We prove most of Lusztig's conjectures from the paper "Bases in equivariant
K-theory II", including the existence of a canonical basis in the Grothendieck
group of a Springer fiber. The conjectures also predict that this basis
controls numerics of representations of the Lie algebra of a semi-simple
algebraic group over an algebraically closed field of positive characteristic.
We check this for almost all characteristics.