  We consider the problem of defining an action of Hecke operators on the
coherent cohomology of certain integral models of Shimura varieties. We
formulate a general conjecture describing which Hecke operators should act
integrally and solve the conjecture in certain cases. As a consequence, we
obtain $p$-adic estimates of Satake parameters of certain non-regular self dual
automorphic representations of $\mathrm{GL}_n$.
