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We survey a new approach towards Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic. The main result that we review is that Lusztig's conjecture is implied by a multiplicity conjecture on the stalks of certain sheaves on a moment graph. This latter conjecture is known to hold if the underlying field is of characteristic 0. From this one can almost directly deduce the conjecture for fields of large enough characteristics; but using a Lefschetz‐type theory on the moment graph, we can give an upper bound on the exceptions. Moreover, one can prove the multiplicity 1 case of the conjecture in full generality. In addition to a survey of the above results, we prove the equivalence between the original conjecture of Lusztig and its generic version, that is, the multiplicity conjecture for baby Verma modules for the corresponding Lie algebra.

Lusztig's conjecture as a moment graph problem