Support varieties of tilting modules over GLn
Cooper, Bobbe Jane
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Let G be a reductive algebraic group scheme defined over the finite field Fp, 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 over the Frobenius kernel G1 of tilting modules with regular highest weight 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 = GLn. Our conjecture is equivalent to Humphreys’ conjecture for p h = n 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.