproducing kernel that allows us to define a (unique) harmonic extension of every analytic function in Kn. It is shown that the spaces K3 and K4 can be imbedded isometrically in the Hilbert spaces B2 and B4 associated with the representations of SU(2) and SU(2) ×SU(2) [⊇SU(4)].