  We prove a character formula for some closed fine Deligne-Lusztig varieties.
We apply it to compute fixed points for fine Deligne-Lusztig varieties arising
from the basic loci of Shimura varieties of Coxeter type. As an application, we
prove an arithmetic intersection formula for certain diagonal cycles on unitary
and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad
conjectures. In particular, we prove the arithmetic fundamental lemma in the
minuscule case, without assumptions on the residual characteristic.
