Asymptotics of unitary matrix elements in canonical bases

We compute the asymptotics of matrix elements in canonical bases of irreducible representations of the unitary group as the highest weight goes to infinity, in terms of the symplectic geometry of the associated coadjoint orbit. This uses tools of Berezin-Toeplitz quantization, and recovers as a special case the asymptotics of Wigner's d-matrix elements for the spin representations in quantum mechanics.