We give a criterion to determine when the cycle class of a locally symmetric subvariety \(\) of a compact locally symmetric variety \(\) generates a non-trivial module under the action of Hecke operators, and give several examples where this criterion is satisfied. We also exhibit examples of subvarieties \(\) which do generate the trivial module under the action of Hecke operators. We show that all Hodge classes (in degree \(\)) on the locally symmetric variety \(\) associated to certain arithmetric subgroups Γ of \(\) are algebraic (provided that \(\)).
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