  The W-set of an element of a weak order poset is useful in the cohomological
study of the closures of spherical subgroups in generalized flag varieties. We
explicitly describe in a purely combinatorial manner the W-sets of the weak
order posets of three different sets of involutions in the symmetric group,
namely, the set of all involutions, the set of all fixed point free
involutions, and the set of all involutions with signed fixed points (or
"clans"). These distinguished sets of involutions parameterize Borel orbits in
the classical symmetric spaces associated to the general linear group. In
particular, we give a complete characterization of the maximal chains of an
arbitrary lower order ideal in any of these three posets.
