Szego kernels, Toeplitz operators, and equivariant fixed point formulae

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 $k=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$; 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 $k\to +\mathrm{\infty }$, and compute its leading term.