On support varieties and the Humphreys conjecture in type A

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Let G be a reductive algebraic group scheme defined over Fₚ and let G₁ denote the Frobenius kernel of G. To each finite-dimensional G-module M, one can define the support variety VG₁(M), which can be regarded as a G-stable closed subvariety of the nilpotent cone. A G-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when p≥ h, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verify this conjecture when G=SLₙ and p > n+1.