I'm a sheaf theorist who knows a little representation theory but is not familiar with it. According to my memory, I was once told that one virtue of the Beilinson-Bernstein localization theorem is that it breaks representations to D-modules and the later, being sheaves, enjoy the local-to-global principle. Thus, one can reduce difficult questions about representation theory to simpler questions about geometry which can then be studied locally.
My question is what is an example of such a strategy? Or more generally, what is an example of a question which is hard to study as a representation theory question but becomes more trackable when studied as a D-module question through this theorem?