In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in \cite{GHKR2}. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.

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