  We observe that the join operation for the Bruhat order on a Weyl group
agrees with the intersections of Verma modules in type $A$. The statement is
not true in other types, and we propose a conjectural statement of a weaker
correspondence. Namely, we introduce distinguished subsets of the Weyl group on
which the join operation conjecturally agrees with the intersections of Verma
modules. We also relate our conjecture with a statement about the socles of the
cokernels of inclusions between Verma modules. The latter determines the first
Ext spaces between a simple module and a Verma module. We give a conjectural
complete description of such socles, which we verify in a number of cases.
Along the way, we determine the poset structure of the join-irreducible
elements in Weyl groups and obtain closed formulae for certain families of
Kazhdan-Lusztig polynomials.
