Integral exotic sheaves and the modular Lusztig-Vogan bijection

Pramod N Achar (1) , William Hardesty (1) , Simon N. Riche (2, 3)

1 LSU Mathematics - Department of Mathematics [Baton Rouge]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]

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.

Domaines: Théorie des représentations [math.RT]

Pramod N Achar, William Hardesty, Simon N. Riche. Integral exotic sheaves and the modular Lusztig-Vogan bijection. Journal of the London Mathematical Society, 2022, 106 (3), pp.2403-2458. ⟨hal-01901593⟩