  Consider a compact K\"ahler manifold endowed with a prequantum bundle.
Following the geometric quantization scheme, the associated quantum spaces are
the spaces of holomorphic sections of the tensor powers of the prequantum
bundle. In this paper we construct an asymptotic representation of the
prequantum bundle automorphism group in these quantum spaces. We estimate the
characters of these representations under some transversality assumption. The
formula obtained generalizes in some sense the Lefschetz fixed point formula
for the automorphisms of the prequantum bundle preserving its holomorphic
structure. Our results will be applied in two forthcoming papers to the quantum
representation of the mapping class group.
