In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra $\mathbb{C}[W]$ and the universal enveloping algebra $U(sl_n)$ have been realized as the cohomology ring $H^\bullet(Z)$ for some certain variety $Z$. Nevertheless, in the paper the author says: "Of course, given an algebra $A$, there is no a-priori recipe helping to find a relevant geometric data $(M,Z)$; ...It is fair to say, however, that it is still quite a mystery why a geometric realization of the algebras $A$ that we are interested in is possible at all."

In the recent years, is there any progress in this area? For example is there any theory that tells us when this $Z$ exists and how to find it? For the universal enveloping algebra $U(g)$ for $g$ other than $sl_n$, does the geometric realization exist?