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It has a geometric side which is a sum of distributions $J_{\\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the \"spectral side\" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\\mathbb{A})$. 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