On the support varieties of tilting modules

Let G be a reductive algebraic group scheme defined over the finite field F_p, with Frobenius kernel G₁. The tilting modules of G are defined as rational G-modules for which both the module itself and its dual have good filtrations. In 1997, J.E. Humphreys conjectured that the support varieties of certain tilting modules for regular weights should be given by the Lusztig bijection between cells of the affine Weyl group and nilpotent orbits of G, when p>h, where h is the Coxeter number. We present a conjecture for the support varieties of tilting modules when G=GL_n. Our conjecture is equivalent to Humphreys’ conjecture for p≥h and regular weights, but our formulation allows us to consider small p or singular weights as well. We obtain results for several infinite classes of tilting modules, including the case p=2, and tilting modules whose support variety corresponds to a hook partition. In the case p=2, we prove the conjecture by S. Donkin for the support varieties of tilting modules.