Combinatorics of Coxeter groups and affine Deligne-Lusztig varieties
Joint work with Liz Milicevic and Anne Thomas.
1. Coxeter groups
Data: a symmetric matrix with entries and . Group is .
- Dihedral group : 2 generators and , . Spherical, rank 1.
- : Affine, rank 2.
Question. Take as generating set all reflections (i.e. conjugates of standard generators) in . What is the diameter of the resulting Cayley graph ?
Answer is known in the spherical case, Dyer 2001, see below.
McCammond-Peterson give an upper bound rank in the affine case
Definition 1 In the tiled sphere or Euclidean space, a gallery is a sequence of contiguous chambers. Its type is the word collecting the reflection used to pass from each chamber to the next. Say a gallery is folded if
Dyer’s bound equals the minimum number of folds of a gallery that comes back to where it started and uses all .
Buildings are covered with apartments which are Coxeter complexes. Galleries in the building, when mapped by Tits retractions, give rise to folded galleries in apartments.
Orienting a Coxeter complex amounts to choosing a chamber of the spherical complex of links. This allows to speak of positive or negative folds.
2. Afine Deligne-Lusztig varieties
where is the algebraic closure of a finite field . The affine flag variety is where is a reductive group over and an Iwahori subgroup (stabilizer of a chamber).
Affine Deligne-Lusztig varieties involves twisting with the Frobenius operator. They are indexed by pairs where and . The issue is wether they are empty or not.
Theorem 2 Let be a Coxeter group. Let be a translation in . Let be a word in . Then the Deligne-Lusztig variety is nonempty iff there exists a positively folded gallery joining to of type . If so, its dimension is essentially the number of positive folds plus the number of positive crossings of the gallery.
As a corollary, we get a bound on rank for elements of the form translation composed with Coxeter element of the link spherical Coxeter group.