Consider homogeneous G/H and G/F,
for an S-algebraic group G.
A lattice L acts on the left
The following rigidity results are obtained:
morphisms, factors and joinings defined apriori
only in the measurable category
are in fact algebraically constrained.
Arguing in an elementary fashion
we manage to classify
all the measurable maps
commuting with the L-action:
assuming ergodicity, we find
they are algebraically defined.