Kazhdan-Lusztig theory provides a pattern of applying tools of algebraic geometry,
such as the theory of Frobenius or Hodge weights, to numerical problems of representation theory.
These techniques have been used in representation theory over a field of positive characteristic
via reduction to characteristic zero, which yields a proof of the desired answer for indefinitely large p.
I will survey some of these results; time permitting I will mention a hope for future connection
of these questions to p-adic Hodge theory