  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.
