  Let G be a reductive group over an algebraically closed field k of very good
characteristic. The Lusztig-Vogan bijection is a bijection between the set of
dominant weights for G and the set of irreducible G-equivariant vector bundles
on nilpotent orbits, conjectured by Lusztig and Vogan independently, and
constructed in full generality by Bezrukavnikov. In characteristic 0, this
bijection is related to the theory of 2-sided cells in the affine Weyl group,
and plays a key role in the proof of the Humphreys conjecture on support
varieties of tilting modules for quantum groups at a root of unity.
  In this paper, we prove that the Lusztig-Vogan bijection is (in a way made
precise in the body of the paper) independent of the characteristic of k. This
allows us to extend all of its known properties from the characteristic-0
setting to the general case. We also expect this result to be a step towards a
proof of the Humphreys conjecture on support varieties of tilting modules for
reductive groups in positive characteristic.
