Arithmetic fundamental lemma for the spherical Hecke algebra

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 \(\textrm{U} (1)\times \textrm{U} (2)\) $\text{U}(1)\times \text{U}(2)$ .