  In this paper, we define and study the arithmetic of the ring of
$\mathbb{U}$-operators for reductive $p$-adic groups. These operators
generalise the notion of "successor" operators for trees with a marked end. We
show that they are integral over the spherical Hecke algebra. This integrality
intervenes crucially in the construction of Euler systems obtained from special
cycles of general Shimura varieties and the generalization of the famous
Eichler-Shimura relation.
