Lusztig's conjecture as a moment graph problem

Abstract

We show that Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic is equivalent to the generic multiplicity conjecture, which gives a formula for the Jordan-H"older multiplicities of baby Verma modules over the corresponding Lie algebra. Then we give a short overview of a recent proof of the latter conjecture for almost all base fields via the theory of sheaves on moment graphs.

17 pages; revised version with minor changes

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• New bases of some Hecke algebras via Soergel bimodules
• Kazhdan-Lusztig combinatorics in the moment graph setting
• Higher-order congruence relations on affine moment graphs: The subgeneric case
• Equivariant $K$-theory of even-dimensional complex quadrics