Silting complexes of coherent sheaves and the Humphreys conjecture

Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p ≥ 0, and let N be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent G-orbit C and each indecomposable tilting vector bundle T on C a certain complex $S(C,T)$ of G × Gₘ-equivariant coherent sheaves on N. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-t-structure. We then show that if p is larger than the Coxeter number, then the hypercohomology H^•(S(C,T)) is identified with the cohomology of a tilting module for G. This confirms a conjecture of Humphreys on the support of the cohomology of tilting modules.