I hope this question has not been asked before.

I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdan–Lusztig polynomials can be expressed via intersection cohomology groups of Schubert varieties?

Is there nowadays more insight why intersection cohomology groups (or perverse sheaves) appear in representation theory? (I should mention that I know about Koszul-Duality and localisation, so if there is another viewpoint I would appreciate to hear about it)