** 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 .

**Examples**.

- 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.

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