  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.
  To this end we construct a non-commutative resolution of the nilpotent cone
which is derived equivalent to the Springer resolution. On the one hand, this
noncommutative resolution is shown to be compatible with the positive
characteristic version of Beilinson-Bernstein localization equivalences. On the
other hand, it is compatible with the t-structure arising from the equivalence
of Arkhipov-Bezrukavnikov with the derived category of perverse sheaves on the
affine flag variety of the Langlands dual group, which was inspired by local
geometric Langlands duality. This allows one to apply Frobenius purity theorem
to deduce the desired properties of the basis.
  We expect the noncommutative counterpart of the Springer resolution to be of
independent interest from the perspectives of algebraic geometry and geometric
Langlands duality.
