  Let $\gamma$ be an automorphism of a polarized complex projective manifold
$(M,L)$. Then $\gamma$ induces an automorphism $\gamma_k$ of the space of
global holomorphic sections of the $k$-th tensor power of $L$, for every
$k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the
trace of $\gamma_k$ in terms of fixed point data. More generally, one may
consider the composition of $\gamma_k$ with the Toeplitz operator associated to
some smooth function on $M$. Still more generally, in the presence of the
compatible action of a compact and connected Lie group preserving
$(M,L,\gamma)$, one may consider induced linear maps on the equivariant
summands associated to the irreducible representations of $G$. In this paper,
under familiar assumptions in the theory of symplectic reductions, we show that
the traces of these maps admit an asymptotic expansion as $k\to +\infty$, and
compute its leading term.
