We prove a comparison isomorphism between certain moduli spaces of $p$-divisible groups and strict ${\mathcal{O}}_{K}$-modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized $p$-divisible groups and polarized strict ${\mathcal{O}}_{K}$-modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.