This is report on work in progress with Nigel Higson. We are exploring an idea which comes from a very simple observation: The Bruhat cells of various flag manifolds are exactly the orbits of the action by a nilpotent matrix group. So one might try to use the apparatus developed for singular foliations in order to address representation theory problems. Making a start with this, we look at the case of $CP^n$ and the action by triangular matrices. It turns out that the nilpotency of this group allows us to shed some geometric light in the well-known K-theory group of $CP^n$, using index theory and techniques developed with Georges Skandalis to split singularities. Using these techniques we also construct interesting $K$-theory elements.