In crystal field theory, I would like to know how to calculate the basis function of a sub group of O(3) in fucntion of the initial spherical harmonics basis. 

For instance, in the Figure 7.2 example (d-state splitting in a crystal field), it is possible to express the basis of the B2g representation of the D4h symmetry, in function of the T2g representation of the Oh symmetry thanks to the GTClebschGordanTable. But how to do the same for Oh and O(3) so: how to express the basis functions of the T2g representation of the Oh symmetry in function of the basis (that are the spherical harmonics of J=2) of the D2 representation of O(3)?

I did not find an easy way even using  GTAngularMomentumChars or GTWignerProjectionOperator ??

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daniel comparat Selected answer as best 27. June 2022