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Kazhdan-Lusztig theory provides a pattern of applying tools of algebraic geometry,
such as the theory ofÌýFrobenius 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