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We define Hecke correspondences and Hecke operators on unitary RZ spaces andstudy their basic geometric properties, including a commutativity conjecture onHecke operators. Then we formulate the Arithmetic Fundamental Lemma conjecturefor the spherical Hecke algebra. We also formulate a conjecture on theabundance of spherical Hecke functions with identically vanishing firstderivative of orbital integrals. We prove these conjectures for the case$U(1)\times U(2)$.

Arithmetic Fundamental Lemma for the spherical Hecke algebra