Draft Summary: Coxeter Elements and Centers

We show that the abstract group structure of the Weyl group often forces the structure of the center of the corresponding Lie group or p-compact group. For example, if G is a connected simple compact Lie group and Center(WG) is nontrivial, then Center(G) is an elementary abelian 2-group. Generalizing to p-compact groups, if WX is irreducible as a Qp reflection group and Center(WX ) is non-trivial, then the center CX defined by Dwyer-Wilkerson is contractible if p is odd. The existence of a non-trivial center for WX is detected by the rational cohomology type of X, so this principle is easy to apply. If W has a trivial center, one can still bound CX in terms WX . The role of central elements in WX must be replaced with that of elements analogous to the Coxeter elements of Lie Weyl groups. This gives sharp results – for any irreducible example X with Weyl group not that of a Lie group the center CX is trivial. In the case of a Lie-like Weyl group, the possibilities for CX are bounded by the analogous Lie examples. Similar triviality results and bounds hold for the fundamental group π1(X) and the k-invariant of the BW → BT → BNT sequence.