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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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