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We consider the problem of defining an action of Hecke operators on thecoherent cohomology of certain integral models of Shimura varieties. Weformulate a general conjecture describing which Hecke operators should actintegrally and solve the conjecture in certain cases. As a consequence, weobtain $p$-adic estimates of Satake parameters of certain non-regular self dualautomorphic representations of $\mathrm{GL}_n$.

Hecke operators and the coherent cohomology of Shimura varieties