Dimension formula for the affine Deligne–Lusztig variety $$X(\mu , b)$$

Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

“…Admissible subsets of W . A useful description of the admissible set is given in [12] in terms of weight of minimal length paths in the quantum Bruhat graph. This relies on the characterization of covering relation in W as established in [21,Proposition 4.2] and hence as such it necessarily brings into picture the superregularity condition, cf.…”

“…This characterization is of independent interest, and more recently it has been utilized by He and Yu in [12] to deduce a partial description of admissible sets in W . This latter description is a crucial ingredient in proving the main result in their paper, which provides a formula for the dimension of X(µ, b).…”

“…In section 7, we improve upon a result about dimension of X(µ, b). Based on recent discovery of the dimension formula for single affine Deligne-Lusztig variety X w (b) for "sufficiently large" w in [13], He and Yu provide a formula in [12] for the dimension of X(µ, b) under the assumption that µ is superregular. We show that this formula in fact holds true for all dominant regular µ.…”

“…sit. We use this additivity to show that certain steps in the existing proof in [12] can be carried out in a way that superregularity is no longer required. We point out that the proof philosophy has a resemblance to that of theorem A, in the sense that we leverage the results already established in loc.…”

Shadows in the Wild - Folded Galleries and Their Applications

“…Later Görtz, He and Nie [18] explained the nonemptiness pattern for all x and all basic b. Finally He [22] and He and Yu [23] gave dimension formulas for all x and (eventually) all b. See also He's ICM notes [21] for a general exposition of the grouptheoretic methods and results.…”