## Notes of Petra Schwer’s Cambridge lecture 12-01-2017

Combinatorics of Coxeter groups and affine Deligne-Lusztig varieties

Joint work with Liz Milicevic and Anne Thomas.

1. Coxeter groups

Data: a symmetric matrix ${M}$ with entries ${m_{ij}=1,2,3,\ldots,\infty}$ and ${m_{ii}=1}$. Group is ${\langle s_i|(s_is_j)^{m_{ij}}\rangle}$.

Examples.

1. Dihedral group ${D_3}$: 2 generators ${s}$ and ${t}$, ${s^2=t^2=(st)^3=1}$. Spherical, rank 1.
2. ${\tilde A_2}$: Affine, rank 2.

Question. Take as generating set all reflections (i.e. conjugates of standard generators) in ${W}$. What is the diameter ${\ell_R}$ of the resulting Cayley graph ?

Answer is known in the spherical case, Dyer 2001, see below.

McCammond-Peterson give an upper bound ${2(}$rank${-1)}$ 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

${F=k((t))}$ where ${k}$ is the algebraic closure of a finite field ${F_q}$. The affine flag variety is ${G(F)/I}$ where ${G}$ is a reductive group over ${F_q}$ and ${I}$ an Iwahori subgroup (stabilizer of a chamber).

Affine Deligne-Lusztig varieties involves twisting with the Frobenius operator. They are indexed by pairs ${(x,b)}$ where ${x\in W}$ and ${b\in G(F)}$. The issue is wether they are empty or not.

Theorem 2 Let ${W}$ be a Coxeter group. Let ${b}$ be a translation in ${W}$. Let ${x}$ be a word in ${W}$. Then the Deligne-Lusztig variety ${X_x(b)}$ is nonempty iff there exists a positively folded gallery joining ${1}$ to ${b}$ of type ${x}$. 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 ${\ell_R\leq}$ rank ${-1}$ for elements of the form translation composed with Coxeter element of the link spherical Coxeter group.