Integral exotic sheaves and the modular Lusztig–Vogan bijection

Abstract

Let 𝐺 be a reductive algebraic group over an algebraically closed field 𝕜 of pretty 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.