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s . Further, let h $^{∗}$( s ) = l ( s ) + α (1 + a ( s )) and h$_{∗}$ ( s ) = l ( s ) + 1 + αa ( s ) . Finally, let c ( κ, α ) = ∏ s ∈ κ h$_{∗}$ ( s ), c $^{′}$( κ, α ) = ∏ s ∈ κ h $^{∗}$( s ), and w ( κ, α ) = c ( κ, α ) c $^{′}$( κ, α ). We will need the following lemmata. 

Lemma A1 . For the torus scalar product of C ( α ) κ with itself, we have 

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Proof: Macdonald (1995, Chapter VI, § 10) establishes the orthogonality of “ P ” normalizations of Jack polynomials, P ( α ) κ , with respect to the torus scalar product. His formula (10.37) gives an explicit expression (up to a constant that can be evaluated using (10.38)) for 〈 P ( α ) κ , Q ( α ) κ 〉 α , where Q ( α ) κ = c ( κ,α ) c ′ ( κ,α $_{)}$P ( α ) κ (see (10.16)). On the other hand, 

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(see, for example, Table 6 of Dumitriu et al, 2007). Substituting this expression in Macdonald’s formulae, we get (34). □ 

