No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

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

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

http://aip.metastore.ingenta.com/content/aip/journal/jmp/18/6/10.1063/1.523383

Article metrics loading...

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

2008-08-26

2016-02-12

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content