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