  For a particular class of pseudo manifolds, we show that the intersection
cohomology groups for any perversity may be naturally represented by extended
weighted $L^2$ harmonic forms for a complete metric on the regular stratum with
respect to some weight determined by the perversity. Extended weighted $L^2$
harmonic forms are harmonic forms that are almost in the given weighted $L^2$
space for the metric in question, but not quite. This result is akin to the
representation of absolute and relative cohomology groups for a manifold with
boundary by extended harmonic forms on the associated manifold with cylindrical
ends. As in that setting, in the unweighted $L^2$ case, the boundary values of
the extended harmonic forms define a Lagrangian splitting of the boundary space
in the long exact sequence relating upper and lower middle perversity
intersection cohomology groups.
