In the study of singular spaces, topological methods like intersection cohomology have played an important role. They have led to the development of technology like perverse sheaves and these find widespread application in, for instance, representation theory. There are, however, another approach to singular spaces that is analytical in nature (uses the Riemannian metrics defined on these spaces obtained with the singularity removed) and is called $L^2$ cohomology since it involves the study of differential forms that are $L^2$ complete and their associated chain complexes, cohomology etc. Now, there are standard conjectures of Cheeger-Goresky-Macpherson (back to 80s, I think), that relate the two theories. I would like to know the current status of these conjectures. In particular, are there some special cases for which it is already proven ? My interest is specifically in the case of the singular spaces that arise in representation theory. For a plethora of examples that satisfy this description, Lusztig's ICM (found [here](http://www.mathunion.org/ICM/ICM1990.1/)) is a good place (ex include Schubert varieties, Nilpotent Orbits etc.)