Arithmetic fundamental lemma for the spherical Hecke algebra

Source abstract

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