Edge Transport from Parabolic Subgroups of Type $D_4$

This paper is part of the program to classify Kazhdan-Lusztig cells for Weyl groups of type $D_n$. We prove analogous results to those of section 4 of Kazhdan-Lusztig’s original paper, this time related to a parabolic subgroup of type $D_4$. We also show how this is used in the definition of the generalized $\tau$-invariant.