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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 polynomialfunctions on the complex cone K_{ n }={Z= (Z_{1}⋅⋅⋅Z_{ n }) ε C^{ n }, Z^{2}=Z^{2} _{1}+ ⋅⋅⋅+Z^{2} _{ n }=0} (*n*=3,4,⋅⋅⋅). Its subspaces*K* ^{ l } (=*K* ^{ l } _{ n }) 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 Z_{μ} (μ=1,...,*n*) maps *K* ^{ l } into *K* ^{ l+1}. We define its adjoint as an interior differential operator on K_{ n } which maps *K* ^{ l+1} into *K* ^{ l } and transforms as an *n*‐vector. We show that the lowest order differential operator with this property is proportional to *D* _{μ}= (*n*/2−1+√∂) ∂_{μ} −(1/2) Z_{μ}Δ. We define a scalar product in *P* with respect to which the operators Z_{μ} and *D* _{μ} are Hermitian adjoint to each other and consider the Hilbert space completion *K* _{ n } of *P* with respect to this scalar product. The spaces *K* _{ n } are imbedded for all *n* (=3,4,⋅⋅⋅) in the Fock type spaces *B* _{ n }, studied earlier by Bargmann. The space *K* _{ n } possesses a reproducing kernel that allows us to define a (unique) harmonic extension of every analytic function in *K* _{ n }. It is shown that the spaces *K* _{3} and *K* _{4} can be imbedded isometrically in the Hilbert spaces*B* _{2} and *B* _{4} associated with the representations of SU(2) and SU(2) ×SU(2) [⊇SU(4)].

© 1977 American Institute of Physics

Published online 26 August 2008

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/content/aip/journal/jmp/18/6/10.1063/1.523383

2008-08-26

2016-10-23

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