Lusztig's conjecture as a moment graph problem

TL;DR

This paper links Lusztig's conjecture to the generic multiplicity conjecture and discusses a recent proof using sheaves on moment graphs, advancing understanding of representation theory in positive characteristic.

Contribution

It establishes the equivalence between Lusztig's conjecture and the generic multiplicity conjecture and reviews a recent proof for almost all base fields.

Findings

Lusztig's conjecture is equivalent to the generic multiplicity conjecture.

A recent proof of the generic multiplicity conjecture applies to almost all base fields.

Sheaves on moment graphs are instrumental in proving the conjecture.

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.