We show that the Hecke rings act on the $l$-adic cohomology groups of suitable non-singular projective toroidal compactifications of the higher dimensional modular varieties. We extend the fixed point theory of Lefschetz to the correspondences for the Hecke rings on those compactifications. We treat here the Siegel modular case.
"Correspondences for Hecke Rings and (Co-)Homology Groups on Smooth Compactifications of Siegel Modular Varieties." Tokyo J. Math. 13 (1) 37 - 62, June 1990. https://doi.org/10.3836/tjm/1270133003