  We establish various results on the large level limit of projective quantum
representations of surface mapping class groups obtained by quantizing moduli
spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved
that these projective representations lift to asymptotic representations. We
show that the operators in these representations are Fourier integral operators
and determine explicitly their canonical relations and symbols. We deduce from
these facts the Egorov property and the asymptotic unitarity, two results
already proved by J.E. Andersen. Furthermore we show under a transversality
assumption that the characters of these representations have an asymptotic
expansion. The leading order term of this expansion agrees with the formula
derived heuristically by E. Witten in "Quantum field theory and the Jones
polynomial".
