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Let $\gamma$ be an automorphism of a polarized complex projective manifold$(M,L)$. Then $\gamma$ induces an automorphism $\gamma_k$ of the space ofglobal 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 thetrace of $\gamma_k$ in terms of fixed point data. More generally, one mayconsider the composition of $\gamma_k$ with the Toeplitz operator associated tosome smooth function on $M$. Still more generally, in the presence of thecompatible action of a compact and connected Lie group preserving$(M,L,\gamma)$, one may consider induced linear maps on the equivariantsummands associated to the irreducible representations of $G$. In this paper,under familiar assumptions in the theory of symplectic reductions, we show thatthe traces of these maps admit an asymptotic expansion as $k\to +\infty$, andcompute its leading term.

Szegö kernels, Toeplitz operators, and equivariant fixed point formulae