Affine Deligne–Lusztig varieties and quantum Bruhat graph

In this paper, we consider affine Deligne–Lusztig varieties $$X_w(b)$$ and their certain union $$X(\mu ,b)$$ inside the flag variety of a reductive group. Several important results in study have been established under so-called superregularity hypothesis. Such include description generic Newton points Iwahori double cosets loop groups, covering relation associated Iwahori–Weyl group dimension formula for . We show that one can considerably weaken hypothesis sometimes eliminate it, thus strengthening these existing results.