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) is a good place (ex include Schubert varieties, Nilpotent Orbits etc.)

An important special case is Zucker's conjecture: the intersection cohomology of the minimal compactification $\bar{X}$ of an Hermitian locally symmetric variety $X$ is the L^2 cohomology of $X$ (with the metric coming from the symmetric space above). It was proved independently by Looijenga and Saper-Stern.

See this very nice survey by Goresky (which also gives some indications about the general problem raised in your question).

Zucker's conjecture is one ingredient (among many) of the computation by S.Morel of the L-function of non-compact Shimura varieties in automorphic terms, see this other survey.

There are some other cases where the relation between intersection cohomology and L^2 cohomology are well understood. For example, see Hunsicker's "Hodge and signature theorems for a family of manifolds with fibre bundle boundary". Hunsicker mostly treats only the two-stratum case, but she has some nice results about what different intersection homology groups arise as the metric near the lower stratum varies.